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Jaakko Hintikka
REFORMING LOGIC (AND SET THEORY)
1. Frege’s mistake
Frege is justifiably considered the most important thinker in the development of our
contemporary “modern” logic. One corollary to this historical role of Frege’s is that his
mistakes are found in a magnified form in the subsequent development of logic. This
paper examines one such mistake and its later history. Diagnosing this history also
reveals ways of overcoming some of the limitations that Frege’s mistake has unwittingly
imposed on current forms of modern logic.
Frege’s mistake concerns the semantics (meaning) of quantifiers. The mistake is
to assume that this semantics is exhausted by the quantifiers’ (quantified variables’)
ranging over a class of values. These values are the members of the domain (universe of
discourse) of the language to which the quantifiers belong. The entire job description of
the quantifiers is to indicate whither or not at least one member of the domain has a
certain (possible complex) predicate (existential quantifier) and to indicate whether all of
them have one (universal quantifier). In other words, quantifiers are higher order
predicates indicating whether or not a given lower-order predicate is nonempty or
exceptionless. This is in fact precisely how Frege proposes to treat quantifiers in his
logical theory. (See Frege 1984, pp. 153-154, pp. 26-27 of the original.)
This is obviously part of the semantical task of quantifiers. However, it is not the
only one. Quantifiers have another function in language. There is a task that any
language must be capable of fulfilling if it is to serve as a language of science and for that
matter as a language suitable for innumerable purposes in everyday life. This task is to
C:\Hintikka.Reforming logic.and set theory.0408.doc.7/30/2008
indicate what depends on what, more explicitly, to express relations of dependencies and
independencies between variables. It is easily seen that the only way of expressing such
dependencies in an ordinary logical language on the first-order level is through formal
dependencies and independencies between quantifiers. That the variable y depends on x
(in the sense of ordinary-life dependence) is expressed by the fact that the quantifier
(Q1y) to which formally depends on the quantifier (Q2x) to which x is bound. Thus in an
(interpreted) sentence of the form
(1.1) (∀x)(∃y)F[x,y]
the variable y depends on the variable x, as is seen e.g. from the fact that the truth-making
value (“witness individual”) of y depends on the value of x. (Re witness individuals, see
also sec. 9 below.)
Such dependence can be expressed on the second-order level by quantifiers
asserting the existence of a function that embodies this dependence. For instance, (1.1) is
equivalent with
(1.2) (∃f)(∀x)F[x,f(x)]
Here f picks out as its value b=f(a) a truth-making value b of y that corresponds to the
value a=x of each x. It will turn out that this way of expressing the dependence of
variables can also be expressed on the first-order level by means of the dependence
relations of first-order quantifiers. This can be done in IF logic; see section 2 below.
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This independence of the two aspects of the semantics of quantifiers of one another is
vividly seen in many-sorted quantification theory. The two quantifiers can range over
different and even exclusive domains, and yet be either dependent or independent of each
other, as the case may be.
It is not anachronistic to call Frege’s neglect of the role of quantifiers as
expressing such dependencies a mistake. Frege’s own co-discoverer of the logic of
quantifiers, C.S. Peirce, was fully cognizant of this dimension of their semantics. In
practice, its most basic manifestation is the importance of quantifier ordering. In Peirce,
this ordering comes up in the form of the distinction between the two players of the
semantical games and quantifiers of whose importance Peirce was aware. Peirce’s pen-
pal Ernest Schröder struggled with the problems of coping with the same aspect of the
meaning of quantifiers in less vivid terms. (See here Hintikka 1996 (b) and the references
given there.)
2. IF logic and scope
One consequence of Frege’s mistake has been pointed out earlier and corrected, at least
in part. (See e.g. Hintikka 1996.) Since part of the task of quantifiers is to express
dependencies between variables, our logic should be able to do this job completely. In
other words, we should be in a position to express any possible pattern of dependencies
and independencies between variables. These interpreted dependencies between
variables are expressed by the formal dependencies between the quantifiers to which they
are bound. Now how are these formal dependencies codified in the usual logical
notation? The obvious answer is: By the nesting of quantifier scopes. But this nesting
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relation is of a rather special kind. It is among other features transitive and
antisymmetric. Furthermore, it is linear in the sense that the scopes of two quantifiers
cannot overlap only partially. Hence only such dependence patterns can be formulated in
the received logic of quantifiers when the dependence relation has these special
properties. As a consequence, only some of all possible patterns of dependence and
independence can be expressed in the received first-order logic. Hence this logic does
not fulfill its whole job description. Frege’s mistake thus gave rise to a flaw in the
received first-order logic.
This flaw is corrected in what has come to be called IF logic (For it, see e.g.
Hintikka 1996 (a), Hintikka and Sandu 1996.) This can for most purposes be
accomplished by introducing an independence-indicating / (“slash”) that makes a
quantifier (Q2y/Q1x) (replacing (Q2y) independent of another quantifier (Q1x) even when
it occurs in the syntactical scope of (Q1x).
It is thus seen that IF logic is not a special logic alternative to the received logic of
quantifiers. On the contrary, it is our usual Frege-Russell first-order logic that is
unnecessarily restricted in its expressive power and hence should be considered a special
logic among alternatives. In contrast, IF logic is the unrestricted logic of quantifiers.
In this essay, IF logic is not discussed further and is not relied on, either, except as
an object lesson. It is nevertheless in order to point out some consequences of its very
existence.
Once we realize that the nesting of syntactical scopes is not an ideal method of
expressing dependence and independence, we realize also that we have to be careful of
the traditional notion of scope as an explanatory notion in semantics. (Cf. here Hintikka
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1997.) The traditional notion combines two things that per se have nothing to do with
each other. Syntactical scope is used to indicate the dependence and independence of
quantifiers and other logical operators of each other. (This might be called dependence
scope or priority scope.) But it also makes the syntactical segment of a sentence (or
discourse) where a variable is bound to a given quantifier. (Binding scope.)
Once the difference between these two is understood, certain problems in the
semantics of natural language are solved. A case in point is the semantics of the so-
called donkey sentences.
(2.1) If Peter owns a donkey, he beats it.
(2.2) If you give each child a gift for Christmas, some child will open it today.
The meaning of (2.1)-(2.2) cannot be expressed in the notation of the received first-order
logic. But if a binding scope is expressed by parentheses ( ) and dependence scope by
brackets [ ], the logical form of these two will be
(2.3) [(∃x)(O(p,x)] ⊃ B(p,x))
(2.4) [(∀x)((∃y)G(x,y)] ⊃ (∃z)O(z,y))
The apparent difficulty with such “donkey” sentences as (2.1) – (2.2) is largely
due to the very same mistake we saw Frege committing. What distinguishes expressions
like (2.3)-(2.4) from familiar ones is conspicuously the use of the dependence-indicating
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brackets [ ]. A failure to use them is accordingly not to give the dependence-identifying
role of quantifiers their full due.
Much of what has been said of dependence relations between quantifiers can be
said of dependence relations of other logically active notions, including propositional
connectives, epistemic and modal operators etc.. For instance, epistemic logic was held
back for years before it was realized that wh-knowledge can only be adequately
expressed by means of quantifiers that are independent of clause initial epistemic
operators, as e.g. in “It is know who is F” whose logical form turns out to be
(2.5) K(∃x/K)F[x]
where the stroke / expresses independence. (See here Hintikka 2003.)
In general, by freeing the conventions governing the scope we can achieve the
same result as by introducing an independence indicator. In this way, we will be able to
express patterns of dependence and independence between quantifiers (and propositional
connectives) and constants that cannot be expressed in the received first-order logic.
(Constants may also have to be included in the arguments of Skolem functions.) The fact
that we can thus carry out the liberation of quantifiers by changing only the punctuation
of logical sentences is vivid evidence for the naturalness and indeed indispensability of IF
logic.
It is even possible in this way to turn Tarski’s T-schema into a truth definition.
Let us assume that x is a variable for the Gödel numbers x = g (S) of sentences S.. Then
Tarski’s T-schema summarizes all sentences of the form
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(2.6) T(a) ↔ S[a]
where T(x) is a truth predicate. Tarski is right in that we cannot have
(2.7) (∀x)(T(x) ↔ S[x])
As I have pointed out on other occasions, this failure is due to the fact that quantifiers and
other logical operators in S[x] should not depend on the variable x, which has a purely
syntactical role in S[x]. Such dependencies can be ruled out by writing instead of (2.7)
(2.8) (∀x)([T(x)] ↔ S[x])
Of course, this is no longer equivalent to any ordinary first-order sentence. The same
thing can be expressed in IF logic by making all the quantifiers and propositional
connectives in (2.8) (other than (∀x)) independent of the initial universal quantifier (∀x).
Either way, our liberated notation enables us to do what Tarski proved impossible to do
by means of the received Frege- Russell first-order logic: convert the T-scheme into a
genuine truth definition.
3. From existential instantiation to functional instantiation
Another consequence of Frege’s mistake that is (perhaps unwittingly) repeated by later
logicians looks so insignificant that it has not attracted much attention. It concerns the
formulation of the rules of inference for our basic first-order logic. There it looks very
much as if the meaning of quantifiers is done full justice to (in a context of deduction) by
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the usual rules of instantiation. The rule of existential instantiation applies to a formula
(∃x)F[x] with an initial existential quantifier. It allows the replacement of this formula
by F[β] where β can be thought of as standing for a possibly unknown individual of the
kind the given formula says is instantiated. This obviously captures the force of the
existential quantifier as expressing non-emptiness.
Intuitively, the term β operates just like the “John Does” and “Jane Roes” of
lawyers’ jargon. (Wallis thought that historically such legal usage was the historical
model for algebraic symbols; see Klein 1968. p. 321.) Formally, the term β can be a
“dummy name” or in our deductive practice simply a new individual constant.
Likewise, the usual rule of universal instantiation might seem to capture
adequately the semantical force of a universal quantifier as expressing universality
(exceptionlessness).
But even though these instantiation rules express truth and nothing but the truth
about the meaning of quantifiers, they do not tell us the whole truth. One at first sight
inconspicuous feature of theirs is that they apply only to sentence-initial quantifiers.
They do not apply to quantifiers inside a formula, not even if this formula is assumed to
be in the negation normal form. (This assumption is routinely made in this paper.) Every
logic instructor who has taught to her students the usual rule of existential instantiation is
likely to find herself later correcting students who are proposing to apply it to quantifiers
inside a formula, perhaps within the scope of universal quantifiers. At this point, a clever
student could try to embarrass the instructor by asking: “Since the rule of existential
instantiation is obviously based directly on the meaning of the existential quantifier,
surely it ought to be applicable independently of the context. What happens in such an
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application is that we merely choose one individual of a certain kind among existing ones
for our attention.”
If the instructor is up to her task, she will point out that the choice of the
“arbitrary individual” β is not absolute, not a once-and-for-all matter, but depends on
other individuals. More specifically, it depends on the values of the universal quantifiers
within the scope of which the existential quantifiers occurs (in a sentence that is in the
negation normal form).
This answer points to an important truth. Existential instantiation can take place
inside larger formulas, if we use as an instantiating term a function term that takes into
account the dependence of the existential quantifiers to which it is applied on other
quantifiers in the same sentence. If we heed those dependencies, we can generalize the
rule of existential instantiation. The generalized formulation might run as follows:
Assume that S is a sentence in the negation normal form and that the formula
(3.1) (∃x)F[x]
occurs somewhere in S=S[(∃x)F[x]]. Then S may be replaced by
(3.2) S[F[f(y1,y2,…)]
where (∀y1), (∀y2),… are all the universal quantifiers which the scope of which (∃x)
occurs in S, and f is a new function constant. If there are no such universal quantifiers,
the function term f(y1, y2, …) is replaced by a new individual constant. The old rule of
existential instantiation is thus a special case of the new one, viz. the case of sentence-
initial existential quantifiers.
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More generally, we can stipulate that (Q1y1), (Q2y2), … are all the quantifiers in S
on which the quantifier (∃x) depends on there. This formulation can be used also in IF
logic.
Notice that this is a first-order rule in the crucial sense that no quantification over
higher-order entities is involved. The reason why we have considered instantiation by
functions rather than individuals should be obvious. It reflects the fact that witness
individuals may depend on other witness individuals.
By the same toke the rule of existential generalization has to be liberated. It will
allow the replacement of any function term of the form f(x,y1,y2,…) to be replaced by a
variable z bound to an existential quantifier (∃z). This quantifier must occur within the
scope of all the quantifiers (∀y1), (∀y2),… . Otherwise its location is free, assuming only
that we are dealing with a formula in the negation normal form
4. Uses of the rule of functional instantiation
The relative neglect of the generalized rule existential instantiation can be taken to be an
instance of the same mistake as has been here attribute to Frege. But is it a mistake in the
present context? Defenders of status quo can try to claim that the rule of functional
instantiation is dispensable, and that its neglect is therefore justified, perhaps in the
interest of theoretical economy.
Admittedly, the rule of functional instantiation is redundant in the received
treatment of first-order logic. In this logic, we can let an existential formula wait in our
logical argumentation until by means of applications of other rules it has been brought to
the surface of our formulas, in other words until it has been brought to a sentence-initial
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position. But in principle we have to ask whether this dredging process affects the
semantics of an existential quantifier, including its dependence relation to other
quantifiers. Logicians have been victims of bad luck in that the process of bringing an
existential quantifier to the surface of a sentence does not affect its deductive function in
the received first-order logic. This is bad luck in that it has directed their attention away
from those aspects of the logic of quantifiers that are due to dependence and
independence relations between them, thus making this instance of Frege’s mistake a
mistake.
An example can illustrate the way in which functional instantiation helps to make
logical proof s shorter and more natural. Consider the conditional
(4.1) (∀x)(∃y)(∀z)(F(x,y) & G(y,z)) ⊃ (∀x)(∀z)(∃y)(F(x,y) & G(y,z))
Its proof e.g. by the tableau method would involve six instantiations, three layers of
formulas and two branches on the right side. In contrast, consider an application of the
rule of functional instantiation to the antecedent of (4.1). It yields
(4.2) (∀x)(∀z)(F(x,f(x)) & g(f(x),z))
An application of the rule of existential generalization yields the consequent.
This proof is not only simpler than e.g. a tableau proof. It is obviously far closer to the
ways in which mathematicians actually think. If you do not see this at once, think of the
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ways in which you would express the functional instantiation proof in the jargon of
mathematicians. The antecedent would be read somewhat as follows:
Given (only) x, there is an object y such that for any z, F(x,y) and G(y,z) But if
so, since this object depends only on x, it will trivially satisfy for any x and z the same
conjunction.
This inference would be considered completely trivial. Yet in reality it involves
an appeal to a principle of reasoning too strong in its general form to be accommodated in
the current first-order axiom systems of set theory, as we will see.
We can use functional instantiation systematically and obtain a huge
simplification of many first-order logical proofs. What one can do is to turn a proposition
into a negation normal form and eliminate all existential quantifiers by means of the rule
of functional instantiation. The remaining quantifiers are all universal. They can all be
moved to the beginning of the sentence and largely neglected. The reason is that all the
variables bound to them admit arbitrary substitutions. Without any great loss of
generality, we can assume that all predicates have been replaced by functions, perhaps by
their characteristic functions. (The characteristic function of a one-place predicate A(x)
is a function f(x) such that A(x) iff f(x) = O. This is easily generalized.)
When all this is done, all usual formal first-order logical proofs become literally
symbolic calculations in which all of the logic of quantifiers is reduced to substitutions of
terms (usually function terms) for free variables in equations combined with each other
truth-functionally. It would be interesting to see what a proof theory for such a logic of
equations might look like.
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A logic developed along these lines does not have theoretical interest only. It
yields a proof method which often is in practice incomparably handier that the usual first-
order proof methods. In order to see this, consider an example. Suppose that we have to
prove the proposition about Abelian groups that would usually be expressed as follows
(4.3) x o (z o y) = (x o y) o z
where o expresses the group operator. (The symbol o expresses a two-argument
function.) Even to express (4.3) by means of quantifiers would require seven of them:
(4.4) (∀x)(∀y)(∀z)(∀u)(∀v)(∀w)(∀t)
(((z o y=u & (x ou)=v & (x y)=w & (wo z=t) ⊃ v=t) o
The associative and commutative laws would likewise require several quantifiers. To
deduce from them (4.3) by means of ordinary first-order logic would be a messy
enterprise. In contrast, the functional deduction is trivial:
(4.5) x (z o y) = xo(y o z) = (xoy) o z o
The first identity is justified by the commutativity of , the second by its associativity.
But not only does functional instantiation facilitate formal logical proofs, tacit
instantiation plays a pervasive role in ordinary human reasoning. Take, for instance the
old chestnut of a puzzle that I have used earlier to illustrate reasoning in ordinary life:
o
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(4.6) A gentleman and his sister are sitting on a bench in a park. Pointing to a child
playing nearby, he says: “That’s my niece. His sister says, “But not mine.” How is it
possible for both of them to be right?
How do we solve in real life such problems? Let us try to do so, and watch
ourselves in process. The child is the brother’s niece if and only if she is female and
(4.7) (∃x)(S(b,x) & P(x,c))
Here c = the child, s = the sister, b = the brother, S(x,y) = x and y are siblings, and P(x, y)
= x is a parent of y. Obviously, (4.7) is tacitly obtained from the definition of a niece.
Our singular terms b and c are tacitly instantiating certain variables (say y and z) in such
a definition. In order to argue further, we obviously have to instantiate the x in (4.7). This
should introduce a function term p(y,z) for the so far unidentified parent of c. But of
course our reasoning practice suggests its dependence of y and z, and argue in terms of it
as if it were a simple term p, and argue simply as follows: Both s and p are siblings of b,
while p and s are not siblings. This is possible only if p=s.
Here it is seen how we spontaneously argue in terms of function terms as if they
were constants. In contrast, a conventional first-order proof would be so complicated as
to tax severely one’s patience, and would not be halfway as übersichtlich.
In more general terms, the rule of functional instantiation thus allows an
automatic concrete interpretation of what is going on in a purely formal or “symbolic”
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proof. It can be viewed as a codification of the idea behind mathematicians’ time-
honored locution for an existential quantifier: “One can find.” This verbal formula leaves
unexpressed the crucial question: What has to be known before one can find it?
This interpretability is relevant to the philosophical problem of understanding
formal logical proofs. Wittgenstein was especially keenly attuned to this problem, but
never found a solution that would have satisfied him. Here we can see what kinds of
interpretations of logical arguments might have satisfied him. (I can imagine Frank
Ramsey surviving and forcing Wittgenstein to see the point.)
5. Functional instantiation is a first-order rule
It is worth emphasizing that a first-order logic amplified with a rule of functional
instantiation is still a first-order logic. Considered alone, such logic is precisely as
strong as the received first-order logic, not any stronger. Moreover, it is first-order in the
crucial sense that it involves no quantification over any higher-order entities is involved.
We all know Quine’s quip “to be is to be a value of a bound variable”. In the present
context, it is much more that a clever slogan. I am convinced Hilbert was right in
thinking that our difficulties in the foundations of mathematics are due to problems
concerning the existence of higher-order entities. (Those problems are e.g. instantiated
by a problem of choosing the axioms of set theory.) A first-order logic that includes
functional instantiation is free from all such problems. The fact that in the rule of
functional instantiation we introduce function constants over and above individual ones
merely reflects the trivial fact that the witness individuals that show (in the sense of
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displaying) the truth of a quantificational sentence can depend on other such witness
individuals.
By the same token, we need not worry about the consistency of the rule of
functional instantiation.
Another indication of the first-order status of the rule of functional instantiation is
that this rule is a valid logical principle of independence-friendly first-order logic. Even
the dispensability of the rule of functional instantiation the received first-order logic is
interesting in the present context. It can be considered a proof of the fact that the rule of
functional instantiation expresses a purely logical principle, and a first-order one to boot.
6. Functional instantiation and the axiom of choice
It might nevertheless seem that the main role of the rule of functional instantiation is to
provide us with a way of improving first-order logic, but not anything relevant to
foundational issues. This can perhaps be said if first-order logic is considered only by
itself. When it is used in wider context, it turns out to have remarkable powers.
For one thing the rule of functional instantiation is no longer dispensable in IF
logic. The dependence and independence relations admitted there are more sensitive than
those to which received logic confines us, so sensitive that they can be disturbed in the
process of bringing an existential quantifier to a sentence-initial position.
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7. Axiom of choice — the axiom of choice
The rule of functional instantiation might still seem to be only a handy tool in improving
the theory and practice of our basic first-order logic. In reality, its most striking
repercussions lie in the foundations of mathematics, especially in set theory.
In dealing with these foundations, we have to go beyond first-order logic. The
received first-order logic is too weak for the purpose, and therefore has to be considered
as a part of a larger enterprise, be it set-theory or higher-order logic. Now what happens
if our modified first-order logic that now includes the rule of functional instantiation
operates as a part of second-order logic? Obviously, we have to assume that this second-
order logic includes the usual unproblematic second-order quantifier rules, including
universal instantiation. Consider then, a sentence of the following form
(7.1) (∀x)(∃y)F[x,y] ⊃ (∃f)(∀x)F[x,f(x)]
Given the rule of functional instantiation, (7.1) is logically true. In order to see that it is,
consider its negation
(7.2) (∀x)(∃y)F[x,y] & (∀f)~(∀x)F[x,f(x)]
An application of the rule of functional instantiation to the first conjunct of (7.2) yields a
formula of the form
(7.3) (∀x)F[x,g(x)]
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Universal instantiation as applied to the second conjunction yields
(7.4) ~(∀x)F[x,g(x)]
which contradicts (7.2). Hence (7.1) is logically true.
In a similar way we can obviously prove any conditional of the form
(7.5) (S ⊃ S(sk))
where S is a first-order sentence and S(sk) the second-order sentence that asserts the
existence of a full array of the Skolem functions for S.
What is remarkable about (7.1) is that it is an application of what is usually called
the axiom of choice. Indeed the schema instantiated by (7.1) is sometimes used as a
formulation of the axiom of choice. Since (7.1) are provable by using only first-order
principles including the rule of functional instantiation (over and above trivially valid
ones), it follows that the (so-called) axiom of choice is a valid first-order logical
principle.
This conclusion is reinforced by the fact that the axiom of choice is valid in first-
order IF logic. For instance, it is easily seen that the counterpart of (7.1) in IF logic is a
logical truth there. In IF logic, the consequent of (7.1) becomes
(7.6) (∀x1)(∀x2)(∃y1/∀x2)(∃y2/∀x1)(((x1=x2) ↔ (y1=y2)) & F[x1,y1] & F[x2,y2))
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This is logically equivalent with the second-order sentence
(7.7) (∃f1)(∃f2)(∀x1)(∀x2)((x1=x2) ↔ (f1(x)=f2(x)) & F[x1,f1(x1)] & F(x2,f2(x2)])
Here the first conjunct says that f1 and f2 are the same function. Hence (7.7) is equivalent
with
(7.8) (∃f)(∀x)F[x,f(x)]
which is the consequent of (7.1).
In short, the rule of functional instantiation is tantamount to a strong form of the
axiom of choice. In the rest of this paper much of the discussion is formulated in terms
of the axiom of choice. It should not be forgotten that we shall be in effect talking about
the first-order rule of functional instantiation.
In view of what has been found about the first-order status and the consequent
indispensability of this rule, the nature and status of the axiom of choice have to be
reconsidered. Indeed the first-order status of the axiom of choice is in stark contrast to the
ways it is usually dealt with. Usually, it is considered a set-theoretical principle. Often,
this principle is codified into the axiom system of set theory. This is where the term
“axiom” in “axiom of choice” comes from. Even though this term will be seen to be
inappropriate, it will nevertheless be used in what follows.
What has been seen is that the rule of functional instantiation has the effect of
turning the “axiom” of choice into a first-order logical truth. This is interesting also in
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view of the history of foundational studies. The ideal that the great Hilbert had was to do
mathematics entirely on the first-order level. (He blamed all the ills in the foundations of
mathematics on the use of higher-order conceptualizations.) (Hilbert 1922, pp.162-163.)
The first and foremost example of an indispensable higher-order mode of
reasoning is the axiom of choice. Hilbert’s ε-calculus was an attempt to bring the axiom
of choice down to the first-order level. (Se Hilbert and Bernays 1934-39.) It was not a
complete success in this respect. For one thing, it did not facilitate consistency proofs for
elementary arithmetic.
One can even pinpoint the crucial shortcoming of Hilbert’s epsilon-technique. He
was on the right track in using choice terms, but he failed to indicate explicitly what the
choices in question depend on. It is a variant of the mistake we found in Frege: A failure
to appreciate fully the role of dependence relations in first-order logic.
We have now seen that this mistake is not inevitable. By showing that the “axiom” of
choice is a first-order logical principle, we have realized an important part of Hilbert’s
hopes. This has repercussions for the evaluation of Hilbert’s foundational work in
general. For instance, when elementary number theory is based on IF first-order logic
instead of the received one, it becomes possible to prove its consistency by arguably
elementary means. (See Hintikka and Karakadilar 2006.)
8. Axiom of choice vs. axiomatic set theory
The first-order character of the axiom of choice means that it is inappropriate to construe
it as an axiom of a nonlogical mathematical theory, viz. axiomatic set theory. It ought to
be instead a part of the logic which is used in set theory and in terms of which proofs in
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set theory are being couched. The failure of logicians to do that is attributable to the
disregard of dependence relations between quantifiers that has been called Frege’s
mistake. Indeed, from (7.1) one can see how the axiom of choice is a matter of spelling
out what the dependence of an existential quantifier or a universal one means.
It might at first look as if it were merely a matter of terminology whether the
assumption we are dealing with is called a first-order logical principle or a set-theoretical
axiom. However, calling it an axiom of a mathematical theory has a point only if this
axiom makes a difference in the sense of ruling out otherwise conceivable alternatives.
The claim that is made here is therefore that a set theory without the axiom of choice
involves serious interpretational problems. Later in this essay, it will be discussed how
these difficulties are manifested in the foundations of set theory.
But what is the difficulty here? A version of the axiom of choice is included in
the usual axiom systems of set theory. And there does not seem to be any difficulties in
formulating the first-order logic that is used as a basis of set theory so as to include a rule
of functional instantiation. Here the issue seems to be merely a matter of philosophical
emphasis.
Things are not so simple, however. Here we meet the feature of the problem
situation that has not been completely unknown but whose full significance has not been
appreciated. Logicians are here facing a dilemma. On the one hand, the form of the
axiom of choice that is used in first-order axiomatic set theories does not capture the
same full force of the principle that is among other formulations captured by (7.1). On
the other hand, if we try to incorporate assumptions codifying this force in the usual first-
order axiom systems of set theory, they become uninterpretable and even inconsistent.
21
This happens independently whether the strong axiom of choice is introduced by a
separate set-theoretical axiom or whether it is introduced by strengthening the underlying
logic used in set theory by incorporating the role of functional instantiation in it.
The fact that the full force of the axiom of choice cannot be stated in first-order
axiomatic set theories without making them uninterpretable as set theories can be seen in
different ways. Any set theory AX that can serve as a basis of mathematical theories
should allow the reconstruction of elementary arithmetic. Hence we can use Gödel
numbering or an equivalent technique to discuss the syntax of AX in the very same set
theory based on AX. Among other things, we can then formulate a numerical predicate
K(g(s)) = K(x) that says that the sentence with the Gödel number x=g(S) does not have
all its Skolem functions. By the diagonal theorem there is then a sentence S of the form
K(n)=K(g(S)) with the Gödel number n. Here n is the numeral expressing n. Intuitively
(although slightly inaccurately) S could be taken to say, “My Skolem functions do not all
exist” in the same sense as the famous Gödel sentence which says, “I am unprovable.”
Thus S must be true, for if it were false, its Skolem functions would exist. Such existence
is enough to guarantee the truth of S. Hence S will be true but without its Skolem
functions, which is violates the notion of truth.
Moreover, the existence of S is easily proved formally in the set theory in
question. This does not mean that the set theory in question is inconsistent, But it means
that it does not admit of the intended kind of interpretation, that is, an interpretation
where the objects quantified over are sets. For the allegedly true sentence S would be
false in such a model.
22
Furthermore, an incorporation of the full axiom of choice in conventional axiom
systems of set theory would make them inconsistent. In the self-applied set theory we
could form a predicate P that applies to the Gödel number n=G(S) if and only if the
Skolem functions of S all exist. Such a predicate would be a truth condition for set
theory. Alas, from Tarski’s impossibility theorem it follows that such a truth predicate is
impossible on the pain of inconsistency, as it would allow for a truth definition for a first
order theory in the same theory. This observation turns out to touch some of the most
important presumed uses of set theory; see sec.10 below.
The use of a restricted form of the axiom of choice in axiomatic set theory is
sometimes motivated by reference of the distinction between sets and classes that is made
in some set theories. The axiom of choice is taken to be applicable to sets only, not to
proper classes. This is not very satisfactory theoretically, either. There does not seem to
be anything intrinsic to a collection of objects that would make it a proper class instead of
a set. For instance, what is it about the class of all unit sets that makes it a proper class?
Frege even identified this class with the number one. Surely the number one should be
capable of serving as a value of any set-theoretical variable x even in a context like x ∈ c.
9. Set theory vs. model theory
Is there an explanation of this tremendous strength of the rule of functional instantiation?
Yes. Its strength is not accidental. It is based on the very nature of quantificational
discourse. More specifically, it is based on the fact that the existence of a Skolem
function for a quantificational sentence S is the natural truth condition for S.
23
One way of seeing this is in terms of the idea of “witness individuals”
vouchsafing the truth of S. For a sentence of the form (∃x)F[x], a witness individual b is
one satisfying F[x], that is, making F[b] true. For a sentence of the form (∀x)(∃y)F[x,y],
witness individuals a, b must satisfy F[a,b]. But here the choice of b depends on the
choice of a. Hence the existence of suitable witness individuals means the existence of a
function f(x) such that, for each a, a and f(a) can serve as witness individuals. This is
generalizable as a matter of course to the existence of Skolem functions as guaranteeing
the existence of the appropriate witness individuals.
This truth condition is equivalent to any other adequate truth condition. This
explains the significance of the rule of functional instantiation, for it is what provides for
the existence of Skolem functions for any true sentence. If those Skolem functions do not
always exist for a true sentence, truth is not expressible in the language in question. In
this sense, the ultimate reason why the strong form of the axiom of choice which is
codified in the rule of functional instantiation is not available in first-order
axiomatizations of set theory is Tarski’s impossibility theorem: such a strong form of
axiom of choice would make truth expressible in those axiomatizations.
Some philosophers have earnestly tried to find “truthmakers”, that is entities of
some kind or other that serve to make true sentences true. The search has not revealed
unproblematic truthmakers. Now we can see what the true truthmakers of a
quantificational sentence S are. They are the Skolem functions of S.
The persuasiveness of this answer is enhanced by the game-theoretical
interpretation of first-order logic. There Skolem functions are codifications of those
strategies that enable a verifier in a semantical game always to win. Such a win may be
24
considered as a tentative verification of the sentence S which is the object of the
semantical game G(S) associated with S.
The failure of the rule of functional instantiation in an axiomatized set theory
therefore means that truth is not definable in it. It may look as if truth may be definable
in a suitably formulated axiomatic set theory on the first-order level. Such appearances
are deceptive, however. What happens in such cases is that the pseudo-definition yields
sometimes wrong results. In particular, these will be in any model (in the first-order
sense of a model) of first-order axiomatic set theory where allegedly true sentences
whose Skolem functions do not exist and others therefore are not true on a set-theoretical
interpretation of the model. Now the availability of a truth predicate is a condition sine
qua non for any realistic model theory. The failure of all truth predicates in a first-order
axiomatic set theory therefore means that first-order axiomatic set theory is an inadequate
framework for model theory of itself.
This is a striking result in that it contradicts the widespread idea of axiomatic set
theory as the natural medium of all model theory. This idea is simply wrong. For any
halfway adequate model theory you need the notion of a truth, which just is not available
in a set theory using traditional first-order logic. First-order axiomatic set theories are
poor frameworks even for their own model theory. A fortiori, there are likely to be poor
frameworks for any theory formulated in their terms.
The inadequacies of first-order axiomatic set theory as a framework of model
theory are made especially serious by the role of metatheoretic conceptualizations in
modern mathematical practice. Philosophers often seem to entertain an oversimplified
picture of a mathematician as a chap who sets up axiom systems and then draws logical
25
conclusions from them. Perhaps this oversimplification is not peculiar to philosophers
only. Even some practicing mathematicians think that all mathematics can do is to draw
conclusions from the axioms of ZF set theory. (Cf. Ruelle, 2007, pp. 63, 73-74.) Model-
theoretic questions are on this view a superstructure that may perhaps be the business of
logicians and philosophers rather than mathematicians per se.
This is a radical misrepresentation of current mathematical practice. Not only has
the line between mathematical theories and their model theories become inconspicuous.
Much of what counts as actual mathematical theorizing is in fact model-theoretical.
Consider, as an example, group theory. Only a miniscule part of any work in set theory
consists of deductions from the axioms of the theory. The bulk of the actual work in
group theory is metatheoretical, consisting largely in such things as classifications of
groups of different kinds, representation theorems, and other ways of gaining an
overview of the models of group theory (i.e. groups) of different kinds. Particular
deductive consequences of the axioms do not play a much bigger role in the real theory
than particular numerical equations like Kant’s 5+7=12 play in actual number theory.
This feature of mathematical practice explains a curious episode in the history of
twentieth-century logical theory. (Cf. Hintikka 2004.) Tarski’s preferences in logic were
algebraic rather than geometric or set-theoretical. In the forties, he ganged up with Quine
to criticize Carnap’s attempts to build a model theory in the form of “logical semantics”.
This makes it prima facie surprising that it was Tarski who in the fifties and sixties led
the development of the present-day model theory. The solution lies in the fact that Tarski
was virtually forced to develop a model theory by his pursuits in the theory and
metatheory of different algebraic systems. It was not initially thought of as a separate
26
branch of logical studies, comparable to proof theory or recursion theory. It was created
as part of the metamathematics of algebra.
10. The meaning of quantifiers and the foundations of mathematics
The rule of functional instantiation does not presuppose any particular “standard”
conception of logic, either. In fact, it offers means of implementing such “nonstandard”
variants of logic as constructivistic and intuitionistic ones and also bringing out their
precise differences from the “classical” logic. Indeed, all we have to do is to restrict the
interpretation of the function constants introduced in functional instantiation in some
desired way, for instance to constructive functions or to known functions.
The problems discussed in this paper are thus likely to come up in any reasonable
approach to the foundations. In view of the role the axiom of choice plays in the
arguments marshaled here, it is therefore instructive to see that, for all the lip service to
the contrary, some of the most prominent constructivists among philosophers of logic (for
some reason they call themselves intuitionists) have ended up endorsing the axiom of
choice. They include Michael Dummett (1997, pp. 52-59) and Per Martin-Löf (1984, pp.
50-52). This strikingly illustrates the fact that what is at issue in the axiom of choice is
the meaning of quantifiers, not the interpretation of mathematical truth in general.
This can be generalized. The introduction of the rule of functional instantiation
has striking consequences for the understanding of what is referred to as “mathematical
practice” and what has recently become a revered holy cow in semi-popular philosophy
of mathematics. In spite of the attention ostensibly paid to this practice, some of its
significant features have not been noted. One of them is the fact that mathematicians
27
routinely use functional instantiation in their reasoning. As soon as objects of a certain
kind exist, mathematicians introduce symbols for them, even though those objects depend
on others,. Often that dependence is not explicitly indicated.
For one simple example, one of the axioms of group theory could be expressed as
(10.1) (∀x)(∃y)(x o y = e)
But nobody in actual practice (other than a logic student) starts a proof from (10.1). A
mathematician immediately introduces a symbol, e.g. x-1 for the y. This is but an
application of functional instantiation.
Likewise, in defining the continuity of a function f(x) at the value xo in an interval
x1 ≤ xo ≤ x2 by the usual ε–δ method, textbooks write out only one symbol for ε and δ,
respectively, even though δ in reality is a function δ(ε) of ε. Moreover, δ depends also on
xo so that it should strictly speaking be expressed as δ(εi xo), even though in introductory
texts this is never expressed. (If δ actually can be chosen independently of xo, we have a
definition of uniform continuity, instead of continuity simpliciter.)
In many, probably most case, such functional instantiations can b treated as
expository tricks. But this does not change the fact that mathematicians routinely rely on
a rule of inference that is (in suitable contexts) extremely strong, in fact so strong that it is
incompatible with the usual axiom systems of set theory. This in turn refutes the
commonplace belief that first-order axiomatic set theories can be considered a lingua
franca of all mathematics.
28
The fact that axiomatic set theory does not capture certain obviously acceptable
modes of inference must also be considered a serious limitation to the uses of axiomatic
set theory. It seems to me that we should pay much more attention to these limitations.
Thus we should for instance consider Gödel’s and Paul Cohen’s unprovability results as
warning signs, as symptoms of shortcomings, rather than informative achievements
concerning the continuum hypothesis or the axiom of choice. (Cf. Cohen 1966.)
In sum, what the logic is that practicing mathematicians in effect use is a version
of first-order logic that includes functional instantiation. This logic is easily confused in
its applications with ordinary first-order logic. The reason is that when mathematicians
instantiate their (usually tacit) quantifiers, the dependence of the instantiating “arbitrary
object” on other objects is often, perhaps typically, left unexpressed. This is not merely a
matter of exposition. Since mathematicians are frequently using functional instantiations
in contexts involving sets on other higher-order entities, their logic is in fact much
stronger than the received first-order logic and in fact stronger than the usual first-order
axiomatized set theories. This shows how unrealistic these first-order axiomatic set
theories are as frameworks of mathematical practice.
It is no excuse for this failure that its roots may lie in the nature of literally
hardwired human preferences in logic reasoning. In general, human reasoners like to
operate with free variables or other symbols that behave like constants in that their
dependencies on other objects can be disregarded, rather than bound variables or other
symbols whose dependence on others is spelled out. The exception is a variable bound to
a sentence-initial universal quantifier. Such a variable can be thought of as representing
“an arbitrary individual” or perhaps “an unknown individual” about which we can reason
29
in the same way as ordinary known ones. This instinctive preference may be due to the
hardwired characteristics of the human information-processing faculty. (See von
Neumann 1958, Hintikka 1990.) Now we have seen that this preferred method of
reasoning can be made possible by the rule of functional instantiation. This rule has
therefore an important role in any humanly natural system of reasoning.
Perhaps we can from the vantage point that has been reached also put the large-
scale history of modern logic into an interesting and perhaps ironic perspective. One
characteristic feature of the entire logicist enterprise of Frege, Russell and Whitehead and
their ilk is that they tried to reduce mathematical reasoning to purely logical reasoning.
For instance, in Frege’s axiomatization of his Begriffsschrift there are no
characteristically higher-order assumptions. Frege thought he could formulate the crucial
assumption in terms of the identity conditions of extensions and value-ranges of
propositional functions (Cf. the Basic Law V of his Grundgesetze.) This enterprise might
seem hopelessly unrealistic, in view of the apparent limitations of first-order reasoning.
However, it is now seen that logicists’ reliance on first-order logic is not entirely
misplaced. Unfortunately, what later logicians and mathematicians did was look for the
sources of the missing greater strength outside logic, mainly in set theory, instead of
making the most of what they already had in first-order logic.
If the idea behind the rule of functional instantiation is as simple as it has been
seen to be and yet so consequential, how come it has not been used and studied before?
It does not seem unfair to blame it on the same neglect of the role of quantifiers as
dependence indicators as we initially diagnosed in Frege.
30
One form of this neglect is a failure to pay attention to the assumptions that are
actually made in the reasoning used in mathematical practice. When the axiom of choice
was first formulated, it turned out that it had unwittingly been used frequently in accepted
mathematical arguments, sometimes by the very critics of the axiom. It seems that this
self examination should be continued. It has turned out that what looks like a simple
first-order inference may in fact be an appeal to a strong version of the axiom of choice.
This is important in a foundational perspective for the purpose of understanding
mathematical practice. This practice may involve assumptions that go well beyond, not
only our usual first-order logic, but our usual axiomatic set theory. This provides an
interesting perspective on projects like the “reverse mathematics” of Harvey Friedman. It
is of great interest to see precisely what assumptions an actual mathematical argument
presupposes.
11. Quo vadis?
Where should foundational studies be headed after Frege’s mistake has been rectified?
This is too large and too sweeping a question to be dealt with in one paper. Some
observations nevertheless seem pertinent. For one thing, set theory should in the future
be based on some logic that allows the formulation of a truth predicate for set theory by
the means of set theory itself. One such logic is IF first-order logic. But whatever logic
can serve this purpose presumably must dispense with the law of excluded middle, as IF
logic does. This would necessitate giving up of Frege’s well-known requirement on sets,
viz. that the membership in one of them is well defined, not allowing indeterminate cases.
This would mean a significant change in our very notion of set.
31
One can also ask: In the light of these results, where should the study of set theory
be heading? Or should we rather ask: What should set theory be taken to be? There is an
age-old debate as to what logic really is, a theory (alias “science”) or a conceptual tool
for all sciences, an organon. This question is still very much alive. For instance, are the
axiomatizations of this or that part of logic on a par with the axiomatizations of scientific
theories? The deep differences between the two are sometimes overlooked.
The same question should be asked about set theory. It is often taken to be like
any other mathematical theory. But if so, how can it be a way of codifying logical
principles of reasoning, such as in the axiom of choice? In any ordinary axiomatic theory,
we need some logic by means of which we reason about its models. Now the axiomatic
assumptions in set theory are assumptions concerning those models. How can they at the
same time codify modes of reasoning about those models?
The interpretation of axiomatic set theory as a normal mathematical theory leads
to other strange results. For what are the objects which it theorizes about? All actually
existing sets? But how do we know what there actually exists? Either we have to
postulate an upper floor of our universe populated by abstract Platonic objects or else we
have to envisage a super-universe of possible structures, some sort of “model of all
models”. Neither conception can be easily disproved, but neither has much appeal to a
thinker who takes set theory to claim to have a special foundational role. For surely we
need some subject-independent logic in order to reason about such entities.
What has been found in this essay suggests an unpopular answer. It has been
found that one of the kingpins of set theory, the axiom of choice, must be considered a
logical principle, even a first-order one. This strongly suggests looking at the entire set
32
theory in the same way, as a part of logic rather than as a separate mathematical theory.
This suggestion is supported strongly, virtually conclusively by developments starting
from IF logic. Many mathematical conceptualizations and modes of reasoning that go
beyond the resources of the received logic and hence were typically considered set-
theoretical rather than logical, for instance equicardinality and Kǔnig’s lemma, are
captured by means of IF logic. Indeed, if we are willing to use very strong forms of
tertium non datur, the entire force of second-order logic an be captured in a suitably
enriched first-order logic. Since second-order IF logic arguably catches all the inferences
needed in normal mathematics, set theory becomes dispensable as a foundational
enterprise, unless it merges with the strengthened first-order logic.
A revision of set theory along the lines sketched here is not a retreat. On the
contrary, it opens new opportunities. It was seen that the full force of the rule of
functional instantiation cannot be realized within the framework of first-order axiomatic
set theory. Small wonder, therefore, that important problems such as the truth of the
continuum hypothesis cannot be solved in a system like ZF set theory. With the help of a
logic incorporating the rule of functional instantiation these problems become more
easily accessible already on the first-order level. Hence even if you do not want to give
up first-order axiomatic set theory tout court, you may be interested in examining what
can be done in an alternative approach.
33
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