Found Sci (2013) 18:625–640DOI 10.1007/s10699-012-9294-7

The Unexpected Applicability of Paraconsistent Logic:A Chomskyan Route to Dialetheism

Nicholas D. McGinnis

Published online: 6 June 2012© Springer Science+Business Media B.V. 2012

Abstract Paraconsistent logics are characterized by rejection of ex falso quodlibet, theprinciple of explosion, which states that from a contradiction, anything can be derived.Strikingly these logics have found a wide range of application, despite the misgivings ofphilosophers as prominent as Lewis and Putnam. Such applications, I will argue, are of sig-nificant philosophical interest. They suggest ways to employ these logics in philosophicaland scientific theories. To this end I will sketch out a ‘naturalized semantic dialetheism’ fol-lowing Priest’s early suggestion that the principles governing human natural language maywell be inconsistent. There will be a significant deviation from Priest’s work, namely, theassumption of a broadly Chomskyan picture of semantics. This allows us to explain naturallanguage inconsistency tolerance without commitment to contentious views in formal logic.

Keywords Chomsky dialetheism inconsistency · Tolerance language paraconsistent priest

1 Introduction

1.1 Paraconsistency and Dialetheism

Paraconsistent logics are characterized by rejection of ex falso quodlibet, the principle ofexplosion: from a contradiction, anything can be derived (e.g. classically, via disjunctivesyllogism). There are various means of avoiding trivializing inconsistency. Relevant log-ics require that some appropriate connection exist between antecedent and consequent, orpremises and conclusion (Read 1988). Priest’s ‘Logic of Paradox’ is a three-valued logic,where the third value is interpreted as ‘both true and false’ (Priest 1987, 2001, 1991). DaCosta has developed logics of formal inconsistency which isolate contradictions in an effort topreserve as much of classical logic as possible (Da Costa 1974). Adaptive logics distinguish

N. D. McGinnis (B)Department of Philosophy, University of Western Ontario, Stevenson-Hall 2150,London N6A 5B8, Canadae-mail: nmcginni@uwo.ca

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between a full ‘upper-limit’ and restricted ‘lower-limit’ logic that retracts unwanted infer-ences (Batens 2000). There are many more.1

There are some purely philosophical motivations for such logics. We may want a “deonticlogic that does not trivialize conflicting obligations,” or again an “epistemic logic that doesnot trivialize inconsistent beliefs” (Brown 2002, p. 629). There is, of course, a cost: some veryintuitive inferences are blocked, or we may only regain the inferential strength of classicallogic at the price of non-monoticity (Priest 1991).

Dialetheism is a separate issue. A dialetheist asserts that there is one, or it is possible thatthere is one, sentence S such that S and the negation of S is true. To avoid triviality, the diale-theist must adopt a logic that rejects or limits explosion, i.e., a paraconsistent logic (thoughone need not be a dialetheist to see the value in paraconsistent logic). The clearest example ofa candidate ‘dialetheia’ is the liar paradox. Priest argues that the paradoxes of self-referenceare in fact sound arguments whose most natural interpretation is that they are both true andnot true (Priest 1987). Truth-value gap accounts—that such sentences are neither true norfalse—fall prey to ‘extended liar paradoxes.’2

1.2 Dialetheism ‘Naturalized’

In what follows I will argue that the myriad successful applications of paraconsistent logicsare suggestive of ways to integrate paraconsistency in scientific theories (and not merelyas descriptions of actual scientific reasoning). More specifically, logics where explosion iscurtailed appear to better capture natural language processing in the ‘Chomskyan’ sense.They provide an empirically adequate description of non-trivial (‘appropriate’) lexical itemselection and the internalist computations that enter into this process; they explain the creationand interpretation of inconsistent sentences; and they satisfy the optimality conditions of the‘minimalist’ program in linguistics. While at least part of the motivation for these claims isthe successful and widespread use of paraconsistent logics in analogical engineering contexts(discussed in Sect. 2), descriptive adequacy is also a major consideration. Moreover separat-ing out natural from formal languages, as per the Chomskyan view, allows us to explain agreat deal of phenomena without committing ourselves to contentious views in logic.

The Chomskyan picture rejects the thesis that a scientific account of natural languagesemantics can be ‘externalist,’ that is, given in terms of reference or truth-conditions. Instead,as Chomsky summarizes, “natural language consists of internalist computations, and per-formance systems that access them, along with much other information and belief” (Chom-sky 1995a, p. 27). ‘Naturalized’ dialetheism would characterize the computational systemas paraconsistent and the candidate inconsistent objects as ineliminable explanandum ofan abstract computational theory “that describes and relates internal (mental) states andprocesses” (McGilvray 1998, p. 230).

Is this really an argument for dialetheism proper? Two obvious problems present them-selves. First, inconsistent information is not typically seen as a compelling candidate fordialetheia. Second, and more worrisome, a non-truth-conditional semantics seems to leaveno room for the notion of a ‘true contradiction’ for truth and falsity is no longer at issue.

In response to the first problem, we can ask the following question: to the extent that weare committed to the existence of the objects, properties, laws, and so forth, that our besttheories describe, might we not be led from a successful inconsistent theory of natural lan-guage processing to the dialetheic position? We would require two separate presuppositions:

1 An accessible general introduction is Bremer (2005).2 “This sentence is either false or neither true nor false.” See Priest (1987, pp. 19–31).

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first, that ontological commitment is determined by methodological naturalism; and second,that our inconsistent theory is not merely a device of convenience, where the ‘true contradic-tions’ can be re-described in consistent terms but it is unwieldy to do so. I will argue that aninconsistent theory of natural language can plausibly satisfy the desiderata for ’metaphysicaldialetheism’:

There is (or at least there might be) no way of accurately and completely describing theworld without committing ourselves to contradictions. If there is some fact such thatboth it and its negative correlate obtain, then any consistent description of the worldwill miss describing at least one fact. (Mares 2004, p. 270).

What counts as a ‘fact’ is contentious, of course; here the purported ‘facts’ include mentalstates, events, and properties. I will again follow Chomsky’s lead: “unless offered some newnotion of body or material or physical, we have no concept of naturalism apart from method-ological naturalism” (Chomsky 1995a, p. 37). In other words, there is no principled reason toprivilege some ill-defined notion of the materially physical when defining a ‘fact’—facts arewhat appear in well-supported theories, even if abstract or higher-order. This ‘naturalized’argument for dialetheism is not meant as an alternative position to the ‘metaphysical’ view,but as a scientifically-minded (instead of philosophical) defense of it.3 And it seems clearthat facts about mental states can conflict with one another; all that is left to show is that suchconflicts arise during the regular operations of the language faculty.

The second objection can be answered by considering the important difference betweennatural language and scientific languages. While it is not the case that natural languageexpressions have truth-conditional semantics, the theoretical description of the processesthat produce expressions is not itself a product of natural language:

I-Languages, as instantiations of natural language, are not scientific symbol systems.Unlike natural languages, these … reach beyond innate endowment; learning a sci-entific language is notoriously late and laborious, and people differ greatly in theirabilities. Scientific systems are interesting, for unlike natural languages, they actuallycome quite close to allowing one to define the meanings of expressions in terms oftheory-internal truth-role or function (McGilvray 1998, p. 242).

Scientific languages have stipulated properties serving specific goals or ends. Among theseproperties are robust notions of reference, truth, and predication. As would be expected,they are difficult to learn and play little role in our day-to-day lives. By separating scien-tific from natural language we can deny the externalist, truth-conditional approach to naturallanguage semantics while still affirming that a naturalistic scientific theory of the computa-tional processing that underpins the faculty is inconsistency-tolerant and genuinely refers tocontradictory external entities. We’ll have occasion to return to this point in Sect. 3.

1.3 Canonical Versus Naturalized Dialetheism

Consider Priest’s claim, in In Contradiction, that

The inconsistency of our linguistic principles is the very thesis I am affirming … thenatural presupposition is that of inconsistency. For language and the principles thatgovern it have developed piecemeal and under no central direction. (Priest 1987, p. 5).

3 I am indebted to an anonymous reviewer for clarification on this point.

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What I propose is to place this thesis within the context of a broadly Chomskyan theory ofnatural language. Certainly the most provocative and controversial version of dialetheismaffirms the possibility of true contradictions ‘in nature’, of the kind described in Priest’sarticle Sylvan’s Box—direct perceptual acquaintance with an ‘impossible object’ such as abox that is both occupied and empty at the same time (Priest 1997). Inconsistent theories aresomehow less alluring than inconsistent things out there, even if we internalize the stricturethat we read off our ontology from science. After all, theories can be mere theories.

So-called ‘weak’ paraconsistency is the view that paraconsistent models are “merely math-ematical tools that prove to be useful but, in the end, not representative of real possibility.”(Beall 2004b, p. 6). What ‘real possibility’ means here can be spelled out in more detail. If theidea is that we require paraconsistent logics to model inconsistent theories, the weak paracon-sistentist nevertheless maintains that such inconsistent theories are artefacts of limited humancapacity; they do not claim that “the logic of completed science has to be a paraconsistentone” (Bremer 2005, p. 16). Weak paraconsistency is certainly compatible with the accountpresented here, though I will present some considerations against such an interpretation.

This is in contrast to the strong view, that there are provably, unavoidably true contradic-tions. Here Priest’s canonical version fares better at first glance. The argument proceeds fromsemantic considerations. A rough sketch can be given quickly. Any general account of lan-guage—that is, of any language—must serve as its own metalanguage (otherwise it wouldn’tbe perfectly general). This desiderata, semantic closure, delivers seemingly unavoidableantinomies such as the liar paradox. Together with Tarski’s universally accepted conven-tion-T we arrive at a conclusion just as puzzling as any ‘impossible box’: provably truecontradictions and no obvious guilty party to convict.4

Priest’s ‘canonical’ account differs from the present proposal in that it relies on truth-functional semantics, where the Chomskyan picture does not; a dialetheic theory of naturallanguage processing would not rely on Tarski’s convention to generate dialetheia, sincethe semantics proposed is not truth-functional. Rather, the dialetheia figure as ineliminableaspects of a theory of internalist computation requiring non-explosive treatment to be empir-ically adequate. For Priest, meanwhile, it is taken for granted that

At the heart of a theory of meaning for a language is a theory of truth … [which] spellsout in a systematic way the truth conditions for all the sentences of that language.(Priest 1987, pp. 71–72).

The dialetheic proposal about the meaning of antinomies such as the liar paradox then com-mits us to inconsistent truth-conditions. Priest tries to convince us that such conditions canobtain, and not only in semantics (though the semantic paradoxes remain the crucial wedge):“there are several areas where very natural considerations push us towards the conclusion thatsomething may be both true and false,” such as the truth-conditions of sentences involvingchange and conflicting norms (Priest 1987, p. 85).

The common response to the strong view is to insist that, somehow, the liar paradoxand other related antinomies are unsound; and that, in any event, acceptance of dialetheismhas worse consequences yet, consequences unacceptable even to the dialetheist. Perhapsexplosion cannot be avoided after all; or deleterious irrationality awaits.5

4 The convention understood in the usual fashion: [(T) ‘p’ is true (in L) if and only if p.]. Substitute ‘Thissentence is false’ for p into the convention. A basic outline of the argument from provability to truth is givenin Bremer (2005). Priest goes into far more detail in Priest (1987).5 The collection of papers in section V [‘For the LNC’] of Beall et al. (2004a) capture the important positionsin the literature. They will not be discussed in what follows.

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It is important to notice that Priest’s above-quoted claim from In Contradiction, ofthe inconsistency of our linguistic principles, stands in tension with the canonical argu-ment for strong paraconsistency. Priest’s observation that language has developed “piece-meal” and without “central direction” is likely correct as a matter of evolutionary fact,but this is an observation about natural language for humans, not about languages ingeneral.6 Formal languages simply do not develop piecemeal. I am inclined to read theproposal of natural language inconsistency as an essentially naturalistic proposal con-cerning the nature of the human language capacity, i.e., language considered as a naturalobject.7

The literature on dialetheism, while important in its own right, is largely concerned withformal languages, on the assumption that natural language is sufficiently akin to the former—a claim explicitly defended in Bremer’s introductory text (Bremer 2005, p. 25–26), andoften implicitly assumed elsewhere.8 This seems incorrect, for reasons that will be given inSect. 3.

So I wish to bracket the issue of dialetheism in relation to formal languages and explorethe possibility that natural human languages should be considered separately. To sepa-rate natural and formal languages we adopt a broadly Chomskyan understanding of nat-ural language processing—as strictly internalist and non-representationalist. This leavesopen the possibility of accepting dialetheism with regards to either natural or formal lan-guages; or with both; or with neither. The argument will be that if we find compellingreason to adopt a paraconsistent model of human natural language processing, there will beno reason to believe that any inconsistencies present will be an ideally eliminable part ofthat theory, whatever else we may conclude about formal languages. The mental phenom-ena will not be ‘redescribable’ in consistent terms because they aren’t taken to representexternal things, but mental events, states and processes that can be genuinely contradic-tory.

In the next section I will motivate paraconsistent logics as viable candidates for thekind of internalist computation Chomskyan linguists take language to be; then I will fur-ther outline and defend the Chomskyan view. In the concluding section I will outline somelimitations of the current proposal but note the relative unimportance of a priori scruplesin scientific methodology, a key difference between the investigation of natural and formalphenomena.

6 This may not be fair to Priest, who claims that formal language should be understood as a model of naturallanguage. But I am not arguing that the tension is internal to Priest’s view; it is rather about the rejectionof externalist truth-conditional semantics in mainstream linguistics on similar evolutionary and naturalisticgrounds, too often ignored to the detriment of our philosophical theories. I am again indebted to a carefulanonymous reviewer here.7 Which is to say that I am merely bringing in Priest’s suggestion to mainstream linguistics. The methodo-logical point should be uncontroversial: “A naturalistic approach to linguistic and mental aspects of the worldseeks to construct intelligible explanatory theories, taking as ‘real’ what we are led to posit in this quest,and hoping for eventual unification with the ‘core’ natural sciences: unification, not necessarily reduction”(Chomsky 1995a, p. 1). The substantive view will, of course, have to be defended.8 The following observation is entirely correct: “It may be the default view in the philosophy of language thatnatural languages are, at least in key semantic respects, rather like the formal languages invented by math-ematical logicians. (That the logical languages are invented, with the properties being explicitly stipulated,is meant to be an unimportant difference.) This is the first plank of the view to be rejected” (Stainton 2006,p. 916).

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2 The Case for Paraconsistency

2.1 Two Prominent Critics

The most persistent objection to dialetheism is that there can be no such inconsistent truth-conditions: it simply “violates our intuitions that there are no true contradictions [ …] there areno ineliminable contradictory facts,” metaphysically or semantically (Mares 2004, p. 271).Indeed some philosophers, such as Lewis and Putnam, have this intuition quite vociferously:

The reason we should reject this proposal is simple. No truth does have, and no truthcould have, a true negation. Nothing is, and nothing could be, literally both true andfalse. This we know for certain, and a priori, and without any exception for especiallyperplexing subject matters. The radical case for relevance should be dismissed justbecause the hypothesis it requires us to entertain is inconsistent (Lewis 1982, p. 434).

To clarify Lewis’ remarks here it should be noted that in a later letter he writes that heis becoming “increasingly convinced” that we can indeed “reason about impossible situa-tions,” indicating the remarks above are meant to apply equally to the thesis that there couldbe inconsistent facts and the related thesis that the principles of our reasoning might wellbe at times paraconsistent (Lewis 2004, p. 176). The latter is an important observation thatmust be accommodated by any adequate theory, as Priest notes: in interpreting the story ofSylvan’s Box, one does not “infer from the description of the box that it was … shot offinto space,” or infinitely many other possibilities. “Not everything happens in the story,” andthat’s exactly right (Priest 1997).

How is it that we are able to draw correct inferences, including the production of appropri-ate linguistic output, from plainly inconsistent data—particularly if we understand the mindas essentially a sophisticated modular computational system? The beginnings of an answercan be found in consideration of similar tasks performed by similarly modular computingarchitectures. Non-explosive logics have found an important place in software and com-puter engineering in precisely the place the dialetheist would expect: large-scale informationintegration across more or less autonomous modules requiring discriminatory treatment ofmultiple conflicting inputs to create ‘safe’, appropriate output. This is exactly the problemPriest singles out in Sylvan’s Box: how do we handle inconsistent information?

The practical uses of paraconsistency in this context belie Putnam’s comment on the apriori nature of the law of non-contradiction. In the same vein as Lewis, Putnam writes

But to convince me that it is possible to imagine the falsity of “For all statementsP,¬(P ∧ ¬P) is true”, you have to put an alternative logic in the field. … I am awarethat some people think such a logic—paraconsistent logic—has already been put inthe field. The lack of any convincing application of that logic makes them, at least atpresent, a mere formal system in my view (Putnam 2000, pp. 220, 230).

Lewis and Putnam offer similar complaints: both argue that true contradictions, howevercharacterized, are irreconcilable with our intuitive notion of truth. Yet Putnam, by char-acterizing paraconsistent logics as ‘mere’ formalisms, appears at least open to the idea thatfinding ‘convincing application’ might change his mind. These applications do exist and lendcredence to the suggestion that the principles of our language might well be paraconsistent.

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2.2 The Applicability of Paraconsistent Logic

Is Putnam right? Are paraconsistent logics empty formalisms without convincing application?No.

Fascinating practical uses for paraconsistent logics have been developed that reflect thelarge-scale integration of computing resources made possible by improvements in network-ing and multi-core processing, resulting in distributed ‘modular’ architectures with multipleinterface points. Commenting on the scale of the information-processing problems that result,MIT computer engineer Carl Hewitt notes that

Because of the scale and concurrency of the information systems involved, inconsis-tency is the norm … there is no practical way to reason about this information withoutusing paraconsistent logic because no one knows where most of the inconsistenciesare located. At a given time, only a small subset will have been identified and most ofthese will be unresolved (Hewitt 2008, p. 98).

Previously stand-alone components are coming together to form modular, non-hierarchicalorganizations that offer up heterogeneous and often conflicting information that requirescareful management. The inconsistencies that plague such large-scale information structuresare not trivially removable, either, for obvious reasons of computational tractability.

There may be a normative presumption that ideally issues of inconsistency ought to beresolved because there are, in fact, no true contradictions. Two considerations should be keptin mind here. First, inconsistency is, practically speaking, an ineliminable and even desir-able feature of sufficiently complex systems. It is expected that inconsistencies arise andwe accommodate it. Epistemologists have long known that consistency-verification for largesets of beliefs is a computationally intractable task. The practically-minded folk that populateengineering departments work with inconsistency as a given; so should we. Second, thereis every reason to think that when it comes to human minds, inconsistencies are not evenin principle eliminable. It is a simple fact of psychology that our internal mental states canconflict in fundamental ways, with each other and with themselves. Any theory of mind thatis committed to the existence of real mental events is ipso facto committed to inconsistency,as is any theory of language that concerns itself with internal (mental) events as well.

These points are not controversial outside philosophy; this is reflected in the matter-of-fact approach taken to practical problems of inconsistency processing. Here’s a representativequote from a collection of papers titled Inconsistency Tolerance:

All seem to agree that data of the form q ∧¬q cannot exist together, and that the conflictmust be resolved somehow. This view is too simplistic for developing robust softwareor intelligent systems, and furthermore fails to use the benefits of inconsistent informa-tion in intelligent activities, or to acknowledge the fact that living with inconsistencyseems to be unavoidable. Inconsistency in information is the norm in the real world,and so should be formalized and used, rather than always rejected … Inconsistency isuseful in directing reasoning and instigating the natural processes of argumentation,information seeking, multi-agent interaction, knowledge acquisition and refinement,adaptation, and learning (Bertossi et al. 2004, p. 2).

The authors add that

The central position is that the collapse of classical logic in cases of inconsistencyshould be circumvented. In other words, we need to suspend the principle of absurdity

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(ex falso quodlibet) for many kinds of reasoning. A number of useful proposals havebeen made in the field of paraconsistent logics (Bertossi et al. 2004, p. 6).

There are many situations where inconsistency tolerance is critical. Any context where infor-mation is incomplete, vague, or needs to be discarded for efficiency; where complex queriesto multiple unverified databases must be answered within reasonable response times; whererules, norms, demands or prioritization conflict; where tractability precludes consistency-checking; where consistency isn’t a desiderata. Database management has pioneered theparaconsistent approach (e.g. for spatial location databases (Rodriguez 2004), health-caseontologies (Imam et al. 2007), and logistical planning (Kassoff and Genesereth 2007), cur-rently being extended to ‘logical spreadsheets’ that output relevant information even whengiven inconsistent information.9)

Da Costa has helpfully summarized a great deal of the practical research done:

Annotated logics … have been developed and applied to fields like robot control, airtraffic control, control systems for autonomous machines, defeasible deontic reasoning,information systems and medicine … non-monotonic and defeasible forms of reasoninghave been approached by paraconsistent logics, leading to the development of softwaresthat are being used in traffic control; the hardware counterpart was presented as a chip.(Da Costa et al. 2004)

Artificial intelligence research has also produced a sizeable literature on the uses of para-consistency, far too voluminous to recount here. A representative example can be found inthe work of Jair Abe and colleagues, who have developed an autonomous robot (‘Emmy’)running on a paraconsistent logic controller “which can be applied to resolve conflicts andto deal with contradictions and/or paracompleteness, by implementing decision-making inthe presence of uncertainties” (Abe et al. 2006, p. 851).

These contexts, and applications, mirror natural language production, where appropri-ate linguistic output is the result of a ‘weakly’ modular but highly efficient computationalsystem. Inconsistency tolerance is an important feature fulfilling the information processingrequirements of computational systems where information can’t be discarded, consistencycannot be verified, and efficiency, understood in terms of response time and appropriatenessof output, is a central concern. We have every reason to believe that an internalist computa-tional system dedicated to language production would have the same requirements. It is tothis system we turn.

3 Naturalization and Language

3.1 Why Natural Language Semantics Can’t Be Externalist

Naturalized semantic dialetheism proposes to incorporate Priest’s suggestion, that “the prin-ciples of our language” are inconsistent, into mainstream Chomskyan linguistics. This has theconsiderable advantage of explaining our ability to handle ‘inconsistent’ sentences withoutbeing further committed to any particular views about formal logic: the semantic antino-mies are generated by an internalist computational engine with no regard to external-world

9 Currently being used by Stanford University for classroom assignments. The U.S. military has been fundingfurther research via DARPA, the ’Defense Advanced Research Projects Agency,’ to investigate the use ofinconsistency-tolerant logical spreadsheets for organizing e.g. ‘troop deployment and training’ (Young 2007).

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‘truth-conditions.’ Nevertheless, this engine, among its computational properties, must beinconsistency-tolerant.

Separating formal from natural language is controversial, as I’ve noted already. Philoso-phers have generally been committed to a truth-conditional, externalist semantics followingthe influential work of Davidson (1967), Putnam (1975) and Burge (1979). As we’ve seen,this means accounts of language must then be logically equipped to deal with the semanticantinomies. Yet this orthodoxy is at odds with the views of most linguists, who think that

A comprehensive science of language cannot (and should not try to) describe rela-tions of semantic reference … if there is to be a genuine science of linguistic meaning(yielding theoretical insight into underlying relativities, aiming for integration withother natural sciences), then a theory of meaning cannot involve assigning external,real-world, objects to names, nor sets of external objects to predicates, nor truth-values(or world-bound thoughts) to sentences. (Stainton 2006, p. 913).

This may strike philosophers as a radical view, but it follows directly from a commitmentto methodological naturalism—for externalism is scientifically intractable. This naturalisticstance is in contrast to an a priori ‘philosophical’ project such as ‘rational reconstruction.’The externalist project is beset with unaddressed problems. Inter alia Stainton (2006) notesthe difficulty of rigorously individuating ‘public language words’ meant to stand in theright word–world relation.10 He argues further that even if they could be individuated, com-mon-sense concepts are not amenable to systematization.11 Finally he argues that there isno evidence that the language faculty does assign truth-conditions to sentences and muchevidence that it doesn’t.

This last point is worth expanding. As Stainton notes,

On the one hand, there is no empirical reason for thinking that what the language fac-ulty assigns to a sentence would be capable of being true or false, even given contextualparameters like time, place, speaker, hearer, etc. (There’s lots of empirical reason forthinking that people can say, and think, things that are true or false; but that is anothermatter.) The only thing which drives one to this expectation is, at bottom, a dubiousanalogy between natural objects and artifacts whose properties are stipulated (e.g. thepredicate calculus). For the methodological naturalist, that in itself is damning. On theother hand, there is lots of empirical evidence that the language faculty alone doesn’tassign thoughts (or propositions, or truth conditions, or what have you). In particu-lar, very many sentences either lack truth conditions altogether, or are assigned truthconditions only via the rich interaction of different mental faculties (Stainton 2006,p. 930).

10 An argument echoed by Chomsky: see Chomsky (1995a, pp. 47–48). It should be noted Stainton does notnecessarily endorse all the arguments presented in the paper, which are summarized for exegetical purposes;moreover (Stainton 2011) explicitly defends the notion of ‘public languages.’11 See Chomsky’s discussion of the name ’London’: “We can regard London with or without regard to itspopulation: from one point of view, it is the same city if its people desert it; from another, we can say thatLondon came to have a harsher feel to it through the Thatcher years, a comment on how people act and live.Referring to London, we can be talking about a location, people who sometimes live there, the air above (butnot too high), buildings, institutions, etc., in various combinations. A single occurrence of the term can serveall these functions simultaneously, as when I say that London is so unhappy, ugly, and polluted that it shouldbe destroyed and rebuilt 100 miles away … no object in the world could have this collection of properties.”Quoted in Stainton (2006, p. 925).

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These ‘interaction’ effects would quickly swamp a theory of ’language’ and turn it into awhat McGilvray calls an intractable “theory of everything” (McGilvray 1998, p. 237). Toillustrate this Stainton produces a pair of sentences:

1. Poems are written by fools like me.2. Mountains are climbed by fools like me.

The difficulty is that the first sentence has intuitive truth conditions which implies that allpoems are written by fools, but the second sentence, in order to be true, does not require thatall mountains are climbed by fools. The truth-conditions differ for these otherwise syntacti-cally identical sentences on account of background knowledge and beliefs the linguisticallycompetent user has about ‘poems’ and ‘mountains.’ But this information has nothing to dowith the language faculty: to incorporate these would be to have a ‘theory of everything.’Thus, “not being solely an aspect of the language faculty, it follows that the truth conditionswhich people assign to sentences do not fall within the domain of the science of language”(Stainton 2006, p. 932).

These ‘interaction effects’ would rapidly overwhelm any serious scientific theory of lan-guage purporting to treat semantics truth-conditionally. Tractability would be lost in the het-erogeneity of things and predicates treated. One can re-introduce systematicity in a formalor scientific language, but this has nothing to do with the natural language faculty. Scientificterms and languages are austere, stipulative, difficult to master and often deeply-counterintu-itive, directly because they aim to “construct symbolic systems in which certain expressionsare intended to pick out things in the world” (Chomsky 1995a, p. 46). Nevertheless theseendeavours do not “inform us about ordinary language or common-sense understanding”(Chomsky 1995a, p. 46). The difficulties in learning scientific languages in fact provideevidence that natural language processing can’t be made responsible for ‘going outside thehead’.

This point ties into the observation that the first thing that needs to be explained in regardsto natural language is the so-called ‘poverty of the stimulus’. Humans acquire and use lan-guage in ways that far outstrip their exposure to it, indicating the presence of a significantinnate endowment. Evidence for poverty of the stimulus is convincing. A “typical property ofthe lexicon” is, for example, that a sentence such as ‘I painted my house brown’ is automati-cally taken to mean its outside was painted; the interior dimension of usage can be ‘marked’,but the default is to understand the exterior (Chomsky 1995b, p. 20). Similarly one is not‘near’ a house one is inside of, even though it is treated as an one-dimensional ‘exteriorsurface’ in the unmarked case (Chomsky 1995b, p. 20).12 All this is so quickly understoodby a child acquiring a language that she can only be said to be ‘learning’ it in the loosestsense; to explain these abilities it is necessary to assume “that knowledge of language [is] insubstantial measure innately determined, hence virtually uniform among languages” despitesurface dissimilarities (Chomsky 1995b, p. 20). An extreme case of the ‘poverty of stimulus’is found in creole languages:

That the complexity of a native speaker’s competence vastly exceeds the complex-ity of the linguistic environment is transparently shown by the emergence of creoles,which have all the properties of natural languages but take a drastically impoverishedlinguistic environment, a pidgin, for input (Hornstein et al. 2005, p. 4).

12 While the understanding that ‘in’ precludes ‘near’ is so intuitive as to almost seem trivial, geometricallyspeaking a point is just as ‘near’ a surface if it is inside or outside it. These are the kinds of anthropocen-tric features human languages are replete with, creating severe poverty of the stimulus issues—and makingnon-anthropocentric, objective (‘scientific’) languages deeply counter-intuitive.

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If a general-purpose learning algorithm without pre-established innate constraints and struc-tures was responsible for language acquisition, children exposed to ‘pidgins’ would notdevelop creole languages with properties not present in the input data (but which are found inother natural languages). These innate features make learning possible. A rather more formalcharacterization of the problem of language ‘learning’ is given in Gold (1967); the resultshows that the linguistic data available to children radically underdetermines the grammar inthe absence of pre-established constraints, for any finite set of linguistic exemplars is com-patible with an indefinite number of potential grammars. Thus the emergence of a ‘normal’language from impoverished input is evidence for the existence of a dedicated language fac-ulty that humans are both with, which has an genetically-determined initial state, matures,and stabilizes into a steady state, in a similar fashion to other human faculties (e.g. facialrecognition) and organs (Chomsky 1997, pp. 13–16).

Under the ‘principles and parameters’ approach, natural language acquisition is largely acombination of lexical acquisition and parameter setting (idealized as ‘switches’ with two ormore settings). The faculty begins in an initial state (the ‘universal grammar’) and, as a resultof linguistic exposure, settles into the individual’s ‘I-Language,’ the only form of ‘language’visible to science:13.

An I-language is a computational system that generates infinitely many internal expres-sions, each of which can be regarded as an array of instructions to the interface systems,sensorimotor (SM) and conceptual–intentional (CI) (Chomsky 2007, p. 5).

Universal grammar “provides a fixed system of principles and a finite array of finitely valuedparameters” so that a“selection � among these options determines a language” (Chomsky1995b, p. 170). The computational system of the language faculty then selects and arrangesitems from the lexicon, interacting with only two ‘external’ interfaces: an articulatory–per-ceptual system and a conceptual–intentional system (Chomsky 1995b, p. 2, 6). As Chomskynotes, the combination of fixed principles and parameter settings accounts for the rich surfacevariety of human language while acknowledging the deeper similarities between them. Thetotal space of available communication protocols given our physical capacities is much, muchlarger than the space of actually available natural languages; this observation is explained bythe hypothesis of linguistic nativism or universal grammar.

On this view, meanings are syntactically individuated, derive and recombinate from aninnate stock, are generatively infinite, and are strictly internalist. They reflect particularlyhuman interests and perspectives: meanings, Chomsky writes,

focus attention on selected aspects of the world as it is taken to be by other cogni-tive systems, and provide intricate and highly specialized perspectives from which toview them, crucially involving human interests and concerns even in the simplest cases(Quoted in McGilvray 1998, p. 256).

These ‘perspectives’ interface with phonetic production systems, of which some details areknown and understood; they are generally of little interest to philosophers. The conceptual–

13 As opposed to the social, political, geographical, etc. congeries called ‘English’, ‘Chinese’, ‘Greek’, andso on. Peter Ludlow contrasts I-Languages to the common view: “[I-languages] are data structures in a kind ofinternal computational system with which humans are born and which they have co-opted for communicationand other purposes … from the [typical philosophical] perspective, on the other hand, a natural language is akind of social object the structure of which is purported to be established by convention (however ‘convention’is to be understood) … on Chomsky’s view, such social objects do not exist and would be of little scientificinterest if they did” (Ludlow 1999, pp. 17).

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intentional systems running alongside are only beginning to be understood, and it is here thatinconsistency-tolerance becomes a plausible constraint on the faculty.

3.2 Inconsistency Tolerance as Optimality Constraint

Human language is then a combination of parameter settings and fixed principles. The ‘mini-malist program’ adds that investigation into the functioning of the language faculty should beguided by design optimality considerations; these optimality considerations form part of therationale for specific constraints and principles. Anything that can be dispensed with, is. Onesignificant fixed constraint is “efficient computation,” which is of “particular significance forgenerative systems” (Chomsky 2007, p. 3). In particular,

Optimality considerations … require comparison of computations to determine whethersome object is a valid linguistic expression. Unless sharp constraints are introduced,the complexity of such computations will explode, and it will be virtually impossibleto know what is an expression of the language (Chomsky 1997, p. 29).

The minimalist program’s basic stance is to ask how little can be attributed to universalgrammar (UG) while still retaining descriptive adequacy in regards to language acquisi-tion and use—in other words, to limit the complexity of UG by paring it down to thefewest operations (e.g. ‘Merge’) and relying, as far as possible, on general biological andphysical constraints not particular to the language faculty (Chomsky 2007, pp. 3, 4). Theability to generate ‘usable,’ that is, interpretable sentences, can be explained in terms ofa simple set of operations and properties (e.g., it seems clear that once interpreted lateroperations cannot return to modify that interpretation, as the ‘simplest’ solution would pre-dict). So it is established that the computational properties of natural language will beexceedingly well-suited to the task it performs, and not at all reflective of philosophicaldesiderata.

At this point the sense that a fundamental category mistake is being made might appear.If dialetheism means anything at all, it is in the context of an inferential system concernedwith truth-preservation. A ‘language faculty’ with a proprietary computational system andinternalist semantics cannot, in any relevant sense, be understood as a ‘logic’. The languagefaculty is not concerned with ‘truth-preservation’ and the generative computational proce-dures associated with it do not look anything like those of formal logic.

This criticism is not wrong, exactly. It does ignore the potential separation of ‘logic’from specific computational application. Early integrated circuits, after all, were composedexclusively of NOR and NAND gates, the only truth-functionally complete operators in thesentential calculus. Still, the computational operations permitted by such primitive circuitsare not always part of a truth-preserving inferential system: they may be used for any numberof purposes. One response would therefore be to characterize the fundamental or underlyinglogic of the language faculty’s computations to be ‘paraconsistent’, for example implementedas a ternary-valued logic running ‘underneath’ the system (perhaps at the neural level), sim-ilar to the paraconsistent logics implemented in various hardware applications; and that thiswould explain the ability of the language faculty to handle inconsistent input.

While this might function as an answer of sorts to the objection, it does not reallyrecover ‘dialetheism,’ which is, after all, a claim about interpretation. If the candidateinconsistent facts are to be mental events involved in the production of language theinconsistency-tolerance cannot be shoved down to the neural level (after all, presumablya neuron does not both fire and not fire). In any event, little is understood about this aspect ofthe human brain, and the suggestion remains mere speculation. There could be a wider chasm

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than expected between the neurobiological facts and the higher-level semantic phenomenaphilosophers wish to explain, and so the prevalent view, which is that formal language areuseful models for explaining natural language, cannot simply be assumed to be true.14

A more principled answer can be found in the idea of inconsistency-tolerance as a conse-quence of optimality conditions at the interface. As Chomsky notes,

The language faculty is part of the overall architecture of the mind/brain, interact-ing with other components: the sensorimotor apparatus and the systems that enterinto thought, imagination, and other mental processes, and their expression and inter-pretation. The language faculty interfaces with other components of the mind/brain(Chomsky 1997, p. 29).

It is at the interface that inconsistency tolerance is paramount. As Stainton has noted, whilenatural language semantics is decidedly not truth-functional, still “people can say and thinkthings that are true and false” (Stainton 2006, p. 930); not to mention things that are trueand false. At the least, we can imagine true contradictions, as Sylvan’s Box has shown. Our‘other mental processes’ also have no presumption of consistency built-in: our beliefs mightconflict; our desires might conflict; our very percepts might conflict; and our judgments abouttruth-conditions might take contradictory form. There is no limit to the potentially inconsis-tent information that could find its way to the interface points the language faculty has withthe components of the mind. Contradictory or conflicting input at the interface (incompatiblefeatures, affirmations and denials of a state of affairs, and so on) must be handled in somediscriminate way. The information need not be explicitly characterized as ‘truth-functional’(e.g. as a conscious judgement about truth-functional properties) in order to be inconsistentfrom the language faculty’s ‘point of view’. It is enough that, at the interface points, thefaculty receives conflicting input. For instance, a person could have conflicting desires aboutsome course of action, such that she both wants and does not want to undertake it, producingin response the appropriate and grammatically correct sentence ‘I want to and I don’t wantto at the same time’.

The ‘appropriateness’ of linguistic output is crucial. As McGilvray notes, “language useis stimulus-free, unbounded, and yet, in use, typically appropriate to any number of tasks”(McGilvray 1998, p. 234). While McGilvray sees this as evidence that intentional meaning-individuation is hopeless, the observation also has a clear positive consequence: whateverprocesses underpin ‘appropriateness’ are non-explosive. The language faculty, in interactingwith the many components of the mind/brain, always outputs appropriate linguistic forms, nomatter the situation. A person can observe an M.C. Escher drawing; contemplate the liar par-adox; feel torn about a job offer; imagine an impossible situation; and the appropriateness oflinguistic output is never threatened. All these situations involve components of the brain thelanguage faculty interacts with in order to produce relevant, appropriate output. Moreover,they are genuine inconsistencies. Taking mental events as real, any scientific description ofthe language faculty must make reference to inconsistent mental events; while natural lan-guage is not committed to external truth-conditions, scientific languages, being genuinelydescriptive, are.

14 To see why this might be the case, consider the following analogy: while the set of possible behaviours acomputer can exhibit is certainly fully explainable by, and in a sense reducible to, its logical architecture, thevariety of available behaviour at this higher level is vast enough that a hardware-based ‘theory’ (descriptions oflogic gates, electron flow, and so on) purporting to be about the software available at your local shop (games,word processors, and so on) would not only be deeply unsatisfying, but rather ridiculous. There is no properlydefinable set of ‘electron activity in logic-gates’ that corresponds to the perfectly intelligible activity ‘editinga document’.

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This is important when considering the optimality conditions that render humans able togenerate interpretable sentences. On the output side the meanings generated by the semanticinterface of the language faculty have been described as “a configuring instruction to con-ceptual–intentional systems,” individuated solely by “intrinsic, broadly syntactic features”(McGilvray 1998, p. 238, 239). These instructions are not “defined in terms of interpreta-tion”; instead, they “provide specific and essentially human structure and texture” to “cogni-tive domains” where they are taken up and used (McGilvray 1998, pp. 239–240). Chomsky,discussing this feature of meaning, notes in particular that all the ‘normal’ lexical features(such as exterior and interior marking) appear in the interpretation of inconsistent scenarios:

The same is true of impossible objects. If I tell you that I painted a spherical cubebrown, you take its exterior to be brown in the unmarked case, and if I am inside it,you know I am not near it (Chomsky 1995b, p. 20).

The natural language faculty, in other words, re-packages the inconsistent object—here, aspherical cube—into a interpretable sentence that has all the normal properties we’d expect.At first, this may seem unsurprising, as the sentence is in a very typical form. But the form(an action being performed to an object) doesn’t capture the marked/unmarked usage, whichis a semantic property of the lexical item. Further, even if one focuses on the impossibilityof the object (that it is both a cube and not a cube, if it is a spherical cube), painting itstill carries the presumption of its exterior surface being modified. Any number of questionsabout impossible objects can be appropriately answered—as Priest noted in Sylvan’s Box.The explanation for inconsistency-tolerance is, of course, very different in the present case.

The Chomskyan line also does explain how we understand paradoxical sentences withoutmaking any claim about which is the ‘correct’ logic. It certainly can’t be that we think weunderstand some sentence but realize, upon sober reflection, that it is in fact nonsense. Nat-ural language paradoxes, like the Barber’s, can’t be nonsense.15 It’s in virtue of the words,and the ways in which the words are arranged that we come to see that the situation describedis paradoxical. Internalist semantics allow us to make a further determination that the truth-conditions of a sentence are inconsistent. And when asked whether the story’s barber shaveshimself or not, whatever retrieval function is engaged responds in a discriminate fashion tothe query: the system does not ‘crash’. Instead a salient configuration is picked out, processed,and sent back to conceptual–intentional systems.

We can conclude that lexical item selection is clearly not trivialized by inconsistencyin either direction: in the input direction, since the language faculty, when interacting withother components of the mind, manages to choose relevant and appropriate lexical itemsto compute over; and in the output direction, since the ‘configuring instructions’ given tothe conceptual–intentional systems as the result of processing are non-explosive to the com-ponents they are outputed to. These are the places where inconsistency-tolerance matters;within the derivational computation it is indeed too quick to characterize the process as ‘para-consistent’ system (although there are interesting issues surrounding quantifier raising). Atthe interfaces, however, where interaction with other cognitive systems occurs, output, whilefree, is never arbitrary. In order for lexical retrieval to do this in an efficient manner—perhapsin an optimal manner, to maintain computational tractability—the underlying process mustbe discriminatory with regards to contradictions. That is, choice of which lexical items willgo into the derivation is paraconsistent.

15 A barber shaves only and all in town who do not shave themselves. Do he shave himself?

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4 Conclusion

It is clear by now that paraconsistent logic, as a means of reasoning about inconsistentinformation, forms a maturing research program with demonstrated practical application inprecisely the area the naturalistically-inclined dialetheist would expect: in modularized, dis-tributed databases where consistency can’t be assumed or verified and where sophisticatedcomputations must be rendered in an efficient manner. Hypothesizing that the mind—itselfa modular, naturally evolved computational system—would require some measure of incon-sistency tolerance is not only analogically obvious, but necessary for reasons of descriptiveadequacy. We’d expect the brain/mind to be capable of both the production and handling ofinconsistent information. ‘Dialetheia’, or something like them, figure in a theory of inconsis-tent natural language processing to the extent that they appear as an explanandum. All this ismeant as a series of empirical claims, to be confirmed or refuted. Metaphysical or ‘intuitive’a priori considerations simply do not apply, any more than Kantian intuitions by themselvescould disconfirm general relativity.

The great virtue of this proposal is that it explains natural-language inconsistency toler-ance while leaving open questions of formal logic. Even in the absence of a truth-conditionalsemantics, inconsistency-tolerance would be required. Any serious scientific theory of natu-ral language must take into account the fact that modularity of mind will lead to ineliminableinformational inconsistencies. What is far less obvious, and a harder problem for futureresearch, is what exact form this inconsistency-tolerance takes.

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Author Biography

Nicholas D. McGinnis (1979) is a Ph.D. student in philosophy at the University of Western Ontario,Canada. He completed his M.A. and B.A. (Honours) in philosophy, at Concordia University, Montreal, witha thesis on the influence of Wittgenstein on the modern revival of virtue ethics. His research interests includeexperimental philosophy, metaphysics, semantics and political philosophy. The provisional title of his doc-toral dissertation is ‘Reference and Experiment.’ He has also published on Merleau-Ponty (Phenomenology,Interrogation and Biopower: Merleau-Ponty on ‘Human Resources Exploitation’, Review Journal of Politi-cal Philosophy 8.1:125–137, 2011) and presented his work at conferences in Canada, the United States, andEurope.

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