In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical (...) justification, have attracted discrete attention among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

In this note we describe a discussive set theory which is related to a certain modal set theory as discussive propositional logic is related to the system S5 of Lewis.

We outline the rather complicated history of attempts at axiomatizing Jaśkowski’s discussive logic $$\mathbf {D_2}$$ D2 and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing $$\mathbf {D_2}$$ D2 in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of $$\mathbf {D_2}$$ D2 in the language Jaśkowski suggested in his second paper on (...) discussive logic, by following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of $$\mathbf {D_2}$$ D2, introduced by Ciuciura, in which negation is also taken to be discussive. (shrink)

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as $$ {\textsf {D}}_{\textsf {0}}$$ D 0. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic $$ {\textsf {D}}_{\textsf {2}}$$ D 2. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic $$ {\textsf {D}}_{\textsf {2}}$$ D 2 but with the (...) help of the deontic normal logic $$\textbf{D}$$ D. Although we focus on the smallest discussive logic and its correspondence to $$ {\textsf {D}}_{\textsf {2}}$$ D 2, we analyse to some extent also its formal aspects, in particular its behaviour with respect to rules that hold for classical logic. In the paper we propose a deductive system for the above recalled discussive logic. While formulating this system, we apply a method of Newton da Costa and Lech Dubikajtis—a modified version of Jerzy Kotas’s method used to axiomatize $$ {\textsf {D}}_{\textsf {2}}$$ D 2. Basically the difference manifests in the result—in the case of da Costa and Dubikajtis, the resulting axiomatization is pure modus ponens-style. In the case of $$ {\textsf {D}}_{\textsf {0}}$$ D 0, we have to use some rules, but they are mostly needed to express some aspects of positive logic. $$ {\textsf {D}}_{\textsf {0}}$$ D 0 understood as a set of theses is contained in $$ {\textsf {D}}_{\textsf {2}}$$ D 2. Additionally, any non-trivial discussive logic expressed by means of Jaśkowski’s model of discussion, applied to any regular modal logic of discussion, contains $$ {\textsf {D}}_{\textsf {0}}$$ D 0. (shrink)

Jaśkowski's discussive (discursive) logic D2 is historically one of the first paraconsistent logics, i.e., logics which 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages, and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D2p.

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

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. (shrink)

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

S. Jaśkowski introduced the discussive propositional calculus D₂ as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories. D₂ has afterwards been extended to a first-order predicate calculus and to a higher-order logic. In this paper we present a natural version of D₂, in the sense of Jaśkowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus. A semantics for the new (...) calculus is also presented. (shrink)

Jaśkowski’s discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7, 8]): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in {D_{2}}}$$\end{document} iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}$$\end{document}, where (–)• is a translation of discussive formulae from Ford into the modal language. We say that a modal logic L defines D2 iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}$$\end{document}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (†): have the same theses beginning with ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond}$$\end{document}’ as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property (†). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (†). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2. (shrink)

We expose the main ideas, concepts and results about Jakowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. (...) We also give syntactic characterizations of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found (...) in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

In 1995 N. C. A. da Costa and F. Doria proposed the modaltype elegant axiomatization of Jaśkowski’s discussive logic D2. Yet his ownproblem which was formulated in 1975 in a following way: Is it possible toformulate natural and simple axiomatization for D2, employing classical disjunction and conjunction along with discussive implication and conjunctionas the only primitive connectives? — still seems left open. The matter of factis there are some axiomatizations of D2 proposed, e.g., by T. Furmanowski, J. (...) Kotas and N. C. A. da Costa , G. Achtelik, L. Dubikajtus,E. Dudek and J. Konior , satisfying da Costa’s conditions, but they arerather looking very complicated and unnatural. An attempt is made to solveda Costa’s problem. The new axiomatization of D2 is proposed essentiallybased on da Costa’s-Doria axiomatization from 1995. (shrink)

In the paper, we tackle the matter of non-classical logics, in particular, paraconsistent ones, for which not every formula follows in general from inconsistent premisses. Our benchmark is Jaśkowski’s logic, modeled with the help of discussion. The second key origin of this paper is the matter of being tabular, i.e. being adequately expressible by finitely many finite matrices. We analyse Jaśkowski’s non-tabular discussive (discursive) logic $ \textbf {D}_{2}$, one of the first paraconsistent logics, from the perspective of (...) a trivalent tabular logic. We are motivated to step down from the ongoing modal $ \textbf {S5}$-perspective of developing $ \textbf {D}_{2}$ both by certain mysteries that have been surrounding it and by gaps in Jaśkowski’s arguments contra the multivalent tabular perspective. Although Jaśkowski’s idea to use $ \textbf {S5}$ in order to define $ \textbf {D}_{2}$ is very attractive since it allows one to benefit from the tools and results of modal logic, it also gives a ‘non-direct’ formulation and, as it appeared later, is superfluous with respect to what is meant to be achieved since one can define the very same logic but using modal logics weaker than S5. It is also known, due to Kotas, that discussive logic is not finite-valued. So, in light of Kotas’s result that $ \textbf {D}_{2}$ is non-tabular, we propose to associate it with a few dozen discussive formulae that Jaśkowski unequivocally and illustratively suggests to be its theorems or non-theorems rather than with axioms of its modern axiomatizations (one of which Kotas employs) in order to be capable of performing a computer-aided brute-force search for suitable trivalent matrices in the cases of one and two designated values. As a result, we find trivalent matrices with two designated values that might be dubbed ‘discussive’ because they meet Jaśkowski’s suggestion to validate and invalidate the litmus theorems and non-theorems, respectively, despite the fact that none of them validates all the negation axioms in the modern axiomatizations of $ \textbf {D}_{2}$. The matrices found are then analysed along with highlighting the ones that were previously mentioned in the literature. We conclude the paper with a comparative analysis of Omori’s results and a test of Karpenko’s hypothesis. (shrink)

In the late forties, Stanisław Jaśkowski published two papers onthe discursive sentential calculus, D2. He provided a definition of it by an interpretation in the language of S5 of Lewis. The knownaxiomatization of D2 with discursive connectives as primitives was introduced by da Costa, Dubikajtis and Kotas in 1977. It turns out, however,that one of the axioms they used is not a thesis of the real Jaśkowski’s calculus. In fact, they built a new system, D∗2 for short, that differs from (...) D2 inmany respects. The aim of this paper is to introduce a direct Kripke-type semantics for the system, axiomatize it in a new way and prove soundness andcompleteness theorems. Additionally, we present labelled tableaux for D∗2.1. (shrink)

The discussion on the principle of non-contradiction (1946–1957) between Marxist and non-Marxist philosophers was one of the major philosophical discussions in Polish philosophy of this period. In my text, I carefully reconstruct this discussion and outline its relation to Soviet debates on the subject. I show that the change in Schaff’s position happened in the early 1950s under the combined influence of the Lvov–Warsaw School and the changes in the official Soviet position regarding formal logic. I discuss the aftermath (...) following Schaff’s change in attitude towards the analytic tradition for the development of Polish philosophy, as well as the critique of this change by Jarosław Ładosz. In my reconstruction of the latter, I focus on the problem of the historical development of science. I refer to Ilyenkov’s critique of Schaff, opposing synchronic (“positivist”) and diachronic (“dialectical”) concepts of knowledge. As I argue, these opposing concepts of science can be seen as a genuine issue at stake in the Polish discussion as well, especially in the polemic between Schaff and Ładosz. (shrink)

In connection with the discussion on the problem of logic, both within our country and abroad , it has already turned from the general, comparatively abstract, and difficult-to-resolve problem of the "relationship between formal logic and dialectics" to the more concrete problem of the functional scope of formal logic. Some people have even utilized special articles to inquire about a certain law in formal logic: the functional scope of the law of identity or the law of (...) contradiction. The purpose of these philosophers or logicians is to try to define the functional scope of a certain law in formal logic as a starting point, and then to proceed to study the functional scope of formal logic in the entire thinking process of mankind. If the functional scope of formal logic in mankind's entire thinking process is clearly and precisely defined, the solution of this complicated problem of the "relationship between formal logic and dialectics" would certainly not be very difficult. The basic purpose of Comrade Chu-ko Yin-t'ung's article on the problem of "whether or not formal logic's law of contradiction can be contravened in the dialectical thinking process" of men, which appeared in issue No. 82 of the "Philosophy" supplement of Kuang-ming Daily on April 24, is like this. It should be considered as a rather practical method. But I do not quite agree with his basic points of argument and with his analysis of certain examples of dialectics. I intend to present my superficial views here and to study them with Comrade Chu-ko. (shrink)

Recently I have read in succession the four articles on formal logic and dialectics in the current year's Hsin chien-she: Mr. Chou Ku-ch'eng's "Formal Logic and Dialectics" in the second issue, Mr. I Chih's "A Criticism of Confused Concepts on Problems of Logic" in the fourth issue, Mr. Shen Ping-yuan's "A Discussion of ‘Formal Logic and Dialectics,’ " and Mr. Chou Ku-ch'eng's "Further Discourse on Formal Logic and Dialectics," both in the seventh issue. In my (...) opinion, in Mr. Chou's articles, several main theses on formal logic and on the relationship between formal logic and dialectics are erroneous; basically, I agree with the criticisms of Messrs. I and Shen. Nonetheless, I believe that Mr. Chou's errors have had their source in the theory of cognition in that he has unilateralized and absolutized certain aspects that were originally correct, thus leading himself in the wrong direction. In criticizing Mr. Chou, both Messrs. I and Shen failed to proceed from this premise, so they could not make others see where Mr. Chou's errors basically lie. Based on this approach, I am merely offering some personal views on several main problems concerning formal logic and the relationship between formal logic and dialectics as mentioned in Mr. Chou's article. It is hoped that Mr. Chou and other comrades will correct me. (shrink)

In this paper, I present the discussive adaptive logic DLI r . As is the case for other discussive logics, the intended application context of DLI r is the interpretation of discussions. What is new about the system is that it does not lead to explosion when some of the premises are self-contradictory. It is argued that this is important in view of the fact that human reasoners are not logically omniscient, and hence, that it may not (...) be evident to discover the inconsistencies in one's beliefs. In addition to this, DLI r can handle cases in which different participants contradict each other. It is shown that, in both kinds of cases, DLI r leads to an interpretation of the discussion that is as rich as possible (even though no discussive connectives are introduced). (shrink)

The research into patient involvement is seldom concerned with the significance of cultural and structural factors. In this discussion paper, we illustrate our considerations on some of the challenges in implementing the ideal of patient involvement by showing how such factors take part in shaping the ways in which the intentions to involve patients are converted to practical interventions. The aim was to contribute to the approach dealing with contextual and structural factors of significance for patient involvement. With the idea (...) of “institutional logics,” borrowed from the Danish scholar, Erik Riiskjær, we first demonstrate, with examples from our own research, how patient involvement is interpreted differently within the different logics. Then, we show how the different interpretations of patient involvement meet and conflict in mutual competition as the ideals are sought to be converted to practical interventions. At last, we argue that an adequate theoretical model for the development in the future health care system should be expanded with a “patient logic.”. (shrink)

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a (...) clear sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainferential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper. (shrink)

This paper is a critical discussion of A.N. Prior’s contribution to the modern understanding of indeterminism and human freedom of choice. Prior suggested that these ideas should be conceived in terms of his tense logic. It can be demonstrated that his approach provides an attractive formalization that makes it possible to discuss indeterminism and human freedom of choice in a very precise manner and in a broader metaphysical context. It is also argued that Prior’s development of this approach was (...) closely linked to his very personal struggles with fundamental religious and metaphysical questions. In his opinion, holding the doctrine of divine foreknowledge together with the doctrine of human freewill gives rise to difficult logical and philosophical problems. It appears that Prior, rather early on, decided to support what was later known as the Peircean solution, although he also analysed what he considered to be the most important alternative solution—the Ockhamist solution. This paper offers a discussion of some of Prior’s published books and papers as well as some of the papers in his Nachlass. (shrink)

The solution of this problem is extremely important to the process of the development of logic. Its solution is of great theoretical significance in upholding the methodology of dialectical materialism; criticizing the sophistry of metaphysics, skepticism, and relativism; developing formal logic and symbolic logic; forming the scientific system of dialectical logic in the strict sense; and conducting research to strengthen all the concrete sciences. It also has enormous significance for the criticism of "leftist" and right-wing opportunism (...) and revisionism, and especially for criticizing the counterrevolutionary revisionist line of the "gang of four.". (shrink)

In this paper, I present the modal adaptive logic $AJ^{r}$ (based on S5) as well as the discussive logic $D_{2}^{r}$ that is defined from it. $D_{2}^{r}$ is a (nonmonotonic) alternative for Jaśkowski's paraconsistent system D₂. Like D₂, $D_{2}^{r}$ validates all single-premise rules of Classical Logic. However, for formulas that behave consistently, $D_{2}^{r}$ moreover validates all multiple-premise rules of Classical Logic. Importantly, and unlike in the case of D₂, this does not require the introduction of (...) class='Hi'>discussive connectives. It is argued that this has clear advantages with respect to one of the main application contexts of discussive logics, namely the interpretation of discussions. (shrink)

Thought-expressions are not simply good; instead, they become good for us when they make sense, empower action, and support health. From time to time, we may need to consider the difference between thought-expression and discourse, or thought-expression that really makes sense, and the difference between discourse and discussion, or a discourse-situation that makes genuine agreement or disagreement possible for us. In this essay, I explore a problem that D. Z. Phillips and Randy Ramal have termed “logical inversion,” and I argue (...) that wherever logical inversions have gained a foothold in our thought-expressions, they threaten to render us incapable of authentic discourse and discussion by ensuring that we misunderstand the understandings of others such that we deceive ourselves and others by picturing the knowledge and truth of our perspective against an unreal background of conceivable falsehoods which we have falsely attributed to the perspectives of others. Where this has taken place, practices of philosophical attention and conceptual clarification are needed to help us move from a situation of endless thought expression toward authentic discourse and discussion. (shrink)

It is supposed that any scientific theory (here we consider physical theories only) has an underlying logic, even if it is not made explicit. The role of the underlying logic of a theory T is mainly to guide the proofs and the accepted consequences of the theory’s principles, usually described by its axioms. In this sense, the theorems of the underlying logic are also theorems of the theory. In most cases, if pressed, the scientist will say that (...) the underlying logic of most physical theories is classical logic or some fragment of a set theory suitable for accommodating the theory’s mathematical and logical concepts. We argue that no physical theory and no philosophical discussion on the basis of the theory should dismiss its underlying logic, so the arguments advanced by some philosophers of physics in that certain entities (the considered case deals with quantum entities) can be only weakly discerned or be just ‘relationals’ and that they cannot be absolutely identified by a monadic property, are fallacious if one remains within a ‘standard’ logico-mathematical framework, grounded on classical logic. We also discuss the claim that some properties (as those that came from logic) would be ‘illegitimate’ for discerning these entities is not justifiable. Thus, this paper may be interesting for logicians, physicists, and philosophers. (shrink)

A non-Fregean framework aims to provide a formal tool for reasoning about semantic denotations of sentences and their interactions. Extending a logic to its non-Fregean version involves introducing a new connective $$\equiv $$ ≡ that allows to separate denotations of sentences from their logical values. Intuitively, $$\equiv $$ ≡ combines two sentences $$\varphi $$ φ and $$\psi $$ ψ into a true one whenever $$\varphi $$ φ and $$\psi $$ ψ have the same semantic correlates, describe the same situations, (...) or have the same content or meaning. The paper aims to compare non-Fregean paraconsistent Grzegorczyk’s logics (Logic of Descriptions $$\textsf{LD}$$ LD, Logic of Descriptions with Suszko’s Axioms $$\textsf{LDS}$$ LDS, Logic of Equimeaning $$\textsf{LDE}$$ LDE ) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic $$\textsf{D}_2$$ D 2, Logic of Paradox $$\textsf{LP}$$ LP, Logics of Formal Inconsistency $$\textsf{LFI}{1}$$ LFI 1 and $$\textsf{LFI}{2}$$ LFI 2 ). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of $$\textsf{LP}$$ LP, $$\textsf{LFI}{1}$$ LFI 1, $$\textsf{LFI}{2}$$ LFI 2. Furthermore, we show that non-Fregean extensions of $$\textsf{LP}$$ LP, $$\textsf{LFI}{1}$$ LFI 1, $$\textsf{LFI}{2}$$ LFI 2, and $$\textsf{D}_2$$ D 2 are more expressive than their original counterparts. Our results highlight that the non-Fregean connective $$\equiv $$ ≡ can serve as a tool for expressing various properties of the ontology underlying the logics under consideration. (shrink)