It is shown that:(a) classicality is connected with various criteria some of which are fulfilled by TIL while some other are not;(b) some more general characteristic of classicality connects it with philosophical realism whereas (radical) anti-realism is connected with non-classical logics;(c) TIL is highly expressive due to its hyperintensionality, which makes it possible to handle procedures as objects sui generis.Thus TIL is classical in obeying principles of realism and non-classical in transcending some principles taught by textbooks of classical logic

Materna, Pavel

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Logic and Logical PhilosophyVolume 23 (2014), 47–55DOI: 10.12775/LLP.2013.032Pavel MaternaIS TRANSPARENT INTENSIONAL LOGICA NON-CLASSICAL LOGIC?Abstract. It is shown that:(a) classicality is connected with various criteria some of which are fulfilledby TIL while some other are not;(b) some more general characteristic of classicality connects it with philo-sophical realism whereas (radical) anti-realism is connected with non-classical logics;(c) TIL is highly expressive due to its hyperintensionality, which makes itpossible to handle procedures as objects sui generis.Thus TIL is classical in obeying principles of realism and non-classical intranscending some principles taught by textbooks of classical logic.Keywords: TIL, bivalence, classicality, construction, expressivity, exten-sions, intensions, hyperintensionality, procedure, realism, anti-realism1. CriteriaPrinciple of bivalence. The classical logic evidently obeys this principlebut only one of its formulations:• The formulation A:Every sentence/proposition is either true or false.• The formulation B:There are just two truth-values: T (true), F (false).Evidently A implies B but not vice versa.Received May 25, 2012. Published online Sepember 27, 2013© 2013 by Nicolaus Copernicus University48 Pavel MaternaFor T(ransparent) I(ntensional) L(ogic) B holds: TIL is not a many-valued logic. A cannot hold because there are not only total functionsin TIL. Sentences that express (a) improper constructions or (b) denotepropositions whose Strawsonian1 presuppositions do not hold cannot ac-cept any truth-value (which does not mean that they accept a ‘thirdvalue’).Examples. Ad (a) The sentence The greatest prime is odd expresses aconstruction that does not construct anything: it is an improper con-struction.Ad (b) The sentence Charles knows that the Moon is greater thanthe Earth denotes a proposition one of whose presuppositions is not true(viz. that what is known is true). It cannot be true but it cannot befalse either.Thus TIL accepts truth-gaps (but does not accept truth-gluts).Extensionalism. Classical logic(s?) is extensionalist in that it obeys prin-ciples of extensionality, in particular the principle of compositionality. Inthis sense we can say that“Transparent Intensional Logic flouts none of the principles of exten-sional logic and is, insofar, an extensional logic.”Duží et al. (2010, p. 2)The term “extensionalism” is however connected with a more radicalversion: extensionalist in this second sense are such logical systems whosenotions are not sufficient to tell intensions from extensions, in otherwords: the specific character of intensions cannot be evaluated. TIL hasshown that one can define intensions and, at the same time, continueobeying principles of extensionality. Being not able to deal with logicalproblems induced by specific character of intensions means limitation ofexpressivity, leads to a form of reductionism, which treats properties asclasses, propositions as truth-values, magnitudes as numbers etc.TIL is classical in that it obeys principles of extensionality, it is notclassical due to treating intensions as not reducible to extensions.First-order paradigm. Formal systems connected with classical logic arefirst-order. They can be sound and complete. Only propositional (truth-functional) logic is however decidable.1 See Strawson (1950).Is Transparent Intensional Logic a non-classical logic? 49Observe: with increasing expressivity some “good property” is lost:first-order predicate logic is more expressive than propositional logic butis no more decidable. Second-order predicate logic is more expressivethan first-order predicate logic but is no more complete. Note: we cannotprefer decidability: otherwise we would be happy with propositional logiconly.TIL is not based on first-order logic of relations. It is a functionallogic using (interpreted) typed λ-calculus instead. Its first stage basedon using procedures (‘constructions’) works with first-order hierarchyof types, which is however more expressive than first-order predicatelogic. (Classes of any order can be treated within this first-order TIL.)The second stage based on mentioning constructions becomes a hyper-intensional logic proceeding top-down and distinguishing three levels ofcontext: hyperintensional (procedures are mentioned), intensional (pro-cedures are used to construct functions) and extensional (procedures areused to construct values of functions applied to arguments). The degreeof expressivity essentially exceeds that one of first-order systems. Whatis lost is just the relative simplicity (including computational simplicity)of those systems.Applied to logical analysis of natural language TIL defines a ‘ne-ofregean’ semantics, where Frege’s Sinn (we will use meaning ratherthan sense) is explicated as a construction, his Bedeutung (denotation)as the result (if any) of the meaning, i.e. as what the construction con-structs. The term reference of the expression E is used as the name ofthe contingent value of the intension that is the denotation of E (when Eis an empirical expression) in the actual world-time2. A desirable featureof the resulting semantics is that the meaning of an expression is totallyindependent of any context.TIL abandons the 1st order paradigm, being in this sense not a clas-sical logic.Total functions only. Using just total functions might be a feature ofclassical logic. This feature essentially limits a potential expressivity oflogic.3 Not only classical logic avoids treating partiality (intuitionists2 To adduce an example: while the denotation of an empirical sentence is aproposition, its (contingent) reference is its truth-value in the actual world. (Andthe meaning of the sentence is the procedure that constructs that proposition.)3 See Duží (2003) for some convincing arguments.50 Pavel Maternado not like it as well). Definability of partial functions is however abrute fact.4 TIL presupposes partial functions and defines the way ofpropagating partiality of a constituent of a complex construction C overC. Thus the construction (concept) that is expressed by the term thegreatest prime is improper, i.e., does not construct anything: the classof the (natural) numbers that are the greatest prime is empty, so thesingularizer (“the only x such that”) is not defined. In the constructionexpressed by the sentence The greatest prime is odd the oddity cannotbe predicated because its constituent is improper, so that the wholesentence expresses an improper construction and it does not denote anytruth-value.If being classical presupposes dealing just with total functions, thenTIL is not a classical logic.Excluded middle. This Principle of Excluded Middle (PEM) is charac-teristic of classical logic especially as in contrast to intuitionism. Donot forget however that intuitionists have criticized PEM having rede-fined connectives so that the intuitionist sentences have not denotedpropositions in classical sense. Otherwise we could be a little upset, asKolmogorov has stated in his (1932):“So entsteht diese ganz besondere Art von Aussagen, welche zwar einenmit der Zeit nicht veränderlichen Inhalt haben sollen und doch nurunter speziellen Bedingungen ausgesprochen werden können.”Kolmogorov (1932, p. 64)Kolmogorov’s interpretation is well-known: Heyting’s formalization canbe considered to be a logic of problems.Thus it seems that since classical logic deals (due to “classical in-terpretation”) with propositions rather than with constructions PEMshould be classically a principle whose tautological counterpart would beA ∨ ¬A (1)where A is any sentence that is true or false.Yet when we want to really comprehend PEM as an old principlevery important for distinguishing classical vs. non-classical exposition oflogic then we have to adequately interpret the scheme (1).4 Strawson’s presuppositions are also a brute linguistic fact.Is Transparent Intensional Logic a non-classical logic? 51Actually, we can distinguish two interpretations. According to Int 1,A is just a variable whose range is {T, F}. Then there can be no doubtthat our disjunction is a tautology so that PEM holds trivially.Consider however Int 2, according to which A is just a scheme of any(declarative) sentence. Then  see above Principle of bivalence  we cansee that on this interpretation PEM does not hold as soon as we admitsentences that are neither true nor false. Then , of course, PEM is notuniversally valid.Thus Int 2 shows a way how to escape the “jail of classicality”. Weknow two kinds of this escape:• admitting more than two truth-values (“Łukasiewicz way”) or• admitting partial functions (“Boczwar way”).As we have already stated, TIL is a Boczwar way, i.e., no third, fourthetc. truth-value is admitted but truth-gaps arise due to partiality.If PEM is interpreted in the Int 2 sense then TIL is not a classicallogic.2. Classicality as a vague positive ideaP. Banks (allegedly J. M. Bocheński) in 1950 develops (as a fictive di-alogue of a conservative, a modern and an “Aristotelian” logician) anidea of logic which is based on Aristotelism, which is simply a scientificapproach starting with common sense, believing in reason and experienceand accepting the possibility of an unlimited progress based on tradition.This idea of compatibility of progress with tradition can be perceived(vaguely, of course) as the idea that the only logic which can be called‘logic’ is just what is called classical logic. Indeed, ‘classical’ means‘based on tradition’, and we surely cannot refuse the possibility of devel-oping (classical) logic. Further, Banks believes that the genuine logic isjust a collection of metalogical rules and that such a collection containsonly evident, Aristotelian rules. He states that one cannot find anysuch logical system whose metalogical rules would contain non-evident,non-aristotelian rules not obeying, e.g., the principle of forbidden con-tradiction.In my opinion, this is a strong argument against the trendy spirit ofrelativism that misuses (or exploits?) the fact that there are unbelievablymany particular logical systems. Banks-Bocheński admits that thesesystems may contain interesting results, he only wants to distinguish52 Pavel Maternabetween Logic and these systems. The latter can be interesting anduseful but they remind us of something like counter. Language of thegenuine logic can be found, e.g., in Principia Mathematica.Clearly, some ‘non-classical logics’ could not be representatives ofLogic in Bocheński’s sense, for example many-valued or intuitionist log-ics, surely paraconsistent logics. But to say that e.g. Łukasiewicz didnot accept PEM is not convincing because neither he would admit thatthere could be a third possibility when saying that a proposition canhave or not have the value T.It seems that the choice of “the adequate logic” is dependent on somephilosophical standpoint, or  if you like  on some “metaphysical” as-sumptions. Thus our choice of that “degree of classicality” which charac-terizes TIL should be founded on the choice of some such “metaphysics”.But as Dummett in (1991) rightly stresses,“We must not try to resolve the metaphysical questions first, and thenconstruct a meaning-theory in the light of the answers.”Dummett (1991, p. 338)Thus what is primary here and will eventually “determine the answersto the metaphysical ones” (ibidem) is the choice of “a Correct Meaning-Theory” (p. 339).Two extreme positions determine two incompatible ways of choosingcriteria for deciding which theory should be recognized as a correct mean-ing theory: One of them is called (a radical) anti-realism well-known justfrom Dummett’s works, while the other one could be any form of realism.As for classicality or a “degree of classicality” radical anti-realism un-ambiguously demands a non-classical logic.5 We have shown, however,that being realistic (logic) is not a very determinate criterion. In thebroadest sense the minimum of classicality means that the main thesisof anti-realism  i.e. the claim that e.g. the principle of bivalence is notjustified and that the meaning of mathematical / logical terms attachedto them by classical logic(s) is incoherent  is rejected.Concerning just the incompatibility of classicality vs. anti-realism weshould not acquiesce in stating that there are more logics: we should find“some neutral manner of formulating the rival conceptions of meaningso as to be able to argue their merits without prejudging the issue infavor of one or the other” (p. 17).5 See Dummett (1991, p. 340)Is Transparent Intensional Logic a non-classical logic? 53This problem of finding “some neutral manner” can be formulated inanother way, as the problem of adequate explication.3. TIL epistemic framework as a most adequate explication.HyperintensionalityThe necessity of first explicating notions used in a claim or a theorybefore judging whether the claim / the theory is at least rational oreven true has been recognized by Carnap (see his 19506). Carnap hasmentioned some general principles that a good explication should obey(simplicity, exactness, fruitfulness, similarity to the initial intuitions) andhas applied them in his works. Now we will quote from Tichý’s (1988):“The purpose of theoretical explication is to represent intuitions interms of rigorously defined entities. It is to Frege that we owe theinsight that the mathematical notion of function is a universal mediumof explication not just in mathematics but in general. To explicatea system of intuitive, pre-theoretical, notions is to assign to them, assurrogates, members of the functional hierarchy over a definite objec-tual base. Relations between the intuitive notions are then representedby the mathematically rigorous relationships between the functionalsurrogates.” Tichý (1988, pp. 194–195)Well, TIL has been built up in harmony with this characterization7. Toadduce an example, properties of individuals are explicated as functionsthat map possible worlds to chronologies of classes of individuals (type(((oι)τ)ω), abbr. (oι)τω). We could expect an objection formulated,e.g. by G. Bealer in his (1982), where he objects in the sense that thearoma of coffee is a property (an intension), but certainly not a mapping(mappings having no aroma); hence, properties are not mappings. Theobjections of this kind are misunderstandings. Surely no property is amapping but the above mentioned mapping is just a functional surrogate,which can be logically handled. This explication makes it possible tologically capture the fact that properties (e.g., of individuals) “produceclasses” dependently on the current state of the world.Let us try to accept this characteristic of explication. (Observe how-ever that explication in this sense is not an interpretation defined for6 Already in his (1947) Carnap briefly characterizes explication.7 For details see Duží, Jespersen, Materna (2010).54 Pavel Maternaformal systems.) A question arises: Can such an explication be ‘neutral’w.r.t. the opposition realism (or classicality in a broadest sense) vs. anti-realism?For example: As soon as I decide that there are such entities that areintuitively addressed as properties and seek some functional surrogate forthem I have already voted for realism, or not?Perhaps yes: whether some individual has a property is determined according to our explication  by the current state of the world, whichmeans that any individual has or has not a given property independentlyof our knowledge. This realistic spirit of our explication transfers to han-dling mathematical objects. The type of entities associated with mathe-matical sentences as their meanings is such that these entities constructa truth-value. If no truth-value is associated with such meanings thenonly for objective reasons. Thus the meaning of the Goldbach conjectureconstructs a truth-value independently of the fact that no proof has beenfound till now, while no truth-value is associated with the sentence Thegreatest prime is odd because there is no greatest prime, not because wedo not know which prime is the greatest one.But now can we imagine an explication that would support anti-realism? Reading Brouwer, Heyting, Kolmogorov, Per Martin-Löf etc.we can imagine an explication where there would be explicated logicalconstants in a proof-theoretic manner (but frankly, explication of empir-ical entities would be extraordinarily difficult). Then of course, neithersuch an explication would be neutral.The classical realistic systems had in principle underestimated onekind of entity that had been treated only in meta-languages: abstractprocedures, complexes, constructions that are not simple like set-theo-retic entities.8 Anti-realists, I mean intuitionists, have recognized theessential role played by procedures, mainly proofs, and have reinter-preted entities that the realists of the classical kind have defined asset-theoretical objects.The significance of TIL in this connection consists in the fact thatTIL is a hyperintensional logic, which makes it possible not only to usebut also mention the abstract procedures (constructions). Construc-tions are legitimate logical objects and as possible meanings (senses) ofexpressions they fulfill the ideal of structured meaning. The expressivity8 See Tichý (1995, 2004).Is Transparent Intensional Logic a non-classical logic? 55of the resulting semantics essentially dwarfs the expressivity of exten-sional/intensional logics.Thus TIL is classical in following the principles of realism and non-classical in transcending some principles standardly used by classicallogic (where classical logic is what is taught by textbooks of classicallogic).Acknowledgments. This article has been supported by the project P401/10/792 of Grant Agency of Czech Republic.ReferencesBanks, P., 1950, “On the philosophical interpretation of logic: an Aristoteliandialogue”, pages 139–153 in Dominican Studies, Oxford, III/2.DOI: 10.1007/978-94-010-3649-8_1Bealer, G., 1982, Quality and Concept, Oxford: Clarendon Press.Carnap, R., 1947, Meaning and Necessity, Chicago: Chicago University Press.Carnap, R., 1950, Logical Foundations of Probability, Chicago: Chicago Uni-versity Press.Dummett, M., 1991, The Logica Basis of Metaphysics, London, G. Duckworth.Duží, M., 2003, “Do we have to deal with partiality?”, pages 45–76 in Miscel-lanea Logica, vol. 5, K. Bendová and P. Jirků (eds.), Praha: Karolinum.Duží, M., B. Jespersen and P. Materna, 2010, Procedural Semantics for Hyper-intiensional Logic, Springer.Kolmogorov, A., 1932, “Zur Deutung der intuitionistischen Logik”, Mathema-tische Zeitschrift, 35: 58–65. DOI: 10.1007/BF01186549Strawson, P. F., 1950, “On referring”, Mind, 59: 320–344.Tichý, P., 1988, The Foundations of Frege’s Logic, Berlin, New York: DeGruyter.Tichý, P., 1995, “Constructions as the subject-matter of mathematics”,pages 175–185 in The Foundational Debate: Complexity and Constructiv-ity in Mathematics and Physics, W. DePauli-Schimanovich, E. Köhler andF. Stadler (eds.), Dordrecht, Boston, London, and Viena: Kluwer.DOI: 10.1007/978-94-017-3327-4_13Pavel MaternaInstitute of PhilosophyAcademy of Sciences of Czech Republicmaternapavel@seznam.cz

Is Transparent Intensional Logic a non-classical logic?

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Is Transparent Intensional Logic a non-classical logic?

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