1

ROBERTA BALLARIN

VALIDITY AND NECESSITY

ABSTRACT. In this paper I argue against the commonly received view that Kripke’s

formal Possible World Semantics (PWS) reflects the adoption of a metaphysical

interpretation of the modal operators. I consider in detail Kripke’s three main innovations

vis-à-vis Carnap’s PWS: a new view of the worlds, variable domains of quantification,

and the adoption of a universal notion of validity. I argue that all these changes are driven

by the natural technical development of the model theory and its related notion of

validity: they are dictated by merely formal considerations, not interpretive concerns. I

conclude that Kripke’s model theoretic semantics does not induce a metaphysical reading

of necessity, and is formally adequate independently of the specific interpretation of the

modal operators.

KEY WORDS: Carnap, Kripke, modal logic, necessity, possible world semantics,

validity

1. INTRODUCTION

The history of the model theory of modal logic, commonly known as possible worlds

semantics, did not unfold linearly. As a consequence, it is particularly complex, rich and

interesting in its own terms.1 In this paper I will grossly oversimplify the historical

development of possible world semantics (from now on PWS) in order to focus on a

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theoretical question: Is it true that some crucial formal developments that occurred from

the pioneering work of Carnap to the nowadays commonly accepted Kripkean version of

PWS are the natural formal reflections of philosophical considerations, mainly of the

switch of focus from Carnap’s logical modalities to Kripke’s metaphysical necessity and

possibility?

Most philosophers interested in the formal semantics of modal logic seem

inclined to answer positively the above question. Cocchiarella for example labels the

following (Kripkean) practice a ‘model-theoretic artifice’: “[A]llowing modal operators

to range over only some and not all of the worlds . . .” and claims that such an ‘artifice’ is

“quite appropriate and may in fact be required for operators purportedly representing

non-logical modalities (e.g., temporal or causal modalities) . . . [H]owever, . . . the

employment of such an artifice is inappropriate in the semantics of what one considers to

be a purely formal . . . sign.”2

More recently, Hintikka and Sandu have pressed the same point. The title of the

final section of their “The Fallacies of the New Theory of Reference” states their position

with the boldness of a slogan calling for a much needed reform program: Kripke

Semantics Is Not the Right Semantics of Logical Modalities. It is worth quoting

extensively from this section:

[I]n its usual form, the so-called Kripke semantics is not the correct

semantics for logical modalities either. As has been pointed out repeatedly

. . . Kripke semantics . . . is analogous to the non-standard interpretations

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of higher-order logics, which is not equivalent with the intended standard

interpretation of these logics. In other words, the so-called Kripke’s

semantics does not provide us with the right model theory of logical

(conceptual) necessities in any case.

Hence the New Theorists either have to change the logic they are

basing their discussion on or else admit that they are not dealing with

purely logical (alethic) modalities, but with some kind of metaphysical

necessity and possibility.3

Not everybody calls for a logical reform, nonetheless it is widely assumed that the

kind of necessity one is philosophically interested in determines the formal semantics one

adopts – or at least should adopt. David Kaplan for example speculates on the interaction

between two alternative notions of necessity (logical and metaphysical) and two

corresponding approaches to validity (maximal and universal).4 And very recently Sten

Lindström has argued that Kanger’s formal semantics for modal logic is adequate for the

notion of logical necessity, while Kripke’s PWS is adequate for Kripke’s own

metaphysical view of necessity.5

In this paper I challenge this widespread opinion. My contention is that the formal

development of PWS is best understood as driven by technical considerations intrinsic to

the formal semantics itself, and not by overt or covert philosophical agendas. To support

my claim, I will consider three crucial aspects in which Kripke’s formal semantic

apparatus for modal logic altered the received Carnapian PWS. These three changes are

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crucial both intrinsically and insofar as they might be, and have been, viewed as dictated

by philosophical considerations. I will argue on the contrary that these changes are due to

merely formal reasons. A purely formal explanation of these formal developments is

called for. We can and must detach alternative philosophical views of necessity from

logical considerations regarding (intrinsically) the model theory of modal systems.

Finally, I will argue that Kripke’s (formal) semantics is in some important sense the right

(formal) semantics for modal systems independently of one’s logical or metaphysical

understanding of the modalities.6

2. CARNAP: STATE-DESCRIPTIONS, ANALYTICITY, AND LOGICAL TRUTH

I will start by introducing some of the basic ideas behind Carnap’s formal PWS. In 1946

Carnap published “Modalities and Quantification”, and thus proved Quine de facto wrong

insofar as his early well-known criticisms to quantified modal logic seemed to suggest

some kind of technical unfeasibility. In this work, Carnap presents a quantified modal

system and offers some ideas concerning its proper interpretation, ideas that one year

later he will develop in Meaning and Necessity.7

Carnap states explicitly that the notion of necessity he has in mind is

logical/analytic:

[T]he guiding idea in our constructions of systems of modal logic is this: a

proposition p is logically necessary if and only if a sentence expressing p

is logically true. That is to say, the modal concept of the logical necessity

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of a proposition and the semantical concept of the logical truth or

analyticity of a sentence correspond to each other.8

Carnap endorse the analytic interpretation of the modal operators prevalent at the time,

according to which modal operators represent in the object language the analogue of the

meta-predicate of analyticity. According to such a view, the modal operators reflect at the

object level a semantic predicate, i.e., a predicate of sentences; moreover, such operators

operate on intensional entities, viz., on propositions.9

Carnap speaks of “the logical truth or analyticity of a sentence,” implying their

equivalence. However, already in 1943 Quine pointed out that the class of logical truths

is at best a proper subclass of the class of analytic truths.10 I believe that Carnap’s

disregard of the distinction is a manifestation of what Quine called the ‘epistemologically

biased’ view of logical truth, according to which the key semantic feature of logical

truths consists in their (presumed) analyticity, while their specific difference from

analytic truths in general plays no significant semantic role. In this framework, the

interesting semantic notion is analyticity, and – argues Quine – ultimately a priority.11

Carnap introduces the apparatus of state-descriptions to elucidate the relatively unclear

notions of analyticity and (analytic) necessity. However, the apparatus can be adopted to

represent different semantic interpretations of necessity, i.e., to capture different semantic

properties to which necessity may correspond at the object-language level.

A state-description for a language L is a set-theoretic entity, more precisely a class

of sentences of L such that, for every atomic sentence S of L, either S or its negation, but

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not both, is contained in the class. A state-description does not contain non-atomic

sentences (other than negations of atomic sentences). The semantic values of non-atomic

sentences relative to a state-description are calculated on the basis of the values of atomic

sentences in the customary inductive way, under the basic assumption that an atomic

sentence holds in a state-description if and only if it belongs to it. For example, ∼S (where

S need not be atomic) holds (is true) in a state-description R if and only if S does not hold

in R; (S ∧ T) holds in R if and only if both S and T hold in R; (∀x)Px holds in R if and

only if all the substitution instances of Px hold in R.

By definition, a sentence S of L is L-true just in case it holds in every state-

description for L. Therefore, the proposition expressed by S is necessary just in case it is

true in every possible world (given that state-descriptions are taken to describe possible

worlds).

The appeal to possible worlds should not mislead us into believing that some

metaphysical, rather than semantic, notion of necessity is at stake. As we have seen,

Carnap explicates necessity explicitly in terms of the semantic notion of analyticity,

where this last is represented in terms of L-truth. Possible worlds, through their linguistic

representations (state-descriptions), are just used to elucidate this semantic notion.

To be an adequate formal representation of analyticity, L-truth has to reflect the

basic idea behind analyticity: truth in virtue of meaning alone. Hence, L-truths must be

such that semantic rules alone need to be employed to establish their truth. With this

purpose in mind, L-truth for a sentence S of a language L is defined as truth in all the

state-descriptions of L.12

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According to Carnap, intuitively a state-description is supposed to represent

something like a Leibnizian possible world or a Wittgensteinian possible state of

affairs.13 The entire range of state-descriptions for a certain language is supposed to

exhaust the range of alternative possibilities (describable in that language). Clearly, it is

the fact that state-descriptions represent possible worlds – possible ways things might

have been – that makes Carnap’s formal apparatus intuitively apt to represent necessity

and possibility. Carnap’s appeal to Wittgenstein’s states of affairs adds the further

intuition that all combinatorially consistent combinations (of truth-value assignments to

atomic propositions) are indeed possible.

Carnap’s elucidation of the modalities by means of state-descriptions encapsulates

two distinct ideas. First there is the semantic ascent, the idea that necessity corresponds to

a semantic property. Insofar as state-descriptions serve the purpose of defining L-truth,

they encode the interpretive idea that necessity is ultimately analyzed in terms of a

semantic property. Second, Carnap’s reference to Wittgenstein and to the idea of logical

consistency suggests a combinatorial extension for necessity, i.e., what we might

characterize as a logical understanding of necessity.

However, the Wittgensteinian logical/combinatorial view of necessity based on

logical consistency on the one hand and the analytic interpretation of necessity on the

other are in conflict. The presence of a conflict is witnessed by the conflict in extension

between the two notions. Some consistent combinations of truth-value assignments to

atomic sentences are ruled out by the analytic interpretation. Meaning postulates are then

needed to exclude such combinations. As Quine points out:

8

In recent years Carnap has tended to explain analyticity by appeal to what

he calls state-descriptions. . . . The criterion in terms of state-descriptions

is a reconstruction at best of logical-truth, not of analyticity.14

Quine’s main concern is that the atomic statements of the language may not be

semantically independent. If that is the case, a state-description may verify two

incompatible statements, for example, “John is a bachelor” and “John is married”. Hence,

such a description is not suitable to represent an (analytically) possible world: surely

there is no world where John is both a bachelor and married. To exclude cases of this

kind, the atomic terms of the language must be logically independent from one another

(in a way in which ‘married’ and ‘bachelor’ are not). Alternatively, the logical

connections between atomic terms must be spelled out by means, for example, of

Carnap’s meaning postulates, which rule out those state-descriptions that do not

correspond to authentic analytic possibilities, and so reinstate analyticity as the main

encoded notion.

In sum, state-descriptions can be taken to encapsulate at least two distinct notions:

analyticity (analytic necessity) on the one hand and logical truth (logical necessity) on the

other. To underline the distinction between analytic necessity and logical necessity, we

need just notice the natural different extensions of these two notions. Analytic necessity

excludes some logically consistent state-descriptions (for example, states descriptions

containing both “John is a bachelor” and “John is married”). On the other hand, if we

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disregard the meanings of all but the logical expressions of the language all logically

consistent combinations become possible. From the logical point of view, the natural

extension of necessity includes all combinatorially consistent combinations. Further

considerations may induce us to restrict the class of state-descriptions. These

considerations however are semantic in nature, and not purely logical – in the sense that

they depend on the logical connections that hold between interpreted non-logical terms

(like ‘bachelor and ‘married’).15

It is important to notice that despite Carnap’s focus on (i) analyticity and (ii)

combinatorial/logical necessity, state-descriptions may also be seen as playing the same

role that Tarskian mathematical models play.16 We can view state-descriptions as

logically consistent combinations of truth-value assignments to the uninterpreted non-

logical atomic sentences and predicates of the language. In this way, analyticity, which

has to do with interpreted sentences, is set aside, and a new formal notion of validity

emerges. Such a notion applies to (partially) uninterpreted sentences.17

3. FORMAL VALIDITY AND STATE-DESCRIPTIONS

Once again, let us look at Quine to find a third possible semantic interpretation of

necessity. Quine points out that we may provide a semantic reading of necessity (link

necessity to a property of sentences) by linking necessity to some formal notion of

validity:

10

Something very much to the purpose of the semantical predicate ‘Nec’ is

regularly needed in the theory of proof. When, e.g., we speak of the

completeness of a deductive system of quantification theory, we have in

mind some concept of validity as norm with which to compare the class of

obtainable theorems. The notion of validity in such contexts is not

identifiable with truth. A true statement is not a valid statement of

quantification theory unless not only it but also all other statements similar

to it in quantificational structure are true. Definition of such a notion of

validity presents no problem, and the importance of the notion for proof

theory is incontestable.

A conspicuous derivative of the notion of quantificational validity

is that of quantificational implication. One statement quantificationally

implies another if the material conditional composed of the two statements

is valid for quantification theory.

This reference to quantification theory is only illustrative. There

are parallels for truth-function theory: a statement is valid for truth-

function theory if it and all statements like it in truth-functional structure

are true, and one statement truth-functionally implies another if the

material conditional composed of the two statements is valid for truth-

functional theory.

And there are parallels, again, for logic taken as a whole: a

statement is logically valid if it and all statements like it in logical

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structure are true, and one statement logically implies another if the

material conditional formed of the two statements is logically valid.18

[I]t is at the semantical or proof-theoretic level, where we talk about

expressions and their truth values under various substitutions, that we

make clear and useful sense of logical validity; and it is logical validity

that comes nearest to being a clear explication of ‘Nec’, taken as a

semantical predicate.19

We see here the suggestion that necessity be linked to validity. But what is

validity? Validity might prima facie be confused with logical truth, after all “a statement

is logically valid if it and all statements like it in logical structure are true”. However, this

characterization is too narrow. Validity is system relative, and not all systems are

naturally seen as ‘logical’. Some candidate examples of non-logical systems are second

order logic, first-order set theory, or (closer to home) modal systems. Validity is crucial

to proof-theory in providing a standard to which to compare the theorems of the system,

and if possible obtain soundness and completeness results. But clearly not all truths

provable in any formal system may be plausibly regarded as logical truths (expressible in

the language of that system).

Quine viewed this formal understanding of necessity as a reduction of the obscure

notion of necessity to the clear notion of validity. In fact, Quine thought it illuminating

because reductive:

12

As long as necessity in semantical application is construed simply as

explicit truth-functional validity, on the other hand, or quantificational

validity, or set-theoretic validity, or validity of any other well-determined

kind, the logic of the semantical necessity predicate is a significant and

very central strand of proof-theory. But it is not modal logic, even

unquantified modal logic, as the latter ordinarily presents itself; for it is a

remarkably meager thing, bereft of all the complexities which are

encouraged by the use of ‘nec’ as a statement operator. It is unquantified

modal logic minus all principles which, explicitly or implicitly . . . involve

iteration of necessity; and plus, if we are literal-minded, a pair of question

marks after each ‘Nec’.20

To sum up, we find in Carnap’s definition of L-truth the seeds of three alternative

semantical understandings of necessity. The apparatus of state-descriptions may be taken

to encode the notion of analyticity, and in this sense meaning postulates are a crucial part

of the apparatus itself. Alternatively, it may be taken as a codification of the notion of

logical truth, schematically characterized as truth resistant to substitution of the non-

logical constants, and independently of any further thesis one may hold concerning the

nature of logical truths. In this second interpretation, meaning postulates are best seen as

additions to the basic apparatus. Finally, it can be taken as codifying the validities of

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some formal system. Validities are still schematically characterized, yet we do have

validities of systems that may not be considered part of logic proper.

These three semantic understandings of necessity share a common presupposition.

After picking a relevant semantic property, they treat state-descriptions as a means to

codify such a property. But given that necessary truths are assumed to be extensionally

equivalent with the semantically privileged truths (be they analytic, logical or valid), the

necessary truths of the object language must match exactly the truths across all states

descriptions (questions of iteration aside).21

However, as suggested earlier, state-descriptions are early versions of Tarskian

models, as such apt to capture a formal notion of validity for the object language at hand

– formal insofar as it applies to formal sentences, i.e., uninterpreted sentences. The

question to be considered is the following: What is the proper role of this model theoretic

notion of validity? If it is, as Quine suggests, a notion central to proof theory, its central

role consists in providing a class of truths with which to compare the class of theorems of

the system.

However, insofar as state-descriptions serve this proof-theoretic purpose for

modal systems themselves, they are not meant to provide a class of L-truths extensionally

equivalent to the class of necessary truths. Their role consists rather in providing a model

theoretic match to the theorems of a modal system. But as Quine says: “The notion of

validity in such contexts is not identifiable with truth. A true statement is not a valid

statement of quantification theory unless not only it but also all other statements similar

to it in quantificational structure are true.” In this perspective, there is no guarantee that

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the theorems of a modal system include all the true necessities expressible in the modal

language of the system. In this formal, Tarskian perspective, the purpose of the model

theory of modal logic is to find model theoretic matches to the theorems of the modal

systems, not to provide a reductive interpretation of the object language operators in

terms of a Carnapian match of validities to necessities.

4. FROM CARNAP TO KRIPKE

Quine’s 1947 paper “The Problem of Interpreting Modal Logic” starts by saying,

There are logicians, myself among them, to whom the ideas of modal logic

. . . are not intuitively clear until explained in non-modal terms.22

We may think of Carnap’s work in the forties and of the work of other logicians

in the late-fifties as an attempt to respond to this complaint by putting modal logic on

equal footing with the familiar non-modal systems of logic. The idea was to extend to it a

form of the Tarski-style extensional semantics of first order logic.23 Carnap’s work on the

modalities starts the important model theoretic approach to the semantics of modality.

State-descriptions are the precursors of the model theoretic apparatus of possible worlds

that will be so fruitfully employed in the late fifties and early sixties to provide a proof

theoretically adequate model theoretic semantics for modal logic.24 This, I claim, is a

completely different role than the one played by state-descriptions to provide an

interpretation of the modal operators.

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Carnap’s work attempts both:

(i) To link necessity to the semantic notion of truth across all state descriptions

and explicate it in terms of validity;

and

(ii) To provide a proof theoretically adequate extensional formal semantics of

modal logic.25

I claim that these two attempts are intrinsically at odds with each other. To support my

claim, in what follows I shall analyze the evolution of PWS from Carnap’s work to

Kripke’s 1959 “A Completeness Theorem in Modal Logic” and 1963 “Semantical

Considerations on Modal Logic” and “Semantical Analysis of Modal logic I”. In

particular, I will concern myself with three main points:

(1) The nature of the ‘worlds’;

(2) The domain(s) of quantification;

(3) The notion of validity.

According to a very widespread view – explicitly upheld by Cocchiarella,

Hintikka and Sandu, Lindström, and at least implicitly by Kaplan – Kripke’s

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development of the model theory produces a new notion of validity appropriate for a

metaphysical understanding of necessity. But if the Kripkean formal semantics is to

impose a non-logical notion of necessity, it must be in its changes concerning the worlds,

the domains of quantification, and/or the notion of validity itself.26 I will argue however

that all the above developments are best understood as driven by the Quinean task of

developing a formal notion of validity central to proof theoretic purposes.

My focus on Kripke’s model theoretic innovations is not meant to suggest that

such innovations were suggested and adopted by him exclusively. In the fifties and early

sixties, Hintikka, Montague, and Kanger all modified Carnap’s formal semantics in ways

similar to Kripke’s.27 However, it is Kripke who is standardly characterized as providing

a model theory inadequate for logical necessity. Moreover, Kripke is the only one who

adopts all the three main modifications discussed in this paper.

Concerning the nature of the worlds, Kripke was the only one who characterized

them as simple points of evaluation. Such a characterization is crucial in affording a link

between the model theoretic semantics and algebraic treatments of modal logic.28 On the

other hand, variations of the domain of quantification are considered both by Kanger and

Hintikka. Finally, and interestingly enough, Hintikka like Kripke also adopted a new

notion of validity that required truth in all arbitrary sets of worlds. Despite his own

adoption of this universal notion of validity, in his later work, as we have seen, Hintikka

criticizes Kripke’s model theory as inadequate for logical necessity exactly because of the

notion of validity it endorses.

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5. THE NATURE OF THE ‘WORLDS’

Carnap’s notion of validity for a given language is truth across all state-descriptions for

that language. State-descriptions are collections of sentences. They are taken to represent

possible worlds, or possible states of affairs. But what possible states of affairs there are

is determined by the state-descriptions (unless as we have seen one wants to restrict them

in order to capture an antecedently given idea of possibility, as in Carnap’s case of

analytic possibility). In this sense, the span of possibilities represented by the state-

descriptions is very much bound to the language whose sentences they collect. Both in

the obvious sense that only the (combinatorial) possibilities expressible in the language

can be represented, but also in the less trivial sense that all combinatorially consistent

sets of sentences of the language are taken to describe possible states of affairs.

Later on, in 1963, Carnap himself adopts models in place of state-descriptions.29

Models are assignments of values to the primitive non-logical constants of the language.

In Carnap’s case predicate constants are the only primitive constants to which the models

assign values, since individual constants are given a fixed pre-model interpretation. Value

assignments to variables are done independently of the models.

In 1959, Kripke also uses models, i.e., complete assignments of values, as

representatives of possible worlds. The terminology however can be misleading: what

Kripke in 1959 calls ‘models’ are not such assignments. To avoid confusion, I will

reserve the term ‘model’ for the assignments of values, and use instead ‘M-model’ for

Kripke’s (modal) models.

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Given a domain D of individuals, a model is an assignment of values to the

variables of the language (Kripke’s language has no non-logical constants), such that

each propositional variable is assigned a truth-value, each individual variable is assigned

an element of D, and each n-adic predicate variable is assigned a set of ordered n-tuples

of elements of D. A Kripke M-model instead is an ordered pair (G, K), such that K is a

set of complete assignments (models), G is an element of K, and all elements of K agree

in their assignments to individual variables.

Consider Kripke’s 1959 models, representatives of possible worlds. As seen, they

have evolved from Carnap’s sets of sentences (state-descriptions) to assignments of

values, i.e., mathematical functions that correlate syntactical entities to values.

Nonetheless, such assignments are still very much language driven, in the sense that all

combinatorially consistent assignments of values are possible.

In 1963, a further evolution takes place concerning the nature of the worlds in

PWS. Worlds are not anymore represented by models, rather simply by points of

evaluation. Of the set K of worlds in a model structure, Kripke just says that it is a non-

empty set. Nothing is assumed concerning the nature of the elements of K.30 What is the

significance of this change? Does this technical change in the model theory have

philosophical repercussions? In particular, does it reflect the adoption of a new

interpretation of the modal operators of the object language?31 I claim that it does not.

Rather than reflecting a new philosophical interpretation of the modal operators,

the adoption of points in place of models reflects a more abstract understanding of the

formal semantics. The formal semantics of a modal system does not formally represent or

19

encode some antecedent semantic conjecture (e.g., that “necessarily” means true in all

alternative possible worlds). Instead it provides an algebraic characterization of the

system. Kripke himself recognizes the connection between his formal semantics for

modal logic and work in the algebras of modal systems:

The most surprising anticipation of the present theory, discovered just as

this paper was almost completed, is the algebraic analogue in JÓNSSON

and TARSKI [“Boolean Algebras with Operators”]. Independently and in

ignorance of [“Boolean Algebras with Operators”] (though of course

much later), the present writer derived its main theorem by an algebraic

analogue of his semantical methods . . .32

Moreover, as Kripke notices, the changes in his formal semantics are

generalizations of his own 1959 treatment – “The present treatment generalizes that of

[1959] in the following respects”33 – but surely such a generalization can hardly be

viewed as apt to or designed for capturing independent specific assumptions on the

proper interpretation of the modal operators. The generalizations Kripke is referring to

are the introduction of the accessibility relation R which makes it possible to deal with a

whole range of modal systems, not only S5; and, more importantly for our present

concern, the switch from worlds as models to worlds as points. Here is what Kripke says:

20

For in [1959], we did not have an auxiliary function Φ to assign a truth

value to P in a world H; instead H itself was a “complete assignment”, that

is a function assigning a truth-value to every atomic subformula of a

formula A. On this definition, “worlds” and complete assignments are

identified; so distinct worlds give distinct complete assignments. This last

clause means that there can be no two worlds in which the same truth-

value is assigned to each atomic formula. Now this assumption turns out

to be convenient perhaps for S5, but it is rather inconvenient when we

treat normal [Modal Propositional Calculi] in general.34

In this passage, Kripke connects the switch from worlds-as-models to worlds-as-

points to his current concern with a larger spectrum of modal systems. In S5 the

accessibility relation can be taken to connect every world with every other world

including itself; hence there is no need of world-duplicates. Once the accessibility

relation gets restricted, however, world-duplicates become important in achieving model

theoretic generality. They make it possible to have model structures in which worlds

differ in the way they relate to each other, while being intrinsically indistinguishable –

both in the sense of having intrinsically indistinguishable worlds in the same model

structure, and in the sense of having different model structures differing only for what

concerns the accessibility relation R.

Once the worlds are not identified with assignments, an external function Φ is

needed to assign values to variables relative to worlds. An M-model is now a model

21

structure (G, K, R) plus an evaluation function Φ . There is absolutely no sense in which

it is natural to think of such model theoretic constructions (vis-à-vis the 1959 M-models)

as better suited to represent a non-semantic notion of metaphysical necessity. The

function Φ is there to do the job of the old assignments. The suggestion that if the

assignments are done by independent functions (themselves members of K) then a logical

interpretation of necessity is natural, while if the assignments are all dealt with by a

function Φ on the elements of K, then a restriction on all the combinatorial combinations

of assignments is more natural (and so a metaphysical interpretation of necessity is

implied) is in my view highly implausible.

I believe that the idea behind such a suggestion is the following: While

assignments are language driven, points are given independently of the language; hence

they lend themselves more naturally to excluding combinatorial assignments that do not

correspond to metaphysical possibilities (“There is just no such point!”).

Such a view thrives on the identification of points with worlds and on the further

idea that the worlds are the real possibilities.35 But if worlds are supposed to have any

kind of metaphysical reality, they just cannot be algebraic points to which a function Φ

assigns values. If we are inclined to think of worlds as somehow real possibilities, we

should make sure not to identify them with points of evaluation, and just think of the

points as world representatives. But then why would the points of evaluation be better fit

to represent all the metaphysically possible worlds, rather than all the combinatorially

consistent assignments of values (or anything else for that matter)? If on the other hand,

we are not inclined to think of the worlds as anything other than the elements of K, then

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we may identify the worlds with the points, but then a world is just that: a locus of

evaluation, in no way better fitted to represent one interpretation of necessity over

another.

6. THE DOMAIN(S) OF QUANTIFICATION

In 1946, Carnap assumes a fixed domain of quantification for his quantified system

(functional calculus) and consequently for his modal functional calculus. He is well

aware that such an assumption raises the question of completeness, even before the

addition of modal operators. Gödel proved completeness for the first order predicate

calculus with identity, but he employed a notion of validity as truth in every (non-empty)

domain of quantification. Carnap instead adopts one unique denumerable domain of

quantification.36 The adoption of a fixed denumerable domain of individuals generates

some additional validities already at the pre-modal level. For example (Carnap’s

example), it becomes valid that there are at least two individuals. Carnap raises the

question of the completeness of the functional calculus with identity vis-à-vis his own

notion of validity, nonetheless he adopts a fixed denumerable model. This is a clear sign

that his notion of validity is driven by external, philosophical rather than technical

considerations – he chooses a unique domain of a fixed size despite the consequent loss

of completeness.

In 1959, Kripke adopts what we may in retrospect consider as a middle position

between Carnap and Kripke’s later 1963 work. Domains may vary, and only the empty

domain is excluded, but each M-model is defined on an antecedently given domain. This

23

amounts to positing no variability of domains between the models inside a modal

structure (an M-model), while postulating variability across M-models. This change

consents to regain Gödel’s indifference to the domain of quantification in establishing the

validity of non-modal formulas. Concerning modal formulas, this kind of limited

variability of the domain of quantification already generates some differences from

Carnap.

Assuming Carnap’s fixed domain, it is not only valid that there are at least two

individuals, but also that it is possible that there are at least two individuals, and that it is

necessary that there are at least two individuals. There is no world with fewer than two

individuals to invalidate these claims. In Kripke’s semantics the variability of the

domains across M-models is enough to invalidate all of these three claims. It is not valid

that there are at least two individuals, that possibly there are at least two individuals, nor

that necessarily there are at least two individuals. The kind of domain variability assumed

in 1959 is already by itself, even if no other changes in the model theory are adopted,

sufficient to produce a significant change in the notion of validity, and a first detachment

of validity from necessity. Validity is truth across all Domains (hence across all M-

models on each Domain),37 but necessity is now relative to a given Domain for an M-

model.

This has two kinds of consequences. In the first place, non-modal sentences that

are not valid may still turn out to be necessary in a certain Domain. For example, M-

models with domains with two or more individuals will make “There are at least two

individuals” necessary, while the presence of M-models with one-individual domains will

24

suffice to make this same sentence not valid. In the second place, the validity of modal

sentences is affected too. For example, neither “Necessarily, there are at least two

individuals” nor “Possibly, there are at least two individuals” will turn out valid, but they

will still be true, and necessarily so, in M-models defined on Domains containing more

than one individual.38

In 1963, a further change takes place. Kripke introduces variability of domains

not only across distinct M-models, but also across worlds in the same M-model. Kripke

does support such a change with informal arguments. For example, concerning the

function ψ that assigns domains to the worlds H in an M-model, he says:

Notice, of course, that ψ(H) need not be the same set for different

arguments H, just as, intuitively, in worlds other than the real one, some

actually existing individuals may be absent, while new individuals, like

Pegasus, may appear.39

Despite Kripke’s informal, philosophical motivation, this change too seems in line with

the general trend of moving towards a more general, algebraic model theory, where fewer

and fewer restrictions are placed on the combinations of worlds and domains. Moreover,

there is a particular technical problem with Kripke’s 1959 semantics.

Thanks to the assumption of the variability of domains across (intra-M-model)

worlds, Kripke is able to construct a counter-example both to the ‘Barcan Formula’ and

its converse. Let us just consider the Barcan Formula:

25

(BF) (∀x)[ ]Fx → [ ](∀x)Fx

Kripke considers a structure with two worlds, the actual G point and one possible

world H extending it. The domain of G is the individual a, which is F (and thus all things

in G are F). The domain of H is the set {a,b}. a is still F at H and so we get that the

antecedent of (BF) is true at G. But the consequent is false. It is false because (∀x)Fx is

false at H. And this last is false because the new individual b – a mere possibilium from

G’s point of view – is not-F at H.

However, as Kripke mentions, Prior seems to have proved the Barcan formula in

quantified S5.40 If this is the case, the 1963 model theory invalidates S5-theorems.

However, Kripke suggests that neither the Barcan formula nor its converse is really

provable in S5. He reconstructs an alleged proof for the converse Barcan formula, and

shows how the proof goes through only by allowing the necessitation of a sentence

containing a free variable. But if free variables are to be considered as universally bound,

then necessitating directly an open formula, without first closing it, amounts to assuming

the derivability of the necessitated open formula from the necessitation of its closure,

which is what was to be proved. I.e., from “Fx” we should only derive “[ ](∀x)Fx”,

given that “Fx” is to be read as “(∀x)Fx”. If instead from “Fx” we are allowed to derive

“[ ]Fx”, and understand it as “(∀x)[ ]Fx”, we are implicitly assuming the derivability of

this last from “[ ](∀x)Fx”.

The question to be considered is the following: If the Barcan formula and its

converse are not theorems of S5, how could Kripke have proved completeness in 1959

while adopting a model theory that provided no countermodel to these formulas? Recall

26

that the 1959 model theory does not allow domains to vary inside M-models. Hence, both

the Barcan formula and its converse hold in every 1959 M-model. If so, they are

validities of the 1959 model theory. But they aren’t provable in S5. The fact is that in

giving his completeness theorem in 1959, Kripke had assumed Prior’s alleged result (see

p. 9 of Kripke’s 1959 paper).

The realization of the improvability of (BF) and its converse under the standard

reading of free variables as universally bound is sufficient to justify the 1963 revision of

the model theory. If these formulas are not provable, in order to have completeness we

need model theoretic constructions that provide counterexamples to their alleged

validity.41

Independently of what came first in Kripke’s mind – be it the selection of modal

structures with variable domains, the proof theoretic consideration that Fx must be first

universally closed and then necessitated, or even a philosophical intuition concerning

possible objects – what is ultimately essential is that in 1963 like before in 1959 the

essential task is to provide a match between a certain class of structures and a particular

proof system. Changes in the model theory and in the proof theory proceed hand in hand.

The logician’s formal interest lies in finding the appropriate class of structures

corresponding to a certain system, or the right system to capture a certain class of

structures.

7. MAXIMAL VALIDITY VERSUS UNIVERSAL VALIDITY

27

The above considerations naturally bring us to our next topic: validity. As we have seen,

Kripke claims to have derived on his own the main theorem of “Boolean Algebras with

Operators” by an algebraic analogue of his own semantical methods. The main theorem

of Jónsson and Tarski’s work is a general representation theorem for Boolean algebras

with operators. Such a theorem is the algebraic analogue of a model theoretic

completeness theorem for modal systems. This brings us to a crucial shortcoming of

Carnap’s notion of validity.

We have seen how Carnap takes inspiration from Wittgenstein’s combinatorial

view according to which a logical truth is characterized by holding for all possible

distributions of truth-values. In propositional logic, this amounts to a logical truth having

the value true at all rows of its truth-table. Once modal operators are added, Carnap

assumes an interpretation of necessity according to which [ ]p is true just in case p is

valid, i.e., true at all truth-table rows. Carnap’s notion of validity (and necessity) is

maximal validity, i.e., truth across all the truth-table rows/state-descriptions/assignments

of values.

Given the way state-descriptions are built out of atomic sentences of the language,

it follows that each atomic sentence and its negation turn out to be true at some, but not

all, state-descriptions. Hence, given that a sentence [ ]ϕ is true in a state-description if

and only if ϕ is true in every state-description, it follows that neither [ ]p nor [ ]∼p is

ever going to be true for an atomic p. Hence, their negations ∼[ ]p and ~[ ]~p (◊p) are

validities in Carnap’s maximal sense. Moreover, given any two atomic sentences p and q

there is surely a state-description in which both turn out to be true.

28

This notion of validity has a significant logical consequence: the principle of

substitution fails. If, for example, given atomic p and q we substitute ∼p for q, then no

state-description will assign them the same truth-value. Similarly, we cannot substitute

for p a logically complex sentence which is either logically true or logically false, e.g.,

(r∨∼r) or (r∧∼r). In Carnap’s semantics, for every atomic sentence p both ◊p and ∼[ ]p

are valid. But we do not want ∼[ ](r∨∼r) and ◊(r∧∼r) to turn out valid.

According to Quine’s schematic characterization of logical truth/validity, in order

to qualify as logically true/valid a sentence has to remain true even when its simple

elements are substituted by logically complex elements. Hence, some of Carnap’s

maximal validities do not qualify as schematically valid (for example ◊p for atomic p). If

we accept Quine’s schematic characterization of validity, and we think of the logic of

necessity as aiming at providing the schematic validities of a modal language, we will

have to conclude that Carnap’s model theory is inadequate to capture the schematic

validities of even logical necessity: No matter which interpretation of the modal operators

one adopts, ◊p is not schematically valid.42

Technically, this situation calls for a choice. In a modal propositional system that

adopts as an axiom ◊p for every atomic sentential letter p, the rule of substitution must be

restricted. Alternatively, if the rule of substitution is not restricted (or the axioms are

given as axiom schemes), sentences like ◊p for atomic p cannot be added as axioms. In

other words, we cannot have a system that has both axioms like ◊p with atomic p and a

uniform rule of substitution; otherwise we could derive as a theorem ◊(p∧∼p).43

29

This point was noticed by Makinson who argued that (Carnap’s) naïve logical

understanding of the modal operators leads to failure of substitutivity. If one however

considers the schematic nature of logical truths and revises accordingly the naïve notion

of logical necessity, one will be natural brought to reinstate substitutivity.44

Burgess endorses and develops Makinson’s point. He essentially points out that

for propositional S5, the universal notion of validity matches in extension the schematic

notion. Hence one may well adopt a Kripkean model theory not because of

interpretational questions on necessity, but because one adopts a standard fully

substitutional view of propositional variables.45

Concerning quantified modal logic, Carnap’s notion of maximal validity makes it

impossible to prove completeness. According to maximal validity, the models of a modal

language L* are exactly the models of its non-modal part L.46 Given a first order language

L, all its models are taken to represent possible states of affairs. A necessary truth is a

truth across all such models. So in a modal extension L* of a first order language L, [ ]ϕ

is true (in each model) if and only if ϕ is valid. Hence, for any non-valid first-order

sentence ϕ (any first-order sentence ϕ that is true at most in some but not all of the

models), ∼[ ]ϕ will be true in all the models, i.e., valid. (Carnap’s models all agree in the

evaluation of modal sentences.) But the non-logically true first-order sentences are not

recursively enumerable, hence neither are the validities of the modal language. Hence,

quantified modal logic is incomplete vis-à-vis Carnap’s maximal validity.47

It may reasonably be conjectured that it was primarily the search for a

completeness proof for quantified S5 that led Kripke to revise Carnap’s model theory and

30

introduce his 1959 notion of universal validity. The crucial change consists in the

introduction of M-models. An M-model, remember, is an ordered pair (G, K), with G an

element of K, and K an arbitrary subset of models/assignments of values. Validity is not

anymore defined as truth across all models, rather as truth across all M-models. Once

again, this is a move that adds generality to the model theory. Instead of considering only

the maximal structure that contains all models, all the subsets of the maximal structure

are considered. The maximal structure is now one among others, and not all sentences

that are valid in it (maximally valid in Carnap’s sense) will turn out to be universally

valid:

In trying to construct a definition of universal logical validity, it seems

plausible to assume not only that the universe of discourse may contain an

arbitrary number of elements and that predicates may be assigned any

given interpretations in the actual world, but also that any combination of

possible worlds may be associated with the real world with respect to

some group of predicates. In other words, it is plausible to assume that no

further restrictions need be placed on D, G, and K, except the standard one

that D be non-empty. This assumption leads directly to our definition of

universal validity.48

My account above simplifies matters in not considering the role of the G model.

In fact, to any set of models K there corresponds more than one M-model, according to

31

which element of K is selected as the actual world. This however has no effect on which

sentences turn out to be universally valid.49 Moreover, to be careful, given that in 1959

M-models are defined on Domains, not every subset of models counts as K in an M-

model. Only subsets of models on the same domain form a set K in an M-model. That is

how universal validity is defined as validity in every non-empty domain in 1959. As we

have seen, by 1963 even this last restriction is lifted: now every arbitrary subset of

worlds can play the role of K in a model structure.

Kripke’s model theory disconnects validity from necessity. In 1959, Kripke

defines validity in an M-model as truth in the actual world of the Model, and not as truth

in all the worlds of the Model. Hence, even for one single M-model necessity does not

correspond to validity. In 1963, Kripke revises his terminology and calls a sentence

simply true in an M-model when true in the actual world of the M-model.50 Nonetheless,

the original 1959 terminology suggests that Kripke, insofar as he was willing to speak of

validity for a single M-model (set of worlds), never assumed a correspondence between

necessity and validity. More importantly, universal validity is truth in all M-models (sets

of worlds), while necessity is always relative to an M-model. Carnap’s project of linking

necessity to validity is simply not pursued.

8. CONCLUSION

Given the above considerations, I conclude that all the three main changes in the model

theory considered in this paper – the evolution from state-descriptions to models and then

to points of evaluation; the passage from one fixed domain to variable domains across

32

model structures, and then to variable domains inside model structures; finally, the switch

from maximal to universal validity – point towards a more general, combinatorial,

algebraic model theory, and are justified by technical rather than philosophical

considerations, especially by the search for completeness results.51

This conclusion refutes the widespread claim that it is the switch to metaphysical

necessity and possibility in his philosophical work to dictate Kripke’s adoption of a new

notion of Universal Validity and his other reforms of the received model theory for modal

logic.

Contrary to Cocchiarella, Hintikka, Sandu, and Lindström I believe that Kripke’s

model theory reflects a change in the notion of model theoretic validity itself, a new view

of its role, not a change in the notion of necessity. In my view, Kripke’s model theory is

the right model theory even for the logical modalities.52 Once we abandon Carnap’s

project of linking the interpretation of the modal operators to the notion of validity

adopted for the modal system under consideration, we are free to start judging the model

theory in its own terms. In this sense, the right model theory is not the model theory

whose class of validities matches a preferred class of necessities, rather the model theory

that adequately represents the proof theoretic apparatus; hence that affords soundness and

completeness results. Kripke provided a completeness proof for quantified S5, something

we saw Carnap could not possibly achieve.53

It is worth noticing that the claim that Kripke’s universal validity corresponds to a

metaphysical notion of necessity is not even plausible on the face of it. Kripke’s model

theory reduces the class of validities. For example, ◊p for atomic p is not universally

33

valid, given the presence of (incomplete) structures where all worlds verify ~p. If we

were interested in a notion of validity to be linked to Kripke’s metaphysical necessity, we

would have to pursue the project of increasing the number of validities, i.e. necessities,

rather than cutting them back.54

More importantly, once the project of linking the semantical property of validity

to the interpretation of the modal operators is finally abandoned, one may conclude not

only that Kripke’s Universal Validity does not reflect a particular philosophical stand on

necessity, but also that, from a formal point of view, this is the right notion of model

theoretic validity, regardless of one’s metaphysical or logical interpretation of necessity.

Moreover, those who sympathize with Quine’s schematic characterization of

validity will find a whole bunch of Carnapian logical necessities (for example ◊p for

atomic p and perhaps even “Possibly there exist at least two individuals”) very

implausible candidates to the status of (modal) validities. From this schematic point of

view, there are not only Kripkean metaphysical necessities, but also Carnapian logical

necessities that are not schematic in nature, hence unfit to play the role of (modal) logical

validities.

To sum up, once the project of linking the object language modal operator of

necessity of a certain modal system to the notion of validity for the modal system under

consideration is abandoned, validity starts playing a crucial proof theoretic role and not

an interpretational function. I have argued that it is the proof theory, and the search for a

matching class of structures, which best explains Kripke’s development of PWS. Such

development is technically natural, and interpretationally irrelevant. It surely does not

34

capture as validities the new class of metaphysical necessities. Neither does it capture the

class of Carnap’s logical necessities. Kripke’s validities are at best fit to capture the class

of schematic validities for S5 (on this see the Appendix). By detaching necessity from

validity, and not pursuing the project of an extensional match between the two, Kripke

has freed his hands and made it possible to provide a more austere logic of necessity, one

that does not make all necessary truths of one kind or another valid. His model theoretic

apparatus, designed as it is to correspond to the proof theory, allows him to regain a

schematic notion of validity even for a modal language.55

APPENDIX

The main thesis of this paper is that a model theoretically adequate notion of validity has

to match extensionally the proof theoretic notion of theorem, if anything at all. At the end

of the paper I also suggest that Kripke’s universal validities may well correspond to

Quine’s schematic validities. However, in “Opacity” David Kaplan argues that there are

some schematic (modal) validities that are not universally valid.56 Kaplan’s example is

“Possibly, there exist at least two things”. Given that this is true, it is a schematic validity

(there are no non-logical terms to substitute). However, it is not universally valid, given

the presence of M-models where all worlds contain only one individual. In such M-

models, “Possibly, there exist at least two things” is false. Hence this sentence is not

universally valid, i.e., not true in all Kripkean M-models. If Kaplan is right, my claim that

Kripke’s notion of universal validity corresponds to a (Quinean) notion of schematic

validity for the modal language cannot be correct.

35

Lindström argues that while Kripke models refute the validity of sentences like

“Possibly, there exist at least two individuals” such sentences should be valid for the

logic of logical necessity.57 It should be clear by now that in my view the logic of logical

necessity should be no different from the logic of metaphysical necessity, and that I

regard Kripke models as apt to capture the right logic (notion of validity) in both cases.

Kaplan’s point with which I am taking issue now is a further one: it does not

concern which notion of model theoretic modal validity is the right one. It has to do

instead with a comparison between a schematic notion and Kripke’s model theoretic

universal conception of validity. Kaplan claims that these two conceptions differ in

extension. I claim that they do not, once schematic validity is extended to include full

domain variability.

Questions concerning the cardinality of the domain (from which Kaplan’s

example is drawn) are of a rather special kind, already at the pre-modal level. Even in the

case of a non-modal first-order language, we have a coincidence between the model

theoretic and the schematic notions of validity only if we define schematic validity as

truth for all replacements of the non-logical signs and for all domains. Kaplan says of a

schematic validity that “it would be true no matter how we were to reinterpret its non-

logical signs”, but then he glosses this definition as follows: “i.e., no matter what

grammatically appropriate expressions are substituted for the non-logical signs and no

matter what domain of discourse the variable[s] [are] taken to range over.” (“Opacity”,

p. 275, emphasis added).

36

I regard the addition of domain variability as an adjunct to the purely schematic

characterization of validity. However, if such an addition is welcome, shouldn’t we in a

similar vein add a corresponding change to accommodate modal sentences? If non-modal

schematic validities are independent of how many actual individuals there are, shouldn’t

modal schematic validities be similarly independent of how many possible individuals

there are?

Consider “There are at least two individuals.” This is not by Kaplan’s standards a

schematic validity, even if it is indeed true under all replacements of its non-logical signs

(it contains no such signs to be replaced). It is not a schematic validity because Kaplan

assumes that how many individuals there actually are is not a matter of logic, and hence

adds resistance to domain variability to the definition of schematic validity/logical truth.

But then shouldn’t we in a similar spirit deny schematic validity to “Possibly,

there are at least two individuals” (which Kaplan deems a schematic modal validity),

given that this claim does not remain true if we allow variability not only of the actual

domain, but also of the combination of possible domains connected to it? If non-modal

schematic logical truths are independent of the actual size of the universe, shouldn’t

modal schematic logical truths be similarly independent of its possible sizes? In other

words, shouldn’t we go combinatorial all the way when providing the schematic logic of

necessity? According to such an extreme combinatorialism, it is consistent that any

combination of possible sizes (not just all combinations) may accompany any actual size

of the universe. In calculating the schematic validities of a modal language, we must

consider not only the independence of logic from the actual number of individuals, but

37

also its independence from the possible number of individuals. Not only should we

consider the possibility of the actual world containing only one individual, but also the

possibility that it so does of necessity. If logic has to be indifferent to what exists, it has

to be indifferent to what might or might not exist too. If logic is not concerned with how

many individuals there really are (definitely more than one), it must equally be

unconcerned with how many there could have been. If pure from questions of cardinality,

logic should not claim that given any actual domain all other domains are possible.

Rather, given any domain any other combinations of domains are possible.58

The suggested amendment to the notion of schematic validity develops a full

analogy between the quantifiers and the modal operators. Independently of what the real

cardinality of the world is, to capture the logic of the quantifiers, we make use of domains

of all different cardinalities. Similarly, independently of what the right span of possible

domains is according to one’s preferred view of necessity and possibility, to capture the

logic of the modal operators, we are to make use of all different spans of possible

domains.

I suspect that what drives Kaplan’s intuitions is once again the old Carnapian

assumption of a linkage between necessity and validity. Consider the non-modal sentence

“There are at least two individuals”. In such a non-modal case logical intuitions prevail.

The intuition is that such a sentence is consistent, hence (logically) possible,

independently of how many individuals there actually are. Even the false sentence “There

are at most two individuals” is deemed possibly true, given its consistency.

38

Moreover, any hypothesis about the possible cardinality of the universe is

regarded to be true, independently of the real actual cardinality. This is a form of

maximal combinatorialism concerning necessity: whatever the actual size of the universe,

all sizes are regarded as (logically) possible. And given the possibility that there are at

least two things (or at most two things, or what not), it is then necessarily possible that it

be so. Once necessity is settled, given its assumed link to validity, Kaplan judges it

(schematically) valid that possibly there are two things.

The hypothesis of a necessity-validity link induces first a cut in basic necessities,

i.e., the generation of too many possibilities (all consistent sentences are deemed

possible); but then, given that all possibilities are necessarily possible, we have an over-

generation of validities for the modal language.

I conclude that we should revise the notion of schematic validity for a modal

language to account for the possibility of any combination of possible domains to a given

domain, not just a maximal combination. If the amendment is accepted, the class of

schematic validities for S5 coincides with Kripke’s class of universal validities.59

ACKNOWLEDGEMENTS

I wish to thank Joseph Almog, David Kaplan, Tony Martin, and the SMU Philosophy

Discussion Group for helpful comments and conversations, and an anonymous referee of

this journal for his valuable suggestions.

NOTES

39

1 For an excellent review of the historical development of possible worlds semantics, see

B. J. Copeland, “The Genesis of Possible Worlds Semantics.” See also R. Goldblatt,

“Mathematical Modal Logic: a View of its Evolution” which covers the history of modal

logic from its early beginnings up to its most recent contemporary developments.

2 N. B. Cocchiarella, “On the Primary and Secondary Semantics of Logical Necessity”, p.

26.

3 J. Hintikka & G. Sandu, “The Fallacies of the New Theory of Reference”, p. 281.

4 Cf. D. Kaplan, “Opacity”, Appendix E.

5 See S. Lindström, “Modality Without Worlds: Kanger’s Early Semantics for Modal

Logic” (1996) and “An Exposition and Development of Kanger’s Early Semantics for

Modal Logic” (1998). In the more recent “Quine’s Interpretation Problem and the Early

Development of Possible Worlds Semantics,” (2001), Lindström is more cautious in

linking Kripke’s formal semantics to a metaphysical interpretation of necessity. He writes

there: “One reason for arguing that Kripke’s notion of necessity in 1959 is not logical

necessity is Kripke’s use of non-standard models. . . . This conclusion is however, not

unavoidable . . . Kripke’s reason for allowing non-standard models, in addition to

standard ones, when defining validity, could have been logical rather than philosophical.”

(p. 209).

6 In “Which Modal Models are the Right Ones (for Logical Necessity)?” John Burgess

argues against the idea that Kripke’s models are adequate for metaphysical necessity, but

not apt to represent logical necessity. Burgess’s main argumentative strategy differs from

mine. His central argument against this widespread misconception focuses on the claim

40

that such a misconception arises from a subtle use-mention confusion. My main strategy

instead does not consist in directly attacking such a view, but rather in showing how each

change in Kripke’s PWS is better explained as due to logical reasons, not to interpretive

hypotheses about the nature of necessity. I do however claim that the misconception

depends on the assumption that the notion of validity one adopts is linked to the

interpretation of the object language operator of necessity. The assumption of such a

linkage may be seen as presupposing a use-mention confusion.

7 R. Barcan (Marcus)’s “A Functional Calculus of First Order Based on Strict

Implication” was published just a few months before Carnap’s work. It also contains a

quantified modal system, but no semantic considerations.

8 R. Carnap, “Modalities and Quantification”, p. 34.

9 The two points are separate. For example, in characterizing analytic necessity, Quine

just says that [ ] p is true if and only if p is analytic, with no further talk of propositions

as semantic entities for the operator to operate upon. A semantic view of necessity can be

adopted without reference to intensional entities such as propositions.

10 “At best” because once analyticity is not part of the definition of logical truth, there is

no guarantee that all logical truths be analytically true.

11 See W. V. Quine, “Carnap and Logical Truth”.

12 R. Carnap, Meaning and Necessity, Chapter I, Section 2.

13 Cf. R. Carnap, Meaning and Necessity, p. 9. We will see later how possible worlds on

the one hand and Wittgensteinian states of affairs on the other encapsulate two distinct

intuitions. Carnap seems aware of the distinction: “In my search for an explication I was

41

guided, on the one hand, by Leibniz’ view that a necessary truth is one which holds in all

possible worlds, and on the other hand, by Wittgenstein’s view that a logical truth or

tautology is characterized by holding for all possible distributions of truth-values.” (The

Philosophy of Rudolf Carnap, p. 63. Emphasis added.)

14 Cf. W. V. Quine, “Two Dogmas of Empiricism”, pp. 23-4.

15 I am talking of the natural extension of these notions. This does not rule out the

possibility that further considerations may bring one to conclude that the real extensions

of these notions are different from their natural ones, perhaps even that the two notions

coincide in extension. This would be the case, for example, if one assumed that there is

no logical connection between atomic expressions of the language because atomic,

logically independent meanings are assigned to atomic expressions. This further

assumption makes analytic necessity coincide in extension with logical necessity, but it is

a further assumption.

16 Carnap switches to models in his later work: “In my book on syntax . . . and still in

[Meaning and Necessity], the values assigned by the semantical rules to variables and

descriptive constants were linguistic entities, viz., expressions, classes of expressions, etc.

Today I prefer to use as values extra-linguistic entities, e.g., numbers, classes of numbers,

etc. In an analogous way I now represent possible states of the universe of discourse by

models instead of state-descriptions, which are sentences or classes of sentences.”

(“Language, Modal Logic, and Semantics” in The Philosophy of Rudolf Carnap, p. 891,

footnote 10.)

42

17 Of course, someone can be epistemologically biased toward logical truth or validity,

but the point is that he need not be.

18 W. V. Quine, “Three Grades”, p. 165.

19 W. V. Quine, ibid., p. 168. Quine’s semantic ‘Nec’ corresponds to Carnap’s ‘L-true’.

20 W. V. Quine, ibid., p. 171.

21 For the purpose of this discussion I set aside iterated modalities.

22 W. V. Quine, “The Problem of Interpreting Modal Logic”, p. 43.

23 The use of such extensional techniques is not unique to Carnap and Kripke. In the late

fifties and early sixties, the suggestion was made by S. Kanger “Provability in Logic”, R.

Montague, “Logical Necessity, Physical Necessity, Ethics, and Quantifiers”, and J.

Hintikka, “Modality and Quantification”. I am focusing on Carnap and Kripke partly

because Kripke’s version of the possible worlds model theory has become the standard

one, and also because, as we shall see, the evolution of the formal semantics from Carnap

to Kripke brings to the fore the question of whether the extensional model theory

provides an interpretation of the modal operators.

24 In fact, Carnap himself in his more logical, rather than philosophical, work employs

state-descriptions to tackle logical questions of soundness and completeness, rather than

to provide philosophical clarifications; cf. “Modalities and Quantification”.

25 Carnap faces the question of the soundness and completeness of his modal systems in

“Modalities and Quantification”.

26 I consider the introduction of an accessibility relation R between worlds when

discussing validity.

43

27 See note 23.

28 R. Goldblatt argues convincingly that Kripke’s adoption of points of evaluation in his

model structures was a particularly crucial innovation. Such a generalization opened the

door to different future developments of the model theory and made it possible to provide

model theories for intensional logics in general. Because of this and other features of

Kripke’s model theory, Goldblatt claims that Kripke’s work deserves a special place in

the development of possible world semantics. See “Mathematical Modal Logic: a View

of its Evolution,” part 4. See also J. Burgess, “Kripke Models”, where similar points are

emphasized and the overall superiority of Kripke’s work in formal semantics is argued

for.

29 Cf. “Language, Modal Logic, and Semantics”, in The Philosophy of Rudolf Carnap,

pp. 889-900.

30 In 1963, a model structure contains also an accessibility relation R between worlds,

needed to represent a notion of relative possibility. We are not concerned here with R,

nor with G, which is taken to represent the actual world.

31 J. Almog suggests that this is the case. He says that worlds taken as primitive points,

rather than models, are better suited to represent Kripke’s idea of metaphysical

possibility/necessity, rather than the combinatorial idea of truth across all consistent

assignments of values (cf. “Naming without Necessity”, pp. 217-8). Almog’s suggestion

has recently been supported by S. Neale (cf. “On a Milestone of Empiricism”, p. 321).

32 S. Kripke, “Semantical Analysis of Modal logic I. Normal Modal Propositional

Calculi”, p. 69, footnote 2.

44

See also Kripke’s Reviews of E. J. Lemmon’s “Algebraic Semantics for Modal

Logics”, where Kripke claims of Lemmon’s “algebraic method of proving completeness

theorems for certain modal propositional logics” that it “can be obtained by

straightforward applications of the methods . . . of the reviewer.” (p. 1021). This, by the

way, was exactly the point Lemmon wanted to make.

33 S. Kripke, ibid., p. 69. Emphasis added.

34 S. Kripke, ibid., p. 69.

35 Cf. J. Almog: “But they still lacked the idea of the world-as-a-point, possible worlds as

primitives, each with a structure intrinsic to it.” (“Naming without Necessity”, p. 218).

Points, I would think, are intrinsically structure-less. It must be their identification with

worlds that drives the idea of intrinsic structure.

36 He also assumes a one-to-one relation between individual constants and the objects of

the domain. But this is a further point from the one under consideration.

37 And, as a consequence, across all models/worlds. Kripke defines truth in an M-model

as truth in the actual world of the M-model, not as truth in all the models/worlds of the

M-model. Nonetheless, when it comes to validity across all M-models, the two

definitions make no extensional difference, at least before an operator sensitive to the

choice of the actual world like “Actually” is introduced.

38 Notice that even if Carnap had assumed Gödel’s domain variability, his validities about

the cardinality of the universe would not have coincided with Kripke’s. For any finite

number of individuals, it would still be valid in Carnap’s maximal sense that possibly

45

there are that many things. No such sentence is valid in Kripke’s universal sense (except

for “Possibly, there is at least one individual”, given the exclusion of the empty domain).

39 Kripke, “Semantical Considerations on Modal Logic”, reprinted in Reference and

Modality, p. 65.

40 Cf. A. N. Prior, “Modality and Quantification in S5”.

41 Kripke is very careful, and does not bluntly state that (BF) and its converse are not

provable. We have seen that the provability of the converse of (BF) depends on how one

decides to treat free variables. The notion of provability that one adopts (like the notion

of validity that one chooses) is relative. What is interesting is to find the right match

between a certain notion of provability and a certain notion of validity, in order to find

out which system characterizes which class of structures, whether our main interest is

proof-theoretical (i.e., in the system) or model-theoretical (i.e., in the set-theoretical

structures).

42 Cf. D. Kaplan, “Opacity”, Appendix E “Schematic Validity and Modal Logic”.

43 Interestingly, in his own formal work Carnap adopts a system equivalent to

propositional S5, for which he proves completeness, but that does not reflect his notion of

maximal validity and the philosophical assumption that atomic p’s abbreviate logically

independent sentences (cf. “Modalities and Quantification”).

S. Thomason has proved completeness for a system that adds to the axioms of S5

all formulas ◊p, for atomic p (more precisely, the axioms added state the possibility of

any conjunction of distinct propositional constants or their negations – of which ◊p for

atomic p’s is a limit case). This system of Thomason, and not S5, captures the notion of

46

validity philosophically endorsed by Carnap, i.e., validity across all state-descriptions. Cf.

Thomason, “A New Representation of S5”. David Kaplan has done unpublished work on

this subject, from knowledge of which I have benefited.

44 See Makinson, “How Meaningful are Modal Operators?”

45 Cf. Burgess, “Which Modal Logic is the Right One?”, pp. 86-87.

46 Cf. Carnap, p. 892 of “Language, Modal Logic, and Semantics”, in The Philosophy of

Rudolf Carnap.

47 Things are really slightly more complicated. First of all, given Carnap’s assumption of

a fixed denumerable domain, not all first order models qualify as worlds. Secondly, even

allowing variability of the domains, the result needs to be more carefully stated. For a

full, precise statement, cf. N. B. Cocchiarella, “On the Primary and Secondary Semantics

of Logical Necessity”. Cocchiarella attributes the result to D. Kalish and R. Montague.

See also D. Kaplan, “Opacity”, pp. 253-4, and S. Lindström, “Quine’s Interpretation

Problem and the Early Development of Possible Worlds Semantics”, p. 209.

48 S. Kripke, “A Completeness Theorem in Modal Logic”, p. 3. Emphasis added.

49 At least for the language under consideration, before an actuality operator is

introduced.

50 Cf. Kripke, “Semantical Analysis of Modal Logic I”, p. 70.

51 See E. J. Lemmon’s “Algebraic Semantics for Modal Logics I” and “Algebraic

Semantics for Modal Logics II” for a very clear exposition of the correspondence

between Kripke’s model theory and algebras for modal systems. Lemmon connects

Tarski and McKinsey’s algebraic method to Kripke’s model theoretical method, by

47

showing how Kripke’s 1963 completeness results for various propositional modal

systems can be derived form algebraic completeness results. Lemmon’s main theorem

proves that algebras for modal systems can be represented as algebras based on the power

set of the set K in the corresponding Kripke’s structures. As a consequence, algebraic

completeness translates into Kripke’s model theoretic completeness.

See also D. Makinson, “On Some Completeness Theorems in Modal Logic”,

where completeness results for various modal systems are provided making use of

Lindenbaum sets, the modal equivalent of the Löwenhein-Skolem theorem is proved, and

the possibility of mirroring these results in algebraic form is emphasized.

52 This point is also made by Burgess in “Kripke Models” and “Which Modal Models are

the Right Ones (for Logical Necessity)?”.

53 In my view, while it is the job of philosophers to investigate the notion(s) of necessity,

to evaluate different modal systems as better apt to represent different readings of the

modal operators, and also to inquire into an intuitive notion of logical truth for English

modal sentences (that may be represented in the formal language), it is not their job to

devise the right model theory for the system itself. This last task pertains to the

mathematical logician.

54 One possible view could be that when it comes to metaphysical necessity we are not

looking for an unqualified extensional match between necessities and validities, rather for

a match between necessity in a structure and the validities in that structure. While

extensionally more plausible, this view still suffers of the criticism presented in this paper

48

to the idea that the evolution of the notion of validity corresponds to philosophical

changes in the notion of necessity, not to formal changes of validity itself.

Curiously, S. Lindström recognizes that Kripke’s logic of necessity is relatively

meager and that the class of Kripke’s validities is not meant to capture the class of

metaphysical necessities, but he still argues that Kripke’s validities are appropriate for

metaphysical necessity only. This is so because the new notion of validity is taken to

induce at the object language level a new meaning for the modal operators: “Observe

how the relativization of the definition to C [an arbitrary class of worlds] changes the

meaning of [ ]. Instead of [ ]ϕ meaning that ϕ is logically true, it now means that ϕ is

universally true relative to the given set of state descriptions.” (“Modality without

Worlds”, p. 270.)

In this paper I have argued precisely against this general idea – whatever specific

form it may take – that changes in the notion of validity reflect at the meta-level

corresponding changes in the interpretation of the modal operators.

55 This isn’t surprising considering (Quine’s argument) that validity in terms of

substitution (schematic logical truth) and validity in terms of models (model theoretic

validity) coincide when completeness and the Löwenhein-Skolem theorem hold, and

given that the modal equivalents of these theorems hold vis-à-vis Kripke’s model theory.

See Quine, Philosophy of Logic, part 4 on Logical Truth.

56 Cf. D. Kaplan, “Opacity”, pp. 275-6.

57 See Lindström, “Modality Without Worlds”, pp. 282-3.

49

58 Notice that these are conditional claims. I do not intend to defend the claim that logic

should indeed be indifferent to questions of cardinality. But if such a claim is accepted –

as it generally is and surely so by Kaplan – then cardinality issues should be completely

disregarded and we should not stop halfway.

59 In “Which Modal Models are the Right Ones (for Logical Necessity)?”, Burgess

considers the formula “◊∃x∃y~(x = y)” and argues against some possible arguments to

regard it as valid. Since I do not take such a formula to be valid in some intuitive sense, I

am not going to disagree with Burgess on this. However, Burgess also claims that

questions about the validity of this formula are often treated in too simplistic a manner.

His reason for this last claim is that such a formula should not be translated into English

as saying “Possibly, there are at least two individuals”, but rather as something like: “For

any domain it is (logically) possible that there should be more than two elements in it.”

Moreover, Burgess claims that questions concerning the validity of this last sentence are

not obviously meaningful, because of Quinean reasons about different possible ways of

specifying the domain in question and how these different specifications may alter the

correct answer.

This is not the right place to take issue with Burgess’s claim. Let me only clarify a

few points. First, in this paper I am not specifically concerned with the notion of logical

possibility. My argument is about any notion of necessity one may be interested in. I have

in fact argued that the notion of necessity/possibility at stake should not determine the

notion of validity one adopts. Presumably, Quinean arguments of the kind Burgess is

considering apply exclusively to a logical understanding of necessity. Second, I am not in

50

the appendix arguing that “◊∃x∃y~(x = y)” should indeed be regarded as valid. Such a

sentence is, as a matter of fact, not valid in Kripke models. But, Kaplan claims, it is

schematically valid. Contra Kaplan, my claim is simply that it might well not be

schematically valid given certain adjustments to the notion of schematic validity that one

should want to endorse.

Finally, I do have a lot of sympathy for Burgess’s desire to keep a clear

distinction between formal sentences and English sentences, and I agree that questions

concerning the validity of formal sentences are to be kept apart from questions

concerning the intuitive logical truth of their English counterparts. However, I am not

sure that the right translation, if any, for “◊∃x∃y~(x = y)” is the one Burgess proposes.

But to argue for this last point would take me far beyond the limits of this work.

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