Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a (...) consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. Our dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion. (shrink)

Contrary to what seems to be predicted by a Strawson-inspired view, in presupposition denials the presupposition triggered by, e.g., ‘the king of France’ seems to be cancelled. To explain this puzzling instance of the projection problem, defenders of a Strawson-inspired view have proposed various ad hoc ambiguities. I develop a version of Segmented Discourse Representation Theory that explains the puzzling presupposition-cancelling phenomenon relying only on independently motivated pragmatic processes. Appealing to Kripke’s “test” for the adequacy of ambiguity motivating counterexamples, I (...) argue that my analysis is to be preferred over proposals that posit ad hoc ambiguities. Two key insights of my analysis are, first, that the “cancellation” of presuppositions in presupposition denials depends upon the illocutionary effects of rejection and correction, and, second, that the process of presupposition accommodation often involves the speaker performing an ancillary sort of speech act, viz. the speech act of presupposing. (shrink)

Lewis (Noûs 13:455–476, 1979) claimed that branching-time(-like) models can be derived from his sphere models. However, he did not present any specific construction of branching-time(-like) models from his sphere models formally. Meanwhile, Hosokawa (in: Modern logic of modality and its philosophical range: counterfactuals, Gettier problem, and information flow, Tokyo Metropolitan University, 2018) presented a logico-mathematically strict manner in which sphere models can be reconstructed from branching-time models. Subsequently, Hosokawa (J Logic Lang Inf 32:677–706, 2023) presented a proof-theoretically refined version of (...) hybrid tense logic for a certain type of conditionals, which is referred to as hybrid tense logic for temporal conditionals ( $$\textbf{HTL}_{TC}$$ ). In this paper, we interpret a hybrid version $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ of Lewis’s counterfactual logic $${\textbf{V}}$$ into $$\textbf{HTL}_{TC}$$, and then prove that $$\textbf{HTL}_{TC}$$ is more expressive than $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ on the class of temporal sphere models, i.e., sphere models derived from a type of branching-time models. (shrink)

Cirquent calculus is a proof system with inherent ability to account for sharing subcomponents in logical expressions. Within its framework, this article constructs an axiomatization $$\text{ CL18 }$$ CL18 of the basic propositional fragment of computability logic—the game-semantically conceived logic of computational resources and tasks. The nonlogical atoms of this fragment represent arbitrary so called static games, and the connectives of its logical vocabulary are negation and the parallel and choice versions of conjunction and disjunction. The main technical result of (...) the article is a proof of the soundness and completeness of $$\text{ CL18 }$$ CL18 with respect to the semantics of computability logic. (shrink)

In this text, we consider Gärdenfors’ conceptual spaces that are separable Hilbert spaces. In particular, the results we obtained apply to finite-dimensional Euclidean spaces. Our main contribution can be formulated as a combination of the theory of opinion dynamics with the theory of conceptual spaces. This combination, in turn, leads us to propose a new model for the time evolution of conceptual spaces. To achieve this goal, we propose some extension of the multidimensional opinion dynamics model of Parsegov, Proskurnikov, Tempo (...) and Friedkin to opinions with values in Hilbert spaces. (shrink)

We define a Kripke semantics for a conditional logic based on the propositional logic $$\textsf{N4}$$ N 4, the paraconsistent variant of Nelson’s logic of strong negation; we axiomatize the minimal system induced by this semantics. The resulting logic, which we call $$\textsf{N4CK}$$ N 4 CK, shows strong connections both with the basic intuitionistic logic of conditionals $$\textsf{IntCK}$$ IntCK introduced earlier in (Olkhovikov, 2023) and with the $$\textsf{N4}$$ N 4 -based modal logic $$\textsf{FSK}^d$$ FSK d introduced in (Odintsov and Wansing, 2004) (...) as one of the possible counterparts to the classical modal system $$\textsf{K}$$ K. We map these connections by looking into the embeddings which obtain between the aforementioned systems. (shrink)

Term Functor Logic is a term logic that recovers some important features of the traditional, Aristotelian logic; however, it turns out that it does not preserve all of the Aristotelian properties a valid inference should have insofar as the class of theorems of Term Functor Logic includes some inferences that may be considered irrelevant (e.g. ex falso, verum ad, and petitio principii). By following an Aristotelian or syllogistic notion of relevance, in this contribution we adapt a tableaux method for Term (...) Functor Logic in order to avoid irrelevance and we offer some sort of intuitive semantics in terms of traditional inferential situations (e.g. propter quid, quia, non causa ut causa, and non sequitur). (shrink)

In Entities and Indices, M. J. Cresswell argued that a first-order modal language can reach the expressive power of natural-language modal discourse only if we give to the formal language a semantics with indices containing infinite possible worlds and we add to it an infinite collection of operators $${{\varvec{actually}}}_n$$ actually n and $$ Ref _n$$ R e f n which store and retrieve worlds. In the fourth chapter of the book, Cresswell gave a proof that the resulting intensional language, which (...) he called $${\mathscr {L}}^*$$ L ∗, is as expressive as an extensional variant of it, called $${\mathscr {L}}$$ L, which has full quantification over worlds. In both linguistics and philosophy, Cresswell’s book has been viewed as offering a compelling argument for preferring extensional systems in the study of natural language. In this paper, after providing a model-theoretic definition of the relation being as expressive as that can be applied to Cresswell’s languages $${\mathscr {L}}$$ L and $${\mathscr {L}}^*$$ L ∗, we show that the intensional language $${\mathscr {L}}^*$$ L ∗ is not as expressive as the extensional language $${\mathscr {L}}$$ L. This result, we claim, undermines Cresswell’s argument to the effect that English modal discourse has the power of explicit quantification over worlds. Additionally, we show that $${\mathscr {L}}^*$$ L ∗ does become as expressive as $${\mathscr {L}}$$ L when we add Cresswell’s operator of universal modality $$\square $$ □ to $${\mathscr {L}}^*$$ L ∗, which provides an extra amount of expressive power. Recently, I. Yanovich has advocated a view that is similar to ours in important respects. At the end of the paper we offer a short discussion of his formalism. (shrink)

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized based (...) on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing’s connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and complete with respect to the presented algebraic structures.

This paper is about sentences of form To be human is to be an animal, To live is to fight, etc. I call them ‘infinitive sentences’. I define an augmented propositional language able to express them and give a matrix-based semantics for it. I also give a tableau proof system, called IL for Infinitive Logic. I prove soundness, completeness and a few basic theorems.