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.

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.

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$$\varphi $$φand$$\psi $$ψinto a true one whenever$$\varphi $$φand$$\psi $$ψhave 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$$\textsf{D}_2$$D2, Logic of Paradox$$\textsf{LP}$$LP, Logics of Formal Inconsistency$$\textsf{LFI}{1}$$LFI1and$$\textsf{LFI}{2}$$LFI2). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of$$\textsf{LP}$$LP,$$\textsf{LFI}{1}$$LFI1,$$\textsf{LFI}{2}$$LFI2. Furthermore, we show that non-Fregean extensions of$$\textsf{LP}$$LP,$$\textsf{LFI}{1}$$LFI1,$$\textsf{LFI}{2}$$LFI2, and$$\textsf{D}_2$$D2are 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)

The aim of this paper is to give a general background and a uniform treatment of several notions of mutual interpretability. Sentential calculi are treated as preorders and logical invariants of adjoint situations, i.e. Galois connections are investigated. The class of all sentential calculi is treated as a quasiordered class.Some methods of the axiomatization of the M-counterparts of modal systems are based on particular adjoints. Also, invariants concerning adjoints for calculi with implication are pointed out. Finally, the notion of interpretability (...) is generalized so that it may be applied to closure spaces as well. (shrink)

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 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)

In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculusD 2 . In [9] J. Kotas showed thatD 2 is equivalent to the calculusM(S5) whose theses are exactly all formulasa such thatMa is a thesis ofS5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other thanS5 and modalities other thanM. This paper generalises the above work. LetA be an arbitrary modality (i.e. string ofM''s,L''s and negation signs). Then (...) theA-extension of a set of formulasX is {¦A X}}. Various properties ofA-extensions of normal modal systems are examined, including a problem of their axiomatizability. (shrink)

The aim of this paper is to propose a criterion of finite detachment-substitutional formalization for normal modal systems. The criterion will comprise only those normal modal systems which are finitely axiomatizable by means of the substitution, detachment for material implication and Gödel rules.