Logical and Philosophical Ideas in Certain Formal Approaches to Language

This paper reminds, puts in order, sketches and also initiates some researches from the field of logic and philosophy of language. It lays emphasis on the logical-linguistic and ontological developmental lines originated with Polish researchers. The author discusses two opposite orientations of the former line in the process of formalization of language, called here nominalistic and Platonistic. The paper mentions the author's result (1989; 1991) concerning theoretical equivalence of two axiomatic approaches to language syntax which take into consideration these two different philosophical orientations, respectively. It also introduces certain formal-logical foundations of language semantics, discusses and develops the problem of the syntactic and semantic categorial agreement, i.e. the agreement of the syntactic category of each language expression with the ontological category assigned to the reference of this expression. The author, in particular, gives a solution to this problem for quantifiers. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Sequent Calculus for a Negative Free Logic

This article presents a sequent calculus for a negative free logic with identity, called N. The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.     

A modal sequent calculus for a fragment of arithmetic

Global properties of canonical derivability predicates (the standard example is Pr() in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite model property, interpolation and the fixed point theorem.The second author holds a grant from the Consiglio Nazionale delle Ricerche, gruppo G.N.S.A.G.A.     

Game Semantics for the Lambek-Calculus: Capturing Directionality and the Absence of Structural Rules

In this paper, we propose a game semantics for the (associative) Lambek calculus. Compared to the implicational fragment of intuitionistic propositional calculus, the semantics deals with two features of the logic: absence of structural rules, as well as directionality of implication. We investigate the impact of these variations of the logic on its game semantics. Presented by Wojciech Buszkowski     

A new formulation of discussive logic

S. Jakowski introduced the discussive prepositional calculus D ₂as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D ₂has afterwards been extended to a first-order predicate calculus and to a higher-order logic (cf. the quoted paper). In this paper we present a natural version of D ₂, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented.     

AI,ME AND LEWIS (ABELIAN IMPLICATION,MATERIAL EQUIVALENCE AND C I LEWIS 1920)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). Meyer was supported in this work as a Visiting Fellow in the College of Engineering and Computer Science, ANU.     

Logic as a Universal Medium or Logic as a Calculus? Husserl and the Presuppositions of “the Ultimate Presupposition of Twentieth Century Philosophy”

This paper discusses Jean van Heijenoort's (1967) and Jaakko and Merrill B. Hintikka's (1986, 1997) distinction between logic as a universal language and logic as a calculus, and its applicability to Edmund Husserl's phenomenology. Although it is argued that Husserl's phenomenology shares characteristics with both sides, his view of logic is closer to the model‐theoretical, logic‐as‐calculus view. However, Husserl's philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl's position shows that holding a model‐theoretical view of logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher's views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy.     

Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and s, it follows constructively that the K-completeness of s implies MP(S), a form of Markov's Principle. If s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when s is many-one complete, MP(S) implies the usual Markov's Principle MP.An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implics MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP.These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Gödel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article.     

On Intermediate Predicate Logics of some Finite Kripke Frames,I. Levelwise Uniform Trees

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.     

Light Affine Set Theory: A Naive Set Theory of Polynomial Time

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL.In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over {0, 1}* is computable in polynomial time if and only if it is provably total in LAST.     

Quantum logical calculi and lattice structures

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T _eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the value-definiteness of propositions is not postulated, the calculus T _eff represents a calculus of effective (intuitionistic) quantum logic.Beginning with the tableaux-calculus the equivalence of T _eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T _eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.     

The Tense Logic for Master Argument in Prior’s Reconstruction

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K _t 4 plus a new axiom (P): ‘p Λ G p ⊃ P G p’. This formula was used by Prior in his original analysis of Master Argument. (P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by (P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. Presented by Jacek Malinowski     

Relevant implication and the weak deduction theorem

It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.I am grateful to the referees; their comments enabled me to correct a mistake and to clarify several passages.     

On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.     

On Magari's concept of general calculus: notes on the history of tarski's methodology of deductive sciences

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn _S , called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (?o?, Suszko, S?upecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic.

Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section.     

Inexact Knowledge with Introspection

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

Semantic trees for Dummett's logic LC

The aim of this paper is to provide a decision procedure for Dummett's logic LC, such that with any given formula will be associated either a proof in a sequent calculus equivalent to LC or a finite linear Kripke countermodel.     

The Role of Mathematics in Deleuze’s Critical Engagement with Hegel

Abstract

The role of mathematics in the development of Gilles Deleuze’s (1925–95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze’s interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770–1831) presents in the Science of Logic. Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz’s account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908–44), which plays a significant role in Deleuze’s project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense.     

On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer _L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr _L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr _L (m) for some linear modal and tense logicsL.