ontology and conceptual frameworks part ii

H A I M G A I F M A N

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S

P A R T I I

1. INTRODUCTION

The aim of this essay is to establish a new method of analysing ontological questions. The method also involves an analysis of conceptual frameworks and of translations (and partial translations) from one framework into another. Some of the basic ideas were presented in Part I which appeared in Erkenntnis, vol. IX, No. 3. For the reader 's convenience they are presented in this introduction in summary form under the title 'Main Theses'. I am indebted to Hillary Putnam for calling my attention to observations by Kreisel, Dummet t and himself to the effect that realism in mathematics amounts to a belief in the objective truth values of mathematical statements. My main theses can be seen as an unfolding of this theme; degrees of ontological affirmation are to be measured by the comprehensiveness of the class of objectively true-or- false sentences; reductions, equivalences and partial equivalences are to be established via recursive translations between frameworks, and the whole machinery is to be applied not only in mathematics but everywhere. Some detailed applications are given as examples in w where, inter alia, a logical reconstruction of the Leibniz-Newton controversy about absolute space and time is carried out, using the proposed

method. For the sake of precision I introduced a suitable terminology and

outlined in w a general formal system which can serve as a frame of reference for the sort of analysis intended. The basic concepts of the proposed machinery are further and more informally elaborated in w in which context Tarski's convention (T) appears in a new light. This chapter also contains a tentative classification of frameworks, which leads to an explication of the concept of open texture. In w some detailed examples are worked out as illustrations of the method. The chapter takes about half of the second part. Its length necessitated the shifting of other chapters to still another, third, and, hopefully, last part of this essay.

Erkenntnis 10 (1976) 21-85. All Rights Reserved Copyright �9 1976 by D. Reidel Publishing Company, Dordrecht-Holland

22 H A 1 M G A I F M A N

Among these are the chapters in which open frameworks and the general concept of translation are treated. w consists of some general observations on problem-situations and the role of objects.

M A I N T H E S E S

Ontology is concerned with the relationships between conceptual frameworks and objective reality. In an ontological question one asks whether, or how far, a given well-understood language describes something that is objective and real.

II

The extent to which a given conceptual framework describes an objective reality is determined by the class of sentences in the framework's language which have objective truth values, that is to say, those sentences for which the question of their being true is factually meaningful, having an objective yes-or-no answer which is independent of our particular state of knowledge. One's ontological acceptance of a given framework is to be measure by the class of sentences which one regards in this way, i.e., which one believes to be objectively true or false. The more comprehen- sive the class, the greater is the ontological commitment to the framework, or, in my terminology, the ontological affirmation of the framework. (I use this term since 'commitment ' has been used consis- tently by Quine and his followers to indicate commitment to particular objects - a concept which to my mind is not very fruitful in the explication of ontology.)

I I I

Affirmation of a conceptual framework presupposes comprehension of its sense, but not vice versa. One can have as good an understanding of a framework as one can wish, and yet reject its factual meaningfulness just as one rejects the factual meaningfulness of a well-understood dream or fantasy. Ontological controversies take place where both sides grasp the

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , II 23

same sense of the framework, while disagreeing about its objective status. Denying factual meaning to a sentence is to be expressed, as in Frege's way, by denying it any truth value. Truth values can be denied not only because some noun phrase fails to denote, as in the standard examples, but for reasons which cannot be expressed within the framework under discussion. These are the ontologically interesting cases whose analysis calls for the application of a two-valued logic with truth value gaps.

(Our actual frameworks constitute changing open textures whose historical development is determined, among the rest, by their shifting ontological status. Thus, in historical examples the aspects of sense and of factual meaningfulness are often intermingled. Yet the distinction should be made, not only for the sake of clarity, but also because through it we note those crucial junctures where questions concerning a framework's factual meaning merge into questions concerning its sense.)

I V

Conceptual frameworks cannot be defined, but they can be pointed out. They can be treated as systems consisting of uninterrupted languages and indicators which serve as guidelines to the intended sense of the framework. One can use as indicators, inter alia, the so-called obvious truths of the framework or, in the terminology of this essay, its evident conceptual expectations.

The laws of logic and derivation rules are to be reckoned among them. A sentence is analytic with respect to the given framework if it is derivable from its evident conceptual expectations. Being an evident conceptual expectation is generally a matter of degree, and in a more detailed or accurate account one would refer to degrees of analyticity.

I shall use 'L' to refer ambiguously to the framework as well as to its underlying uninterrupted language. Where necessary I shall use the phrases 'L-framework' and 'the language of L ' to avoid confusion.

v

Among the sense-determining factors of a given framework there are rules which stipulate that such and such sentences have truth value; these sentences are those which, according to the framework (that is, from the

24 H A I M G A I F M A N

point of view of one who affirms it), should be objectively true or false.

Such stipulations are necessary because truth value gaps, or the possibility of their emergence, may follow from the conventions of the

f ramework or may be part of its very sense structure. A full ontological

affirmation of a f ramework means an agreement that the sentences which, according to the rules of the f ramework, have to be true-or-false, are

indeed true-or-false. [(1I) should therefore be qualified accordingly.] The inclusion of stipulations that certain sentences are objectively t rue-or-

false among the sense-determining factors of the f ramework does not

contradict the distinction made in (IID between sense and factual mean-

ing. Such stipulations are but claims which can be understood without

being accepted. They indicate what the f ramework pretends to describe.

To affirm a f ramework ontologically means to grant it its ontological

pretensions. Partial affirmation consists in accepting only part of these claims; it can be regarded as an affirmation of a different, though comparable, f ramework whose ontological pretensions are more modest.

So far we have summarized some of the main points of Part I. The

following is new.

2. THE L+-SYSTEMS TRUTH FROM THE VIEWPOINT OF L)

When someone who affirms the f ramework L asserts that a sentence A, in

L, is neither true nor false, he employs a concept of truth which is a correlate of the f ramework; in his assertion ' t rue ' should be indexed by 'L ' , for it indicates the concept of being true as conceived from the aspect

of L. ~ The rules of L, which determine the occasions on which sentences do or do not have truth values, are to be stated using expressions such as

" ' A ' is true1." or equivalent ones. Let us therefore extend the language of L to a language L + in which such rules can be explicitly stated. This extension should n o t amount to an addition of a full truth predicate to L, for - as argued in Part I - such an addition would constitute an ontologi-

cal enlargement of the f ramework in question. L § should serve as a convenient tool for displaying explicitly the point of view of those who affirm L. The enrichment should have no ontological significance, and the rules formulated in L + are to be reckoned as the conceptual expectations of L (i.e. of the framework). (That L + is not essentially richer than L is to be established, using the forthcoming criterion of translatability. In those

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , 11 25

exceptional cases where L + turns out to be an essential enrichment, we

may conclude that our original construction of the L- language was

inadequate for expressing the conceptual expectations of L. Therefore L should have been enlarged. As we shall indicate, no essential enrichment

will take place if we iterate the extension and pass from L + to L++.) One

way of constructing L + is to add, for each sentence, A, in L, a new

individual constant ' A ' which is to serve as its name. Then add a monadic

predicate 'TrueL ( )' which takes as arguments only names of sentences of L (if other terms and, in particular, variables are used as arguments - the

resulting expressions are not well-formed formulas). Another , preferable

though equivalent, way of constructing L + is the following one: Add 'TrueL' to the language L , not as a predicate but as a sentential

opera tor which is applicable to sentences of L, thereby producing sen-

tences of L+; grammatically it functions like a negation sign or as a modal

operator . On this reading, the English analogue of 'True~. (A) ' would not

be The sentence ' A ' is true

but

It is true that A.

Clearly, one can iterate the opera tor to any depth (producing expres-

sions such as 'True~ ( T r u e L ( A ) ~ (A)) ' ; if 'TrueL' is t reated as a predicate, new names are needed for each increase of depth). However , I shall

not need these iterations (mainly because TrueL(TrueL(A)) will be

equivalent to TrueL(A)). There is also nothing to prevent one from

applying 'TrueL' to formulas containing free quantifiable variables. In the

present t reatment it is, however, enough to apply it only to sentences.

Thus, the sentences of L + are obtained by first adding to L all the sentences TrueL(A), where A is a sentence in L, and then by using the logical operat ions of L to form compounds. ( ' A ' is used here as a

schematic variable ranging over sentences. Since my intentions are clear,

I need not be pedantic about the distinction between use and mention; see note 16, Part I.)

Now we can define falsity, by

FalseL (A) = of TrueL ( ~ A )


and we can express the statement that A is true of false by

TrueL (A) v FalseL (A).

' ~ ' and ' v ' are the negation and disjunction signs of L. We assume that L has these connectives and that they can be used in this way to express in L + the required notions. (I do not mean that the concept of negation is philosophically prior to the concept of falsity. The equivalence is an evident expectation which ties together the concepts of truth, falsity and negation.)

In general, whever we employ connectives of L to state rules in the language L +, our usage should be in accord with the meaning of these connectives in the L-framework. To give an extreme example, if in the L-framework it is not evident that a sentence and its negation cannot hold

simultaneously, then the definition given above should be abandoned, and 'False' should be introduced as an additional primitive. The rules which we have in L + must not introduce any new derivable sentences in the original language L, that is to say, the L+-system should be a conservative extension of the L-system.

To provide a general treatment I shall start with the most general assumptions concerning the framework L. I shall assume that in this framework we have a concept of implication; if A and B are sentences of L, then 'A ~--L B ' is to be read 'A implies (in L) B' . We have also what

might be called a concept of a priori, or unconditional, validity; '~-LA' states, roughly speaking, that A is unconditionally valid in the framework L, meaning that it is an evident expectation or derivable from such - what I earlier described as analytic with respect to L. ( 'Analytic' has several connotations which I like to avoid in this context, hence the change in terminology.) If L has implication as a sentential connective, then 'A ~--LB' is equivalent to 't-LA-->B'. But not every logical calculus admits implication as a sentential connective; for the sake of generality I start with two basic concepts instead of one.

[When the framework is approximated by some formalized version, 't--L' plays the usual role of that sign in present-day logic; that is to say, representing L as a deductive theory 'A f-LB' means that B is derivable in the theory from A, and '~-LA' means that A is a theorem of L. Note, however, that the precise meaning of 'A ~L B' is not conveyed b.y saying that if we assume A we can derive B. Such a phrasing blurs the cases of truth value gaps, for by assuming A we also assume TrueL (A). If

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , l I 27

we have implication inside L, then somebody who affirms L and regards A as neither true nor false will regard TrueL(A) as false, and consequently he will not regard A -~ TrueL (A) as true.

The expressions 'A b-/~ B ' and 'b-LA' belong to a certain restricted metalanguage in which we can describe the knowhow of employing the language L. Whereas statements in L are supposed to describe states of affairs relating, say, to people or stars or elementary particles or what not, sentences of the metalanguage to not represent any knowledge that that the user of the L-system might have (or pretend to have); they represent his knowledge how to employ the L-system. The rules formulated below are in this metalanguage;, all of them fall under the schematic forms: 'b-LA' or 'A b-L b' or 'if so and so and so and s o . . . etc., then so and s o . . . etc.' where each 'so and so is of the form 'A b-L B ' or 'b-LA'. Note that there is no call for expressions of the form 'not ~-LA ', for what can be the meaning of such an expression? If we intend to state that one cannot infer from the rules that ~-LA (meaning that there is no finite sequence

which constitutes a proof of 'b-LA' from the rules) then this involves quantification over all finife sequences of expressions of the above- mentioned metalanguage; it is, essentially, a statement in a r i thmet ic- a framework in its own right, far exceeding the metalanguage in which the rules are stated. If, on the other hand, we want to say that, viewed within the L-framework, A is not a priori valid, then we have to assume that in the L-system we have the means of stating just this. That is to say, there should be a sentence B in L such that ~-LB, and such that B states (in some way) that A is not a priori valid (or not necessary, or that A has prior probability smaller than 1). Analogous observations apply to 'not A ~-L B'.

Although the rules are not stated in the language L itself, they should be considered among the evident conceptual expectations of the L- framework.]

The rules are intended to represent a common core of almost all systems one can think of. The most notable exception is the system of Quantum Logic proposed by Kochen and Specker 1 in which the conjunction (or disjunction) of two sentences of L need not always be defined.

'A ' , 'B ' , 'C ' are schematic variables ranging over sentences of L

(1) (2)

A b-L A, and if A b-L B and B b-L C, then A b-L C.

If b-LA and A b-L B, then b-LB.


We assume that L has conjunction, disjunction and negation as sentential connectives

(3a) (3b) (4a) (4b) (5) (6)

A ^ B ~-LA a n d A ^ B b-LB If C~-c A and CF--L B, then C~-c A ^ B A ~-cA vB and B F-LA v B If A ~--t C and B ~-L C, then A v B ~-c C If A ^ B t - t CA -nC, then A ~-t. ~ B If F-LA, then B ~--t A and ~ A ~-c B

(i.e., any sentence implies any unconditionally valid sentence and the negation of an unconditionally valid sentence implies any sentence.2 Note

that (5) implies B ~-L 7 (A ^ ~ A ) for all A, B.)

(7) ~ ~ ( A ^ ~ A )

((7) may seem doubtful when A has no truth value; of this later on). Define A to be contradictory if ~ t n , then it is not difficult to show that

t--LA if and only if -hA is contradictory. That A ~-c ~ A , and that if A ~ c B, then ~ B ~-L ~ A follow from (1)-(5), as do several other well-

known laws. L e t ' A --- LB' stand for 'A ~ t B and B ~-L A '. Then ---~_ is an equival-

ence relation and if we divide by it (i.e. we 'identify' every A and B for which A------LB) we get a set on which ~ c induces a partial ordering. L-equivalence is a congruence relation with respect to -7, ^, v (i.e., if A ~LA' then -~A---L~A', etc.), and thus, each connective induces an operation on the partially ordered set. The resulting structure is the Lindenbaum-Tarski algebra. (3) and (4) can be rephrased as stating that that algebra is a lattice in which the greatest lower bound (or meet) and least upper bound (or join) operations correspond to a and v, respectively. The lattice has maximal and minimal elements which correspond to the unconditionally valid and the contradictory sentences.

The distributivity of the Lindenbaum-Tarski lattice is equivalent to:

(D) A ^(BvC)~-L (A ^ B ) v(A ^C)

Except for certain interpretations of quantum mechanics which are based on a nondistributive lattice (see notes 10 and 1 1 in Part I) (D) seems to hold in all other systems.


If L has implication, ~ , as a sentential connective, then it is charac- terized by the following rule:

(8a) A ^ (A ~ B) F-c B (8b) If A ^ C ~--t B, then C ~--t A ~ B.

In terms of the Lindenbaum-Tarski algebra this means that the lattice is relatively pseudo-complemented. (That is to say, given any two members, there is a maximal element whose meet with the first member is smaller than, or equal to the second; if the first member corresponds to A and the second to B, then this maximal element corresponds to A ~ B). This property implies distributivity (and, in the case of a finite lattice, is equivalent to it). If the lattice has this property, then one can introduce implication as a sentential connective in L if it was not there already. (One can do so, essentially, even if the lattice is only distributive.) But if the lattice is not distributive, L does not admit implication in the role of a sentential connective.

If L has -~, then it follows easily that

A ~-L B if and only if ~-cA ~ B.

The rules can then be formulated so that 't--t' Occurs only in compounds of the form '~-c A ': replace every 'A ~--t B ' by '~L A ~ B' . This will yield either rules of the form '~-L. A ' or stipulations of the form ' I f . . . t h e n . . . ' . Regarding those A ' s for which '~--t A ' , is a rule as axioms, and treating the stipulations of the second kind as derivation rules, we get a deductive system in the usual sense.

If the system does not admit ~ , we can still eliminate compounds of the form 'A ~z~ B' , provided that we have a uniform way of associating with every two sentences A, B a sentence C(A, B) such that

A ~--t B if and only if r--LC(A, B) .3

In addition to (1)-(7) any particular framework L will have many additional rules of a more specific character (including, probabily, rules concerning quantifiers, function symbols and what not). Among these there will be rules stating that ~-cA, where A is an evident expectation in the language L.

Now extend L to L § by additing 'T rue t ' as a sentential operator which acts on sentences of L, as explained before. We apply 'P-r' to names of


sentences of L § as well. We shall omit the explicit mention of L in 'TrueL' ; thus 'True ' will stand, unless otherwise indicated, for ' T r u e t ' ,

and 'False ' (defined as ment ioned above) for 'Fa lse t ' . Whenever we use 'T rue (A) ' it is implicitly assumed that A is a sentence in the original

language L. For L § we have the following:

(9) Every general schema which is a rule of L is a rule of L +,

where the sentential schematic variables range over the sentences of L +.

(10a) If F-LA then F-I True(A)

(10b) If A ~ t B then True(A) F-- t True(B) (11) True(A)~-t A (12) (True(A) v False(A)) ^ A J-L True(A)

(13) True(A) ^ True(B) WL True(A ^ B)

The converse implication True(A ^ B) ~--t True(A) ^ True(B) follows

already from the rules which we have. Fur thermore, the rules entail

that True(A)vTrue(B)~-LTrue(AvB) and, if L has ~ , that

True(A --> B) F--L True(A) ~ True(B). None of the converse implications follows from the rules. (Which is as it should be, for on many occasions the best way to represent a f ramework is to allow for the possibility of a true

disjuction in which none of the disjuncts has a truth value.)

Among the consequences of these (together with (1)-(7), we have:

True(A) WL ~Fa l se (A) ,

False(A) ~--t ~ T r u e ( A ) ,

True(A) ~L F a l s e ( ~ A )

(Recall that, by definition, False(A) is True(~A)) .

Note that we get ~-LFalse(A ^ ~ A ) . This might be objected to, on the grounds that, if one who affirms L holds a sentence A and its negation to be neither true nor false, then he may regard A ^ ~ A as neither true nor false as well. My answer is that the logic of the f ramework is not a three valued logic; one can regard sentences which are neither true nor false as having a third value- 'undefined' , but then the value of a sentential compound will not in general be a function of the values of its components. There is no harm in treating ~ ( A ^ ~ A ) as valid (including the cases where A has no truth value) as long as this will not limit us in an essential way when we formalize or reconstruct existing frameworks. On

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R ' a K S , I1 31

the other hand, by committing ourselves to the validity of A v -tA we rule out in advance intuitionistic theories, and we might also run into difficulties with non-intuitiorIistic situations, where truth value gaps arise. The validity of ~(A ^ --hA), where A has no truth value, is a harmless quirk and not too high a price for achieving some simplicity in the general rules.

(14) If True(A) I--L B thenTrue(A) ~L True(B).

(The argument for (14 is that if we allow the application of 'True' to iterate, then, by (10b), if True(A) t-L B then True(True(A )) ~-L True(B) and later we shall see that True(True(A)) should be equivalent to True(A).)

Note that if A---L--nTrue(A) then, by (11) and (1), we have: True(A) ~-L -nTrue(A), from which, using (3), (5), (7) and (1), we get: ~L --n True(A); since, by our assumption ~True(A ) I--L A, we get by (1): ~-LA. If we employ (10a) we get: I--L True(A) which, in view of the fact that F-L ~True(A), is a contradiction. This state of affairs will obtain if 'True' is construed as a predicate. An analysis of the argument will throw some light on attempts to allow unrestricted self-referential applications of a truth predicate, while avoiding the paradoxes by utilizing a three- valued logic and by weakening Tarski's convention (T). It shows that much more than convention (T) will have to be given up, to the extent that the 'truth predicate' with which one eventually ends up cannot be considered as a truth predicate on quite elementary grounds. 4

Belief in the truth-or-falsity of a sentence A can be expressed (by somebody who affirms the framework L) by asserting True(A)v False(A). Note that from our rules it follows that:

True(A) v False(A ) ~L True(~ A ) v False(~ A)

(The converse implication is not as inevitable as it seems at first. It does not hold in our proposed interpretation of intuitionistic theories, where we may have ~-L -n-hA, implying ~-LFalse(~A), without having I-LA v ~A.)

It is natural to require that a sentential combination of true-or-false sentences should itself be true-or-false. Indeed, if the logic is distributive (i.e., if (D) holds), this follows already from the rules we have. That is to

say: (True(A) v False(A) ^ (True(B) v False(B)) I--L True(A * B) v False(A * B))


where '*' may stand for any of the connectives. [In the non-distributive case we should include it explicitly among the rules.] Furthermore, the rules entail that False(A)~--t False(A ^ B), True(A)~--t True(A v B), and, in the distributive case, False(A) ^ False(B) t--t False(A v B). [Let us add the last implication as an explicit rule for the non-distributive case.] If we have -~ in L, then it follows also that False(A) ~--t True(A -~ B), True(B) ~-L True(A -~ B) and True (A)^ False(B) ~-L False(A -~ B).

The upshot of all this is that, when the sentences which we combine are true-or-false, the sentential connectives function (in the distributive case) exactly as they do in the standard two-valued logic. This is what one would expect. For the very fact that A and B are regarded as objectively true or false enables us to define the truth value of a sentential compound

5 A * B in the standard way. Component sentences of true-or-false sentences need not, of course,

be true-or-false. For example, A ^ ~ A is (by (7)) false, whatever A is. But the making-up of true-or-false sentences out of components that are not considered true-or-false goes much deeper than this and other trivial examples (which may be thought of as mere technical conventions). It is a wide-spread phenomenon appearing in diverse frameworks, such as intuitionistic mathematics, physical theories, and even in everyday dis- course. Occasionally it can be avoided at a high price by increasing substantially the system's complexity. But sometimes it cannot be eliminated in principle (I shall later discuss some examples). In particular,

somebody may assert A1 ~A2 because he understands A~ and A2 and sees the link by which one can pass from A1 to A2 without being committed to the truth-or-falsity of either. [As I shall later point out, the framework L can express the point of view of someone who understands the framework L0 but does not affirm it. The L-language will include the L0-1anguage but not all the Lo-valid sentences will be valid in L. Thus we may have F--L o True(A;) v False(A;), i = 1, 2, without either of these sentences being valid in L ; and yet F-L A ~ --> A2. One can argue that in such cases --> plays the role of some intensional or modal implication. But the fact is that I need not introduce here any operators of the modal sort, just as I do not need them in intuitionistic mathematics.]

In addition to the rules stated so far and any other special ones concerning the sentences in a particular L, any particular framework will have some specific rules (i.e., not necessarily general schemes) of the


following forms:

(I) True(A) v False(A) t-- L True(B) (II) True(B) t-L True(A) v False(A) (III) A F-L True(A)

(I) determines necessary conditions of the form True(B) for the truth- or-falsity of some given sentence A. (II) determines sufficient conditions of the same form. For example, if A = 'The present king of France is bald', then in (I) we can have B = 'There is a country France', whereas 'There is a person who is presently the king of France' can serve as B both in (I) and in (II). Note that the case of t--z, True(A) v False(A) reduces to (II) by taking as B a valid sentence.

I shall call any B which satisfies (I) a necessary presupposition for A, and any B which satisfies (II) a sufficient presupposition for A. A necessary and sufficient presupposition for A is a sentence B, satisfying both (I) and (II). (Of course, A is trivially a sufficient presupposition for itself.) Note that any necessary (or sufficient) presupposition for A is also a necessary (or sufficient) presupposition for ~A. Note also that our rules entail that

(i) If B1 is a necessary and sufficient presupposition for A, and B2 is a necessary and sufficient presupposition for B1 then B2 A B1 is a necessary and sufficient presupposition for A and that

(ii) If B is a necessary and sufficient presupposition for A, then True(A) -~L True(B)A A. (Use (use (13) for (i) and (12) for (ii).)

(III) determines the class of those sentences in L for which Tarski's convention (T) holds (i.e., holds unconditionally). For, combining (Ill) with (II) we get: A ---c True(A). This class plays a significant role in the framework, which is the subject of the next section. Note that if B is a necessary and sufficient presupposition for A, and if convention (T), holds for B, then True(A)=--LBAA, and furthermore, under this assumption we have also:

(iii) Convention (T) holds for B A A

(iv) If ~l~ True(B) v False(B) then I--L True(A A B) v False(A A B).

3. T H E S I G N I F I C A N C E O F C O N V E N T I O N (T)

To simplify the formulation, I shall assume that the language L has implication, -~, as a sentential connective. (The reader can easily


rephrase the text if ~ is not available; e.g., replace ' ~ - L A ~ B ' by 'A F-L B' , or 'l--t A --> (B --> C)' by ' A ^ B ~--t C', etc.) 'A ~ B ' stands for '(A --> B) ^ (B -> A) ' . For a given sentence, A, of L, convention (T) can be expressed in L + by A +->true(A); more precisely, convention (T) means that this bi-implication is valid in the L-framework:

(T) I--L A ~--> True(A)

(7") has been regarded as a guide for setting up a suitable semantics. Employing any grammar one analyses sentences into components and gets a notion of an atomic formula. Then the concept of truth (or satisfaction) is to be defined in a metalanguage (which includes the given language) for the atomic formulas and, by recursive rules, for all sentences, so that the bi-implication becomes a theorem of the metalanguage. This has been considered a necessary criterion, or at least a desirable feature, for any proposed truth definition. Thus, although convention (T) has been regarded by some as trivial and uninformative, it

became significant when it was considered as a regulative criterion for possible semantics. Now, my point of departure is different. I am not interested here in setting up a semantics, and I take the concepts of truth

and falsity as given primitives by which ontology is to be explicated. Convention (T), I shall argue, does not always hold, and this very fact makes it ontologically interesting. Furthermore, the sentences for which convention (T) does not hold are not necessarily compounds; they may be

atomic sentences. There is a sense in which the equivalence of A and True(A) is, indeed,

trivially valid. Say that an assertion of A is obviously equivalent to an assertion of B if anyone who asserts A should agree to assert B and vice versa; that is to say, a rejection of B while asserting A would be construed as a misunderstanding of the language in question. I use 'assertion' here in Frege's sense: an assertion of A is a display of A in circumstances which indicate the displayer's belief that A is true. Now, an assertion of A is obviously equivalent to an assertion of True(A). This is trivial, and if convention (T) is interpreted as stating just this, then indeed it is valid and uninteresting. But, of course, the validity of A ~-~True(A) amounts tO much more than the equivalence of the assertions of A and of True(A). Assume that someone who works in the L-f ramework regards the

sentence A as neither true nor false. Then, he should regard True(A) as


false and, consequently, he cannot accept A ~ T r u e ( A ) as true (for this

would force him to regard A as false). Thus (T) fails in the case that A,

when regarded from the point of view of the L-framework, is neither true nor false. It may be argued that one can still maintain the truth of A ~ T r u e ( A ) if one regards also True(A) as neither true nor false. But this, I claim, would mean that one is not reasoning within the framework of L. For if regarding A as neither true nor false is a result of thinking in terms of L, then the conclusion that A is not true should be expressed by the sentence -nTrue.(A) where 'True' stands for 'Truer ' ; hence one should regard --nTrue(A) as true and True(A) as false.

For example, let A be of the form 'el preceded e2' where each 'ei', i = 1, 2, stands for some definite description, or proper name, of an astronomical event. Let L1 be a framework which is based on Newtonian physics. Mr. Alpha who works within L 1 may regard A as neither true nor false, if he thinks that 'el ' or 'e2' fail to denote an event (like, say, the failing of 'the present king of France' to denote a person). This failure of denoting should be expressible or derivable from something expressible, in L~ (otherwise Alpha's position regarding A could not be explained without resorting to another framework - namely, the framework which he employs when he concludes that el fails to denote an event). Another reason for regarding A as neither true nor false could be the impossibility of associating a precise time-point with a given event. The time intervals

during which el and e2 can be said to have taken place may overlap, and it may not be possible to narrow them down sufficiently so as to have, in

principle, a clear-cut answer to the question which event preceded which. Again, this sort of vagueness should be expressible, or derived from something expressible, in L 1 . But barring these contingencies, Mr. Alpha will regard A as true-or-false and A ~--~True(A) as true. Now consider Ms. Omega who works within the framework L2, which is based on

relativistic physics. The L2-1anguage can include the Ll-language as a sublanguage, but the evident expectations will, of course, be different. Now, Ms. Omega may find out that the particular events el and e~ determine what is known in physics as a space-like interval, that is to say, the temporal ordering of e~ and e2 is not absolute and can be reversed by changing the frame of reference 6. If so, Omega will regard A as neither true nor false and True(A) as false; thus, convention (T) will not apply. The state of affairs which constitutes the ground for regarding A as


neither true nor false (namely, that the temporal ordering of el and e2 depends on the frame of reference) can be stated in L2 but not in the L1- framework - where it is either indescribable or contradicts evident expectations. It is for this reason that, in this example, (T) holds in L1 but not in

L2.

The 'True ' which Omega uses is, in fact, 'TrueL2', or, for short, 'True2', whereas the 'True ' which Alpha uses is 'True1'. This, of course, does not mean that the ontological difference between them can be solved simply by adding indices and splitting the truth operator. Omega will maintain that Alpha's concept of truth, inasmuch as it reflects the L r f r a m e w o r k , is not the right concept; it does not relate to objective reality in the way it pretends to, but to some deluding fantasy. She might understand his point of view and even include 'True1' as an operator in L2 which operates on sentences of L~ (where the L~-language is a sublanguage of L2). Her understanding of Alpha's conception will be expressed by regarding

Truel (A) o A as true (i.e., she will assert True(True1 (A) ~-~ A )), but then she will regard both sides of the bi-implication as neither true nor false; in particular, she will assert -1True(True~(A)) and ~ T r u e ( ~ T r u e l ( A ) ) ,

where 'True' stands here for the " t rue" truth operator, namely 'True2'. 7 What are the sentences for which ~-LA o T r u e ( A ) (i.e., for which

convention (T) applies unconditionally in the given framework)? If, for some non-contradictory B (i.e., B for which ~ B is not valid in L) the

truth of B implies (in L) that A is neither true nor false, then we cannot have F - L A o T r u e ( A ) . For knowing the state of affairs which B describes, anyone working in L would have to admit that A is neither true nor false, which - as I argued - would prevent him from accepting A True(A) as true. I therefore propose the following criterion for determining the a priori scope of convention (T):

Convention (T) applies, unconditionally, to a sentence A (in the framework L) if and only if one cannot describe, by any sentence of L, a state of affairs the knowledge of which would induce those working in L to regard A as neither true nor false.

Expressed formally, the criterion is: (CrT) ~-LA~-~True(A) if and only if for every B, if I-LTrue(B)-~ ~(True(A) v False(A )) then ~- i. ~True (B) 8.

Note that (CrT) is not a rule of the same sort as those which we had in w 2, but a regulating criterion for setting up the L+-system.


If ~-LTrue(A)v False(A), then (by (11) and (12)) WLAoTrue(A). The converse implication might seem a natural consequence of (CrT); for

if we cannot imagine a state of affairs in which the assertion that A is neither true nor false is justified, does it not follow that A is t rue-or-false?

However , this reasoning need not apply in general. To justify an assertion that A is true or false may require more than the impossibility of

asserting that it is neither true nor false. This, I shall argue, is the way to

read intuitionistic mathematics. In order to assert that A is not true, an intuitionist requires a construction which shows the impossibility of any

construction by which an assertion of A can be justified. But such a construction would then constitute, by itself, sufficient ground for assert-

ing ~ A , i.e., that A is false. Hence the intuitionist knows that no state of

affairs is possible in which he can justifiably assert that A is neither true

nor false. But for him this does not constitute sufficient ground for

asserting that A is true-or-false. Concerning those A ' s for which A v - l A

has not been intuitionisticaily proved, the intuitionist reserves any judgment. In my proposed reading of intuitionistic mathematics convention

(T) applies to all sentences. Hence T r u e ( A ) v False(A) is equivalent to

A v ~ A . By adding 'True ' to an intuitionistic theory we do not achieve any additional clarification concerning the status of objectively t rue-or-

false sentences. An assertion that A is true-or-false can, in such a

f ramework, be justified only on the grounds of an (intuitionistic) proof of

A v ~ A . Intuitionistic theories furnish us with handy ready-made examples. But

situations where we cannot describe in L any state of affairs in which we

will assert that A is neither true nor false, and yet we refrain from

asserting that A is true or false are possible in other f rameworks and are highly significant. In these situations our judgment is, in principle, reserved, which indicates a certain openness in the f ramework in ques-

tion. Before proceeding to elaborate this point, let me take care of some technical details concerning convention (T).

One would expect that the sentences for which convention (T) holds are closed under sentential combinations. Indeed, a sentential combination of true-or-false sentences is (as indicated in w true-or-false; consequently, if there is no sentence which describes a state of affairs in which any of the components is not true-or-false, the same is true for the whole combination. That sentences satisfying (T) should be closed under


sentential combinations is, therefore, a consequence of (CrT). Thus, we

should add a rule to the effect that if ~ t A ~-*True(A), then t-L ~ A ~, True ( -nA) and similar rules for binary connectives. Let us go one step

further and stipulate:

(15a) F-L (A ~-~True(A))-~ (-qA ~-~True(~A))

(15b) ~--t [(A ~-~True(A)) ^ (B ~-~True(B)] (A * B ~--~True(A * B))

where * ranges over sentential connectives. Note that (15b) follows already from the other rules if * is either ^ or v. (I do not know if this is

true where * is -~ and I did not check whether it can be deduced from simpler plausible rules.)

(CrT) can and, I think, should be generalized to cases where convention (T) does not apply a priori but follows from the truth of some

sentence, C. The criterion is obtained by formulating (CrT) for the f ramework which one gets by adding C to the set of valid sentencesg:

(CrT) For every A and C, ~t_ True(C) -~ (A ~-~True(A)) if and only if, for every B, such that ~- tTrue(C) ^ True(B)-~-q(True(A) v False(A)), we have t -LTrue(C) ~ ~True(B) .

Now back to the problem of "open situations". Consider a sentence A

in L. If ~ r True(A) v False(A), then the factual meaningfulness of A is

unconditionally guaranteed. This not being the case, we may still try to

characterize that state of affairs where A is true-or-false. Working in the

L- f ramework , we should accomplish this by using a sentence in L. Here we look for a necessary and sufficient presupposition for A, i.e., for a

sentence B such that F-r (True(A) v False(A)) ~ True(B). We can, of course, express in L + the truth-or-falsity of A, by True(A) v

False(A). But this would miss the p o i n t - w h i c h is to characterize the

truth-or-falsity of A using the conceptual apparatus of L. That is to say, we have to describe it in terms of people, stars, e lementary particles or what not - all those things treated in the framework. Thus, we look for a sentence, B, in L. However , a necessary and sufficient presupposition, B, need not constitute a satisfactory solution inasmuch as the factual meaningfulness of B itself may be unguaranteed, i.e., True(B) v False(B) need not be valid. If so, we can continue and look for a necessary and sufficient presupposit ion for B, and so on. If after finitely many steps we get a necessary and sufficient presupposit ion which is true-or-false uncondi-


tionally, we have reached our goal. For let B1 be necessary and sufficient

for A, and let Bi+l be necessary and sufficient for Bi, i = 1 . . . . . n - 1, and assume that ~LTrue(B , ) v False(Bn). Then letting C -- B1 ^ . . . ^ B,, we conclude by applying (iv) of w that I--LTrue(C) v False(C) and, by (i) of w C is necessary and sufficient for A. Here is an illustration:

Let A be 'The supernova which has been observed by the youngest son of the present mayor of Tegucigalpa preceded the crucification of Jesus Christ'. Presumably, our first step in securing the truth-or-falsity of A would be to guarantee the existence of the two events in question. Thus we try B1 = ' T h e youngest son of the present mayor of Tegucigalpa observed one, and only one, supernova, and Jesus Christ was crucified.' Now, trying to characterize the truth-or-falsity of B1 we put B 2 = 'The present mayor of Tegucigalpa has at least one son, and there was a human being who is presently called "Jesus Christ" ' . Then we go on and put B3 = 'There is a human being who is the present mayor of Tegucigalpa',

and, to be on the safe side, we might add B4 = 'There is at present, a municipality named "Tegucigalpa" ' . Feeling that B4 is unconditionally true-or-false we stop there. Now we argue, if B4 is true, then B3 is true- or-false, and if B3 is true, then B2 is true-or-false, etc. that is, if we are working in the framework of Newtonian physics. We can therefore choose as our necessary and sufficient presupposition, C, the conjunction

B1 ^ B2 ^ B3 ^ B4. (Here we rely on the fact that, by our rules, a conjunction is false if one of the conjuncts is. "~) If our framework is based on relativistic physics, we will add another conjunct, namely, 'The temporal ordering of the supernova which has been o b s e r v e d . . , and the crucifix- ion of Jesus Christ does not depend on the frame of reference' (or something to this effect, stating that the interval of the events in question is time-like). This part of the presupposition is inexpressible in a framework based on Newtonian physics.

But are we really sure that B4 is unconditionally true-or-false? Or that, if B4 is true, then B3 is true-or-false, and so on? May there not be circumstances in which we will hesitate when having to make one, or any, of these assertions? Of course there are. Trying to insure ourselves against circumstances of this sort we may add all kinds of details to our necessary and sufficient C. We may specify, for example, certain conditions for being the mayor of Tegucigalpa, ruling out, say, cases where the post is held simultaneously by two people. Or we may go into what

40 H A | M G A I F M A N

constitutes an observation of a supernova. But all that we can achieve are better and better (and at the same time more complex) approximations to the elusive description we are after. Regarding in certain cases the task as if it had been, or could be, accomplished, is a necessary idealization for

the purposes of systematic analysis, and for classifying sentences and conceptual frameworks. For frameworks behave differently in this respect: conceptual vagueness is always present to some degree (unless we work in a purely mathematical frame), and it causes truth-value gaps. But there are varying shades and degrees of vagueness. Some frameworks involve elements of indeterminacy, vagueness or open texture, in an essential way which cannot be ignored. Not only can we not regard, in these settings, every A as true-or-false, but we cannot even characterize by a true-or-false condition, the exact scope of A ' s truth-or-falsity. In other frameworks vagueness may be ignored or dealt with in such a way that we can, in principle, assume that the scope of the truth-or-falsity of any sentence can be determined in the framework itself by a true-or-false condition. (This, of course, includes frameworks in which every sentence is true-or-false a priori.) In the following definition I try to explicate formally (using the L +-terminology) this intuitive idea.

DEFINITION:

(i) Call framework L closed if we have an effective procedure ~1 of associating with every sentence, A, a sentence, B, such that ~--L (True(A) v False(A) ~ True(B) and P--L True(B) v False(B) (in words, B is a necessary and sufficient presupposition for A, which is true-or-false unconditionally).

(ii) Call L open if it is not closed. (iii) Call L weakly closed if we have an effective procedure of associat-

ing with every sentence A a sentence B such that ~--L (True(A)v False(A)) ~--~True(B) and ~'-L B ~--~ True(B).

(iv) Call L wide-open if it is not weakly closed. Obviously, every closed framework is weakly closed. But not vice

versa. The definition classifies frameworks into closed, open but not wide-open, and wide-open. This is but a first step towards an analysis of the various degrees of openness, or indeterminedness, which are built into frameworks. Let us see what distinctions are captured here.

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , I1

First note that:

41

C O R O L L A R Y 1. A framework is closed if and only if one can associate effectively with every sentence A a sentence A ~ such that ~ - L T r u e ( A ) ~ A ~ and ~-LTrue(A*) v False(A*), A similar necessary and sufficient condition for being weakly closed is obtained if we require that f--L A ~ ~--~ True(A *) (instead of I--L True(A *) v False(A *). [For if L is closed get B such that ~--L (True(A)vFalse(A))~--~True(B) and ~-LTrue(B) v False(B), then put A ~= B ^ A and use (ii) and (iv) of w Conversely, if we have sentences A ~* and (-1A) *~ which satisfy the above mentioned conditions for A and -~A, respectively, put B = A ~* v (-hA) ~. The argument for the second part is analogous, using (ii) and (iii) of 2.]

C O R O L L A R Y 2. (I) If L is closed, then, for every A in L

(C1A) ~L (A "~-~ True(A)) -~ (True(A) v False(A))

(II) If L is weakly closed, then the following are equivalent: (i) L is closed

(ii) For every A, (C1A) holds (iii) For every A, if F--L A ~--~True(A) then F-LTrue(A)

v False(A). [By (I) we have ( i ) ~ (ii); ( i i )~ (iii) is trivial and ( i i i )~ (i) follows directly from the definition. To prove (I), get B such that ~--L (True(A)v False(A)) ~--~ True(B) and ~-LTrue(B) v False(B). Then show that, assuming A ~ True(A), we get True (B)~ True(A) v False(A) and False(B) False(A). We leave the rest for the reader.]

Thus the distinction between closed and weakly closed but open frameworks rests on the difference between an assertion of True(A)v False(A) and an assertion of A *--~True(A), i.e., between objective truth- or-falsity and the holding of convention (T). In general, the first is the stronger and narrower concept, it entails the second, but not vice versa. In closed frameworks the two are identified; in frameworks which are only weakly closed they are not. First we consider examples of closed frameworks.


4 . E X A M P L E S A N D O B S E R V A T I O N S

C O N C E R N I N G C L O S E D F R A M E W O R K S

If every sentence is unconditionally true-or-false, then the framework is obviously closed. Such are mathematical frameworks which are based on a realistic conception of the theory in question. Say, the theory is first-order arithmetic. A realistic position here amounts to regarding every sentence in the language of first-order arithmetic as objectively true-or-false. (The reason for regarding them so and the ways by which we come to know the truth values of sentences do not concern me here.) Let us now look into some non-mathematical examples where truth value gaps may occur.

I start by discussing examples where the truth value gaps have a semantical but not an ontological significance. Then I shall go into some detailed examples where truth value gaps indicate ontological controversies. The chapter provides opportunities of applying my proposal in a precise way to some problem situations.

D E F I N I T E D E S C R I P T I O N S

Consider a first-order language, L, having variables, predicates, sentential connectives, as well as the ~-operator for forming definite descriptions (txc~ = the unique x such that ~b). L is supposed to be not a mere formalism but a well-understood language.

Let L0 be the sub-language obtained by excluding from L the ~- operator and with it, all definite descriptions (but retaining all the rest). We assume that in the L-f ramework every sentence of Lo is true-or-false. The truth-or-falsity of sentences containing definite descriptions depends, in general on the legitimacy of these descriptions, the legitimacy of Lx~b being expressible in L by 3 !x~b where this stands for ::Ixda(x) ^ Vx, y(q~(X) A qb(y)-~ x =y) .

As stated in w the closedness of the framework is equivalent to having an effective way of associating with every sentence A, a sentence A * such that ~-cTrue(A*)vFalse (A *) and ~--t T r u e ( A ) o A * . In the present case we assume that A * is in L0, which guarantees its truth-or-falsity. Once the mapping * is defined, everything about the L-framework is determined by the L0-framework. Since ~LTrue(A)*-~A , we have the following way of determining the L-validity of any sentence of L § (that is to say, any sentential combination of sentences of L and sentences of the

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , 1I 43

form True(A), where A is in L). Replace every True(A) by A ~; this will yield a sentence B of L ; then, ~-LB if and only if ~-L True(B), which is equivalent to I-LB ~. Thus, the set of L-valid sentences of L + is completely determined by the set of L-valid sentences of L0.

There are many ways of defining A ~ and not all of them lead to logically equivalent sentences. By employing different definitions, we may get different closed frameworks. But (assuming that, as far as L0 is concerned, they agree) they are all ontologically equivalent. For, as we shall see when treating of translations, the mapping which sends A to A ~ is an ontological translation of the L-f ramework into the L0-framework (i.e., the sub-framework whose language is L0). Hence these two are ontologically equivalent. (The mapping which leaves unchanged every sentence of Lo is a translation of Lo into L). If L ' is another framework, having the same language as L, such that in L ' True(A) is equivalent to A '

(where A ' is in L0), then also the U-f ramework is equivalent to the L0-framework. Thus the L and U-f rameworks are equivalent. In fact, ~ can be regarded as a translation of L into L', and ' can be regarded as a translation of L ' into L.

This observation regarding equivalence is also true in the following much more general situation:

Two frameworks L1, L2 share a common sub-framework Lo and we have two recursive mappings of the sentences of Li into those of Lo ( i = 1 , 2 ) such that ~-L,True(A)~--~True(A ~1) and ~-L2True(B)~--~

True(B~ZJ), where A and B are in L1 and L2 respectively, and A tll and B E2J are their values under the corresponding mappings. Then the L,- frameworks, i = 0, 1, 2 are all equivalent. (Note that we need not assume here that in L0 every sentence is true-or-false, or even that the frameworks are closed. If ~-t~,True(A llJ) v False(A El1) then, of course, A Ell can replace True(A~11); similarly for B.)

In the case of our present L and L' the frameworks have the same quantification domain (assuming some Tarskian interpretation of Lo). But in general, ontologically equivalent frameworks need not have any common domain for interpreting quantifiers. Neither is there any need to base an ontological translation on some matching of objects from one domain against objects from another.

The problem of translating from L into Lo can also be viewed as the problem of eliminating definite descriptions. In the Principia this is


treated via the concept of scope introduced for that purpose by Russell and Whitehead. The recipe which can be extrapolated from the Principia runs, roughly, as follows: With every occurrence of a definite description (in some sentence, or formula) one has to associate a subformula (or more precisely - an occurrence of a subformula) containing the description - to

be referred to as the description's scope. Choose an occurrence of a description tx4' (x) such that 4' contains no definite descriptions. (Without loss of generality assume that x does not occur in any other part of our sen tence -o the rwise replace x by some new variable.) If

6 ( . . . ~x4"(x) . . . ) is the scope of ~x4'(x), (i.e., an occurrence of 6 is the scope of this particular occurrence of tx4"(x)) replace this occurrence of

by 3!x4'(x) ^ Vx[4'(x)-~ 6 ( - - - x . . . ) ] . [Here we assumed that all the free variables of Lx4"(x) are free in 6; if n o t - l e t ul . . . . . Uk be the free variables of Lx4'(x) which are not free in 6. Then, there is a maximal

subformula o-(. . . Lx4'(x) . . . ) of 6 such that Ul . . . . . Uk are free in o-. Obtain ~,' from ~, by changing tr to Vx[4'(x)-~ ~r(... x . . . ) ] ; then, in the

whole sentence, replace ~ by (VUl . . . Uk 3 !X4") ̂ t~'. The free variables of and of its replacement are, in all cases, the same.] In this way an

occurrence of a definite description is eliminated. The scopes of definite descriptions in the resulting sentence are determined, in a natural way, as those which are ' inherited' from the scopes in the original sentence 72. The procedure may now be repeated until we get a sentence in L0. It can be

shown that the order in which we, thus, eliminate descriptions does not matter, in as much as all the final outcomes in L0 are logically equivalent (in the standard first-order logic). To be sure, any particular definition will have to fix somehow the order of elimination (say, by eliminating first the left-most description Lx4' in which 4' is description-free).

In languages of the Principia-type the concept of scope is essential. Any occurrence of a definite description in a sentence must be accompanied by a scope-marker which indicates the description's scope. Let LRw be the language obtained from L by adding scope-markers and making their employment obligatory. If B nw is the L0-sentence obtained by eliminat-

ing descriptions from a sentence B of LRW, then in the LRw-framework B Rw is not only the equivalent of True(B), but of B itself; that is to say, F-I~R~B ~--~B RW. The whole point of using scope-markers in B is that this

equivalence can be used to define the sense of B. Consequently, in LRw every sentence is true-or-false unconditionally and there are no gaps.


Sentences containing descriptions without scope-markers such as those in L (or, for that matter, in the natural languages) can be converted into

sentences of the LRw-type by employing some scope-convention, that is to s a y - a convention by which scopes are attached to occurrences of definite descriptions in sentences. If a sentence, A, in L is converted in this way into a sentence, B, in Lnw which, in turn, is translated into B Rw in Lo, then we can put A ~ = B Rw and get a mapping * from L into Lo.

This mapping is said to be based on the scope convention by which A is translated into B. However, in the Russell-Whitehead view the original sentence A is ambiguous and the scope-convention is simply a way of fixing its precise meaning. They would regard A * not merely as the equivalent of True(A) but as the equivalent of A itself; which makes a great difference. It guarantees the truth-or-falsity of all the sentences in L. (Furthermore, by the scope-convention of the Principia for sentences not displaying scope-markers we get (-qA) * = -q(A ~), for all A ; which leads to A ' s being true-or-false and equivalent to A ~, even if we do not assume this explicitly.)

Contrary to what Russell and others have thought, the avoidance of truth value gaps is no merit in itself. In point of being definite, clear-cut, sharply outlined, or what have you, frameworks without gaps are not any better than frameworks which are closed. What matters is not the absence of gaps but the possibility of defining exactly, by true-or-false sentences in the framework language, the circumstances in which gaps occur. Given any closed framework in which A * is a true-or-false

equivalent of True(A), one can set up an ontologically equivalent framework without gaps, simply by including all the equivalences A A ~ among the valid sentences. (The mapping ~ constitutes an ontological translation from the first into the second, as well as a translation from the second into the first.) Simplicity, conformity with existing intuitions, suggestiveness in pointing out analogies and similar heuristic factors determine the strong and weak points of ontologically equivalent

frameworks; these will depend on the context and the purpose of the framework's use. Whether it is preferable to regard A ~ as the equivalent of A or only of True(A) should be judged along these lines.

The problem of eliminating definite descriptions is not ontological, but a question of semantics. I would not have gone into such details, were it not for the fact that the question of defining the mapping ~ arises


whenever we have a closed framework. The present case shares certain basic features with other, more complicated and, ontologically speaking, more interesting problem-situations; for example, where the legitimacy of using certain predicates is at stake. Even when we are concerned solely

with definite descriptions, the basic concept should not be that of scope but that of the mapping + or, what is equivalent to it, that of the necessary and sufficient presuppositions for the truth-or-falsity of given sentences. The notion of scope is a guide by which we can, sometimes, define this mapping. It focuses our attention on the way in which the necessary and sufficient presuppositions for components (i.e. sub-formulas) of a given sentence affect the presuppositions of the sentence as a whole. A scope convention amounts to a relatively simple way of constructing the presupposition for the sentence from the presuppositions for its components, so that the structure of the sentence is preserved. Viewed thus, the notion generalizes and we can speak of scopes of predicate-occurrences or other expressions, where the possibility of "illegitimate" formations has to be

reckoned with; the scope of an expression is, roughly speaking, the maximal component whose truth presupposes the legitimacy of the expression as a necessary condition. There are translations, +, which do not derive from any conventions concerning scope; that it to say, even if a scope convention can be correlated with +, it will be some ad hoc construction of scopes incorporating explicitly the definition of + and more complicated than it; so that treating + in terms of scope would be quite useless.

Here are some ways of defining the mapping + for the language L. Let L ~1) be the framework in which A + = A ~ , where A ~1 is the translation of A into L0, based on the P r i n c i p i a scope convention, by which the scope of a description is the atomic formula containing it, By an atomic formula we mean a formula of the form R ( t l . . . . . tn) where R is a predicate and the t~ are terms (i.e., variables or definite descriptions). The above-mentioned procedure of eliminating descriptions yields in this case the following inductive definition:

(i) If ~b is in Lo then ~b ~1) -- 4~

(ii) ( R ( . . . . ~xl~bt . . . . . ~XkC~k . . .))r

=flX~ k l~!Xi~)(i I) A V X l . . . . . Xk( i=1 ~ R ( . . . x , . . . . . Xk)).


(We assume, without loss of generality, that, in the atomic formulas, x,

occurs only in the ith description. Variables which occupy argument places of R are left untouched.)

(iii) (-q~b) (1)=~(~b(1)), (~b * ~)( l )=4~(l) , ~b(l), (Ox4~)(1)= Ox(~b(l)), where * is any sentential connective and O is any qualifier.

In L (1) every sentence is true-or-false a priori because of the following

general observation: If ( ~ A ) * = -q(A *) then True(A) v False(A) is equi-

valent to A ~* v ~ ( A *) which, in turn, is valid because A * is true-or-false. The other extreme is to let the scope of any description be the whole

sentence in which it occurs. Roughly speaking, it means that the truth of any sentence presupposes the legitimacy of all the descriptions contained

in it. Let A <2) be the value of A under the mapping which is based on this

scope convention, and let L (2) be the f ramework in which True(A) ~-+ A (2) is valid. (The reader may construct by himself the inductive definition of (2).)

Note that L (2) does not satisfy the rules (1)-(14) of w 2. For no sentence

in which an illegitimate description occurs can be true in L(:); thus the

truth of A need not imply the truth of A v B and instances of tautological

schemas such as A -> A may be neither true nor false. This by itself should

not rule out L (2) as a working framework. In rules ( 1)-(15) I tried to set up

some definite workable frame without sacrificing too much of the gener-

ality of the treatment. (Thus, ~ ( A ^ q A ) has been included as a valid

sentence, and the truth of A was made to imply the truth of A v B; the validity of A v - q A was not included, for it would have been too restrictive.) Nonetheless an analysis along the lines of this essay can be

carried out for f rameworks not satisfying these rules. Violations of the

rules indicate that sentential connectives do not function as expected. It is easy to see that L (1) satisfies all the rules. As far as simplicity and technical

convenience are the main goals, L (1) is preferable. But, as Strawson has pointed out, a faithful representat ion of natural languages calls for

truth-value gaps. The trouble with L (2) is that it makes for gaps in a simple

wholesale manner. It is enough that B should include an illegitimate description, for A v B to be neither true nor false. For example, let A1 = 'The present king of France is bald' (construed as an application of the predicate ' . . . is bald ' to a definite description) and let A2 = 'snow is black' . The truth values in L (1) and L (2) for various compounds involving


A 1 and A 2 a r e as follows, where ' - - ' indicates a truth value gap.

A1 ~A1 A l v - n A I AI-~A1 AI-->A2 A I V - I A 2 L ~1) F T T T T T L (2) . . . . . .

We should perhaps take a middle course between L ~1) and L ~2). I find

each of the following two assignments of truth values more intuitive than those of L ~1) o r L (2).

A1 -hA1 A l v ~ A 1 Aa-->A1 A1->A2 AlV-nA2 - - - - T T - - T

- - - - - - T - - T

Roughly speaking, the idea is that instances of tautologies should be true even if they involve illegitimate descriptions, and a disjunction

should be true if one of the disjuncts is true; but except for t h a t - illegitimate descriptions should cause truth value gaps. In the first

assignment, ' tautologies ' means classical tautologies, whereas in the

second the tautologies are of an intuitionistic type. (Since the motivation

has to do with natural languages, one must treat in this context the problem of the natural reading of formal connectives. Does -nA~ corres-

pond to 'The present king of France is not bald' or to ' I t is not the case that

the present king of France is bald '? To discuss it here will carry us too far astray.) Let me define, as an illustration, a mapping * by which this

intuitive idea is precisely accomplished, where the tautologies are taken in the classical sense. The second intuitionistic interpretation leads to a

much more complicated mapping. In each case the resulting system

satisfies all of our rules (1)-(15).

Call a formula basic if it is either atomic or a negation of an atomic formula. Define * as follows:

(i) If 4) is in L0 then 4) * = 4). (ii) If L has the equality sign and 4) is t = t, where t is a definite

description, then 4)* = Vx(x = x) and (~4))* = ~Vx(x =- x). (Here ' = ' is used in a systematic ambiguous way, according to the context - it is either the equality sign of L or the equality sign of our metalanguage.) If t has

free variables and we want 4)* to have the same free variables as 4), we can add to it conjuncts of the form v = v.

(iii) If B ( . . . ~xl4)1 . . . . rXk4)k...) is a basic formula (i.e., ' B ' stands


for 'R ' , where R is a predicate, or for ' ~ R ' ) , which is not of the forms

treated in (ii), then B ( . . . ) * = / ~ = 1 ::l!xi057 AVXI. . . Xk(//~k=l 057--') B(. . . x l . . . . . Xk . . . . )).

(.iv) If 05 = B1 v . . . v Bn where the Bi are basic and if for some i and/', ~Bi = Bj then 05* = Vx(x = x); otherwise 05* = B~ v . . . v B2.

(v) If 05 is a Boolean combination of atomic formulas, transform it, using the standard technique of classical logic, into a normal conjunctive form, that is to say, into a conjunction 051 ̂ �9 �9 ̂ ~b,~ where each 05~ is a disjunction of basic formulas. .Then 05 ~ = 05 ~ ^ . . . ^ 05~.

(vi) If 05 is not of the form treated in (v), transform it, using the standard techniques of classical first order logic, into prenex normal form, i.e. into a formula Qlx . . . . Qmx,,,O where ~0 is a Boolean combination of atomic formulas and the Q~'s are quantifiers. Then 05 * =

Olx . . . . QmxmO ~. (Note that * is defined inductively, where the induction is first on the

depth of the descriptions occurcing in it and second on the complexity of the formula.)

Let L (3) be the framework in which True(A),~-~A* is valid. It can be proved that L (3) satisfies (1)-(15) and, furthermore, all the rules of the classic sentential logic. However, Vx05(x)~ 05(~x~) need not, in general, be valid in L (3). (By our rules, the truth of this sentence is equivalent in L (3) to 3x(-n05(x)v05(~xO)) *~, and the reader can easily construct a counter-example in which 05 and qJ are atomic.) Note that the mapping ~ is not based on any scope convention.

One can make up a suitable semantics for L (3) along the following lines:

Assume that we are given a Tarskian interpretation of L0, that is to say, a certain relational structure Mo for the language L0. We can associate with Mo a class, C(Mo), of structures of a certain type and define a certain (non-standard) concept of satisfaction for formulas of L in structures of C(Mo), such that the following holds for every sentence, A, of L : A * is satisfied in M0 if and only if A is satisfied in every structure of C(Mo). Thus, A is true if and only if it is satisfied in every structure of C(Mo), and

it is false (i.e. ~ A is true) if and only if it is not satisfied in every structure of C(Mo). In particular, we get: A has a truth value (in a world which is described by Mo) if and only if its truth value is invariant in a certain class (i.e. C(Mo)) of structures. This is one of many examples in which having a truth value can be formally explicated as having an invariant truth value

5 0 H A I M G A I F M A N

in a certain class of structures whose definition may involve set theoretical concepts. In fact, all closed frameworks I could think of can be analysed in this manner, including cases where, unlike the present example, truth value gaps are ontologically significant. This semantic analysis does not settle any ontological controversies. It is merely a very useful device, constructed within the framework of set theory (or a fragment of it), by

which various formal aspects of the formalism can be clarified. Of this later on.

Our present example, where truth value gaps are due to definite descriptions, is ontologically uninteresting, but formally it exhibits certain structural features which appear in other closed frameworks (such as the mapping*, scope conventions and the just mentioned characteriza-

tion of truth value gaps in terms of classes of structures). It is for this reason that I treat it in some detail as an illustrating example. The precise definition of the class C(Mo) and the concept of satisfaction which is used

13 for structures in C(Mo) are, however, relegated to a note Variants of L ~3) are obtained by trying to incorporate into the

framework additional intuitions concerning definite descriptions. Thus, one could argue that rx~b = Lx4, should be true whenever ~b and 4J are logically equivalent, or equivalent in some other sense. I will not go here into the ramifications to which intuitions of this sort lead 14

None of the frameworks L ~), i = 1, 2, 3, represents faithfully actual

usage in natural languages. Better approximations require much more complicated rules for fixing truth values and truth value gaps. ~ 5 Further- more, interpretations of sentence tokens in the natural languages depend on the wider contexts in which these tokens occur. No mapping, ", which is defined for sentences can do justice to such phenomena. A natural language should, therefore, be represented not by a single formalism but by a bunch of ontologically equivalent formal frameworks (unless the dependence of the sentences on wider contexts is incorporated into the formalism itself). The problem of fixing formal rules which faithfully represent actual usage of definite descriptions is part of the general program of the formal analysis of natural languages and cannot be solved apart from it. All this is, however, a question of semantics, not of ontology. Is there any possible fact which our ordinary usage of descriptions enables us to describe, but which cannot be described in systems

ONTOLOGY AND CONCEPTUAL FRAMEWORKS, II 51

such as L? The answer is no. But our usage is just too complex to be

faithfully reflected in L or similar systems.

P R O P E R N A M E S

Truth value gaps can also be due to non-denoting proper names. Assume

that L contains proper names ( 'Kurt G6del ' , 'Tegucigalpa', 'Arcturus ' ,

'~-'). A name without denotation is the analogue of an illegitimate

description. In order to define the mapping ~, we have to express in L the

fact that a given proper name has a denotation. Let a be a proper name

and x a variable. The only value of x which can satisfy x = a is the

denotation of a. Assume that any object which a may denote is a member

of the range of x. By the customary (referential) interpretation of the

existential quantifier 3 x ( x = a) is true if and only if a has a denotation.

Using this device ~ can be defined in ways completely analogous to the

definitions underlying L ~i~, i = 1, 2, 3, and other variants, with 3 x ( x = a)

in the role played previously by 3!x& Again we can use the notion of

scope, namely - the scope of an occurrence of a proper name. The scope

convention will determine the manner of incorporating the conditions

3 x ( x = a) into the sentence A ~ (where A includes the proper names in

question). The mapping * will, of course, deal simultaneously with proper

names and definite descriptions (as well as any other possible source of

truth value gaps). It may involve different policies concerning proper

names and definite descriptions (such as different scope conventions).

Furthermore, the treatment of proper names may be parallel to the

treatment of descriptions in L ~3~, or a similar variant, and thus be

uncorrelated with any scope convention. The precise details of the

possible definitions can be easily drawn. For example, extending the

policy of L ~11 to proper names would result in defining (R (a , b)) ~

(where R is a binary predicate) as 3 x ( x - a ) ^ : : l y ( y = b ) ^

Vx, y ( x = a ^ y = b ~ R ( x , y)) , and in adopting the same clauses for sen-

tential compounds and quantifiers as in the definition of ~ Note that the sentence A * will be true or false a priori. The interpreta-

tion of quantifiers on which the device rests implies it. Let us add to the

formulas of Lo all equalities of the form x = a, where x is a variable and a a proper name, and then close it under sentential combinations and

52 H A I M G A 1 F M A N

quantification; let L~ be the language whose set of formulas is the set thus obtained. Then every sentence of L~ is true-or-false and ~* is a mapping into L~. The treatment of proper names is much simpler, here, than that of definite descriptions because there is no analogue to the phenomen of nested applications of the L operator.

By Russell's theory, proper names are representatives of descriptions. One could in this view eliminate names by substituting suitable descriptions for them. If the substituting descriptions contain proper names, one could repeat the process. Assuming that in each case, after finitely many steps the remaining names, are unconditionally guaranteed to have denotations, we can proceed and eliminate the descriptions as before. All this would lead to another definition of ~. Kripke, Kaplan and others, reviving an idea of J. S. Mill, proposed a theory by which proper names are labelling devices. They are passed on along human communication channels and historical traditions. The denotation of the name is, roughly speaking, that object which is linked to it by a casual communication chain (or bunch of chains) along which the name has reached us (possibly undergoing changes along its way). There is no reason why L should not be rich enough to describe all this. Descriptions referring to possible communication chains along which a particular name may have reached us could be formulated inside L. Conditions for denoting can then be formulated and used in the construction of A ~. After all, when a scientist wants to find out if 'Tully = Cicero' is true, he should check (according to this theory) whether the main traditions by which these names have reached us lead back to the same individual. The truth of this sentence is equivalent to the obtaining of a certain state of affairs. The conceptual framework employed by that scientist being, among the rest, about such state of affairs, the language should have a vocabulary to describe them. Kripke, Kaplan, et al. (if I understood them correctly) would prefer it o the rwise - they would exclude from L the necessary machinery for describing the account which they themselves give of proper names. Instead of which they would use the device of ' 3 x ( x = a) ' . Here we see how different explication goals may result in different ways of parcelling our conceptual stock. The philosopher prefers the parcelling which lends greater clarity and brings into sharper focus that point of view which he aims at explicating. ~6 Needless to say, the difference between the various theories concerning the denoting of proper names is semantic and not

ONTOLOGY AND CONCEPTUAL FRAMEWORKS, II 53

ontological. It gives rise to different closed frameworks. But obviously two-way translations render the frameworks ontologically equivalent.

M A N Y - S O R T E D L A N G U A G E S A N D

C A T E G O R Y M I S T A K E S

L may be (in addition to what has been said until now) be a many-sorted language. That is to say, all terms (variables, descriptions and proper names) are classified into several sorts. With every k-place predicate, P, a k-tuple of sorts is associated: whenever P(tl . . . . . tk) occurs, it is required that the k-tuple of the sorts of tl . . . . . tk should be the one associated with P. (This can be generalized by allowing to associate with any predicate more than one tuple of sorts.) Violations of these restrictions are described as category mistakes (e.g. 'Julius Caesar is smaller than 7r'). If category mistakes are not ruled out as ungrammatical, they may give rise to truth value gaps. We assume that the language, including the classification into sorts and the association of sorts with predicates, is given recursively. Hence we have a recursive procedure for recognizing category mistakes. In the rules defining ~ we can include a stipulation that if

P(tl . . . . tk) is a category mistake, then P ( t , , . . . , tk) ~ is some fixed contradictory sentence (e.g. ~ V x ( x =x) ; if required, we can add as dummy variables the free variables of the atomic formula). The complete definition of ~ determines, inter alia, the effect which a category mistake has on larger compounds containing it. Each of the policies underlying L (i~, i = 1, 2, 3, and other variants, has its obvious analogue for the treatment of category mistakes. In particular, one can introduce the notion of the scope of a category mistake, and use some scope convention to define the mapping ~. More involved policies, like those of L (3), cannot

be traced to scope conventions. The details, I trust, are by now obvious and are left to the reader.

If L combines all the features discussed so far (i.e. has the ~-operator, proper names which may lack denotation, and is many-sorted), the mapping # will have to take care of all of them simultaneously. It will incorporate policies for definite descriptions, proper names and category mistakes (as well as any other factors which may cause truth value gaps). The interplay of the different factors may also give rise to additional degrees of freedom, besides those of choosing a policy separately for each


factor; (e.g. special additional stipulations may be used for certain expressions which simultaneously involve category mistakes as well as illegitimate descriptions). There is no point in going any further into technical details. So far, all the different mappings ~ result in ontologically equivalent systems. Let us now discuss an entirely different case where truth value gaps indicate different ontologies.

S O M E O N T O L O G I E S O F S P A C E A N D T I M E

The following language, K, can serve as a formal system through which certain prerelativistic ontological views concerning space and time are explicated. By means of K the customary Euclidean spatio-temporal structure is described, time points are associated with events, and spatio- temporal locations with material bodies. The nature of events or material bodies is left completely unspecified; they figure as idealized abstract elements. I chose this restricted language because it is simple and yet adequate for illustrating certain ontological questions.

K is a many-sorted language. Its terms (i.e. constructs in the category of variables or names) are classified into the following five sorts:

(i) Terms of sort TP which are supposed to range over, or denote, time points.

(ii) Terms of sort SP which are supposed to range over, or denote, points in a three-dimensional Euclidean space.

(iii) Numerical terms ranging over, or denoting, real numbers. (iv) Proper names of particular events, so r t - Ev. (v) Proper names of particular material bodies, so r t - MB.

We have quantifiable variables of each of the first three sorts. These three sorts belong to the sublanguage, Ko, of K which describes the "pure" spatio-temporal structure. Ko has the equality sign for each of the three sorts, that is to say, 't~ = t2' is a well-formed formula, whenever tl and t2 are of the same sort (alternatively we could use three equality signs). In addition, Ko has the following predicates and function symbols:

(1) A binary predicate, ' < ', for temporal precedence; 'tl < t2' asserts that the time point t~ precedes the time point t2.

(2) A 4-place function symbol ' r / which takes terms of sort TP as arguments; r,(vl, v2; v3, v4) is the ratio of the time interval (vl, v2) to the time interval (v3, v4). It is a real number and thus 'rt(v~, v2; v3, v4)' is a


numerical term. (We recall that the ordering < imparts a direction to the

intervals; speaking informally, if v ~ v 2 and v 3 ~ v 4 the ratio rt(vl, re; v3, vn) is positive if and only if v~ < vz~--~v3 < v4. Reversing the order of either the first pair of arguments or the second, reverses the sign of the ratio.) If the last two arguments denote the same time point, the ratio is undefined. In order to avoid meaningless terms, a value can then be assigned by an arbitrary convention, say the value if 0 (i.e., we have

'/)3 =/)4--> rt(t~l, 1)2; /)3, /)4)~--0 ' as an axiom). (3) A 4-place function symbol 'rs' which takes arguments of the sort

SP. The term 'rs(xl, x2; x3, x4)' is a numerical term and denotes the length ratio of the line segment (Xl, x2) to (x3, x4). Since the intervals in this case are not directed, the ratio is always non-negative. The above mentioned

convention is also adopted for the case x3 -- x4. (4) In addition, we have in K0 the function symbols ' + ' and ' . '

denoting, respectively, addition and multiplication of real numbers. Other numerical functions and constants (such as subtraction, or 0 and 1). as well as the ordering, < , of the real line can be defined in terms of these and, therefore, their names can be used freely as a convenient shorthand (e.g., define 'x < y ' as ' 3 z ( z + z r z ^ x = y + z �9 z ) ' ) . K . has quantifiers and the usual sentential connectives.

All the concepts of elementary geometry can be expressed in Ko. (For example, to say that point y is on the same line and between points x and z we use ' x ~ z ^ ^ y ~ z ^ x C y ^ rs(x, y ;x , z ) + r ~ ( y , z ; x , z) = 1'). Had we explicitly included in K0 the machinery of elementary geometry, we could

have dispensed with the number-theoretical terms altogether. For, as is well known, elementary real number theory can be expressed within the framework of elementary geometry. Having numerical terms is, however, much simpler; besides, the reduction to elementary geometry is not feasible if we increase the number- theoret ic language by including, say, the constant 'Tr', or by adding a predicate to denote the subset of natural numbers. (Anyway, a curtailment of the quantification domains does not, in itself, signify any ontological gain.) There is also a converse reduction of geometry to real number theory. It can be used to eliminate geometrical terms, as long as we are interested in K0 only. But in the wider context of K it does not result in a framework with the same expressive power.

K is obtained by adding to K0 the following items:

56 H A I M G A 1 F M A N

Names, say el, ez . . . . , of particular events, names, say 'a ' , 'b', 'c', . . . . of material bodies, and two function symbols, a unary ' r ' and a binary 'p' denoting, respectively, the mappings which associate time points with events and spatial locations with material bodies and time points. Thus, 'z(e) ' , which is of sort TP, denotes the time of the event e, and 'p(a, t)', of sort SP, denotes the centre of gravity of the body a at the time t.

[The second stroke of Big Ben at 5 p.m., December 21, 1975, is an example of an event which may have a name in K. Usually an event is regarded as taking place during a certain time interval. The reader is

advised to think of r(e) as the middle point of the duration interval of e. Actually, the interval is not completely determined'; there is always an unavoidable residual vagueness. In K this is ignored. We shall see in another chapter what happens if we incorporate the vagueness in the framework. The same applies to the inherent indeterminateness in defining the precise boundaries or the centre of gravity of a particular

body, such as the moon, or a particular animal. We also ignore the fact that, usually, a given material body does not exist at all times. We do so for the sake of simplicity; K can easily be modified so as to take care of this point.]

In Ko one can formulate the axioms of elementary real number theory, as well as axioms stating that the space (over whose points range the variables of sort SP) is a three-dimensional Euclidean space, and that the time line is a Euclidean line directed by < . All these will be referred to as

the classical axioms of space and time. In present-day terms Newton's space-time ontology can be explicated

as follows:

(i) The usual rules of first-order logic apply in K. (ii) The classical axioms for space and time are valid.

(iii) Every sentence of K is objectively true-or-false.

Note that the applicability of standard first-order logic does not entail the truth-or-falsity of every sentence in the system. (The validity of A v-qA does not necessarily imply that of T rue (A)v False(A). The framework L ~3~ of the previous sections satisfies all the rules of the classical sentential calculus but contains sentences without any truth value. Likewise, truth value gaps can occur in systems satisfying the rules of the classical predicate calculus.)


Newton believed in an absolute space. (iii) is a precise expression of this view. To explain the absoluteness of space as the existence of space points is of no help, unless this vague and ambiguous statement is meant as a heuristic outline to be further explicated by means of (iii), or similar specifications. Neither is the quantification over space points of much significance here. Quantifiable variables of sort SP are already included in K0, but all the statements of the Ko-framework can be completely interpreted in terms of real number theory. (One can set up a model of K0 consisting of a directed geometrical line, a three-dimensional Euclidean

space and the field of real numbers; the first two items can be reinter- preted in terms of the real number field so that, all in all, Ko covers just the

real number field.) Any sentence of K0 can be translated into a number- theoretic sentence and, as is well known, the elementary theory of real numbers is decidable. We can enrich the number theoretic machinery of Ko, so as to obtain an undecidable theory, but we still are in the domain of pure mathematics�9

It can be argued that K0 should be regarded as a description of the actual (physical) space and time, which is quite different from treating it as a purely mathematical construction. But how, then is this difference made manifest? Believing that actual space is not Euclidean, one would reject the classical space-time axioms and arrive at a K0 framework which violates (ii). One can, however, accept the classical picture of space while rejecting its absoluteness. Leibniz held that "space is nothing else than the order of existing things", "something merely relative", " the order of bodies among themselves", or "an order of things which exist at the same t ime" and thus, it is not "something in itself"; it has no "absolute reality". 17 Geometry, according to Leibniz, describes the possible mutual relations of bodies, not a self-existing empty space�9 All the same, Euc- lidean three-dimensional geometry is a true description of objective reality. He also attributes to geometrical truths a necessity of the highest order. Thus, he argues, it is inconceivable that actual space should not be three-dimensional because it is a theorem of geometry that there can be no more than 3 mutually perpendicular lines; a similar objective and necessary status is accorded to the mathematical apparatus which deals

�9 �9 1 8 with ratios of segments or other quantmes. That the sentences of Ko are objectively true-or-false and that the classical space-time axioms are true accords fully with Leibniz's conception. A Ko sentence of the form 'Bx&',


where x is of sort SP, would be interpreted by Newton as asserting the existence of a space point having a certain property; Leibniz would read it

as a statement about possible body configurations. But as long as both are working within Ko no discongruity will arise. Is there any difference that matters between their conceptions? Is there any real issue to the problem of a self-existing empty space? The answer is yes. But the difference emerges only upon passing from Ko to K. Leibniz would reject (iii). The

rejection is not related to the role of the quantifiers but to the employment of certain atomic constructs (either by themselves or as constituents in bigger compounds). In particular, certain atomic sentences of K will be regarded by him as meaningless.

Is the (spatial) location of the body b at the time t~ the same as its location at the time t2? In Newton's conception the question has an objective yes or no answer, but in Leibniz's ontology it is meaningless (except for the trivial case where tl = t2). One can ask whether the distance between two bodies changes when we pass from time tl to time t2; or, more generally, whether a body's location relative to a given system of bodies is the same at two different times; but one cannot, argues Leibniz, speak of absolute (i.e., non-relative) changes of locations. ~9 Here is the real issue of the debate: it bears, among the rest, on the nature of motion. 2~ Using K, it can be pinned down as follows: In Leibniz's ontology, the sentence

p(al, z(e l ) )= p(a2, ~-(e2))

is meaningless, unless it happens that z(el)= "r(e2), in which case the sentence can be rephrased as: p(al, ~-(el))=p(a2, "c(eO). Furthermore, there is no place for expressions such as

rs(p(al, tO, p(a2, t2); p(a3, t3), p(a4, t4))

where 'ti', i = 1, 2, 3, 4, are different terms of sort TP; for there can be no meaning in asking, say, whether the interval between body a l at time tl and a2 at time t2 is twice the interval between a3 at time t3 and a4 at time t4. One can, however, compare the interval between al and a2 both at time t, with the interval between the same or another pair, both at time t'. That is to say, the expression

rs(p(al, t), p(a2, t); p(a3, t'), p(a4, t'))


is legitimate (thus, the preceding expression would be meaningful if we

presuppose that t~ = t2 and t3 = t4). To define precisely the sentence constructions of K which are true-or-

false in Leibniz's ontology, it is technically convenient to eliminate first

the variables of sort SP. For, using quantified SP variables, sentences can 4

be made which do not explicitly involve the above-mentioned illegitimate constructions but are equivalent to sentences which do; for example:

3x[p(al, ta)=x ^p(a2, t2)=X], is equivalent to p(al, t l )=p(a2 , t2). Not that quantifiable SP variables have any significance for marking the

difference between the two ontologies. The fact is that Newtonian ontology, which is committed to absolute space, is fully expressible in a sublanguage in which no variables of sort SP exist. Let /s be the sublanguage obtained by omitting from K the variables of sort SP, then we have:

T H E O R E M : There is a primitive recursive mapping which associates with every sentence A, of K, a sentence ,4, of K, such that the hi-implication A ~ , 4 is provable from the classical space-time axioms. (This is also true for formulas, O, whose free variables are of sort TP or numerical; the translation ~ will have the same free variables.)

Thus, every fact expressible in K is expressible in/~, and Newton's ontology amounts to accepting all sentences o f /~ as true-or-false. Note that if we replace K by K0, the result is the well-known reduction of pure geometry to real number theory; the whole point of the claim is that this can be done also for the "non-pure" K which includes names for material

21 bodies and the location function p The a priori valid sentences of /~ are those of the form ft,, where A is a

priori valid in K, as well as the sentences which are derivable from them in first-order predicate calculus. Assume that the a priori valid sentences of K are the sentences derivable from the classical space-time axioms (i.e. we do not regard any particular factual statement concerning bodies and events as a priori valid). Then /~ can be axiomatized as well; that is to say, there exist a recursive set of a priori valid sentences in/s which implies all the others. It can be shown that the following axioms will suffice:

(i) The number theoretic axioms and the axioms concerning time (these are the same i n / ( as in K) and


(ii) The sentences of the form fi,(al . . . . . an) where A ( a l . . . . . an) is the following sentence of K

Vtl . . . . . tn3vl . . . . . vn[ /~p(a i , ti) = vi]

Here the 'v~' are of the sort SP. Note that, since the 'p(a~, ti)' are of the sort SP, A ( a l , . . . , an) is a theorem of first-order logic. But its translation, fi,, is not. We have to add it to the list of axioms in / ( . Roughly speaking, A(a l . . . . . an), expresses in / ( the fact that the orbits of the bodies a l , . . , an lie in one three-dimensional Euclidean space (see Note 21).

The axioms which state that the SP variables range over a three- dimensional Euclidean space, where 'rs' is interpreted as the distance- ratio function (including the convention rs(vl, v2;v, v)= 0) are stated in the 'pure' sublanguage K0. They all become derivable number theoretic theorems upon being translated into/( . If additional axioms (e.g., statements expressing some particular facts concerning certain events and bodies) are included in the K-framework as a priori valid, then we will include their translations into / ( as axioms in the corresponding /( framework.

Now let /(L be the sublanguage of /( obtained by imposing the following restrictions: An equality whose sides are occupied, by terms of sort S P can figure in a formula only if it is of the form 'p(a, t) =p(b , t)' (where t is of sort TP); furthermore, 'rs' can be used only in terms of the form ' r~ (p ( a, t ), p ( b, t); p ( a', t'), p ( b ', t') )'. Leibniz's space-time ontology can now be expressed precisely as an affirmation of the truth-or-falsity of all sentences of/(L. Sentences o f / ( which are not in/(L are, in this view, true-or-false only if by some agreed convention they can be translated and reread as sentences of/(L. Only by virtue of such a translation do they acquire an objective status. In short, all the facts which one can describe i n / ( are describable in/(i.. As for sentences in K, they can be translated into/( , and then this criterion applies. Concerning a priori validity in/(L we have the following:

Suppose that a priori validity in the K-framework coincides with derivability from the classical space-time axioms (then the a priori valid sentences o f / ( are those derivable from the above-mentioned/(-axioms, (i) and (ii)). The a priori valid sentences of/(L will be all the a priori valid sentences o f / ( which are in/(L- One can show that they are axiomatizable inside /(L- The axioms are obtained by replacing the axioms of the above-mentioned (ii) (of which none is in/(L) by the following groups:

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , I1 61

(iia) Axioms stating, roughly speaking, that for every true t the axioms of three-dimensional Euclidean geometry holds, where 'p(al , t)', . . . . 'p(an, t)' are interpreted as geometrical points and 'rs' is interpreted as the distance-ratio function. (Besides the universal quantification over t we need here only the language of real numbers).

(iib) p(a ' , t ' )~p(b ' , t ' ) -~rs (p (a , t ) , p(b,t); p(a' , t ' ) , p(b ' , t ' ) ) . rs(p(a', t'), p(b', t'); p(a", t"), p(b", t")) = rs(p(a, t), p(b, t); p(a", t"), p(b", t")) as well as the convention p(a', t') = p(b', t ' )~ r~(p(a, t), p(b, t); p(a', t'), p(b', t')) = O.

If a statement describing in K some particular fact concerning events and bodies is included as a priori valid, then, in order to qualify as factually meaningful in the /~L-framework, it should belong to, or be reconstructed in,/~L. Then we can add it to the axioms in the corresponding/~L-framework-

The situation becomes even clearer if we eliminate SP terms altogether. This can be done without diminishing the expressive power of /~ (and, hence, of K). Omit f rom/~ the function symbols 'p' and 'r~' and add, instead, a 4-place predicate EN and an 8-place function symbol 'rN' ( 'N' stands for 'Newton') which are interpreted as follows:

'EN'(al, a2, t~, t2)' is interpreted as the equivalent of 'p(a~, tl) = p(a2, t2)'

and

'rN(al, a2, a3, a4, tl, t2, t3, t4)' as the equivalent of 'rfip(al, tl), p(a2, /2); p(a3, t3), p(a4, t4)) ' .

Let the language thus obtained be /(N. Every formula of /~ can obviously be translated into/~N and vice versa. (The only uses of 'p' and 'rs' in / ( are to make up compounds of those two forms which are translatable into/~N. The sentences of/~N are, thus, trivial reformulations of the sentences of/~. Note, however, that when we translate an axiom system f r o m / ( into/~N we must include the equality axioms for terms of sort SP. For, equalities of this kind are now replaced by expressions of the form 'EN(. . . ) ' which, formally, are not equalities. The logical axioms of equality must be reformulated in terms of 'EN' and included in whatever system we have. A similar observation holds for the for thcoming/~ . )

62 HAIM GAIFMAN

Now, let/~L be obtained by replacing the 4-place 'EN' by a 3-place 'EL' and the 8-place 'rN' by the 6-place 'rL ', where the intended interpretation of these in terms of KN is:

EL(a, b, t ) ~ Df Eu(a , b, t, t)

rL (al, a2, a3, a4, tl, 12) ~-Df rN(al, a2, a3, a4, tl, tl, te, /2)-

Thus, /s can be regarded as a sublanguage of /~N. In Newton's ontology every sentence of /~N is true-or-false, whereas in Leibniz's ontology /~u sentences are true-or-false only if they can be reread as sentences of/<L.

If/s is chosen as the language of the framework expressing Leibniz's space-time conception, then every sentence is true-or-false and the framework is, trivially, closed. If it is/s then we have to set conditions for truth-or-falsity. The conditions can be arrived at as follows:

Terms of the form 'rN(...) ' and formulas of the form 'EN(. . . ) ' are legitimate (in Le'ibniz's view) only when certain presuppositions hold. The presupposition for 'rN(al . . . . . a4, tl . . . . , t 4 ) ' is 'tl = t2 A t3 = t 4 ' and that for 'EN(al , a2, t~, t2)' is tl = t2'. These conditions can now be used in the definition of the mapping # which assigns to every sentence, A, of/~N an a priori true-or-false sentence A * which is equivalent to True(A), The a priori truth-or-falsity of A ~ is guaranteed if it is a sentence of /s (considered as a sublanguage of KN). The situation can be viewed in analogy to the case of definite descriptions, as discussed at the beginning of this chapter, with/s in the role of Lo. Legislations concerning L have counterparts for/~N. In particular, the notion of scope can be applied to occurrences of 'rN' and 'EN', so that various scope conventions give rise to various mappings 4~. As in the case of the framework L, scope conventions cannot express the fine points of the more intuitively appealing translations. These are too complex, or subtle, to derive from scope conventions. The present situation is furthermore complicated by the numerical function symbols. How does the legitimacy of 'rN(a, t)' (where a = a l . . . . . a4 and t = tl . . . . . t 4 ) affect the status of ' r N ( a , t)+rN(a', t')'? Does 'rN(a" t) �9 0' unconditionally denote 0 including the cases where 'rN(a, t)' is illegitimate? These are questions which the policy for defining 4~ has to settle one way or another. There are several alternatives. For example, we may decide to regard Boolean combinations of numerical


equalities as true, if they are true for every assignment of numerical values to the illegitimate expressions 'rN(...) ' (where every legitimate expression denotes the number corresponding to it in our actual world). Thus, for example, 'ru(a �9 t) �9 0 = O' would always be true. A similar policy may be applied to occurrences of 'EN(a 1, a2, tl, t2)'. Namely, a sentence is true if it is true for every assignment of truth values to the illegitimate expressions 'EN(. . . ) ' occurring in it. Using these two ideas, one can set up a precise definition of #e (somewhat in analogy to the definition of ~ in the Lr We leave the details to the reader.

The different mappings, ~ , of/~N into/ (L give rise to different but ontologically equivalent closed frameworks representing Leibniz's ontology. There is no point in asking which represents better the historical Leibniz, for these distinctions belong to a semantical analysis which did not exist at his time. On the other hand, the distinction between the conception that every sentence of/(N is true-or-false, and the conception that sentences of/(N can be regarded as true-or-false only by virtue of a translating convention in to/ (L is a modern reconstruction of an actual historical controversy. Unlike the differences which arise due to varia- tions in the translating function, ~ , this one is a genuine ontological disagreement.

How is one to know, the reader might ask, whether the difference is genuinely ontological? The final answer will be provided through the general criterion of translatability. The essence of it, in this particular situation, can be presented right now. Let us confront the pair of languages L0 and L, as defined at the beginning of this chapter, with the pair /(1. and /(N. In both, the first of the two is a sublanguage of the second. But the L-framework is, by definition, no richer than the Lo-framework, in that every fact describable in L is also describable in L0. Various translations of L into Lo amount to different interpretations of certain constructs in L. But there is an agreement concerning Lo, and there is an agreement that L0 is sufficient for describing every state of affairs which is describable in L. Now, Newton and Leibniz agree as t o / ~ but there is no agreement on the second point. Suppose that Leibniz uses the language/~N together with a mapping, # , by which (like the mapping ~) of L !1)) every sentence of /(N comes out true-or-false. For example, assuming that 7(el) ~ r(e2) the sentence ' r s ( a l . . . . . a4,

r(eO . . . . . ~'(e4)) = l ' is regarded by him as false and its negation as true

64 HAIM G A I F M A N

(just as Russell regards 'The present King of France is bald'). Could this constitute a common language for the two conceptions? By no means. Newton would object to Leibniz's interpretation of/~N in terms of/s not, however, because he prefers a different interpretation, bus because tie maintains that certain states of affairs describable in /s cannot be described in/~L at all. Whether 'r~(al . . . . . a4, ~-(el), � 9 r(e4)) = 1' is true cannot, in his view, be settled simply be observing that ~'(el) ~ ~'(e2). The proposed Leibnizean interpretation would make the following sentence a priori valid

T(el) # T(e2)~False(rN(al . . . . . a4, r(e2) . . . . . ~-(e4)) = 1),

which Newton would, of course, reject. Any proposed recursive translation of KN - as conceived by Leibniz, into KN - as conceived by Newton will lead to discrepancies of this kind. This fact is the precise expression of the ontological difference.

Non-equivalent ontologies can, and generally do, have 'overlapping' or 'common' parts. For example, the /s is common to both

A

Newton and Leibniz; the controversy is about whether KN has any factual meaning over and above/~L. In general, a common part can be defined as a partial equivalence, that is to say, as a pair of partially recursive translations from one framework into the other and vice versa, where 'partial' indicates that the translation need not be everywhere defined. For example, the identity mapping of every sentence of/s to itself is both a partial translation from the/~L-framework into Newton's/~N and from Newton's/~N into/~L. (The partialness comes out in the second direction, where the mapping is not defined in the whole of/~N.) Common parts make feasible communications between upholders of non-equivalent ontologies. (One of Feyerabend's errors has been to overlook the role of partial translations. Communication was conceived by him as a zero-one affair, where non-equivalent frameworks are completely disconnected. In fact, communication is graded.) A partial translation indicates a certain overlap of usage which, in turn, establishes an overlap of meaning.

Now, the same ontology can be represented by different but ontologi- caily equivalent frameworks (such as L ~i), i = 1, 2, 3, or K, I~ and/s Newtonian conception). If two non-equivalent frameworks, say L1 and L2, have a common part, then the "same" common part will be shared by any other pair, say L'~ and L~, which represent, respectively, the same

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , I1 65

ontologies. By combining the recursive mappings which establish the equivalences of Li and LI, i = 1, 2, with the partial translations which constitute the common part of L1 and L2, we get the induced partial translations constituting the "same" common part of L'~ and L;. The following commutative diagram gives the picture:

partial translations L i , ' L2

equivaence I I equv ence induced partial

translations

The induced translations are, of course, recursive, but they may be quite different from the original ones in point of simplicity or obvious- ness. An apt choice of the representing frameworks may display a common part which would have been obscured in other representations. Thus, the change from K to/~, or to/~N clarifies considerably our view of the relation of Leibniz's ontology to Newton's. The partial translations become the identity mapping over /(L, and Leibniz's conception is displayed as an affirmation of a simply defined sublanguage of the Newtonian framework. His view can therefore be regarded as a partial affirmation of Newton's ontology. This sort of relationship is a property of the two ontologies in question, independent of the representing frameworks; but our choice of /~N and /(L reveals it in a perspicuous manner.

The clear display of certain common parts (or, in other words, partial equivalences) is an argument for preferring some representations of given ontologies to others. To enhance communication and to establish common patterns of usage is a goal in itself. Here the way of presenting the ontology becomes very important. Certain common parts which are obscured, perhaps even unsuspected, in one representation (because of the complexity of the partial translations) may emerge clearly in another. (For that matter, even full equivalences of frameworks may go unnoticed because the translations are not so obvious.) These considerations can


guide us in choosing the mapping # which determines the representation of Leibniz's ontology in terms Of the language/(N. The mapping should be

chosen so as to m a x i m i z e the explicity displayed common patterns o f the

two systems. For this reason it is preferable to construe 'rN(al . . . . . a4, r(e l ) . . . . . z(e4))" 0' as a term denoting 0, even where 'rN(al . . . . . r(e4)' is illegitimate. We may imagine Leibniz arguing as follows: Newton regards, erroneously, rN(.. .) as if it were a real number reflecting some feature of objective reality. Actually, if z(el) # z(e2) or "r(e3) ~ 7"(e4), there is no real number pertaining to the world in the way imagined by Newton; ' rN(. . . ) ' is then meaningless. But there is no harm in stipulating that ' rN(.. .) �9 0' denotes the number 0. We can do so as a

matter of convention. Newton would regard it as denoting the product of some objectively determined unknown and the number 0. But since the outcome would be 0, no matter what rN(.. .) is, we will find ourselves in agreement as to the value of 'rN(. �9 .) " 0'. Acting on the same principle, I can come to an agreement with Newton when evaluating numerical expressions (say polynomials), provided that their values do not depend on what the Newtonian mythical numbers might be. Likewise I can assign truth values to statements, provided that these truth values do not depend on the particular values of Newton's fictions. A mode of communication can be thus established even in some cases where apparently we are involved in Newtonian phantasies. We may agree, for example, that certain equalities are true or that they imply other equalities. An agree-

ment of this sort is restricted, but in the contexts to which it applies the Leibniz-Newton controversy can be ignored.

This argument justifies the construing of the mapping # along the above-mentioned lines. (A similar one can be used in favour of the framework L f3) at the beginning of the chapter.) Actually, we could do much better in this respect if we are prepared to increase somewhat the purely mathematical part of our language. Thus, we can stipulate that a sentence, A, of / (N is true-or-false if a sentence A ' of/(L can be found for which A ~ - ~ A ' is provable from the classical space-time axioms. This condition can be expressed in the framework if we have at our disposal a predicate which denotes the set of natural numbers (as a subset of the reals). Even more sophisticated mappings become available, which enhance considerably the displayed common parts shared by the two ontolgies, but I shall not go into details here.

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , II 6'7

There is another aspect to the Leibniz-Newton controversy which is not representable in the languages so far considered. Among the connotations of an absolute space is the meaningfulness of questions of the following sort: Could the body b at the time z(e) have been located in a different place (i.e. in a place which differs from its actual location at time z(e))? In Newton's view the question is meaningful, at least in as much as similar modal questions are accepted in other contexts. (Could Newton not have discovered the law of gravitation? etc.) In Leibniz's view the question, as intended, is meaningless. It is meaningful only if re- interpreted in the form: I s . . . possible? where ' . . . ' stands for a description (in/(L or an equivalent language) of a state of affairs in which at time z(e) all, or many, of the bodies other than b form their actual geometrical configuration, but the position of b relative to them is different from its

actual one. A precise expression of the two views calls for extending the l anguage / ( by adding modal operators. In Newton's view one can make meaningful modal statements in which terms of the form p(b, z(e)) are converted into rigid designators. (Formally one can do so using a special operator which operates on terms of sort SP. Alternatively, the same

effect can be achieved by including quantifiers over variables of sort SP and by quantifying into modal contexts. The second alternative means going back to K.) In Leibniz's view, terms of sort SP should not be converted into rigid designators; on the other hand, numerical terms of the form 'rs( )' can (provided, of course, that they are legitimate according to the above-mentioned criterion). Thus, to ask if the ratio of two given distances (specified by means of two pairs of bodies at two given times) could have been different from the actual one is, according to Leibniz, meaningful. (His answer to the question is quite a different matter which need not concern us here.) So far, this leaves open the status of questions of the following sort: Could a given interval (specified by a pair of bodies at a given time) have been bigger than it actually is? (Note that intervals are not identifiable with real numbers, only their ratios are.) Leibniz accepted as meaningful ratios of two intervals which are realized at two different times (i.e., the intervals between p(ai, z(ei)) and p(bi, ~-(ei)), i = 1, 2, with r(el) r ~'(ee)). By itself, this does not imply that he should accept as meaningful comparisons of actual intervals with possible ones (both at the same time). Either a rejection of the last-mentioned question, or its acceptance, is compatible with Leibniz's basic view.


The Leibniz-Newton Controversy about absoluteness was concerned not only with space but also with time. Whereas belief in absolute space can be expressed by admitting as meaningful comparisons of positions of a given body at different times, no explication of this sort is feasible for absolute time. Time points are specified by means of events, and the sentences ' z (e0 = r(e2)' and "r(el) < '/ ' (e2) ' (asserting, respectively, that el and e2 are simultaneous, and that el precedes e2), are acceptable both to Newton and to Leibniz. The only way of making precise sense of the difference is by having recourse to modalities. Leibniz rejects the sentence asserting that an event e could have occurred earlier than it actually did, unless it is re-interpreted as asserting the possibility of a different

temporal situation of e relative to other events. This is completely analogous to the above-mentioned spatial modality of bodies. In particular, Leibniz would accept rigid designators denoting ratios of time intervals, but it is not clear whether he would accept or reject statements such as: The time interval between ~'(el) and ~-(e2) could have been twice as long as its actual length. Each course is, in principle, possible.

The rejection of Newtonian absolute space and time can therefore take different non-equivalent forms. Roughly speaking, one's position is determined by that class of sentences in the Newtonian framework which, in his view, consists of the true-or-false ones and which is sufficient for expressing every relevant fact. One can be more restrictive or less restrictive in determining this class; the more restrictive one is, the less

"absolute" the space and time which he will get. This is exemplified not only in the modal contexts which we have just considered. The more important examples do not involve modalities and can be formalized using our original languages K or/s We shall now consider a third major ontology of space and time which is " in-between" those of Newton and Leibniz. Actually, there are infinitely many ontologies between the two extreme positions and they can be partially ordered according to their strength (i.e., the extent of the Newtonian framework that each affirms). The one which we will present sums up a major historical point of view.

In classical mechanics, there is no way by which, in principle, one can know whether or not a given body is moving. Relative changes in position are discernible, but they do not determine those of the bodies to which one should attribute motion. On the other hand, one can determine whether a body undergoes acceleration by noting the effect of inertial


forces in the body's frame of reference; this has been argued by Newton in favour of absolute space. It constituted a very strong point against Leibniz's restrictive ontology which left no place for absolute acceleration (Leibniz tried, not very succcessfully, to cope with it, using his concept of cause; see Note 20). The argument establishes, however, not the Newtonian conception of absolute space but a weaker ontology within which it is legitimate to ask whether a given body is (absolutely) accelerating, but not whether it undergoes (absolute) motion 22. This conception came to be the main framework of classical mechanics 23.

By a rigid 3-dimensional body-configuration I mean any set of at least four bodies whose positions (i.e., centres of gravity) are not in the same plane and whose mutual distances are constant in time. (Since parts of bodies can be considered to be bodies as well, we can associate with every rigid body the rigid 3-dimensional body configuration consisting of its parts.) Physical frames of reference, or, for short, frames, are represented by rigid 3-dimensional body configurations where, by definition, two configurations represent the same frame if and only if their union is a rigid configuration. Given any set which represents a frame, we can choose four bodies not in the same plane and use them as reference points for a cartesian coordinate system (see Note 21). In this way we associate, in a uniform way, with every time t, a coordinate system with respect to which the position, p(b, t), of any body b (at time t) is represented by a triple of real numbers. (The uniformity means that the roles of the four bodies are fixed, and a fixed formula, not depending on t, is used to determine the coordinates of any body with respect to them.) Now we can regard the variables of sort SP as ranging over triples of numbers (i.e., we replace each by three numeric variables) and interpret 'r, (x l , . . . , x4)', where 'x,' are variables, in the standard way of analytic geometry. Each 'p(b, t)' is also replaced by a triple of numerical expressions which denote the coordinates of b at time t; these expressions involve numerical terminology and terms of the form 'r~(p(c~, t), p(c2, t); p(c3, t), p(c.4, t))' where each ci is either the body b or one of our four chosen bodies. Note that this use of 're' falls within the scope of the restricted /s All in all, any sentence, A, of K is translated in this way into a sentence of/s (Note also that the condition that a given finite list of bodies represent a frame is expressible in/~L.) What we thereby did can be described from Newton's point of view by saying that we assumed the bodies of our given frame to


be in state of rest. Now, there are many ways of associating uniformly a coordinate system with a given frame. But any two of these differ in some fixed (i.e., t ime-independent) similarity transformation (that is to say, the coordinates under one assignment are obtained from those of the other assignment by applying a fixed distance preserving mapping followed up by multiplication by some non-zero constant). It is easily established in K~ that the truth value of the translation of A does not depend on the

�9 " ! t ! r choice of a particular coordinate system 0.e., K L F - A ~-~A , where A and A" are translations based on any two ways of associating a coordinate systems with the same frame; here '/(L' indicates the restricted /~t~- version of the classical space-time axioms)�9 This truth value depends only on the frame. Say that A is true in a given frame if its translation relative to a coordinate system associated with the frame is true.

A Galilean frame is one in which the laws of classical mechanics are true. This definition uses the Newtonian concept of force which figures in the laws of classical mechan i c s - a concept rejected by Leibniz. It is possible to extend K by adding the terminology required for discussing forces (the total force acting at time t on a given body b, and the force exerted at time t by one body on another). Furthermore, this can be done

to /(~ without violating Leibniz's restrictions on the use of 'r~' and of equalities of space points. In the enlarged language, the condition that a given list of bodies represent a Galilean frame is definable.

Any two Galilean frames have a constant velocity with respect to each other. That is to say, given a Galilean frame and a (finite) body configuration representing another Galilean frame, the sentence asserting that all the bodies in the configuration move uniformly with the same constant velocity is true in the first frame. Consequently, a change from one Galilean frame to another is effected by adding to the movements of all bodies, as viewed in the first frame, a fixed velocity vector�9 Vice versa, any addition of a fixed velocity vector corresponds, in principle, to a change from one frame to another, even if the other frame is not represented in our actual world by an existing body configuration. The original above- mentioned definition of frame has to be extended in this way. This raises no difficulties for, knowing the picture (i.e., the true sentences) in one frame, we can tell exactly how the picture will look in another frame, obtained by adding a fixed velocity factor, even if this second frame is not represented by an existing body configuration. (The details will be


presented and used later. Actually, this is true for non-Galilean frames as

well, and the velocity vector can even be dependent on time.) There is no preferable Galilean frame, and the following is the basic

criterion for truth and meaningfulness in classical mechanics:

A sentence (in K) is true if and only if it is true in all Galilean frames. Consequently, factual meaningfulness (i.e., being true-or-false) is equivalent to having the same truth value in all Galilean frames.

By this criterion, the classical space-time axioms, as formulated in K,

and all their logical consequences are, obviously, true, Call sentences whose truth value is not affected by changes of Galilean frames classically invariant. The sentence asserting that b's positions at two given different times are the same (or different) is not classically invariant. In the classical system, any facts expressible in K are expressible by classically invariant sentences. An assignment of truth values to other sentences can be made only by virtue of a convention (i.e., recursive mapping) by which they are

reread as classically invariant ones. No convention of this sort would be acceptable to Newton, for it would give rise to certain evident expectations which he would reject. The situation is analogous to the one which we

previously discussed when comparing the frameworks of Newton's and Leibniz's. On the other hand, a statement asserting that the body, b,

undergoes acceleration is expressible in K and is classically invariant, but in Leibniz's view it is meaningless (unless reread, through some convention, as a sentence of/(t~, a convention which must be unacceptable to the classical view).

So far we have used the notion of a Galilean framework; this involves the concept of force, which is outside the scope of K. Yet it turns out that there is no need to extend K, and that the space-time ontology of classical mechanics is representable as a closed framework whose underlying language is K. As axioms we choose the classical space-time axioms, and it remains to define the mapping r such that True(A) ~ A ~ is valid in the framework. A ~ should express, in K, the fact that A is true in all Galilean frames. Now, imagine that I interpret the language K in some fixed Galilean frame. The possible pictures which one will get by viewing the same situation in other Gatilean frames are exactly those obtained by adding to the movements of all bodies (as viewed in my frame) fixed velocity vectors. Consequently, I can express within my interpretation the condition that some given sentence A is classically invariant. The


condition, roughly speaking, is that, for any fixed velocity vector, changing A so as to reflect the addition of the vector yields a true sentence. The

condition itself is exactly the same, no matter what Galilean frame 1 use to

start with.

To facilitate the notation, I shall employ vector terminology. Vectors

are represented in the standard way by ordered pairs of space points (two ordered pairs represent the same vector, if the corresponding directed

segments point to the same direction and have the same length). If v is a

vector and c a real number, then c �9 v is defined as usual, and if x is a space

point, then x + v is the result of shifting x by v (x + v is a space point). This terminology serves merely as a convenient shorthand; whenever it is

employed, the sentence can be translated in the obvious way into a

sentence of K. I shall represent velocities by spatial vectors. The shift at time u of a

body b that results from applying the velocity v is c(u) �9 v, where c(u) is a

real number which measures the time. The way in which v represents

velocity depends, of course, on the choice of time units. If to and tl are two

time points which determine the time scale, then c(u)=rt(to, tl; to, u). (we assume that to<t1). The definition of A ~ can now be stated as

follows: Given a sentence A, choose two variables to, tl of sort TP and a variable

v ranging over vectors which do not occur in A. (In terms of K, v will be

represented as a pair of variables of sort SP.) Replace every term in A of

the form 'p(b, u) ' by 'p(b, u)+rt(to, tl, t0, t) �9 v and let A'(to, h, v) be the resulting sentence. (If no 'p(b, t)' occurs in A, put A ' = A.) Now we put:

A * = Df Vt0, tl (to < tl ~ VvA '(to, tl, v)].

By adding all the instances of True(A)~--~A ~* as additional axioms to

the classical space-t ime axioms we get the desired closed framework, Kc, representing the space-t ime ontology of classical mechanics. (Note that True(A ) ~ A * is a sentence not in K but in K + as defined in w The rules (1)-(15) of w167 3 need not be included as additional axioms, for they follow already from the axioms we have; the rules of the first-order classical predicate calculus hold as well. We also get T r u e ( A ~ ) v False(A ~) as a theorem; thus A'* is a priori true-or-false. Furthermore, the addition of True(A) ~ A * does not add new theorems in the language K to those derivable from the space-t ime axioms. All these claims need,


of course, checking. For example, since we have True(A)-~ A * we get, by (14), True(A) -~ True(A *); but we have True(A *) ~ (A *)* and hence True(A) ~--~ A **, implying A * ,~*A **. This last sentence is in K, and it is not difficult to see that it is derivable from the space-time axioms. The truth-or-falsity of A * amounts to the validity of A * ~ v (-n(A*)) *, and this is derivable as well. Viewing the situation in terms of set-theoretical semantics (where True(A) means that A is true in every model of a certain set) may be helpful in establishing the claims. I have these details to the reader.

5. S O M E R E M A R K S O N P R O B L E M S I T U A T I O N S

A N D T H E R O L E O F O B J E C T S

Is there an empty self-existing time? Is there an absolute entity, consisting of an ordered line of time-points which are there, independent of any events? Newton pictured the world of events and physical bodies as if it moved through a pre-existing spatio-temporal continuum. He would answer the above questions affirmatively. Leibniz rejected this picture and would give a negative answer. But people have many ways of visualizing a given state of affairs or a given problem; to give an extreme example: one might visualize the natural numbers as a horizontal series going from left to right, while another might view them as going up vertically. So why should Leibniz grudge Newton his world-picture?

Indeed, if there is any real issue to the absolute time controversy, then the above-mentioned questions must be given a precise explication. In preset form they can only serve as vague indicators. Moreover, we are not even assured that they do indicate some real problem, unless that precise explication is found. Leibniz himself had to justify his rejection of the Newtonian world-picture by showing that the problem is real and not merely a question of convenient mental pictures. He did so by redefining Newton's position as an affirmation of the meaningfulness of sentences of the form: 'The even e could have occurred earlier than it actually did'; he presented his own position as denying this sentence any meaning, unless 'earlier' is interpreted relative to other presupposed events. Let us call this sentence B(e), where, unless otherwise stated, 'earlier' is understood as not relative to other events but in an absolute sense.

Now, one can argue, our most obvious way of making sense of B(e) is by means of a Newtonian picture of a pre-existing time-line. Or, in


technical terms, construing '~-(e)' (= 'the time of the event e') as a rigid designator is easist understood by presupposing a Newtonian model with time points in the role of objects. So why recast the problem in terms of B(e) and similar sentences rather than in terms of our original questions? It is quite true that B(e) is the natural outgrowth of a Newtonian world-picture, and that the easiest method of explaining the intended sense of this sentence is to represent the world as a Newtonian model. But the only way of determining whether a Newtonian world-picture amounts to more than a convenient visualizing device is to define its import in wider cognitive contexts. By pointing out sentences whose meaningfulness is implied by Newton's view but not by other views, or whose being or not being a priori valid is a mark of his conception, we can show the extent to which his world-picture does, in fact, make a difference. Now, the sentence 'All events take place in a pre-existing empty time' which, presumably, is valid in Newton's conception but not in Leibniz's, cannot, by itself, constitute mark of real difference. For, to take up our previous example, consider the sentence 'The natural numbers are arranged in increasing order from left to right'. Prima facie it pretends to describe a basic and objective state of affairs, which is true in a model in which 'to be to the right o f . . . ' is construed as a predicate applicable to numbers and co-extensional with ' to be greater t h a n . . . ' . But we certainly should not take it as a mark of ontological difference. A particular way of visualizing numbers may enable one to discover new arithmetical theorems. But these, once proved, should be acceptable as mathematical truths independent of one's mental picture. (Mr. Alpha may even rely on Ms. Omega's intuition in accepting certain mathematical claims without formal proof, using inductive judgment - to wit, that on all other similar occasions the checking of Omega's claims led to their formal verification. This in itself would not constitute an ontological difference in Alpha's mathematical framework24.) It is also easily seen that the difference

between the left-to-right arrangement of numbers and their vertical upward arrangement is immaterial, because we can translate everywhere 'to the right o f . . . ' into ' a b o v e . . . ' ; furthermore, each of these expressions is replaceable by the mathematical 'greater t h a n . . . ' , which manif- ests their redundancy. Although trivial, the example suffices to establish the point that the following of certain intuitions, or the employment of certain models, need not have any ontological significance. Other cases

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , 1I 75

may involve much more subtle delusions. A certain proposed model, or

some statement, may look as if it amounted to a substantial claim about the nature of the world, whereas, in fact, it is nothing of the sort. They delude us like a sophisticated conjuring trick which may take quite an effort to unveil; the reducing translations may be far from simple. This is why the ontological significance of accepting a certain statement, or a certain model, should be judged globally, in terms of its effect on conceptual frameworks as wholes. And this is why the shift from the question as to the existence of empty time to the question as to the meaningfulness of B(e), and similar sentences, is an advance. For here at least we have some hope of finding out how pre-existing empty time is going to connect into wider contexts and what fruit, if any, this intuitive picture is going to bear.

Not that this hope has been vindicated. In the original context of the

Leibniz-Clarke exchange, questions of modality had a bearing on the theological aspect of the controversy. Ignoring this aspect, the explication of empty time through the meaningfulness of B(e) has not, to my mind, provided this concept with sufficiently real import. To give my full reasons, I would have to go into details of modal logic beyond the scope of this essay. Let me only state my agreement with Kripke that the truth conditions of modal statements are not determined by objective pre- existing Kripke models, but on the contrary it is up to us to choose that model which reflects our conclusions as to the meaning of the statement. Now, concerning B(e), I cannot see what arguments could be advanced

either for its truth or for its falsity, or for its factual meaningfulness. The only arguments we come up with do not concern B(e) as intended, but the statement in which 'earlier' is interpreted relative to some other basic events. (The situation is different with respect to other modal statements, e.g. 'Newton could have not discovered the law of gravitation', w te re arguments pro and con, leading to a constructive analysis, are available.) It seems that any decision concerning B(e) would be a matter of subjective taste without affecting wider contexts. (Compare this with the different situation concerning the question of absolute space which, through being explicated in terms of absolute motion, acquired high significance.)

By asking whether time-points, or space-points, or natural numbers, or subjective sensations (as distinguished from physical states of the brain)


exist, we do not define a problem but point in a certain direction in which perhaps some real problem can be found. By giving a positive answer to such a question I am indicating my inclination to adopt a richer ontology; 'richer' means nothing else than that the language whose sentences I shall consider objectively true-or-false will be more expressive, that is, more

expressive than the language of the one whose answer to the question is negative. We might say that the question defines a pre-problem. A problem is determined only to the extent that the two languages are outlined and confronted. When philosophers and scientists argue a question of existence, they may already have in mind some tentative outline of the frameworks which are implied by their positions. But generally speaking, there is no guarantee that a pre-problem will evolve into a real problem. (The question of time-points has led, I think, to a pseudo-problem.) The possible explications of a pre-problem depend on our available conceptual stock. Hence an existence question may acquire

real significance only at a later historical stage. A mere heuristic device of one stage may turn out to be a component of the ontology at a later stage.

To ask whether certain objects exist requires a yes-or-no answer. On the other hand, the languages which are candidates for ontological affirmation may have varying degrees of expressiveness. For example, we saw in w that the class of true-or-false sentences of the classical- mechanics-framework is strictly in between Leibniz 's / (L and Newton's

K. If Newton's position is represented as belief in the existence of space points (or empty space) and Leibniz's position as a denial of this belief, then it would seem that classical mechanics, as presented in w commits one to the partial existence of space points - which is surely absurd. This

again goes to show that conceptual frameworks and not objects should constitute our points of departure.

An additional advantage of being framework-oriented is the distinction between more essential and less essential properties becoming plausible. Starting with objects as our atoms or reality, it seems strange that one property of an object should be more essential to it than another. But in the context of a framework it is only to be expected that some concepts will play a greater role than others 2s.

To sum up, objects mark in a very condensed form, basic features of the framework. They can also indicate some of its possible evolutions. In as much as the same objects may figure in more than one framework, they

O N T O L O G Y AND C O N C E P T U A L F R A M E W O R K S , 1I 77

c o n s t i t u t e c o m m o n m e e t i n g - g r o u n d s fo r d i f f e r e n t c o n c e p t i o n s . R e a l i t y ,

h o w e v e r , is a m a t t e r o f a w h o l e c o n c e p t u a l f r a m e . In th is f r a m e t h e

v a r i o u s o b j e c t s o c c u p y c ruc ia l p o s i t i o n s , l ike t h e c e n t r e s o f v o r t i c e s in a

d y n a m i c f low; t h e y a re u n d o u b t e d l y rea l , b u t t h e i r r e a l i t y is t h a t o f t h e

s t r e a m w h i c h c r e a t e s t h e m .

N O T E S

o But see w and note (7) for additional clarifications of this point. 1 Kochen, S. and Specker, E. P.: 'The Problem of Hidden Variables in Ouantum Mechanics', Journal of Ma&ematics and Mechanics, 17 (1967). 2 This is true of all the rich enough formalisms which are presently available; it holds no matter whether 'implies' is construed as material or intuitionistic or some variant of modal implication. In terms of the Lindenbaum-Tarski algebra it means that the partial ordering has maximal and minimal elements, which is the case even in some very non-standard logics (such as those which were proposed for interpreting Quantum Mechanics). Actually I could replace (5) and (6) by several weaker requirements, but this would involve going into too many tedious details. There is much to be said for a logic which avoids the so-called paradoxes of implications (e.g., that '1 + 1 = 1' implies 'snow is not white'). In natural languages, an assertion that A implies B indicates also that there is a "natural" derivation by which we arrive at B, using A in an essential way; A should be relevant to B. The adoption of (5) and (6) does away with this feature. However, the real argument against (5) and (6) is that the formalized system becomes useless once a contradiction has been derived in it, for then any sentence can be derived as well. Now, in practice this is not always so. People often manage to keep of[ the "inflicted area" around the contradiction and employ the system profitably in safer directions (hoping, eventually, to fix the theory up somehow). This happened in analysis before Cauchy and Weierstrass, this is how Cantor was able to shrug off the antinomies of set theory, and this happens continually in physics. It goes to show the essential indeterminacy of conceptual frameworks in general, the element of open texture which is always and irreducibly present. Formalized versions constitute efficient tools of analysis, but at best they are good approximations. Still, the development of formal calculi in which a contradiction need not imply any other sentence might throw some light on the phenomena of inconsistent but useful theories. Such calculi would have to be much richer than those proposed by Ackerman and developed by Anderson and Benlap (see Hughes & Cresswell, An Introduction to Modal Logic, pp. 298-301). The difficulty is not in the sentential level but in incorporating the idea into a system in which sufficiently rich theories can be developed. 3 We can introduce an additional binary connective c, so that in the Lindenbaum-Tarski lattice c induces a binary function which maps every pair of elements (x, y) in which x ~ y to 1 (the maximal member) and every other pair to 0 (the minimal member of the lattice). Then we have: A ~-L B if and only if ~-Lc(A, B). Of course c is not an implication in the usual sense; for example, unless ~ l ~ A or f-Lc(B,A), we will not have: ~-~c(A, c(B,A)), whereas we always have: ~-LA->(B-~A). The introduction of c does not add new consequences of the form 'A ~L B' or 'I-LA' where A, B are in the original language, but the expressive power of the language is nonetheless increased. Since c corresponds in the Lindenbaum-Tarski lattice to a function which assumes only the values 0 and 1, we should be able, given arty two particular sentences A and B, either to deduce from the rules that


~Lc(A, B) or to deduce that ~-LTc(A, B), Otherwise the set of rules cannot be considered as adequate. We cannot solve the problem simply by including a rule of the form '~-Lc(A, B) or ~-LTc(A, B)', for this will not enable us to deduce a particular branch in the disjunction. It is not by accident that (1)-(7) have the restricted forms which they have; they represent the knowhow of operat ing the L-f ramework, and consequent ly they resemble rules of a constructive game. 4 Consider a language (having 7 and ~ among its sentential connectives) whose users have a concept of validity such that, for every sentence A,

(i) If A ~ ~ A is valid, then T A is valid. Assume that in this language there is a predicate 'tr( )' which can be applied to names of sentences and a is a sentence which is identical to 7 t r ( ' a ' ) . If the language constitutes, with respect to the concept of validity, a consistent system (i.e., if there is no sentence A such that A and ~ A are both valid), then a must be counterexample to at least one of the following general requirements:

(ii) t r ( 'A ' ) -~A is valid; (iii) If A is valid then tr( 'A') is valid. (Tile trivial proof amounts to no more than a precise

formulation of the liar paradox argument.) Note that some concept of validity is involved whenever a well-understood language is used for communicat ing facts; at least, it is inevitable to the extent that we consider the language as having, or as incorporating, a logic. A m o n g the (unconditionally) valid sentences will be, first and foremost , those whose assertion would be justified by anyone who unders tands the language. (There may, of course, also be sentences whose validity rests on complicated proofs and, therefore, is not obvious or known.) Validity is thus a primitive concept which concerns the basic rules of the language game. (In formalized systems, to say that a sentence is valid is simply another way of saying that it holds in the theory in question.) And so this concept remains, whether or not we choose (as often we do) to explicate it by using the semantic set-theoretical concepts of model and satisfaction. Changing the logic (i.e., from classical, two-valued logic to some many-valued or non-s tandard logic) may affect the class of valid sentences, but the concept of a valid sentence is to stay.

Recently, Kripke has developed techniques of expanding a given model, M o (which interprets, in the Tarskian sense, a language, including arithmetic) to a model, M, by adding an interpretation of a new predicate 'tr( )', so that 'tr( )' resembles in some respects a truth predicate for the entire language of M. 'tr( )' takes number -denot ing terms, or numerical variables, as arguments , and the encoding of sentences into G6del numbers can be done in such a way that we actually have a sentence a for which a = 7tr('a '). The model M is such that whenever Mgtr ( 'A ') then also M ~ A , where ' 9 ' denotes satisfaction (in the usual sense). Thus, M ~ tr('A ") ~ A. Shall we then say that tr('A ') ~ A holds in the resulting theory? If by ' theory ' we mean the sententences which are satisfied in M - the answer is yes. But then the concept of truth by which this theory is defined is different from the concept denoted by "tr( )'. In fact, tr( 'o~')-~a is satisfied in M but tr('tr('a')~et') is not. Thus, if truth is that which 'tr( )' denotes and satisfaction in M is a mark of validity, we get simple examples of valid sentences which are not true - contradicting (iii). If we regard 'tr( )' as a truth predicate, it is more appropriate to let the theory consist of all, or some of, the sentences A for which Mgtr( 'A ' ) . (For every atomic sentence, B, we have M~B<:e;Mg tr('B') and in particular M ~ tr('A ") <::~, M ~ tr('tr('A ')'), hence our use of ordinary satisfaction in the criterion is consistent with the view that 'tr( )' denotes truth.) In this case, (i) and (iii) are satisfied, but tr('a') ~ a is not valid, contradicting (ii). Thus, 'tr( )' is obviously not a truth predicate; not, however, because it denotes a concept different from the Tarskian concept, or because convention (T) does not hold, but because, no mat ter how we define our

O N T O L O G Y A N D C O N C E P T U A L F R A M E W O R K S , lI 79

' theory' , some requi rement on the syntactical level, much more basic than convention (T), must be given up. Kripke's construction is interesting and useful because through it one can legitimize many (but not all!) self-referential applications of a truth predicate (namely, those A ' s for which M ~ t r ( ' A ' ) ) . It is a tool by which self-reference can be analysed into several grades, and the vicious cases be separated from the non-vicious ones. But it certainly does not institute a concept of truth which admits unrestricted self-referential applications while avoiding the paradoxes. (Not that, to my knowledge, Kripke ever made this claim; I think he will agree with me on this point,) To get such a new concept of truth, one will have to give up one of the requirements (i), (ii), (iii). To my mind, giving up (iii) in unthinkable. The weakest link seems to be (i), and even here the cost would seem exorbitant. If A ~ ~ A is valid but ~ A is not, then A certainly does not have a s tandard truth value; but, of course, a three-valued (or n-valued) logic is not going to accomplish this. Whether a useful calculus can be constructed by giving up (i) remains to be seen; at present it is doubtful, to say the least. 5 In the non-distributive case as well, if each of 'P ' and ' Q ' s tands either for 'True ' or for 'False' , then the conjunction P ( A ) A Q ( B ) will imply the "correct" truth values for A n B, and A v B. Yet this does not mean that in compounds of true-or-false sentences the connectives function in the s tandard way, for we do not know as yet that the following conjunction distributes:

(True(A) v False(A)) ^ (True(B) v False(B))

i.e., that it can be unfolded, according to the distributive law, into a disjunction of the four resulting combinat ions of P ( A ) ^ O ( B ) . It is possible to add a rule to that effect; if so, we may as well go the whole way and stipulate that the conjunction obtained by adding True(C) v False(C) as a third conjunct should distribute into a disjunction of the eight resulting combinations. It will then follow from the rules that if we assume in L that A, B, C are all true-or-false, then (A v B) ^ C distributes. Thus, compounds of true-or-false sentences will behave in the s tandard manner . Failures of distributivity will evidence a reluctance to grant all the involved sentences the status of being objectively true-or-false. In this way the non-distributivity can be made more palatable. There is, however, another way of construing the system, by which the general distributivity of the above-ment ioned conjunction is not granted. Fur thermore , one can even have ~ L T r u e ( A ) v False(A) and similarly for B and C, without allowing to distribute (A v B) ^ C or (A v ~ A ) ^ (B v ~B) . It amounts to a radical conception which makes non-distributivity an essential feature in the realm of objectively true-or-false s e n t e n c e s - s o m e t h i n g which many will reject as an absurdity. This may account for the resistance which the idea of a non-distributive logic for quan tum mechanics has encountered. I have not checked the implications of each of the two approaches which are outlined here.

6 Pick an arbitrary frame o f r e f e r e n c e , F. Let d be the distance between the three- dimensional points at which el and e2 took place, and let t be the time separating el and e2 - all with respect to F. If d > c �9 t (where c = speed of light), then the same inequality holds in every other f rame of reference, and the interval between el and e2 is said to be space-like. Pairs of events which determine a space-like interval are exactly those for which precedence in t ime can be reversed by changing the frame of reference. Using the wealth of available astronomical data, it is not difficult to point out two particular events, el, e2, and two particular frames of reference which are associated with actual body configurations (star clusters of galaxies), such that e 1 precedes e2 in one frame of reference, while e 2 precedes e in the other.


If d < c.t, the temporal ordering is absolute and the interval is time-like. If d = c.t the ordering is absolute except that non-s imul taneous el and e2 may be s imultaneous in certain frames of reference whose relative velocity to the given one is c. 7 1 hope that, by now, it is clear that my construing 'True ' as 'TrueL' does not mark a trivial relativization of the truth concept to a conceptual framework. The display of 'L ' as an index is obligatory if we are to make sense of a f ramework without necessarily accepting it. In the same way, an intuitionist may use 'r--c' to denote validity in the classical sense while claiming that the concept is ill-founded. In an ontological controversy, where the objective t ruth-or- falsity of certain sentences is debated, both sides intend the same generic primitive concept of truth. To claim that the opponent ' s 'True1' is not the "real" concept of truth is nothing but another way of claiming that certain sentences for which True a(A) v False l (A) is valid are not objectively true or false. 8 It is conceivable that in L there are B ' s such that I-- L ~(True(B) v False(B)). For instance, if we allow sentences involving category mistakes, such as 'Julius Caesar is a prime number ' , we may wish to regard them as neither trae nor false, a priori. Then ~ L ~ T r u e ( B ) but not ~-L--IB. Therefore, we cannot replace, in general, ' l -L ~True (B) ' by '~-L ~ B ' . It seems unlikely that such sentences could play any role in deciding whether a given A is objectively true-or-false but, to be on the safe side, I used everywhere 'True(B) ' instead of 'B ' . This also fits the intuitive idea expressed in the non-formal phrasing of the criterion. 9 Adding C as a valid sentence amounts , by (10a) to adding True(C). A sentence, D, is valid in the f ramework obtained by adding C if and only if True(C) F-D. 1o That the conjunction should he false, a l though some of the conjunets are not true-or- false may be regarded by some as an aesthetic flaw. But this is a small price to pay for the generality which we thereby gain. The proposed setup can be used to analyse diverse problem-si tuat ions. Russell 's theory of descriptions yields necessary and sufficient presuppositions whose truth-or-falsity is guaranteed, without assuming anything special about conjunctions. This looks nicer. But then it is applicable only to that narrow class of examples where t ruth-value gaps arise solely because some definite description fails to denote and where the failure can be rectified in the same language by utilizing first-order quantification. Jl That is to say, a procedure which is at least recursive; in all actual cases it will be primitive recursive. All metaconstruct ions should be effective (see end of Note 3). Speaking, as we do in (ii), of f rameworks which are not closed should not be interpreted as simply denying the existence of some recursive function; in all actual cases we will have quite effective reasons for regarding a given f ramework as not closed. 12 Namely, in any scope which, in the original sentence, includes the occurrence of ~b, this occurrence should be substi tuted as described. A scope which, in the original sentence, is a subformula of ~p(... ~x4"(x) . . .) remains the same except that the occurrence of ~x~b is replaced by x. If a scope coincides with ~b then we can let the corresponding scope in the new sentence be either ~b(... x . . . ) or the whole formula which substi tutes 0(. �9 �9 ~x4' . . . ) ; the final outcomes in Lo will be logically equivalent. ~3 Let Mo = (IMol, Ro . . . . . Ra . . . . ) where IMol = the domain of Mo and R , is the interpretation of the predicate Pa. (If Lo has operation symbols, Mo should also include their interpretations.) Let D = set of all terms of the form Lx4', where 4' is in L. C(Mo) is defined as the class of all structures of the form (M, f,)t~o where Mo c M (i.e., M is a structure for Lo which extends, in the usual sense, M0) and whenever t has n free variables ft is a function from IM[ n into IM]; if t has no free variables f, is some fixed m e m b e r of M. Actually, it suffices to choose some fixed set X which includes Mo such that X - M o is infinite and let C(Mo) consist of all s tructures of the above-ment ioned form in which IMI = X; thus, C(Mo)


need not be a proper class. (There is even a certain subset of C(Mo) whose cardinality is l'lo + cardinality of M0, which can replace C(Mo) in the following results.) Let N E C(Mo), say N = (M, ft)t~O- If t ~ D and its free variables are among x l . . . . . x, then t[aL . . . . . a , ]N is the value of t in N with respect to the assignment of a~ to x~(i = 1 . . . . , n) where a t , . . . , a . ~ IMI. This value and the concept of satisfaction are defined simultaneously by induction (where ' ~' is used for our non-standard satisfaction): If t = ~x~b (x, X l . . . . . x , ) and if a l . . . . . a , c ]Mol and for every structure N' ~ C(Mo) there is a unique b in IMo[ such that N'r al . . . . . a , ] , then t[at . . . . . a , ] is this unique b for the structure N. In all other cases t[a 1 . . . . . an]N =f t (a l . . . . . a,) . The clause for satisfaction of atomic formulas is the usual one (namely, every term which occupies an argument place of the predicate is assigned its value in N and the tuple, thus obtained, should stand in the relation which interprets the predicate). The clauses for Boolean combinations are the usual ones. Finally, N~.:Ixcb[x, al . . . . . a , ] ff there exist b~lM0] such that N~4)[b,a~ . . . . . a , ] and N~Vxg)[x, at . . . . . a .] if for all b 6 ]Moj, N~d~[b, at . . . . . a,]. It can be proved that:

Mogd)*[at . . . . . a.]C::>for all N 6 C(Mo), N~,b[at . . . . , a,]

( 'g ' on the left-hand side denotes satisfaction in the usual sense.) Furthermore, Mog (::l.Vx~b) ~ if and only if there is a member b c Mo such that, for all N ~ C(Mo), b is the unique member of ]Mo[ for which Ng*q~[b]. (~b may have additional free variables which are assigned fixed values in Mo.)

14 If the equivalence in question is material, (i.e., if the truth of Vx (4~ o #J) should imply the truth of ~x~ = ~xO) then the following modifications of clauses (iii) and (iv) in the above given definition of * achieve the desired effect. (~x~b = ~xtb)* is defined as (Vx(~b~--~))* and (7(~x~b =~x~O))* as (TVx(~bo~O))*. Furthermore, in (iv) one should take into account possible identificationsxff different descriptions; we proceed by induction on the number, m, of descriptions which occupy argument places in the atomic formulas. For m = 1 the definition is as given; assuming that the descriptions are Lx~l . . . . . ~x~b,,+t and that the formula is 4/, we add to the given definition of ~0" all the formulas (Vx(qSm+ 1 ~--~bl))* ^ ~,* (i = 1 . . . . . m) as disjuncts, where ~b~ is obtained by substituting Lx4~i for Lx~b,,+t in ~0. If logical equivalence is meant, then * can be defined, provided that in Lo one can express the fact that a given formula is logically valid; that is to say, with any 0 in Lo one should associate, by a recursive procedure, a 4t, such that ~0 is true (in the Lo-framework) if and only if 4~ is logically valid. In that case, * can be easily construed along the lines just given. (Logical validity is equivalent to provability in the predicate calculus; hence it can be expressed in Lo if L o contains a certain fragment of arithmetic.) Analogous observations are true with respect to other concepts of equivalence. These and other variants can be provided with suitable semantics modelled along the lines of Note 12. J5 Consider the sentences (1) 'The present king of France is bald' (2) 'The present king of France met Mr Gerald Ford ' (3) 'Mr Gerald Ford was met by the present king of France' (4) 'Ford 's inauguration was at tended by the present king of France' . I feel that our inclination to regard the sentences as false, rather than as neither true nor false, increases as we pass from (1) to (4). Note in particular the leap from (2) to (3) which is caused by our tendency to construe ' . . . . was met by the present king of France' as a predicate. J6 Describing Cicero in terms of his oratorial ability, political aspirations or personal appearance is obviously different from describing him as an individual to whom the present name 'Cicero! is linked by certain oral and written traditions. But how much weight should


one place on this difference? Usually, one does not think that being linked to a certain label is a property of the labelled object. In philosophical tradition, to describe means to produce a description of the first type. In particular, Frege and Russell intended descriptions of this sort when they spoke of describing objects. Now, if one associates proper names with characterizations of the second type and wishes to maintain as great a distinction as possible between the two types, then, to exclude descriptions of the second type from the language under discussion would be the natural thing to do. One would use them only in metalanguage in order to explain how names denote. But the distinction is far from absolute. It is, in fact, a corollary of the Aristotelian distinction between essential and inessential properties. Now, I claim, the notion of being more essential (or less essential) is meaningful only relative to a conceptual framework. To be essential means to play a central role in the organization and employment of the whole setup. Consider a historian who tries to find whether 'a = b ' is true, where ' a ' and 'b ' occur in certain documents and are, allegedly, proper names of persons. Will he not use, interdependently, testimonies concerning personal appearance, deeds attributed to them, etc., as well as the history of the names ' a ' and 'b ' (their possible origins, naming procedures in the society under consideration, the temporal ordering of certain texts, changes due to transcribers' errors, etc.)? Obviously in his framework both types figure interdependently as parts of the conceptual apparatus. We can hold that, in final analyses, the referents of proper names are defined by second-type descriptions (whereas first-type descriptions serve as confirmatory, evidence). But this implies by no means that second-type descriptions should figure only in the metalinguistic explanation of denoting. Not to include them on a par with those of the first type as part of the language of the framework would result in a dogmatic distortion of the scientists' conceptual system.

There is also the issue of the functioning of names in modal contexts. Proper names are rigid designators, definite descriptions not necessarily so. (By rendering them in suitable phrasing they, too, can be made rigid; in 'The man who proved the incompleteness theorem could have not proved the incompleteness t h e o r e m ' - t h e definite descriptior~ is a rigid designator. In formal languages one can include a special operator to make a description rigid, like Kaplan's 'dthat ') . There are modal contexts in which a proper name cannot be replaced by a definite description without changing the truth-value, unless ~omc additional rephrasings are carried out. Since Russel's account of denoting calls for definite descriptions as substitutes for proper names, and since the Kripke-Kaplan account rejects descriptions of the first type as substitutes and tends to exclude the second-type descriptions from the language, it would seem that their account squares better with modal usage. But this impression is wrong. The question of how the reference of a proper name is determined has no bearing whatsoever on the name's functioning as a rigid designator. For, even if a proper name is explicitly introduced by means of a definitite description fi la Frege-Russell, it is no less rigid, and it is not always substitutable by the definite description which served to introduce it? (As Kaplan pointed out, one can define 'Russell ' by: "Let one Russell be the length of your boat" and then go on and state: "I thought that the length of your boat was more than a Russell".) Thus the rigidity of proper names does not depend on whether their reference is singled out by some description in the Frege-Russel manner, or whether they function as labels pointing to objects at the other end of causal information chains. These are two independent aspects. Any explication of modal contexts will have to take into account rigidity and, in particular, rigidity of proper names, in determining the truth-values of sentences. This is a separate issue (which can be dealt with in the general setup which I propose in this essay). No theory of denoting can, in itself, save us this trouble; no theory is


better off in this respect than any other. When it comes to deciding whether second-type descriptions should be incorporated in the very language of the framework, considerations of modality are neutral. 17 Leibniz's correspondence with Clarke. Here are some characteristic passages: "I have said more than once that I held space to be someth ing merely relative, as t ime is; that I hold it to be an order of coexistences, as t ime is an order of successions. For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together, without enquir ing into their particular manner of e x i s t e n c e . . . Space is someth ing absolutely uniform and without the things placed in it one point of space does not differ absolutely from another point of space. Now, it follows froln this, supposing space to be someth ing in itself besides the order of bodies among themselves, that it is impossible there should be a reason why God, preserving the same order of bodies among themselves, should have placed them in space after one certain particular manner and not o t h e r w i s e . . , for instance by changing East into West. But if space is nothing else but that order or relation, and is nothing at all without bodies but the possibility of placing them, then those two states, the one such as it now is, the other supposed to be its reverse, would not at all differ from one another . Their difference takes place only in our chimerical supposit ion of the reality of space in itself; but in truth one is exactly the same thing as the o t h e r . . . " Leibniz's third letter, w167 5. Ano the r key passage is w in his fifth letter. ~s "But with the dimension of matter it is not thus: the ternary number is de termined for it not by reason of the best, but by geometrical necessity, because the geometricians have been able to prove that there are only three straight lines perpendicular to one another which can intersect at one and the same point. Nothing more appropriate could have been chosen to show the difference there is between the moral necessity that accounts for the choice of wisdom and the brute nece s s i t y . . , than a consideration of the difference existing between the reason for the laws of mot ion and the reason for the ternary number of dimensions: for the first lies in the choice of the best and the second in geometrical and blind necessity." (Theodicy, w .) " . . . ordering has also its quantity, there is that which precedes and that which follows, there is distance, or interval relative things have their quanti ty as well as absolute ones. For instance, ratios, or proport ions in mathemat ics have their quanti ty and are measured by logarithms; and yet they are relations. And therefore, though time and space consist in relations, yet they have their quanti ty." (5th letter to Clarke, w 19 Leibniz's 5th letter to Clarke, w 2o Motion, according to Leibniz, is nothing but a change of distance between bodies. An absolute motion cannot be meaningfully defined in spat io- temporal terminology (such as K). Leibniz tries an al together different explication of absolute motion which rests on the concept of cause. To the extent that the cause of the change of distance inheres in any of the bodies in question, that body is in a state of absolute motion. (In Newtonian conception, being in a state of motion and causing it are totally different aspects.) cf. Leibniz's 5th letter,

w 21 Here is the idea of the proof. As in the reduction of pure geometry to real n umber theory, every variable of sort SP is represented by three numerical variables (the 'coordinates ' of the point) and rs(x~, x2; x3, x4) is represented by (E(Vl, , - v2,i)2/Z(v3,,- Va,~)2) 1/2, where vj.~, vj.2, vj,3 are the three numerical variables represent ing xj. The problem now is to treat 'mixed' expressions such as r~(xl, p(a, tl); x2, p(b, t2)). If 'b l ' , . . . . 'bk' are the names of material bodies, then the s ta tement that p(b~, tl) . . . . . p(b,,, tin) lie all on the same line, as well as the s ta tement that they are in the same plane, are expressible in / r for each can be stated solely in terms of numerical equalities and inequalities involving ratios of intervals


formed by these points. The sentence asserting that all the points coincide is obviously in /~ as well. Now let 'b~', 'b2', 'b3', 'b4' be names (not necessarily different ones) occurring in the sentence, A, (of K). Working inside K and assuming that p(bl, tO . . . . . p(b4, t4) are not in the same plane, we can use them to set up a coordinate system, p(bl, q) will get the value (0, O, 0), p(b2, t2) - the value (1, 0, 0), p(b3, t3) - a value of the form (a,/3, 0) where a and/3 are expressed in terms of ratios of the intervals formed byp(b l , tl), p(b2, t2), p(b3, t3), (or and /3 are uniquely de termined if we assume that they are non-negative); p (b4,/4) gets a value of the form (a ' , /3 ' , 3") where these are non-negat ive and expressible in terms of the ratios involving the four points. Every other point p (b, t) can then be assigned a 3-tuple (61, 62, 63) where the '6i' are numerical terms involving the ratios of intervals whose end points are amongp(b , t), p(bl, ta) . . . . . p(b4, t4). In this way one can replace every term of A by a term in/~, involving possibly the additional TP variables ' t l ' , . . . . 't4'. (We assume that the 't~' do not occur in A.) Let * l ( q . . . . . t4) be the formula of /~ obtained in this way. Let Bl(tl . . . . . /4) be the formula of /~ asserting that p(b~, tl) . . . . . p(b4,/4) do not lie in the same plane. Then the classical space-t ime axioms imply:

3tl . . . . . t4Bl(tl . . . . . t4) ~ [A <--~ :::It1 . . . . . t4Bl(tl . . . . . t4) ^ At(t1 . . . . . t4)].

Thus, taking = l t l , . . . , t4B 1 as a presupposit ion, A is equivalent to a certain sentence in/~. Now let ' a l ' , . . . . 'am' be a list of all the names of material bodies occuring in A, in which each name occurs twice. The s ta tement that for all t i , . - . , t m the points p(al, tl) . . . . . p(am, t,,) lie in the same plane is expressible in/s This s ta tement means that there is some fixed plane which includes all the orbits of (the centres) of the bodies ment ioned in A. Working inside K and assuming this as a presupposit ion, and also that p(bl, tl), p(b2, tz), p(b3, t3) are not on the same line (where the 'bi' occur in A) , we can proceed as before. Every p(b, t), where 'b ' occurs in ' A ' will now be assigned a 3-tuple whose last member is 0 and whose first members are expressed in ratios involving intervals between the points p(b, t), p(bl, tl) . . . . . p(b3, t3). Again we get a sentence in /~ which,

under the presupposit ion of this case, is equivalent to A. Now, the classical space-t ime axioms imply that either among the finitely many bodies ment ioned in A, one has bl, b2, b3, b4 such that there exist t l , . . . , t4 for which p(bl, tO , . . . ,p(b4, t4) are not in the same plane or there is some plane in which all (centres of) the bodies ment ioned in A are at all times, but one can find bl, b2, b3 such that there exist tl, t2, t 3 for which p(bl, q) . . . . . p(b3, t3) are not on the same line, and so on. Each of the alternatives is equivalent to a finite disjunction, where every disjunct corresponds to a particular choice of b a , . . . , b4 or of b ~ , . . . , b 3 or of bl, b2, satisfying the above-ment ioned conditions. (The last alternative is expressed by the c o n j u n c t i o n / ~ b.h'Vtl, t2(p(b, tl) = p(b', t2)) and yields one disjunct.) Each disjunct implies the equivalence of A to a certain sentence in/s All in all we get the required ,4 as a disjunction of these sentences. 22 A naive objection which is somet imes raised concerning this point is: Since acceleration is a derivative of motion, how can one speak of absolute acceleration without assuming absolute m o t i o n ? - Actually, there is nothing paradoxical here. We can speak of motion with respect to a f rame of reference and we can characterize a Galilean frame of reference as a f rame in which certain mechanical laws hold. Whenever we speak of motion, we have to speak of a Galilean frame relative to which the motion is defined. The derivative of motion (with respect to time) is likewise evaluated relative to some Galilean frame. However, all Galilean frames exhibit uniform motions with respect to each other. Hence, the time


derivative of motion, unlike the motion itself, is the same in all Galilean frames. This is the absolute acceleration.

23 This is not intended as an historical account, but as a post factum description of what classical mechanics actually amounted to as a conceptual framework. Historically, the concept of absolute motion has been suspended rather than rejected. Speculations concerning the ether and the possibility of defining absolute motion as motion relative to the ether shoulc~be taken into account. On the other hand, 1 also ignored Mach's criticism which casts doubt on absolute acceleration and which was a forerunner of relativity theory. 24 A systematic employment of non-mathematical techniques (e.g., physical experiments) for establishing mathematical truths, techniques not replaceable by mathematical ones, may, however, eventually lead to a change of a mathematical framework. 2_~ Here, I think, lies the way of explicating the rigid designators of modal contexts from the aspect of frameworks.

ontology and conceptual frameworks part ii

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