carnap_ modal logic

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Rudolf Carnap: Modal LogicIn two works, a paper in The Journal of Symbolic Logic in 1946 and thebook Meaning and Necessity in 1947, Rudolf Carnap developed a modal predicatelogic containing a necessity operator N, whose semantics depends on the claim that,where α is a formula of the language, Nα represents the proposition that α islogically necessary. Carnap’s view was that Nα should be true if and only if α itselfis logically valid, or, as he put it, is Ltrue. In the light of the criticisms of modallogic developed by W.V. Quine from 1943 on, the challenge for Carnap was how toproduce a theory of validity for modal predicate logic in a way which enables ananswer to be given to these criticisms. This article discusses Carnap’s motivationfor developing a modal logic in the first place; and it then looks at how the modalpredicate logic developed in his 1946 paper might be adapted to answer Quine’sobjections. The adaptation is then compared with the way in which Carnap himselftried to answer Quine’s complaints in the 1947 book. Particular attention is paid to

the problem of how to treat the meaning of formulas which contain a free individual variable in the scope of a modaloperator, that is, to the problem of how to handle what Quine called the third grade of ‘modal involvement’.

Table of Contents

1. Introduction

2. Carnap’s Propositional Modal Logic

3. Carnap’s (NonModal) Predicate Logic

4. Carnap’s 1946 Modal Predicate Logic

5. De Re Modality

6. Individual Concepts

7. References and Further Reading

1. IntroductionIn an important article (Carnap 1946) and in a book a year later, (Carnap 1947), Rudolf Carnap articulated a system ofmodal logic. Carnap took himself to be doing two things; the first was to develop an account of the meaning of modalexpressions; the second was to extend it to apply to what he called “modal functional logic” — that is, what we wouldcall modal predicate logic or modal firstorder logic. Carnap distinguishes between a logic or a ‘semantical system’,and a ‘calculus’, which is an axiomatic system, and states on p. 33 of 1946 that “So far, no forms of MFC [modalfunctional calculus] have been constructed, and the construction of such a system is our chief aim.” In fact, in thepreceding issue of The Journal of Symbolic Logic, the first presentation of Ruth Barcan’s axiomatic systems of modalpredicate logic had already appeared, although they contained only an axiomatic presentation. (Barcan 1946.) Theprincipal importance of Carnap’s work is thus his attempt to produce a semantics for modal predicate logic, and it isthat concern that this article will focus on.

Nevertheless, firstorder logic is founded on propositional logic, and Carnap first looks at nonmodal propositionallogic and modal propositional logic. I shall follow Carnap in using ~ and ∨ for negation and disjunction, though Ishall use ∧ in place of Carnap’s ‘.’ for conjunction. Carnap takes these as primitive together with ‘t’ which stands foran arbitrary tautologous sentence. He recognises that ∧ and t can be defined in terms of ~ and ∨, but prefers to takethem as primitive because of the importance to his presentation of conjunctive normal form. Carnap adopts thestandard definitions of ⊃ and ≡. I will, however, deviate from Carnap’s notation by using Greek in place of Germanletters for metalinguistic symbols. In place of ‘valid’ Carnap speaks of Ltrue, and in place of ‘unsatisfiable’, Lfalse.α Limplies β iff (if and only if) α ⊃ β is valid. α and β are Lequivalent iff α ≡ β is valid.

http://www.iep.utm.edu/carnap/



One might at this stage ask what led Carnap to develop a modal logic at all. The clue here seems to be the influence ofWittgenstein. In his philosophical autobiography Carnap writes:

For me personally, Wittgenstein was perhaps the philosopher who, besides Russell and Frege, had the greatestinfluence on my thinking. The most important insight I gained from his work was the conception that the truth oflogical statements is based only on their logical structure and on the meaning of the terms. Logical statements aretrue under all conceivable circumstances; thus their truth is independent of the contingent facts of the world. Onthe other hand, it follows that these statements do not say anything about the world and thus have no factualcontent. (Carnap 1963, p. 25)

Wittgenstein’s account of logical truth depended on the view that every (cognitively meaningful) sentence has truthconditions. (Wittgenstein 1921, 4.024.) Carnap certainly appears to have taken Wittgenstein’s remark as endorsing thetruthconditional theory of meaning. (See for instance Carnap 1947 p. 9.) If all logical truths are tautologies, and alltautologies are contentless, then you don’t need metaphysics to explain (logical) necessity.

One of the features of Wittgenstein’s view was that any way the world could be is determined by a collection ofparticular facts, where each such fact occupies a definite position in logical space, and where the way that position isoccupied is independent of the way any other position of logical space is occupied. Such a world may be described ina logically perfect language, in which each atomic formula describes how a position of logical space is occupied. Sosuppose that we begin with this language, and instead of asking whether it reflects the structure of the world, we askwhether it is a useful language for describing the world. From Carnap’s perspective, (Carnap 1950) one mightdescribe it in such a way as this. Given a language £ we may ask whether £ is adequate, or perhaps merely useful, fordescribing the world as we experience it. It is incoherent to speak about what the world in itself is like withoutpresupposing that one is describing it. What makes £ a Carnapian equivalent of a logically perfect language would bethat each of its atomic sentences is logically independent of any other atomic sentence, and that every possible worldcan be described by a statedescription.

2. Carnap’s Propositional Modal Logic

In (nonmodal) propositional logic the truth value of any wellformed formula (wff) is determined by an assignmentof truth values to the atomic sentences. For Carnap an assignment of truth values to the atomic sentences isrepresented by what he calls a ‘statedescription’. This term, like much in what follows, is only introduced at thepredicate level (1946, p. 50) but it is less confusing to present it first for the propositional case, where a statedescription, which I will refer to as s, is a class consisting of atomic wff or their negations, such that for each atomicwff p, exactly one of p or ~p is in s. (Here we may think of p as a propositional variable, or as a metalinguisticvariable standing for an atomic wff.) Armed with a statedescription s we may determine the truth of a wff α at s inthe usual way, where s α means that α is true according to s, and s α means that not s α:

If α is atomic, then s α if α ∈ s, and s α if ~α ∈ s

s ~α iff s α

s α ∨ β iff s α or s β

s α ∧ β iff s α and s β

s t

This is not the way Carnap describes it. Carnap speaks of the range of a wff (p. 50). In Carnap’s terms the truth ruleswould be written:

If α is atomic then the range of α is those statedescriptions s such that α ∈ s.

Where V is the set of all statedescriptions, the range of ~α is V minus the range of α, that is, it is the class ofthose statedescriptions which are not in the range of α.

The range of α ∨ β is the range of α ∪ the range of β, that is, the class of statedescriptions which are either in therange of α or the range of β.

The range of α ∧ β is the range of α ∩ the range of β, that is, the class of statedescriptions which are in both therange of α and the range of β.



The range of t is V.

It should I hope be easy to see, first that Carnap’s way of putting things is equivalent to my use of s α, and secondthat these are in turn equivalent to the standard definitions of validity in terms of assignments of truth values.

By a ‘calculus’ Carnap means an axiomatic system, and he uses ‘PC’ to indicate any axiomatic system which is closedunder modus ponens (the ‘rule of implication’, p. 38) and contains “‘t’ and all sentences formed by substitution fromBernays’s four axioms [See Hilbert and Ackermann, 1950, p. 28f] of the propositional calculus”. (loc cit.) Carnapnotes that the soundness of this axiom system may be established in the usual way, and then shows how the possibilityof reduction to conjunctive normal form (a method which Carnap, p. 38, calls Preduction) may be used to provecompleteness.

Modal logic is obtained by the addition of the sentential operator N. Carnap notes that N is equivalent to Lewis’s ~◊~.(Note that the symbol was not used by Lewis, but was invented by F.B. Fitch in 1945, and first appeared in print inBarcan 1946. It was not then known to Carnap.) Carnap tells us early in his article that “the guiding idea in ourconstruction of systems of modal logic is this: a proposition p is logically necessary if and only if a sentenceexpressing p is logically true.” When this is turned into a definition in terms of truth in a statedescription we get thefollowing:

s Nα iff sʹ α for every statedescription sʹ.

This is because Ltruth, or validity, means truth in every statedescription. I shall refer to validity when N isinterpreted in this way, as Carnapvalidity, or Cvalidity. This account enables Carnap to address what was animportant question at the time — what is the correct system of modal logic? While Carnap is clear that differentsystems of modal logic can reflect different views of the meaning of the necessity operators he is equally clear that, ashe understands it, principles like Np ⊃ NNp and ~Np ⊃ N~Np are valid. It is easy to see that the validity of both theseformulae follows easily from Carnap’s semantics for N. From this it is a short step to establishing that Carnap’s modallogic includes the principles of Lewis’s system S5, provided one takes the atomic wff to be propositional variables.However, we immediately run into a problem. Suppose that p is an atomic wff. Then there will be a statedescriptionsʹ such that ~p ∈ sʹ. And this means that for every statedescription s, s Np, and so s ~Np. But this means that~Np will be Ltrue. One can certainly have a system of modal logic in which this is so. An axiomatic basis and acompleteness proof for the logic of Cvalidity occurs in Thomason 1973. (For comments on this feature of Cvaliditysee also Makinson 1966 and Schurz 2001.) However, Carnap is clear that his system is equivalent to S5 (footnote 8, p.41, and on p. 46.); and ~Np is not a theorem of S5. Further, the completeness theorem that Carnap proves, usingnormal forms, is a completeness proof for S5, based on Wajsberg 1933.

How then should this problem be addressed? Part of the answer is to look at Carnap’s attitude to propositionalvariables:

We here make use of ‘p’, ‘q’, and so forth, as auxiliary variables; that is to say they are merely used (followingQuine) for the description of certain forms of sentences. (1946, p.41)

Quine 1934 suggests that the theorems of logic are always schemata. If so then we can define a wff α as what wemight call QCvalid (Quine/Carnap valid) iff every substitution instance of α is Cvalid. Wffs which are QCvalid areprecisely the theorems of S5.

3. Carnap’s (NonModal) Predicate Logic

In presenting Carnap’s 1946 predicate logic (or as he prefers to call it ‘functional logic’, FL or FC depending onwhether we are considering it semantically or axiomatically) I shall use ∀x in place of (x), and ∃x in place of (∃x). FLcontains a denumerable infinity of individual constants, which I will often refer to simply as ‘constants’. Carnap usesthe term ‘matrix’ for wff, and the term ‘sentence’ for closed wff, that is wff with no free variables. A statedescriptionis as for propositional logic in containing only atomic sentences or their negations. Each of these will be a wff of theform Pa1...an or ~Pa1...an, where P is an nplace predicate and a1,..., an are n individual constants, not necessarilydistinct.

To define truth in such a statedescription Carnap proceeds a little differently from what is now common. In place ofrelativising the truth of an open formula to an assignment to the variables of individuals from a domain, Carnapassumes that every individual is denoted by one and only one individual constant, and he only defines truth forsentences. If s is any statedescription, and α and β are any sentences, the rules for propositional modal logic can be



extended by adding the following:

s Pa1...an if Pa1...an ∈ s and s Pa1...an if ~Pa1...an ∈ s

s a = b iff a and b are the same constant

s ∀xα iff s α[a/x] for every constant a, where [α/x] is α with a replacing every free x.

Carnap produces the following axiomatic basis for firstorder predicate logic, which he calls ‘FC’. In place ofCarnap’s ( ) to indicate the universal closure of a wff, I shall use ∀, so that Carnap’s D81a (1946, p. 52) can bewritten as:

PC ∀α where α is a PCtautology

and so on. Carnap refers to axioms as ‘primitive sentences’ and in addition to PC, using more current names, we have:

∀⊃ ∀(∀x(α ⊃ β) ⊃ (∀xα ⊃ ∀xβ))

VQ ∀(α ⊃ ∀xα), where x is not free in α.

∀1a ∀(∀x ⊃ α[y/x]), where α[y/x] is just like α except in having y in place of free x, where y is any variable forwhich x is free

∀1b ∀(∀x ⊃ α[b/x]), where α[b/x] is just like α except in having b in place of free x, where b is any constant

I1 ∀x x = x

I2 ∀(x = y ⊃ (α ⊃ β)), where α and β are alike except that α has free x in 0 or more places where β has free y.

I3 a ≠ b where a and b are different constants.

The only transformation rule is modus ponens:

MP α, α ⊃ β therefore β

The only thing nonstandard here, except perhaps for the restriction of theorems to closed wffs, is I3, which ensuresthat all statedescriptions are infinite, and, as Carnap points out on p. 53, validates ∃x∃y x ≠ y. It is possible to provethe completeness of this axiomatic system with respect to Carnap’s semantics.

4. Carnap’s 1946 Modal Predicate Logic

Perhaps the most important issue in Carnap’s modal logic is its connection with the criticisms of W.V. Quine. Thesecriticisms were well known to Carnap who cites Quine 1943. Some years later, in Quine 1953b, Quine distinguishesthree grades of what he calls ‘modal involvement’. The first grade he regards as innocuous. It is no more than themetalinguistic attribution of validity to a formula of nonmodal logic. In the second grade we say that where α is anysentence then Nα is true iff α itself is valid — or logically true. On pp. 166169 Quine argues that while such aprocedure is possible it is unilluminating and misleading. The third grade applies to modal predicate logic, and allowsfree individual variables to occur in the scope of modal operators. It is this grade that Quine finds objectionable. Oneof the points at issue between Quine and Carnap arises when we introduce what are called definite descriptions into thelanguage. Much of Carnap’s discussion in his other works — see especially Carnap 1947 — elevates descriptions to acentral role, but in the 1946 paper these are not involved.

The extension of Carnap’s semantics to modal logic is exactly as in the propositional case:

s Nα iff sʹ α for every statedescription sʹ.

As before, a wff can be called Cvalid iff it is true in every statedescription, when satisfies the principle just stated.As in the propositional case if α is S5valid then α is Cvalid. However, also as in the propositional case, (quantified)S5 is not complete for Cvalidity. This is because, where Pa is an atomic wff, ~NPa is Cvalid even though it is not atheorem of S5 — and similarly with any atomic wff. Unlike the propositional case it seems that this is a feature whichCarnap welcomed in the predicate case, since he introduces some nonstandard axioms.

The first set of axioms all form part of a standard basis for S5. They are as follows (p. 54, but with current names and

http://www.iep.utm.edu/russ-met/



notation):

LPCN Where α is one of the LPC axioms PCI3 then both α and Nα are axioms of MFC.

K N∀(N(α ⊃ β) ⊃ (Nα ⊃ Nβ))

T ∀(Nα ⊃ α)

5 N∀(Nα ∨ N~Nα)

BFC N∀(N∀xα ⊃ ∀xNα)

BF N∀(∀xNα ⊃ N∀xα)

The nonstandard axioms, which show that he is attempting to axiomatise Cvalidity, are what Carnap calls‘Assimilation’, ‘Variation and Generalization’ and ‘Substitution for Predicates’. (Carnap 1946, p. 54f.) In our notationthese can be expressed as follows:

Ass N∀x∀y∀z1...∀zn((x ≠ z1 ∧ ... ∧ x ≠ zn) ⊃ (Nα ⊃ N α[y/x])), where α contains no free variables other than x, y,z1,..., zn, and no constants and no occurrences of =.

VG N∀x∀y∀z1...∀zn((x ≠ z1 ∧ ... ∧ x ≠ zn ∧ y ≠ z1 ∧ ... ∧ y ≠ zn) ⊃ (Nα ⊃ N α[y/x]), where α contains no freevariables other than x, y, z1,..., zn, and no constants.

SP N∀(Nα ⊃ Nβ), where β is obtained from α by uniform substitution of a complex expression for a predicate.

None of these axiom schemata is easy to process, but it is not difficult to see what the simplest instances would looklike. A very simple instance, which is of both Ass and VG is

AssP N∀x∀y∀z(x ≠ z ⊃ (NPxyz ⊃ NPyyz))

To establish the validity of AssP it is sufficient to show that if a and c are distinct constants then NPabc ⊃ NPbbc isvalid. This is trivially so, since there is some s such that s Pabc, and therefore for every s, s NPabc, and so, forevery s, s NPabc ⊃ Pbbc. More telling is the case of SP. Let P be a oneplace predicate and consider

SPP N∀x(NPx ⊃ N(Px ∧ ~Px))

In this case α is Px, while β is Px ∧ ~Px, so that, in Carnap’s words, β ‘is formed from α by replacing every atomicmatrix containing P by the current substitution form of β’. That is, where β is Px ∧ ~Px, it replaces α’s Px. If α hadbeen more complex and contained Py as well as Px, then the replacement would have given Py ∧ ~ Py, and so on,where care needs to be taken to prevent any free variable being bound as a result of the replacement. In this case wehave ~N(Pa ∧ ~ Pa), and so ~NPa.

In fact, although Carnap appears to have it in mind to axiomatise Cvalidity, it is easy to see that the predicate versionis not recursively axiomatisable. For, where α is any LPC wff, α is not LPCvalid iff ~Nα is Cvalid, and so, if Cvalidity were axiomatisable then LPC would be decidable. There is a hint on p. 57 that Carnap may have recognisedthis. He is certainly aware that the kind of reduction to normal form, with which he achieves the completeness ofpropositional S5, is unavailable in the predicate case, since it would lead to the decidability of LPC.

5. De Re Modality

What then can be said on the basis of Carnap 1946 to answer Quine’s complaints about modal predicate logic? Quineillustrates the problem in Quine 1943, pp. 119121, and repeats versions of his argument many times, most famouslyperhaps in Quine 1953a, 1953b and 1960. The example goes like this:

(1) 9 is necessarily greater than 7

(2) The number of planets = 9

therefore

(3) The number of planets is necessarily greater than 7.



Carnap 1946 does not introduce definite descriptions into the language, so I shall present the argument in aformalisation which only uses the resources found there. I shall also simplify the discussion by using the predicate O,where Ox means ‘x is odd’, rather than the complex predicate ‘is greater than 7’. This will avoid reference to ‘7’,which is of no relevance to Quine’s argument. P means ‘is the number of the planets’, so that Px means ‘there are xmany planets’. With this in mind I take ‘9’ to be an individual constant, and use O and P to express (1) and (2) by

(4) NO9

(5) ∃x(Px ∧ x = 9)

One could account for (4) by adding O9 as a meaning postulate in the sense of Carnap 1952, which would restrict theallowable statedescriptions to those which contain O9, though from some remarks on p. 201 of Carnap 1947 it seemsthat Carnap might have regarded both O and 9 as complex expressions defined by the resources of the Frege/Russellaccount of the natural numbers and their arithmetical properties. It also seems that he might have treated the numbersas higherorder entities referred to by higherorder expressions. If so then the necessity of arithmetical truths like (4)would derive from their analysis into logical truths. In my exposition I shall take the numerals as individual constants,and assume somehow that O9 is a logical truth, true in every statedescription, and that therefore (4) is true.

In this formalisation I am ignoring the claim that the description ‘the number of the planets’ is intended to claim thatthere is only one such number. So much for the premises. But what about the conclusion? The problem is where to putthe N. There are at least three possibilities:

(6) N∃x(Px ∧ Ox)

(7) ∃xN(Px ∧ Ox)

(8) ∃x(Px ∧ NOx)

It is not difficult to show that (6) and (7) do not follow from (4) and (5). In contrast to (6) and (7), (8) does followfrom (4) and (5), but there is no problem here, since (8) says that there is a necessarily odd number which is such thatthere happen to be that many planets. And this is true, because 9 is necessarily odd, and there are 9 planets. All of thisshould make clear how the phenomenon which upset Quine can be presented in the formal language of the 1946article. Quine of course claims not to make sense of quantifying in. (See for instance the comments on Smullyan 1948in Quine 1969, p. 338.)

6. Individual Concepts

Even if something like what has just been said might be thought to enable Carnap to answer Quine’s complaints aboutde re modality, it seems clear that Carnap had not availed himself of it in the 1947 book, and I shall now look at themodal logic presented in Carnap 1947. On p. 193f Carnap cites the argument (1)(2)(3) from Quine 1943 discussedabove. He does not appear to recognise any potential ambiguity in the conclusion, and characterises (3) as false.Carnap doesn’t consider (8), and on p. 194 simply says:

“we obtain the false statement [(3)]”

In Carnap’s view the problem with Quine’s argument is that it assumes an unrestricted version of what is sometimescalled ‘Leibniz’ Law’:

I2 ∀x∀y(x = y ⊃ (α ⊃ β)), where α and β differ only in that α has free x in 0 or more places where β has free y.

In the 1946 paper this law holds in full generality, as does a consequence of it which asserts the necessity of trueidentities.

LI ∀x∀y(x = y ⊃ Nx = y)

For suppose LI fails. Then there would have to be a statedescription ss in which for some constants a and b, s a = bbut s Na = b. So there is a statedescription sʹ such that sʹ a = b, but then, a and b are different constants, and so, s a = b, which gives a contradiction.

In the 1947 book Carnap holds that I2 must be restricted so that neither x nor y occur free in the scope of a modaloperator. In particular the following would be ruled out as an allowable instance of I2:



(1) x = y ⊃ (NOx ⊃ NOy)

In order to explain how this failure comes about, and solve the problems posed by coreferring singular terms, Carnapmodifies the semantics of the 1946 paper. The principal difference from the modal logic of the 1946 paper, as Carnaptells us on p. 183, is that the domain of quantification for individual variables now consists of individual concepts,where an individual concept i is a function from statedescriptions to individual constants. Where s is a statedescription, let is denote the constant which is the value of the function i for the statedescription s. Carnap is clearthat the quantifiers range over all individual concepts, not just those expressible in the language.

Using this semantics it is easy to see how (9) can fail. For let x have as its value the individual concept i, which is thefunction such that is is 9 for every statedescription s, while the value of y is the function j such that, in any statedescription s, js is the individual which is the number of the planets in s, that is, js is the (unique) constant a such thatPa is in s. (Assume that in each statedescription there is a unique number, possibly 0, which satisfies P.) Assume thatx = y is true in any statedescription s iff, where i is the individual concept which is the value of x, and j is theindividual concept which is the value of y, then is is the same individual constant as js. In the present example ithappens that when s is the statedescription which represents the actual world, is and js are indeed the same, for in sthere are nine planets, making x = y true at s. Now NOx will be true if Ox is true in every statedescription sʹ, which isto say if isʹ satisfies O in every sʹ. Since isʹ is 9 in every statedescription then isʹ does satisfy O in every sʹ, and so NOxis true at s. But suppose sʹ represents a situation in which there are six planets. Then jsʹ will be 6 and so Oy will befalse in sʹ, and for that reason NOy will be false in s, thus falsifying (9). (It is also easy to see that LI is not valid, sinceit is easy to have is = js even though i ≠ j.)

The difference between the modal semantics of Carnap 1946 and Carnap 1947 is that in the former the onlyindividuals are the genuine individuals, represented by the constants of the language ℒ. In the proof of the invalidityof (9) it is essential that the semantics of identity require that when x is assigned an individual concept i and y isassigned an individual concept j that x = y be true at a statedescription s iff is and js are the same individual. And nowwe come to Quine’s complaint (Quine 1953a, p. 152f). It is that Carnap replaces the domain of things as the range ofthe quantifiers with a domain of individual concepts. Quine then points out that the very same paradoxes arise again atthe level of individual concepts. Thus for instance it might be that the individual concept which represents the numberof planets in each statedescription is identical with the first individual concept introduced on p. 193 of Meaning andNecessity. Carnap is alive to Quine’s criticism that ordinary individuals have been replaced in his ontology byindividual concepts. In essence Carnap’s reply to Quine on pp. 198 200 of Carnap 1947 is that if we restrict ourselvesto purely extensional contexts then the entities which enter into the semantics are precisely the same entities as are theextensions of the intensions involved. What this amounts to is that although the domain of quantification consists ofindividual concepts, the arguments of the predicates are only the genuine individuals. For suppose, as Quine appearsto have in mind, we permit predicates which apply to individual concepts. Then suppose that i and j are distinctindividual concepts. Let P be a predicate which can apply to individual concepts, and let s be a statedescription inwhich P applies to i but not to j but in which is and js are the same individual. We now have two options depending onhow = is to be understood. If we take x = y to be true in s when is and js are the same individual then if x is assigned iand y is assigned j we would have that x = y and Px are both true in s, but Py is not. So that even the simplest instanceof I2

I2P x = y ⊃ (Px ⊃ Py)

fails, and here there are no modal operators involved. The second option is to treat = as expressing a genuine identity.That is to say x = y is true only when the individual concept assigned to x is the same individual concept as the oneassigned to y. In the example I have been discussing, since i and j are distinct individual concepts if i is assigned to xand j to y, then x = y will be false. But on this option the full version of I2 becomes valid even when α and β containmodal operators. This is just another version of Quine’s complaint that if an operator expresses identity then the termsof a true identity formula must be interchangeable in all contexts. Presumably Carnap thought that the use ofindividual concepts could address these worries. The present article makes no claims on whether or not an acceptabletreatment of individual concepts is desirable, and if it is whether one can be developed.

7. References and Further Reading

This list contains all items referred to in the text, together with some other articles relevant to Carnap’s modal logic.

Barcan, (Marcus) R.C., 1946, A functional calculus of first order based on strict implication. The Journal of Symbolic Logic, 11, 1–16.



Burgess, J.P., 1999, Which modal logic is the right one? Notre Dame Journal of Formal Logic, 40, 81–93.

Carnap, R., 1937, The Logical Syntax of Language, London, Kegan Paul, Trench Truber.

Carnap, R., 1946, Modalities and quantification. The Journal of Symbolic Logic, 11, 33–64.

Carnap, R., 1947, Meaning and Necessity, Chicago, University of Chicago Press (Second edition 1956, references are to the secondedition.).

Carnap, R., 1950, Empiricism, semantics and ontology. Revue Intern de Phil. 4, pp. 20–40 (Reprinted in the second edition ofCarnap 1947, pp. 2052–2221. Page references are to this reprint.).

Carnap, R., 1952, Meaning postulates. Philosophical Studies, 3, pp. 65–73. (Reprinted in the second edition of Carnap 1947, pp.222–229. Page references are to this reprint.)

Carnap, R., 1963, The Philosophy of Rudolf Carnap, ed P.A. Schilpp, La Salle, Ill., Open Court, pp. 3–84.

Church, A., 1973, A revised formulation of the logic of sense and denotation (part I). Noũs, 7, pp. 24–33.

Cocchiarella, N.B., 1975a, On the primary and secondary semantics of logical necessity. Journal of Philosophical Logic, 4, pp. 13–27..

Cocchiarella, N.B.,1975b, Logical atomism, nominalism, and modal logic. Synthese, 31, pp. 23−67.

Cresswell, M.J., 2013, Carnap and McKinsey: Topics in the pre–history of possible worlds semantics. Proceedings of the 12thAsian Logic Conference, J. Brendle, R. Downey, R. Goldblatt and B. Kim (eds), World Scientific, pp. 5375.

Garson, J.W., 1980, Quantification in modal logic. Handbook of Philosophical Logic, ed. D.M. Gabbay and F. Guenthner,Dordrecht, Reidel, Vol. II, Ch. 5, 249307

Gottlob, G., 1999, Remarks on a Carnapian extension of S5. In J. Wolenski, E. Köhler (eds.), Alfred Tarski and the Vienna Circle,Kluwer, Dordrecht, 243−259.

Hilbert, D., and W. Ackermann, 1950, Mathematical Logic, New York, Chelsea Publishing Co., (Translation of Grundzüge derTheoretischen Logik.).

Hughes, G.E., and M.J. Cresswell, 1996, A New Introduction to Modal Logic, London, Routledge.

Lewis, C.I., and C.H. Langford, 1932, Symbolic Logic, New York, Dover publications.

Makinson, D., 1966, How meaningful are modal operators? Australasian Journal of Philosophy, 44, 331−337.

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Author Information

M. J. CresswellEmail: [email protected]

Victoria University of WellingtonNew Zealand

Article printed from Internet Encyclopedia of Philosophy: http://www.iep.utm.edu/cmlogic/

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carnap_ modal logic

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