JAAKKO HINTIKKA

ON GÖDEL’S PHILOSOPHICAL ASSUMPTIONS

In his lifetime Gödel published relatively little, and almost all of what hedid publish dealt with logic, set theory or mathematical physics. Eventhe few published papers where he presented philosophical ideas wereessentially extensions of his logical and mathematical work. He was never-theless known to have a strong general philosophical interest, but his actualviews were largely unknown, subject to rumors and guesses. Graduallyas more and more information about his philosophical ideas has becomeavailable, thanks to a large extent to the efforts of Hao Wang. A fourthvolume of hisCollected Worksis being planned, containing Gödel’s un-published philosophical writings. Even before its appearance, it is possibleto form an overview of his philosophical thought or at least identify someof the assumptions on which it rests.

To startin medias res, one instructive clue to the mind of Kurt Gödel thephilosopher is offered by a comparison between him and another philoso-pher Gödel greatly admired both as a logician and as a philosopher, Gott-fried Wilhelm Leibniz. Gödel went as far as to express the belief that“Leibniz apparently had obtained a decision procedure for mathematics”(Wang 1987, p. 73). He also admired Leibniz’s metaphysics, and some ofhis own metaphysical speculations are closely related to Leibniz’s mon-adology; see e.g. Wang (1977, 293–97). But one of Leibniz’s ideas wasnever taken up by Gödel, even though for Leibniz it was one of his mostcentral notions, perhaps the most important one. This idea is that of apossible world. I am not aware of any single use of this idea by Gödel. Thisabsence is to some extent obscured by Hao Wang, who evokes the conceptof possible world on a number of occasions (1987, 195, 251, 254, 262, 263;1997, 10, 17, 120). On none of these occasions is Wang reporting Gödel’swords but putting forward his own ideas. Sometimes (see for instance1987, 254) Wang is preoccupied with the question whether set theory isnecessarily true, and expresses this by asking whether it is true in all pos-sible worlds. On other occasions (see e.g. 1997, 120) Wang notes Gödel’sbelief that a maximal selection of beings actually exist, and moves on tointerpret this as a variant of Leibniz’s ideas that the actual world is the bestof all possible worlds. This transition is not obvious; to attribute maximal

Synthese114: 13–23, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

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richness to the actual world does not presuppose a full-fledged concept ofother possible worlds. All it requires is some notion of the compatibility of(existential) propositions. Many logicians and philosophers have had sucha concept without having a full-fledged notion of possible world.

The absence of the notion of possible world is especially revealing inview of Gödel’s closeness to other ideas of Leibniz’s. It is in Gödel, asit typically is, a part of a more general syndrome. I have called it the one-world assumption. It can be characterized as an assumption that all that canproperly be said to exist must exist somewhere in the actual world and thattherefore all that we can meaningfully speak about must in the analysis bedenizens of the actual world.

Gödel does not seem to have asserted the one-world assumption inany explicit manner. His commitment to it is shown by his acceptanceof its consequences. Perhaps the clearest of them is his view of the statusof logical and mathematical truth. For Leibniz, metaphysical truths weretruths in every possible world, and the same idea is applied to logicaltruths by contemporary modal logicians and modal metaphysicians. ForGödel, as for his fellow one-world theorists Frege and early Russell, logi-cal and mathematical truths are about this actual world. Gödel quoted withapproval Russell’s assertion (in Russell 1919, 169) that

Logic is concerned with the real world just as truly as zoology, though with its moreabstract and general features.

As Wang very well brings out, Gödel followed Russell in comparing

the axioms of logic and mathematics with the laws of nature and logical evidence withsense perception.

As Wang also notes, Gödel does not only believe in the objectivity oflogical and mathematical propositions, he also believes that they are aboutactual objects.

For a thinker like Carnap, logical truths are on a categorially differentlevel from ordinary (“material”) truths. They are not truths in or aboutany one world; they are true in all possible worlds. This is not Gödel’sconception of logical truth. For him, as for Frege and Quine, there is nohard-and-fast distinction between factual and logical (and mathematical)truths.

Gödel’s overall position in the philosophy of logic and mathematics isoften called Platonism. This label is ambiguous, however, and can there-fore be misleading. The contrast it is calculated to highlight is between abelief in nonsensible abstract (“Platonic”) entities and a denial of them.But concepts and conceptualizations that go beyond sense-experience andbeyond the concrete objects it gives us can be of two entirely different

ON GÖDEL’S PHILOSOPHICAL ASSUMPTIONS 15

kinds. A many-world theorist thinks of other possible scenarios (possiblestates of affairs, possible courses of events, possible situations, etc.) assomething that play an important role in our conceptual system, in ourthinking and even in our actions. But such a many-world theorist doesnot think of the alternative scenarios as somehow being realized in someshadowy region of our actual universe, be it a Platonic heaven of abstractentities or (Cantor notwithstanding) the mind of God.

To think so would miss the precise point of other scenarios, viz. thatthey are unrealized possibilities whose very possibility nevertheless is anenormously important fact of our conceptual life. Whether such many-worlds thinkers have even been called, or should be called, Platonists, I donot know and I do not care. They certainly reckon with factors which arein some sense abstract, viz. being unrealized, and inaccessible to sense-perception.

But when Gödel is called a Platonist, something entirely different ismeant. He is not emphasizing the conceptual role of unrealized possibili-ties, but the role of actualized abstract entities. Of course they cannot beactualized in the sensible, material or space-time region of our universe.Hence they must be thought of as existing in another abstract region ofour one and only actual world. It would thus be clarifying to separatealso terminologically the possible-world Platonists from the actual-worldPlatonists or perhaps simply the possibilists from the actualists. The pointI am making here is in any case that Gödel belongs to the latter category,not the former. This is connected with his conception of logical truths astruths about the actual world, albeit about “its more abstract and generalfeatures”.

One important place where Gödel’s actualist conception of logical andmathematical truth shows up is in his functional interpretation (also knownas the “Dialectica” interpretation). (See Gödel 1958.) This interpretationcan very roughly be described as associating with each sentence of first-order number theory another sentence, one which says that all the Skolemfunctions (in a modified and extended sense) of the given one exist. Gödelputs his interpretation forward as an interpretation of Heyting’s arithmetic,hence as an axiomatic and deductive treatment of elementary number the-ory. Yet the most natural way of looking at Gödel’s interpretation is theone first noted by Dana Scott (1968) and later emphasized by myself, viz.to think of Gödel’s translation (correlation) rules as defining a nonstandardgame-theoretical sense of truth.

The assumptions that I have been diagnosing led Gödel to look at hisown incompleteness theorem in a special way. This perspective is impor-tant to diagnose explicitly, for it has not usually been distinguished from

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the other viewpoints that are possible here. For Gödel what is given is anobjectively existing structure of natural numbers, and by implication anobjectively determined set of sentences true in it. Many of Gödel’s profes-sions of “Platonism” are best understood as assertions that this notion ofarithmetical truth is an objectively determined one. Then the question ofcompleteness becomes the question whether the set of arithmetical truthsis axiomatizable. Since for any sentenceS in the language of elementaryarithmetic eitherS or ∼ S is an arithmetical truth, this axiomatizabil-ity would imply a decision method. By grinding out a theorem often atheorem, eitherS itself or its negation,∼ S would eventually make itsappearance for eachS, assuming of course the completeness of the axiom-atization. And such decidability is known to be equivalent to the recursive-ness of the set of arithmetical truth, in other words, the possibility of listingall truths (and all non-truths) by a Turing machine.

Now the question as to whether the set of arithmetical truths is re-cursive is an intra-logical one. It concerns the power (or lack thereof) ofrecursive methods to exhaust all the truths of elementary number theory.The kind of completeness or incompleteness that is at issue here might becalleddeductivecompleteness and incompleteness. (For different kinds ofcompleteness and incompleteness in general, see Hintikka 1996, ch. 6.)This is the kind of incompleteness proved by Gödel. (See Gödel 1931. Itis the only kind considered by him in his classical paper.) Accordingly,what Gödelian incompleteness amounts to is a restriction of the power ofrecursive (purely mechanical) methods in capturing arithmetical truths. Itdoes not say anything about other kinds of limitations of logical methodsin elementary arithmetics or elsewhere in mathematics.

From what was said earlier it follows that deductive incompletenesswas the only kind of incompleteness that made sense to Goödel in the caseof nonlogical theories. It was called by him simply incompleteness. Thispractice has been followed by later logicians and philosophers. They haveaccordingly missed, or at least not paid enough attention in their theorizing,to the fact that deductive completeness and incompleteness are not the onlyrelevant notions here. For a many-world theorist, the crucial question iswhether a theory is descriptively complete or not, that is to say, whetherits models comprise all and only its intended models. This is the sensein which we ask for instance whether elementary Euclidean geometry iscompletely axiomatizable.

This notion of completeness might be called a descriptive one. It isquite different from deductive completeness, even thought the two are notunrelated. A failure to keep the two kinds of completeness separated hasresulted in an enormous amount of confused speculation among philoso-

ON GÖDEL’S PHILOSOPHICAL ASSUMPTIONS 17

phers discussing the consequences of Gödel’s results. This is especiallyblatant when Gödel’s incompleteness result has been taken to imply somekind of intrinsic limitation to human thinking or when Gödelian incom-pleteness is compared with the incompleteness of Euclidean geometrywithout the parallel postulate. In reality, as was explained, Gödelian in-completeness for instance shows the limitations of Turing machines ratherthan of human minds in dealing with mathematical truths.

It must be realized, however, that Gödel would have had a plausible-looking line of defense. As one can easily see, deductive incompletenessimplies descriptive incompleteness if the underlying logic is complete inyet another sense of completeness. This sense is tantamount to the re-cursive enumerability of all (and only) logically valid formulas of a sys-tem of logic. It is often referred to as semantical completeness. Indeed,it was precisely Gödel who for the first time proved (in 1930) the se-mantical completeness of first-order logic. Hence Gödel might very wellhave felt justified in not paying special attention to questions of descriptivecompleteness.

Gödel’s completeness proof for first-order logic (1930) neverthelessdoes not close either the historical or the systematic issues. For the seman-tical completeness of first-order logic cannot be extended to all other log-ics. For instance, second-order logic and more generally speaking higher-order logics are not semantically complete. Admittedly, this holds on whatHenkin later (1950) called the standard interpretation of higher-order log-ics. But this is arguably the most natural interpretation of higher-orderlogics, and it was accessible to Gödel in the form of Frank Ramsey’s(1925) reformulation of the type theory of Russell and Whitehead. Second-order logic is not semantically complete (axiomatizable), but it allowsdescriptively complete axiomatizations of various mathematical theories,including elementary number theory. Such a descriptive completeness isperfectly compatible with the deductive incompleteness of these theories.

It may even be the case that Gödel himself missed thetertium daturhere. As against Zermelo, Gödel says that “the essential point of his resultis that we cannot include the whole mathematics in a formal system. . . ;that already follows from Cantor’s diagonal procedure. . . ”. Now a uni-versal axiom system for all mathematics is undoubtedly impossible in thesense that any candidate will be deductively incomplete. But from thatit simply does not follow that such a universal mathematics cannot bedescriptively complete.

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These observations are directly relevant to any discussion of Gödel’swork and its significance. For instance, Dawson (1997, 67) writes that

The incompleteness theorems hold also for higher-order for-malizations of number theory.

This is of course true, but only if the kind of incompleteness in questionis taken to be deductive incompleteness. Higher-order formalizations ofnumber theory do not have to be descriptively incomplete. Again Gödelreportedly told Wang that he first came upon the first incompleteness theo-rem as a result concerning the undefinability of arithmetical truth in arith-metic itself. (See Wang 1997, 82–83; 1987, 90–91.) But Gödel’s actualincompleteness proof does not concern merely first-order arithmetic butPrincipia Mathematicaand related systems. In such systems, there is noreason why one cannot express first-order arithmetical truth on the levelof a higher type. Because the system is deductively incomplete, there isno reason why a contradiction should arise. An impossibility would comeabout only when we try to define truth for the entire system in itself. Andin some cases one can even formulate such general truth definitions for aformal language in that language itself. Axiomatic set theory is apparentlya case in point. All that one needs to do there is to say that the truthpredicate applies to the Gödel number of a sentenceS if and only if theSkolem functions ofS exist. Hence Gödel’s incompleteness theorem is notjust a reformulation of his result about the undefinability of arithmeticaltruth in arithmetic itself. Moreover, it is far from clear that in higher-ordersystems deductive incompleteness implies descriptive incompleteness.

Subsequent philosophers of mathematics have often tended to dismisshigher-order logics from their considerations in favor of axiomatic settheory. For them, second-order logic is in Quine’s words, “set theory insheep’s clothing”. It is therefore of interest to see that self-applied truth-definitions have proved to be possible already on the first-order level (Hin-tikka 1996, ch. 5). All we have to do is to allow quantifiers to interactwith each other free from the needless restrictions that in ordinary first-order logic are imposed on patterns of dependence and independence be-tween quantifiers. The result is an extended basic logic which I have calledindependence-friendly (IF) logic. (See Hintikka 1996, chs. 3–4.) Such truth-definitions are eminently compatible with the deductive incompleteness oftheories formulated by means of IF logic. This logic turns out to dispensewith the law of excluded middle and to be semantically incomplete.

Furthermore, IF first-order logic enables us to pinpoint the reason whytruth is not definable in an ordinary first-order arithmetical language foritself. Such a truth-definition would have to be expressed by means of

ON GÖDEL’S PHILOSOPHICAL ASSUMPTIONS 19

Gödel numbering as a predicate of Gödel numbers of sentences. Now theuse of the technique of Gödel numbering means that we are speaking ofnumbers and quantifying over them in two different ways: sometimes asbona fidenumbers and sometimes as codifications of the expressions ofthe language in question. Such a schizophrenic use of numbers is possiblewithout contradictions. However, when we have two quantifiers used inthe two different ways they must of course be informationally independentfrom each other. In ordinary first-order logic there is no way of implement-ing such independence. But in IF first-order logic, whose very rationaleis to allow such independencies, even when they are excluded from or-dinary first-order logic, there is no difficulty in principle of formulatingself-applied truth-definitions.

Needless to say, these reasons for the failure of truth-definitions (for thesame language) by means of ordinary first-order logic have nothing to dowith the reasons of the deductive incompleteness of elementary arithmetic.In the light of hindsight, it thus seems unfortunate that Gödel sought toassimilate his incompleteness result to the impossibility of formulatingtruth predicates for the language of elementary arithmetic in that languageitself, even if it is true that historically speaking he reached the formerresult via the latter. The question of truth-definitions belongs to the sameorbit as descriptive completeness. These questions did not come close toGödel’s one-world interests, which were firmly focused on the question ofdeductive completeness.

There is another philosophical issue on which Gödel’s views were cru-cially shaped by his one-world thinking. For him, as was seen, logicaland mathematical objects are as genuine denizens of the actual world asphysical objects, the only difference being that they belong to a supersensory part thereof. If so, the next question inevitably is: How can wecome to know them? We can perceive physical objects and thereby cometo know them, but we cannot perceive abstract entities like numbers. Gödelsometimes simply asserts that we have the same justification in postulatinglogical and mathematical objects as a physical scientist has in postulatingphysical objects bodies. But such postulation is not enough. We have tocome to know the properties and interrelations of the objects we are dealingwith. Gödel’s response to such epistemological questions is a kind of lo-gician’s counterpart to naïve realism. He postulates a counterpart to sense-perception for his second world of abstract entities. This is his famous (ornotorious) mathematical intuition. Its nature and status has presented itselfto philosophers as a problem. In the light of what has been found in thispaper, it can be seen that this problem is self-inflicted or, rather, inflicted onus by Gödel’s one-world assumption. If we do not make it, we do not need

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the postulation of such an “extra-sensory perception” – for that is whatGödel’s intuition ultimately is. Kripke (1980, 19, 44) has tried to ridiculepossible-worlds theorists by asking what kind of telescope it is by meansof which they can establish what there is in other possible worlds. This isa bad joke, for unrealized possible scenarios by definition do not exist tobe observed by a telescope any more than by a naked intuition. They areprecisely the possibilities that were not, or will not be, realized. The realbutt of Kripke’s joke is not a possible-world theorist but a philosopher wholike Gödel believes in an actually existing super sensory region of the realworld. It is not Montague or David Lewis who is the real target of Kripke’spointed question, but Gödel.

Gödel’s own comparison between mathematicians on the one hand andphysical scientists on the other shows the weaknesses of his notion of in-tuition as a source of logical and mathematical truths. Gödel describes hisposition as follows (1944, 137):

Classes and concepts may, however, also be conceived as real objects namely classes as“pluralities of things” or as structures consisting of a plurality of things and concepts asproperties and relations of things existing independently of our definitions and construc-tions.It seems to me that the assumption of such objects is quite as legitimate as the assumptionof physical bodies and there is therefore quite as much reason to believe in their existence.

But scientists do not just postulate physical objects. They manipulate phys-ical objects, construct them and experiment with them. The question of theexistence of physical objects does not arise in science or even in the phi-losophy of science. A physicist does not merely observe phenomena andthen try to devise a generalization to capture them. A physicist can take anactive role in the course of events. He or she can create the kinds of systems(as a physicist would call them) that will have to be among the models (inlogicians’ sense of the term) of a sought-for theory or of an already knowntheory – if it is to be true. Likewise, a many-world logician (alias modeltheorist) can construct isomorphic replicas of unrealized possibilities asparts of the actual world, either as parts of its more concrete or its moreabstract regions. But this presupposes a variety of possible models for thelanguage of the theory not just one fixed structure in the Platonic part ofthe actual universe. Thus Gödel’s one-world outlook forces him to defendhis idea of intuition by means of an oversimplified, static conception of themethodology of physical sciences. His notion of mathematical intuitionis a product of this oversimplified conception. Mathematical intuition issupposed to be an analogue to perception. However, the interesting ques-tion here does not concern mathematical counterparts to perception, but

ON GÖDEL’S PHILOSOPHICAL ASSUMPTIONS 21

counterparts to scientific experimentation and generalization in logic andmathematics.

It follows from the job description of Gödelian intuition as giving usaccess to the abstract Platonic region that it cannot be restricted to directcognitive relation to particular objects. This distinguishes Gödel’s conceptof intuition, as he notes himself (Wang, 1997, 217–218), both Kant’s andHilbert’s notion of intuition. Likewise, Gödel differs sharply from Kantas to the reasons why intuitions can give usa priori knowledge. Kantianintuitions can yield such knowledge because their introduction merely re-produces the operations through which we impose the forms of space andtime on objects and through which we have individuated those objects. Wecan in other words intuitively anticipate the applicability of those formsto experience because we have ourselves projected them to objects. Forthis reason a Kantian intuition can yield knowledge even “without anyobject being present, either previously or now, to which it could refer”(Kant 1783, sec. 8). According to Kant, the use ofa priori intuitions is notlike perceiving an object; it is like introducing a representation for someparticular unknown object in anticipation of any perceptual knowledgethereof. This is all in sharp contrast to Gödel for whom intuition couldaccess mind-independent reality. The very point of those aspects of objectsof which Kantian intuitions could enlighten us was that they were for Kantmind-dependent.

One is tempted to suggest that Gödel’s own practice belied his own the-ory of mathematical knowledge as based on mathematical intuition. WhatI have in mind is his work in axiomatic set theory. According to his officialview, what we are doing there is to try to capture the set of sentences truein the universe of set theory as logical consequences of the axioms of settheory. Of course any actual axiomatization of set theory is incomplete.Hence what we must try to do according to Gödel is to hone our intuitionsand hope that they yield new, stronger axioms. Gödel made some inter-esting suggestions as to what such novel axioms might intuitively be like.Among other ideas he surmised that suitable maximality assumptions notunlike Hilbert’s axiom of completeness in geometry (see Gödel 1986, vol.2, pp. 167–168) might be what is needed in the foundations of set theory.It seems to me that Gödel’s intuitions can be sharpened further here andthat the real challenge in set theory is to find an equilibrium of maximalityand minimality assumptions: the set of natural numbers must be minimalat the same time as there must be as many sets as possible (cf. Hintikka1993).

Be these intuitions as they may, Gödel was not content in his own workto stare at the axioms of set theory and at their known consequences and

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wait for his intuition to tell him what further axioms are available in theway one is tempted to imagine Quine as having done when he proposedhis stratified axiom system for set theory. In his consistency proof for thecontinuum hypothesis, he literally experimented with suitable models ofset theory in constructing the constructible model of set theory and bystudying its properties. The constructible universe is not just an ad hocmodel of certain axioms, such as Hilbert might have used to prove theindependence of certain axioms of geometry of each other. At least ini-tially, the famous axiom of constructibilityV = L was thought of byGödel as a serious candidate for the job of a new axiom of set theory.This kind of model-theoretical experimentation is not an integral part ofGödel’s professed intuitionistic methodology. It is much more in the spiritof a model-theoretical many-world conception.

The intuitionistic methodology seems in any case exceedingly difficultto follow in axiomatic set theory. For one thing it has turned out that thereare in any model of axiomatic set theory sentences that are intuitively truebut actually false (see Hintikka forthcoming). Moreover, the intuitions sup-porting their truth are essentially the same ones as the intuitions supportingthe axiom of choice. More explicitly expressed, in any model of first-orderaxiomatic set theory there are true sentences whose Skolem functions donot all exist. Since these Skolem functions are the functions that yield astheir values the “witness individuals” that intuitively speaking vouchsafethe truth of the sentence in question, it is questionable whether current first-order axiom systems of set theory can be interpreted as truths about anygenuine set-theoretical universe, in the way Gödel wanted to view them.

Moreover, the explicit or tacit appeals to intuitions that we find in thefoundations of mathematics typically turn out on a closer examination toinvolve unspoken discursive assumptions. A test case is offered by the intu-itions of the intuitionists followers of Brouwer. I have argued (1996, ch. 11)that they are best understood as pertaining to our mathematical knowledgerather than mathematical truth. All told, Gödel’s emphasis on mathemat-ical intuition seems to me to be one of the least promising foundationalideas in current discussion.

REFERENCES

Dawson, J.: 1997,Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters,Wellesley, MA.

Gödel, K.: 1986–, in S. Feferman et al. (eds.),Collected Works, three volumes to date,Oxford University Press, New York.

Gödel, K.: 1964, ‘Revised and Expanded Version ofGödel 1947’, in Benacerraf andPutnam 1964, pp. 258–273. (Reprinted inGödel 1986–, Vol. II, pp. 254–270).

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Gödel, K.: 1958, ‘Über eine bisher noch nicht benützte Erweiterung des finiten Standpunk-tes’,Dialectica12, 280–7. (Reprinted inGödel 1986–, Vol. II, pp. 240–251).

Gödel, K.: 1947, ‘What is Cantor’s Continuum Problem?’,American MathematicalMonthly54, 515–25; errata55, 151. (Reprinted with corrections inGödel 1986–, Vol. II,pp. 176–87).

Gödel, K.: 1944, ‘Russell’s Mathematical Logic’, in P. A. Schilpp (ed.),The Philosophy ofBertrand Russell, Northwestern University Press, Evanston, IL, pp. 123–153. (Reprintedin Gödel 1986–, Vol. II, pp. 119–41).

Gödel, K.: 1940,The Consistency of the Axiom of Choice and of the Generalised Contin-uum Hypothesis with the Axioms of Set Theory, Princeton University Press, Princeton.(Reprinted inGödel 1986-, Vol. II, pp. 33–101).

Gödel, K.: 1931, ‘Über formal unentscheidbare Sätze derPrincipia Mathematicaund ver-wandter Systeme I”,Monatshefte für Mathematik und Physik38, 173–98. (Reprintedand translated inGödel 1986–, Vol. I, pp. 144–95).

Gödel, K.: 1930, ‘Die Vollständigkeit der Axiome des logischen Funktionenkalküls’,Monatshefte für Mathematik und Physik37, 349–60. (Reprinted and translated inGödel1986–, Vol. I, pp. 102–23).

Henkin, L.: 1950, ‘Completeness in the Theory of Types’,Journal of Symbolic Logic15,81–91.

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Society28, 148–224.Kripke, S.: 1980,Naming and Necessity, Harvard University Press, Cambridge, USA.Lewis, D.: 1986,On the Plurality of Worlds, Blackwell, Oxford, UK.Montague, R.: 1974,Formal Philosophy, in R. H. Thomason (ed.), Yale University Press,

New Haven, USA.Ramsey, F.: 1925, ‘The Foundations of Mathematics’,Proceedings of the London Math-

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Russell, Bertrand, 1919,Introduction to Mathematical Philosophy, Allen & Unwin,London.

Scott, D.: 1993 (original 1968), ‘A Game-Theoretical Interpretation of Logical Formulae’,Yearbook of Kurt Gödel Society 1991, pp. 47–48.

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Department of PhilosophyBoston UniversityBoston MAUSAhintikka@bu.edu