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How to make sense of the modal logic of analytic truth from a linguistic naturalist perspective.


The core tenet of the Chomskyan philosophy of language and linguistics is that language is a part of the natural world. According to this perspective, human language is a result of a language faculty in the human mind, ultimately the human brain. The representation and implementation of language, on this perspective, thus cannot be logically separated from language itself. This language faculty contains an internal system that generates phonetic, formal and semantic representations. These representations, together with the generative system itself, compose an internal language. This resultant internal language is called an "I-language," and I-languages are taken to be the foundation of all human language. The theory of a particular language is its grammar, while that which all languages share at their biological core is called "Universal Grammar" or "UG," "a theory of the initial state S0 of the relevant component of the language faculty" (The Minimalist Program, 167).

While Chomskyan linguists often talk (some think inconsistently) about common sense public languages such as French, Wolof, Swahili, etc., according to their paradigm, analysis of these languages is not the ultimate goal of their science. The ultimate goal of their science is to understand the internalized languages of individuals and the core biological principles that all of these internalized languages ultimately share. When Chomskyans talk about internalized languages and the biological principles they follow, ultimately, they really are talking about the languages particular to individuals.

Chomsky understands language as a system of internal representations in human minds, ultimately in human brains. The brain, according to Chomsky, has a modular component used specifically for language. The study of this human language faculty can be understood as the study of the initial state of this modular component of the mind. It is the language faculty in this initial state that interacts with the input to language learning, (whatever exactly that is), so as to produce words, phrases and sentences. The language faculty takes the input to language learning and from it generates what we would more commonly call a person's language. The faculty is also called a "generative grammar," from which comes the name of the discipline of generative linguistics.

There are (at least) two senses of the term "language" that emerge from this picture that must be sharply distinguished. Chomsky uses the term "language" to refer to the generative grammar itself, but it is also, of course, often used to describe the output of the generative grammar. Recognizing these distinct senses of "language" allows us to see that Chomsky's claims that, for example, "Peter's language 'generates' the expressions of his language" are not circular. It is the "language" in the sense of the grammar that generates the "language" in the sense of a set of sentences (New Horizons..., 5).

Further systematic ambiguity can be found in Chomsky's use of the term "grammar." While, on the one hand, Chomsky considers one's grammar to be the state of the language faculty that combines with language learning input to produce grammatical expressions, on the other, "grammar" is also used as a term for the scientific description of the language faculty put forward by the theoretical linguist. This ambiguity is complicated by Chomsky's use of person-level mental terms such as "knowledge" and "theory" to describe the mental representation of the language faculty itself. He writes, for instance, that one's generative grammar is one's "theory of his language." Chomsky exegesis aside, this person-level terminology I think should be read as metaphorical. The grammar is simply the system of representations that combine with language learning input to determine complexes of instructions for articulation, perception and the systematic organization of thought viz. the performance systems (New Horizons..., 5).

It is important to note from this that the outputs of the generative grammar are also psychological representations. These complexes of instructions for articulation, perception and thought are linguistic expressions of a given person's psychological idiolect. It should be noted that, taking common sense into consideration, there are thus actually (at least) three relevant senses of the term "language." There is the distinction between the initial state of the language faculty and its output when combined with the input to language learning. There is also, however, the distinction between the conception of the output of a generative grammar as mental representations of performance system instruction complexes, and the common-sense conception of language as something abstractly public, rather than psychological at all. The conclusion that this common-sense public conception of language is not a relevant object of study for scientific linguistics is a core philosophical consequence of the Chomskyan Paradigm.

According to the Chomskyan view, the foundation of language is psychological in the sense that it is rooted in the minds, and ultimately the brains, of particular humans. Contrary (perhaps) to common sense, languages are not abstract objects that are shared in any literal sense by linguistic communities. This does not mean that talking about public languages as things that are shared among communities is nonsense. But it does mean that the appropriateness of such talk is merely practical. When it comes to scientifically understanding the core principles of natural language, "English," "Arabic," "German," etc., are not primary, according to Chomskyans, and, indeed, are actually rather metaphysically suspect. This (perhaps) anti-common-sense perspective on language, though standard in linguistics and typical to scientific inquiry in general, has seemed suspect to philosophers of language such as Lewis, Soames and Katz.

One of the primary alternatives to the Chomskyan paradigm is that of Linguistic Platonism. The clearest description of Linguistic Platonism comes from Katz. He writes:
   Grammars are theories of the structure of sentences, conceived of
   as abstract objects in the way that Platonists in the philosophy of
   mathematics conceive of numbers. Sentences, on this view, are not
   taken to be located here or there in physical space like sound
   waves or deposits of ink, and they are not taken to occur either at
   one time or another or in one subjectivity or another in the manner
   of mental events and states. Rather, sentences are taken to be
   abstract and objective ("Platonist Grammar," 173-4).

Here Katz fleshes out the metaphysics of the view that languages are abstract objects. It means that they do not exist in space or time. It means that they are not particular to the minds or brains of individuals. As it happens, I think there is significant evidence from linguistics that gives strong reason to reject this Platonist viewpoint in favor of a biospychological alternative. In this paper, however, I limit myself to a negative response to Katz's primary argument for Linguistic Platonism.

The Necessity Argument for Linguistic Platonism

According to Linguistic Platonism, languages and their grammars are abstract objects. This philosophy is based on an analogy with Mathematical Platonism in the philosophy of mathematics, according to which mathematical objects are equivalently abstract. Languages are taken to be abstract in the sense that they do not have spatial locations, temporal locations or a subjective basis in the mental states of individuals. Languages are considered to be logically independent of all physical and psychological reality. More than that, as with mathematics, they are immaterial and objective. Furthermore, according to Katz, it follows from this that "[languages] are entities whose structure we discover by intuition and reason, not by perception and induction" ("Platonist Grammar," 173-4).

For Katz, linguistics should be conceived as a rational science analogous to mathematics. Both his definition of Linguistic Platonism and his argument for it are rooted in an analogy between linguistics, on the one hand, and logic and mathematics, on the other. Theorists, he notes, do not conflate logical theories of implication with the actual psychology of people making inferences. Nor do they conflate mathematical theories of numbers with the actual psychology of people making arithmetical calculations. Katz sees this as inconsistent with the "exactly parallel case of linguistics, related to which 'conceptualist' theorists regularly identify the psychology of language with the theory of language itself' ("Platonist Grammar," 193).

Katz's argument against what he calls "conceptualism" is directed mainly at Chomsky and his followers, but it can apply as an objection to any philosophy of language according to which natural languages are contingent objects ultimately logically grounded in the spatiotemporal world. Any philosophy of language according to which natural languages are contingent objects ultimately logically grounded in the spatiotemporal (1) world I will here refer to as "Linguistic Naturalism:" the biopsychological theory of natural languages. Linguistic Naturalism is the antithesis of Linguistic Platonism and vice versa.

Katz's argument is fleshed out in relation to the necessity of analytic truth. As Katz sees it, there is a strong analogy between the necessity of analytic truths in natural languages and the necessity of mathematical truths. It is partly because of the necessity of analytic truths that Katz thinks Linguistic Naturalism is an insufficient philosophy with which to fully understand natural languages. In identifying grammatical principles as logically dependent on their psychological representation and biological implementation, he argues, Naturalists cannot understand any of the consequences of such principles as necessary truths. Since, for the Naturalist, the existence of the grammatical principles themselves is logically contingent, any semantic consequences of these principles, Katz argues, must be contingent as well. This, he continues, leaves analytically true sentences, not as sentences that cannot possibly be false, but merely as sentences that we cannot possibly believe to be false. Katz thinks this conclusion to be unacceptable. "What human beings are psychologically or biologically forced to conceive to be true no matter what," he writes, "is a far cry from what is true no matter what" ("Platonist Grammar," 199-200).

By its very nature, Katz thinks, Linguistic Naturalism makes genuine necessary truth in natural language into a logical impossibility. On a Naturalist conception, the closest thing to necessary truth that can arise, he argues, is believability relative to the biological limits of the human mind. Thus, as he sees it, analytic truth is reduced to merely what humans are under compulsion to think is true, entirely independent of what actually is true. "The grammatical principles underlying analytic truth," he continues, "only express laws of human mental or neural processes, and hence, can only determine what human beings have, on the basis of the nature of their linguistic capacities, to conceive of as true no matter what... [but] nothing can follow about what is true no matter what" (Language..., 5-6).

Since Katz considers logical analytical structure to contain "the most important facts" about natural language, he thus thinks Linguistic Naturalism inadequate as a philosophical framework for the foundations of linguistics. How humans are genetically programmed to represent language, he sees as clearly logically irrelevant to anything that could be necessarily true, in the strong sense he cares about. "Real necessary truth," he writes, cannot be "relativized to what is humanly conceivable, even as genetically determined" because what is humanly conceivable and what is genetically determined are themselves all matters of entirely contingent fact (Language..., 94).

To sum it up, Katz believes (1) that analytic truths are logically, not just nomologically, necessary, (2) that (some) analytic truths are logically determined by natural languages, and (3) that something necessary in this sense cannot be logically determined by something concrete and contingent. Thus he thinks natural languages themselves cannot be concrete and contingent, which amounts to them being abstract and necessary, as his Platonism defines them.

One could object to Premise 1, arguing, in Quinean fashion, that the analytic-synthetic distinction is illusory. (2) One could also object to Premise 2, and argue that analytic truths are logically independent of any natural language. (3) My strategy in this paper, however, will be to reject Premise 3.

As Kripke has written, "It would be wrong to identify the language people would have, given that a certain situation obtained, with the language that we use to describe how circumstances would have been in that situation" ("Vacuous Names...," 57). This insight of Kripke's, combined with his insights on a posteriori necessity (Kripke 1980) will be used below to explain how the necessity of analytic truths can be genuine, while also essentially related to the contingent existence of natural languages.

On the Language We Have for the Worlds We Don't

To understand the fallacy in Katz's necessity argument it is important to understand some of the insights on related matters put forward by Kripke. In various places, Kripke addresses the apparent philosophical dilemmas that arise from negative existential statements considered in modal logical "possible world" contexts. The statement "Moses does not exist" is true of many possible worlds in which Moses does not (and never did) exist. Yet in such worlds, no one would be able to make the statement "Moses does not exist" to the same effect because the term "Moses" would not have the same content. (4,5) The paradox of this situation, however, is merely apparent. As Kripke writes in "Vacuous Names and Fictional Entities:"
   Since we can refer to Moses, we can describe a counterfactual
   situation in which Moses wouldn't have existed. It matters not at
   all that in that situation people would not have been able to say,
   'Moses does not exist,' at least using 'Moses' the way we are using
   it here. Indeed, I can describe a counterfactual situation in which
   I would not have existed, even though if that were the case I
   wouldn't be around to say it. It would be wrong to identify the
   language people would have, given that a certain situation
   obtained, with the language that we use to describe how
   circumstances would have been in that situation (57).

In other words, whatever possible world we are describing, we are describing it from the actual world containing our actual language. Since, in the actual world, the man Moses existed and thus the name "Moses" has the content that it has, the content of "Moses" is extendible to situations in which Moses did not exist. What matters for cross-world semantics is not what people in other possible worlds can say, it is what we can say about those possible worlds.

Kripke makes the same point in Reference and Existence:
   One should not identify what people would have been able to say in
   hypothetical circumstances, if they had obtained, or what they
   would have said had the circumstances obtained, with what we can
   say of these circumstances, perhaps knowing that they don't obtain.
   And it is the latter which is the case here. We do have the name
   'Moses,' and it is part of our language whether it would have been
   part of our language in other circumstances or not (30).

The point being, once again, that the content of a term as it applies in other possible worlds is determined in the actual world. We always speak with our own language, even when we are speaking about possible scenarios in which we would be speaking with a different language. It is true that for inhabitants of a world in which Moses never existed, the statement "Moses does not exist" could not arise in the same way. This does not imply, however, that we should describe this world as one in which the statement "Moses does not exist" is neither true nor false. After all, we are describing it as a world in which Moses does not exist to even set up the example!

The analogy between Kripke's insight regarding negative existential statements considered in modal contexts and the matter of the equivalent extension of analytic statements considered in modal contexts is fairly straightforward. In considering the extensions of the contents of the names of our language to other possible scenarios we ignore the language of the inhabitants of these possible scenarios. In precisely parallel fashion we must ignore equivalent linguistic variation in considering the extensions of analytic truths in cross-world contexts.

Let us assume that the stock analytic truth "bachelors are unmarried men" is indeed analytically true. The particular example is, after all, irrelevant. There are clearly possible scenarios in which the sounds uttered in "bachelors," "unmarried" and "men" would have related to different content than they in fact have. (6) There are even more possible scenarios in which these terms, and their contents never would have existed. But this should not lead us into the profound confusion that such possible scenarios, due simply to their linguistic variation from actual reality, would be worlds in which it is false, or neither true nor false that all bachelors are unmarried men. As with "Moses," we speak of the extension of "bachelors" and "unmarried men" to other worlds from our world with our language even when we are speaking about their extensions in other worlds with other languages.

Another way of framing this point, again following from Kripkean insight (Kripke 1980), is in relation to an understanding of a posteriori necessity. The meaning of "bachelor," in many English idiolects, is identical to the meaning of "unmarried man." Likewise, to presume another stock example, water, the substance, is identical to H2O. Further, in so far as water and H2O are identical, they are necessarily identical. If it is true in the actual world, as we believe, that water is H2O, then it is necessarily true. Yet surely it is inappropriate to move from the perfectly reasonable proposition that water is necessarily H2O to the bizarre proposition that water is a necessary entity that must exist in every possible world! Rather, the necessity of the identity of water and H2O is simply taken to imply that water is H2O in every possible scenario in which it exists.

In this respect, identities between the semantic contents of different terms are logically analogous to other identities. To say that all bachelors are unmarried men is to say that in every world in which the content of "bachelors" (in appropriate English idiolects) exists, it is identical to the content of "unmarried men" (in appropriate English idiolects). There are countless possible scenarios in which Moses, water, and the content of "bachelor" (in appropriate English idiolects) do not exist. This is perfectly consistent with the contingency of Moses, the necessary identity of water with H2O, and the analyticity of "all bachelors are unmarried men." To think otherwise is to confuse the language we have with the languages of the worlds we are describing.

Relative Logical Modality

An upshot of these considerations is that, contrary to Katz's apparent assumption, not all types of logical necessity are true in all possible worlds. Just because analytic truths in natural languages are relative to certain biopsychological facts does not imply that the necessity of these truths reduces to mere contingent fact. Relative logical necessity is a perfectly coherent notion.

Priest offers the following example:
   Given how things are now, it is possible for me to be in New York
   in a week's time, 26 January. Given how things will be in six days
   and twenty-three hours, it will no longer be possible. (I am
   writing in Brisbane.) Or, even if one countenances the possibility
   of some futuristic and exceptionally fast form of travel, assuming
   that I do not leave Brisbane in the next eight days, it will then
   be impossible for me to be in New York on 26 January. Hence,
   certain states of affairs are possible relative to some situations
   (worlds), but not others (Non-classical, 21).

When we ask what is possible, in many contexts, we are asking what is possible given certain actual facts. The question "is it possible that Priest will be in New York on January 26th?" can be asked in different senses. It can be asked with the implicit assumption of knowledge that he is in Brisbane on January 25th, or with any other implicit knowledge assumption that happens to be relevant. In logic class the question could be posed more abstractly. Even then, however, implicit assumptions regarding the existence of Priest and New York and the sensible application of the notions of days and months of the year would be unavoidable. Could Priest be in New York on January 26th in a world where the Earth had an entirely different orbital relation to the Sun? One could likely create an artificial language to deal with this question, but in natural language such worlds are simply not within its cross-world extension of consideration.

Some will object that though relative logical necessity exists, "metaphysical" necessity cannot be relative logical necessity, but must be absolute logical necessity. This objection basically comes down to stomping one's foot and defining the type of identity in question so that is has to be the way one says it is. All I can say to this objection is that the notion of "metaphysical" necessity so defined has no obvious philosophical value. As I will show, analyticity in natural languages can be logically defined using entirely clear logical systems that already exist. Further, this can be done without relying on the notions of conceivability or psychological law as logically relevant. Requiring the relevant logical necessity be "metaphysical" where "metaphysical" is defined as absolute, does not contribute to clarity or precision or to explanation in general. It amounts to banging on the table and demanding that one logical system be used for explanation rather than another. If analyticity in natural language can be given a clear logical explanation without relying on conceivability or psychological law, then those who still demand a separate type of "metaphysical necessity," admittedly, cannot be answered, but I see this as unproblematic.

The relativity of the modality of analyticity in natural language is

not exactly analogous to Priest's above example. As I will go on to show, there is a plurality of types of relative logical modalities.

A given modal logic can be semantically represented by a triple <W, R, a>, in which W is a non-empty set, interpreted as a set of possible scenarios or worlds, and R is an accessibility relation defined among a, interpreted as the actual world, and the other members of W. (7) R is binary in the sense that, for any u and v in W, either uRv or not-uRv. This is to say, that either v is accessible from u or it is not. "Intuitively," Priest writes, "R is a relation of relative possibility, so that uRv means that, relative to u, situation v is possible" (Non-classical, 21).

If statement A is true at some world accessible from a world w, we can say that A is possibly the case from w. Likewise, if statement A is true at every world accessible from a world w, we can say that A is necessarily the case from w (Non-classical, 22).

Speaking from the actual world, there is no accessible world in which it is not true that all bachelors are unmarried men. Thus the sentence "all bachelors are unmarried men" (in appropriate English idiolects) is a necessary truth. As I will explain, this necessary logical truth holds about all accessible possible worlds despite the fact that there are some accessible possible worlds from which no statement of it could arise to be either true or false. Understanding this in detail requires an exposition of different modal logics and their philosophical roles.

Modal Logics

Different modal logics are determined by different definitions of the accessibility relation R among possible worlds. There are three primary logical properties by which R is most commonly defined for modal logics. R is reflexive in so far as, for all x in W, xRx. In other words, R is reflexive if every world is accessible from itself. R is symmetrical in so far as, for all x and y in W, (if xRy then yRx). In other words, R is symmetrical if, if one world is accessible from another, then the other is accessible from it. Lastly, R is transitive in so far as, for all x, y and z in W, (if (xRy and yRz) then xRz). In other words, R is transitive if, from every world, every world that is accessible from an accessible world is accessible. The modal logic K[rho][sigma][tau] is defined as that in which R is all of reflexive, represented by "[rho]" (rho), symmetrical, represented by "[sigma]" (sigma), and transitive, represented by "[tau]" (tau). The definitions for the modal logics K[rho], K[sigma], K[tau], K[rho][sigma], K[rho][tau], etc. are relevantly analogous (Non-classical, 38-9).

As it turns out the logic K[rho][sigma][tau] is equivalent to the logic K[upsilon], for which R is defined by "[upsilon]" (upsilon) as simply universal. In other words, in K[upsilon], by definition, every world is accessible from every other world. And, since it is provable (8) that the set of theorems in K[rho][sigma][tau] is identical to the set of theorems in K[upsilon], it follows that the accessibility relation in K[rho][sigma][tau] is also universal. Both of these logics are thus referred to indifferently as "S5" (Non-classical, 478).

As Priest writes of the question, "which modal logic is correct?" there is "no single answer... since there are many different notions of necessity (and, correlatively, possibility and impossibility)... logical, metaphysical, physical, epistemic, [...] and moral" (Non-classical, 48-9).

Formally fleshed out, Katz's implicit assumption in the necessity argument can be understood as the assumption that S5 is the correct modal logic for necessities owing to natural language. S5 may be the best modal logic to apply for many logical purposes, but the Kripke-inspired considerations put forward in this chapter should lead us to reject it in the analysis of natural linguistic analytic truth.

In considering the semantics of analytic truth in natural language, there is no reason to consider the actual world, or any other possible world to be inaccessible from itself. Thus, we can straightforwardly assume that the correct logic for natural language semantics of analytic truth will have a reflexive accessibility relation. That is: for all x in W, xRx.

Likewise, if a given analytic truth A applies from our world a to a possible world x and applies from x to a possible world y, we have no reason to deny that, from our world a, A applies in y. Thus, we can also straightforwardly assume that the correct accessibility relation will be transitive. That is: for all x, y and z in W, (if (xRy and yRz) then xRz).

Formally speaking, the logical property that the accessibility relation on the necessity of analytic truths in natural language seems it should not have is symmetry. From our world a in which the sentence S "bachelors are unmarried men" exists, S is true in a given world b in which there exist no sentences. But it is false that from b, considered as actual, S is true in a, considered as non-actual; from b, S does not arise to be true or false anywhere. That is: it is not the case that (for all x and y in W, (if xRy then yRx)).

If we define possible worlds for the science of linguistics relative to their sets of natural linguistic sentences, (9) we can say that a world y is accessible from a world x precisely when y contains no sentences not contained in x. That is: xRy [iff.sub.df] (y (sentence-wise) [subset or equal to] x). (10) y is accessible from x when y is a (not necessarily proper) sentence-wise-subset of x. For the purpose of natural language semantics, worlds containing sentences that do not actually exist may be ignored just as worlds without Priest or New York or appropriate Earth-Sun relations are ignored in the analysis of "is it possible that Priest be in New York on January 26th." These worlds for us are linguistically inaccessible because the analysis of their non-actual sentences does not arise without the arising of the sentences themselves. Empirically, we cannot access them. The containment definition provided allows the relation of logical accessibility to mirror the relation of empirical accessibility.

Defining possible worlds for linguistics relative to sentence sets actually offers a much more straightforward relative modal-logical analysis for natural-language semantics than is obvious in the Priest-in-New-York example. By this understanding of W for natural-language semantics, the formal properties of R fall out appropriately:

R is reflexive in that no world contains sentences it does not itself contain. That is: (for all x in W, xRx) [iff.sub.df] (for no x: not-(x(sentence-wise) [subset or equal to] x)).

R is transitive in that if a world z contains no sentences that a world y does not contain, while y contains no sentences that a world x does not contain, it follows that z contains no sentences that x does not contain. That is: (for all x, y and z in W, (if (xRy and yRz) then xRz)) [iff.sub.df] (If (z(sentence-wise) [subset or equal to] y & y(sentence-wise) [subset or equal to] x), then z(sentence-wise) [subset or equal to] x).

Lastly R is not symmetrical in that a world y may not contain sentences not contained in a world x while x does contain sentences not contained in y. That is: not-(for all x and y in W, if xRy then yRx) [iff.sub.df] not-(if (y(sentencewise) [subset or equal to] x), then (x(sentence-wise) [subset or equal to] y).

On this understanding the modal logic for natural language semantics should be K[rho][tau], or one similar. Importantly, my fleshing out the K[rho][tau] accessibility relations in no way depends on conceivability or other nomological notions. They are fleshed out using only the concept of a sentence and purely logical inclusion relations, neither of which the Platonist has any grounds with which to take issue. As noted the logical inclusion relations mirror empirical relations, but this is irrelevant to their logical character and function in the logical description of semantics.

Where such a logical system is applied to natural languages, all actual analytic sentences will come out as necessarily true. Even when considering sentences as if other possible worlds were actual, our relevant definitions on <W, R, a> will make every analytic sentence in that world considerable as actual also come out as necessarily true from that world. The only difference from Katz' analysis, is that, following Kripke, sentences that do not exist in a given world will not be considerable as actual from that world, just as nonexistent sentences are not considerable as actual from the actual world.

As I will show in the coming section, this conclusion is consistent with a proposal that the best analogy between philosophy of language and philosophy of mathematics relates the foundations of linguistics not to Mathematical Platonism, as Katz defends, but rather to Mathematical Intuitionism.

Possible Worlds and Linguistic Intuitionism

Platonism is not the only metaphysical position in the foundations of mathematics. Contrary to Platonists, Intuitionists argue that mathematical truth is a mind-dependent phenomenon. To be clear, this chapter does not seek to take any philosophical position whatsoever in the philosophy of mathematics. Since Katz draws a parallel with the philosophy of mathematics in the definition and defense of his Linguistic Platonism, I am simply relating my own counterargument to the context of his framework.

As Priest writes, for the Mathematical Intuitionist, "the meaning of a [mathematical] sentence is to be given, not by the conditions under which it is true [in the classical sense], where truth is conceived as a relationship with some external reality, but by the conditions under which it is proved, its proof conditions--where a proof is a (mental) construction of a certain kind" (Non-classical, 100). This philosophical position is fleshed out in relation to a unique logic, intuitionist logic, defined in relation to mental constructions of mathematical proofs.

In classical logic: "A [conjunction] B (11)" is defined as true when "A" is true and "B" is true; "AvB" is defined as true when "A" is true or "B" is true; "~A" is defined as true when "A" is not true; and "A [right arrow] B" is defined as true when, if A is true then B is true.

Similarly but importantly distinct, in intuitionist logic: "AAB" is defined as proven when A is proven and B is proven; "AvB" is defined as proven when A is proven or B is proven; "[logical not]A" is defined as proven when there exists a proof that there is no proof of A; and "A [right arrow] B" is defined as proven when there exists a construction that, provided any proof of A, may be applied to provide a proof of B (Non-classical, 100).

As it turns out, intuitionist logic may be semantically analyzed as a modal logic. Intuitionist logic is determined by a definition of the standard modal logical triple <W, R, a>, wherein R is reflexive and transitive, but not symmetrical (analogously to the modal logic Kpr), together with a heredity condition according to which, for every possible world in the set of possible worlds, if a simple statement (and, as it turns out, thus any statement (12)) P is true at a world w, and a world w' is accessible from w, then P is also true at w' (Non-classical, 101). That is: for all P in all w & w', (If ((P in w) & wRw'), then P in w').

Priest offers the following elucidation:
   think of a world as a state of information at a certain time;
   intuitively, the things that hold at it are those things which are
   proved at this time. uRu is thought of as meaning that u is a
   possible extension of u, obtained by finding some number (possibly
   zero) of further proofs. Given this understanding, R is clearly
   reflexive and transitive. (For t: any extension of an extension is
   an extension.) And the heredity condition is also intuitively
   correct. If something is proved, it stays proved, whatever else we
   prove" (Non-classical, 102).

On a modal interpretation, the heredity condition states that all proven statements are necessarily proven. The relevant sense of necessity is reflexive in that a statement is proven in every scenario in which it is proven. It is transitive in that, from the scenario in which it is proven, it is proven in every scenario accessible from an accessible scenario in which it is proven. The relevant sense of necessity is not symmetrical, however, in that a statement can perfectly well be proven in one scenario, and thus, by the necessity of proof, extend from that scenario to all accessible scenarios, while it would not be proven were certain of those alternative scenarios in fact the case and thus, from those certain alternative scenarios, it would be neither true nor false of the actual scenario, considered as non-actual, that it is proven.

Regardless of whether the Mathematical Intuitionist is correct or pragmatic to apply intuitionist logic to the foundation of mathematics, it appears this logic has a useful application in the analysis of natural linguistic analytic truth. Addition of the heredity condition to the modal logic K[rho][tau] seems to define the logical difference between the general realm of natural linguistic truths in general and the specific realm of analytic truths.

The sole modal semantic difference between K[rho][tau]and intuitionist logic, as we have seen, is the heredity condition. And the modal semantic interpretation of the heredity condition, as we have also seen, amounts to the relevant notion of the necessity of proofs. Since analytic sentences, like mathematical sentences, and unlike other natural language sentences considered from a purely linguistic perspective, are necessary, the application of the heredity condition to K[rho][tau] seems thus to individuate the natural language semantics of analytic truth from natural language semantics in general.

For the natural language semantics of analytic truth, we may apply intuitionist logic by saying that the sentences that hold in a scenario are those sentences that are true purely in virtue of their mental representation, the sentences that hold regardless of mind-independent fact.

This move may seem to submit to the Platonist the need for a role to be played by conceivability or otherwise merely nomological necessity, but it does not. Granting that analytic truth is grounded in psychological representation does not commit one to saying that the type of necessity that determines it is psychological. Semantic contents under different descriptions are logically identical, when identical at all, in the same way as water and [H.sub.2]O. The identified objects we are talking about happen to be psychological, but the relations of identity we are applying to them are not. There is no way of changing the psychological or other natural laws that would make a given mental representation non-self-identical. Since analytic truths consist in the identity of mental representations under different descriptions, though in this sense they are psychological in their substance, they are not psychologically determined. To say a sentence is true purely in virtue of mental representation is thus an appropriately logical, rather than nomological statement.

The relevant accessibility relations for analytic truth are as follows:

R is reflexive in that every sentence true purely in virtue of mental representation in a possible world is true purely in virtue of mental representation in that possible world. That is: if (MRT)S in w, then (MRT)S in w. (13)

R is transitive in that, from a given world w, there is no possible world w" not containing more sentences than a possible world w' not containing more sentences than w in which something true purely in virtue of mental representation in w is not true. That is: if ((MRT)S in w & (w"(sentence-wise) [subset or equal to] w' & w'(sentence-wise) [subset or equal to] w)), then (MRT)S in w".

R is not symmetric in that, from a given world w, purely in virtue of mental representation in w, a sentence S may be true in a world w' not containing sentences not contained in w, while S does not exist in w', and thus, from w', is neither true nor false in w (or anywhere else). That is: not-if (if ((MRT)S in w, then (if (w'(sentence-wise) [subset or equal to] w), then (MRT)S in w')), then (MRT)S in w'.

Lastly, the heredity condition holds in that once something has been made true purely in virtue of mental representation it remains true purely in virtue of mental representation whatever else may become true purely in virtue of mental representation. (14,15) That is: If (((MRT)P in w) & wRw'), then (MRT)P in w'.

As Heyting wrote of Intuitionism in the foundations of mathematics:
   we do not attribute an existence independent of our thought, i.e.,
   a transcendental existence, to the integers or to any other
   mathematical objects. Even though it might be true that every
   thought refers to an object conceived to exist independently of it,
   we can nevertheless let this remain an open question. In any event,
   such an object need not be completely independent of human thought.
   Even if they should be independent of individual acts of thought.
   Their existence is guaranteed only insofar as they can be
   determined by thought. They have properties only insofar as these
   can be discerned in them by thought. But this possibility of
   knowledge is revealed to us only by the act of knowing itself.
   Faith in transcendental existence, unsupported by concepts, must be
   rejected as a means of mathematics proof ("Intuitionist," 42).

As Heyting sees the mind-dependence of mathematical facts, the Linguistic Naturalist sees the mind-dependence of linguistic facts. If one is to evaluate the philosophical foundation of linguistics by drawing an analogy with the philosophical foundation for mathematics, one must conclude that it is Mathematical Intuitionism, rather than Mathematical Platonism, as Katz wrongly proposes, with which the philosopher of language should identify.

Again, some will object that the constructed logical necessity of mathematical intuitionism is not "deep" enough to count as "metaphysical" necessity. Again however, I believe this objection to be empty. The necessity of analytic sentences in natural language can be explained, not only by natural scientific means, but also by logical means. While analytic truths do have their substance in psychological representation, the necessity that determines them is not itself psychological, or otherwise nomological. It is the logical relation of self-identity that is determined in the same way for mental objects as for any other ontological variety. No further explanation, I hold, can be reasonably demanded by the Platonist.

How to cite: Life, Jonathan J. (2016), "How to Make Sense of the Modal Logic of Analytic Truth from a Linguistic Naturalist Perspective," Analysis and Metaphysics 15: 36-54.

Received 24 December 2015 * Received in revised form 31 January 2016 Accepted 5 February 2016 * Available online 20 April 2016


(1.) Or, for those who think it distinct from the physical, the psychological world.

(2.) The well-known position of (Quine, 1951).

(3.) Something like this approach is considered by Soames in "The Necessity Argument" (Soames, 1991). I would accept the solution entirely, except for the fact it seems to require commitment to senses as abstract objects, a view with which I am uncomfortable.

(4.) Some other person may have the name "Moses," but applied to this person, the utterance "Moses does not exist" would not have the same content.

(5.) In Jackendoff's internalist semantics the sense in which "Moses" must have different content in a world where he does not exist can be fleshed out in respect to the cognitive valuations "external" and "imaginary" (Jackendoff, 318). In our world the semantic content of "Moses" has the valuation external, while in a world where he never existed it must have the valuation imaginary, leading to differences in the semantic nature of the statement it can be used to make.

(6.) It is obvious that the sounds /ba-cha-lur/ and /mehn/ could have different content. It's not obvious that the terms could, if terms are partly individuated on the basis of what they mean. (One sometimes says not that there is one word "bark" but rather that several words are pronounced /bahrk/. Thinking about it this way, no word 'bark' can mean something different than it does.)

(7.) Interpreted as the other possible worlds.

(8.) For proof see Non-classical, 53-4.

(9.) Or semantically truth-evaluable sub-sentential forms, if such there are.

(10.) Where "sentence-wise" inclusion relates the sentences of the worlds rather than all of their objects, since it is only the linguistic objects with which we are concerned in the particular science of linguistics

(11.) Particular letters here may be taken as sentence variables.

(12.) "The proof is by induction on the construction of formulas. Suppose that the result holds for A and B. We show that it holds for [logical not] A, A [conjunction] B, AvB, and A [right arrow] B. For [logical not]: we prove the counterpositive. Suppose that wRw', and [logical not]A is false at w'. Then for some w" such that w'Rw", A is true at w". But then wRw", by transitivity. Hence, [logical not]A is false at w. For AAB: suppose that A [conjunction] B is true at w, and that wRw'. Then A and B are true at w. By induction hypothesis, A and B are true at w'. Hence, A [conjunction] B is true at w'. For AvB: the argument is similar. For A [right arrow] B: we again prove the counterpositive. Suppose that wRw' and A [right arrow] B is false at w'. then for some w" such that w'Rw", A is true and B is false at w". But by transitivity, wRw". Hence A [right arrow] B is false at w" (Non-classical, 102).

(13.) Where "MRT" notates truth in virtue of mental representation is the appropriately logical sense described.

(14.) Unlike synthetic truths, which may become false with changes in the world.

(15.) This is relative to idiolects considered synchronically rather than diachronically.


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JONATHAN J. LIFE Ph.D., The University of Western Ontario
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Date:Jan 1, 2016
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