The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition.

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in (...) this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two (...) or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £. (shrink)

This paper is a continuation of Part I under the same title. Its Chapter III contains results given in the following publications: U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131. G. Bryll, Związki logiczne pomiędzy zdaniami nauk empirycznych (Logical relations between sentences of empirical sciences). Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i (...) Monografie, Nr 22 (1969), 155-216. This chapter contains an attempt to formalize some questions of methodology of empirical sciences. The theory T is enriched here not only by new definitions but also by a new primitive term and new axioms. The most important notion defined in this chapter is the set IndX of all inductive conclusions obtained on the basis of sentences of the set X. The function CnI defined by the equality: CnI X = X u IndX, satisfies the axioms of the general theory of deductive systems. The set CnI X can be understood as a system of empirical sciences in a narrower sense. This paper is a development of the considerations of G. Bryll, included in the above-listed article by this author. (shrink)

Objects of consideration are various non-classical connectives “hidden” in the classical logic in the form of G˛s with ˛ —a classical connective, and s—a propositional variable. One of them is negation, which is defined as G ⇒ s; another is necessity, which is defined as G ∧ s. The new operations are axiomatized and it is shown that they belong to the 4-valued logic of Lukasiewicz. A 2-point Kripke semantics is built leading directly to the 4-valued logical tables.

Charles Peirce incorporates modality into his Existential Graphs by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that arises from overlooking (...) the distinction between the intermediate truth values that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception. (shrink)

Jan Łukasiewicz is known primarily as the founder of the three-valued system of logic. It is also generally renowned that his reason for introducing many-valued systems of logic was an attempt to refute determinism. When he developed the three-valued and n-valued logic, he employed these systems in his arguments against determinism. On the contrary, Łukasiewicz preferred the four-valued system of logic that is not suitable for a refutation of determinism in his latest period. It seems, however, that determinism still interested (...) Łukasiewicz, as he discussed its denial in his last work, the second edition of the book Aristotle’s syllogistic from the standpoint of modern formal logic. The aim of my paper is to introduce the issue and describe the relation of these two topics primarily in the latest period of Łukasiewicz’s work. There are several possible solutions to the issue. However, Łukasiewicz never addressed the issue clearly in his published works. Therefore, the solutions might be supported or contradicted by his unpublished works in the future. (shrink)

Although Łukasiewicz and Quine do not share many common views, they agreed on one important point in the 1950s: they both denied the distinction between empirical and a priori sciences. This agreement might be surprising as this denial was rather controversial at that time. This paper focuses on Quine’s and Łukasiewicz’s denials of the distinction between empirical and a priori sciences, and proposes three possible answers to the question of why both formulated the same conclusion at a similar time. Firstly, (...) it discusses Quine’s possible influence on Łukasiewicz as Łukasiewicz agreed with Quine’s criticism of modality at that time. Secondly, it considers the possibility that Quine was affected by Łukasiewicz via his debates with Łukasiewicz’s student, Tarski. Lastly, it takes into account the possibility that both philosophers were inspired by an external source, namely the rise of quantum mechanics. (shrink)

In the nineteenth century, philosophy was at a crossroads. While the natural and technical sciences were developing in an unprecedented fashion, philosophy seemed to be stalled. Inspired by the progress of the natural sciences, many philosophers attempted to make such progress in philosophy and make philosophy a truly scientific discipline. This effort was also reflected in the philosophy of the Lvov-Warsaw school. While its founder, Kazimierz Twardowski, following his teacher Franz Brentano, promoted psychology as a method of scientific philosophy, one (...) of his first students, Jan Łukasiewicz, was convinced that mathematical logic was such a method. To use mathematical logic as a tool, Łukasiewicz had to, however, argue convincingly that logic is an independent science and hence is not a part of psychology, i.e., arguing for anti-psychologism in logic. He initially adopted the arguments provided by Husserl, then celebrated as a proponent of anti-psychologism, and Frege’s views. When Łukasiewicz developed, however, his systems of many-valued logic, he denied almost all the principles that characterise Husserl and Frege’s anti-psychologism, i.e., the objectivity of the laws of logic, the existence of apodictic propositions, and the distinction between a priori and empirical sciences. He was, however, a proponent of anti-psychologism up to the end of his life. The aim of my paper is to introduce Łukasiewicz’s unique concept of anti-psychologism that significantly affected the views of mathematical logic in the Lvov-Warsaw School, and the views of his colleagues which helped him develop the concept. (shrink)

The present article contains part of results from my lecture delivered at II Flemish-Polish workshop on Ontological Foundation of Paraconsistency.

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for defining truth-functional (...) modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics. (shrink)

A simple, bivalent semantics is defined for Łukasiewicz’s 4-valued modal logic Łm4. It is shown that according to this semantics, the essential presupposition underlying Łm4 is the following: A is a theorem iff A is true conforming to both the reductionist and possibilist theses defined as follows: rt: the value of modal formulas is equivalent to the value of their respective argument iff A is true, etc.); pt: everything is possible. This presupposition highlights and explains all oddities arising in Łm4.

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

The paper is a study of properties of quasi-consequence operation which is a key notion of the so-called inferential approach in the theory of sentential calculi established in [5]. The principal motivation behind the quasi-consequence, q-consequence for short, stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms and their further occurrence in place of conclusions may be justified or not. The main semantic feature of the q-consequence reflecting the idea is that its rules (...) lead from the non-rejected assumptions to the accepted conclusions.First, we focus on the syntactic features of the framework and present the q-consequence as related to the notion of proof. Such a presentation uncovers the reasons for which the adjective inferential is used to characterize the approach and, possibly, the term inference operation replaces q-consequence. It also shows that the inferential approach is a generalisation of the Tarski setting and, therefore, it may potentially absorb several concepts from the theory of sentential calculi, cf. [10]. However, as some concrete applications show, see e.g.[4], the new approach opens perspectives for further exploration. (shrink)

In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative pseudo-complements as in (...) intuitionistic logic. We show that the complete lattices are intrinsically modal, with banal identity modal operator. We define the autoreferential set-based representation for the class of modal algebras, and show that the autoreferential Kripke-style semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truth-values. The philosophical assumption is based on the consideration that each possible world represents a level of credibility, so that only propositions with the right logic value (i.e., level of credibility) can be accepted by this world, then we connect it with paraconsistent properties and LFI logics. The bottom truth value in this complete lattice corresponds to the trivial world in which each formula is satisfied, that is, to the world with explosive inconsistency. The top truth value corresponds to the world with classical logics, while all intermediate possible worlds represent the different levels of paraconsistent logics. (shrink)

We study a range of issues connected with the idea of replacing one formula by another in a fixed context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas {A1,..., Am,...} and a formula B when for every context C, we have C,..., C,... ⊢ C. Section 1 looks at some differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we (...) find that while the inference from A and B to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A \land B$\end{document}, sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3. (shrink)

A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas □α↔α, □α↔¬α, □α↔⊤, and □α↔⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the schema □α→□β. These logics, (...) along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over. (shrink)

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...) implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.

This article traces the development of possible worlds semantics through the work of: Wittgenstein, 1913-1921; Feys, 1924; McKinsey, 1945; Carnap, 1945-1947; McKinsey, Tarski and Jónsson, 1947-1952; von Wright, 1951; Becker, 1952; Prior, 1953-1954; Montague, 1955; Meredith and Prior, 1956; Geach, 1960; Smiley, 1955-1957; Kanger, 1957; Hintikka, 1957; Guillaume, 1958; Binkley, 1958; Bayart, 1958-1959; Drake, 1959-1961; Kripke, 1958-1965.

I argue that a compulsive seeking for just one sense of consistency is hazardous to rationality, and that observing the subtle distinctions of reasonableness between individual and groups may suggest wider, structuralistic notions of consistency, even relevant to re-assessing Gödei's Second Incompleteness Theorem and to science as a whole.

The logic BN4 was defined by R.T. Brady as a four-valued extension of Routley and Meyer’s basic logic B. The system EF4 is defined as a companion to BN4 to represent the four-valued system of implication. The system Ł was defined by J. Łukasiewicz and it is a four-valued modal logic that validates what is known as strong Łukasiewicz-type modal paradoxes. The systems EF4-M and EF4-Ł are defined as alternatives to Ł without modal paradoxes. This paper aims to define a (...) Belnap-Dunn semantics for EF4, EF4-M and EF4-Ł. It is shown that EF4, EF4-M and EF4-Ł are strongly sound and complete w.r.t. their respective semantics and that EF4-M and EF4-Ł are free from strong Łukasiewicz-type modal paradoxes. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

Łukasiewicz has often been criticized for his motive for inventing his three-valued logic, namely the avoidance of determinism. First of all, I want to show that almost all of the critcism along this line was wrong. Second I will indicate that he made mistakes, however, in constructing his system, because he had other motives at the same time. Finally I will propose some modification of his system and its interpretation which can attain his original purpose in some sense.

An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences hold independently of the nature of the statements involved. A negation-inconsistent logic is a logic having at least one model that satisfies both some statement and its negation. A negation-incomplete logic has at least one model according to which neither some statement nor its negation are satisfied. Paraconsistent logics are negation-inconsistent yet non-overcomplete; paracomplete logics are negation-incomplete yet non-overcomplete. A paranormal logic (...) is simply a logic that is both paraconsistent and paracomplete. -/- Despite being perfectly consistent and complete with respect to classical negation, nearly every normal modal logic, in its ordinary language and interpretation, admits to some latent paranormality: It is paracomplete with respect to a negation defined as an impossibility operator, and paraconsistent with respect to a negation defined as non-necessity. In fact, as it will be shown here, even in languages without a primitive classical negation, normal modal logics can often be alternatively characterized directly by way of their paranormal negations and related operators. So, instead of talking about ‘necessity’, ‘possibility’, and so on, modal logics could be seen just as devices tailored for the study of (modal) negation. This paper shows how and to what extent this alternative characterization of modal logics can be realized. (shrink)

This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two systems of (...) natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)