Vit Puncochar

In this paper, we study two operators, inquisitive disjunction and inquisitive existential quantifier, in the context of first-order intuitionistic logic with constant domains. We explain that these operators allow us to express types of intuitionistic propositions. We first provide this language with a relational semantics and formulate a sound axiomatic system. Completeness of the system for the full language is presented as an open problem but we prove completeness for a rich fragment of the language adapting the methods developed by Gianluca Grilleti in the context of classical inquisitive logic. We also develop a general algebraic framework in which we characterize the class of “Kripkean algebras” generated by the relational semantics. In this way, we obtain our first algebraic characterization of the logic of intuitionistic types. We further characterize the class of all homomorphic images of “Kripkean algebras” that we call “inquisitive algebras”, thus obtaining a second, more general algebraic semantics for this logic.

Kit Fine’s truthmaker semantics allows us to distinguish between implying and containing: A implies B iff every truthmaker of A includes (as a part) a truthmaker of B, and A contains B iff every truthmaker of B is included (as a part) in a truthmaker of A. These are two different notions that can both be transformed into a proper object language connective. While the former notion amounts to the usual notion of implication, the latter one leads to a novel connective with interesting logical properties. In this paper we put forward a truthmaker semantics for the implication of the relevant logic R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathsf {R}$$ \end{document} and we accompany it with the corresponding “containing” operator. This framework will allow us to relate two seemingly very different kinds of relevant implication that has so far been studied separately, namely the implication of the logic R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathsf {R}$$ \end{document} and Angell’s analytic containment. Our approach semantically determines a logic that we call RAC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathsf {RAC}$$ \end{document}. We explore its connection to R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathsf {R}$$ \end{document}, the logic of analytic containment AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathsf {AC}$$ \end{document}, Urquhart’s operational semantics and inquisitive semantics.

This paper is a reaction to Christopher Gauker’s book Conditionals in Context. Gauker’s semantics of assertibility and deniability will be reconstructed and one peculiar aspect of the semantics will be pointed to: connectives are sensitive to the syntactic structure of the formulas they connect. Even though this is a well motivated principle which can be supported by many examples from natural languages, one of its unwanted consequences is that the resulting formal semantics is not compositional. In this paper, Gauker’s semantics will be modified and applied to Stalnaker’s concept of context. In this framework, every sentential connective will be replaced with a pair of connectives one of which will be called extensional and the other intensional. This distinction enables us to have an adequate and compositional semantics of assertibility and deniability. We will provide also a syntactic characterization of the logic determined by this semantics.

The paper extends a framework of information-based semantics for intuitionistic logic with a paraconsistent negation and consistency-sensitive epistemic modalities. In this framework information states represent information collected from various sources and as such they can be inconsistent because they receive contradictory information either from a single inconsistent source or from various mutually incompatible sources. The modalities reflect only those sources that are consistent and trusted. For the paraconsistent logic generated by this framework, we present a Hilbert style system, we prove completeness, the disjunction property, the finite model property and various other results.

Fine (J Philos Log 43:549–577, 2014) developed a truthmaker semantics for intuitionistic logic, which is also called exact semantics, since it is based on a relation of exact verification between states and formulas. A natural question arises as to what are the limits of Fine’s approach and whether an exact semantics of similar kind can be constructed for other important non-classical logics. In our paper, we will generalize Fine’s approach and develop an exact semantics for some substructural logics. In particular, we will provide a truthmaker semantics for the Non-associative Lambek calculus and some of its extensions. This generalization will reveal some interesting connections between Fine’s recent work on truthmaker semantics and his early work on relevant logic.

In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics involving also systems that are not closed under uniform substitution. We just require that each logic must be closed under D-substitutions assigning to atomic formulas only ∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vee $$\end{document}-free formulas. For these systems we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic inquisitive logics that validate a schema called Split and have the disjunction property. In these logics disjunction can be interpreted in the sense of inquisitive semantics as a question forming operator. It is shown that a logic is structurally complete with respect to D-substitutions if and only if it includes the weakest superintuitionistic inquisitive logic. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built up from D-substitutions. Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al..

This paper explores the relation between the philosophical idea that logic is a science studying logical forms, and a mathematical feature of logical systems called the principle of uniform substitution, which is often regarded as a technical counterpart of the philosophical idea. We argue that at least in one interesting sense the principle of uniform substitution does not capture adequately the requirement that logic is a matter of form and that logical truths are formal truths. We show that some specific logical expressions can produce propositions of different kinds and the resulting diversity of informational types can lead to a justified failure of uniform substitution without undermining the view that logic is a purely formal discipline.

V tomto článku předkládám některé důvody proti Dvořákovu nedeterministickému pojetí svobodné vůle, které vyložil ve své nedávno publikované knize Kauzalita činitele. Zejména se zabývám Dvořákovým řešením problému náhody, podle kterého je akt svobodné vůle náhodný do té míry, do jaké se nepodřizuje deterministickým zákonům, a do této míry také nemůže být předmětem morálního hodnocení. Proti Dvořákově koncepci stavím vlastní pojetí svobodné vůle, které je variací na to, co Dvořák označuje jako „standardní teorie“. Z hlediska tohoto pojetí je svobodná vůle možná i v deterministickém světě. Analyzuji, v čem spočívá volba mezi alternativními možnostmi, a argumentuji, že je možné mít kontrolu nad vlastním jednáním, i když nemáme kontrolu nad minulostí, která toto jednání jednoznačně determinuje. Zároveň vysvětluji, proč se v ostrém rozporu s Dvořákovou koncepcí domnívám, že člověk v deterministickém světě může nést odpovědnost za své činy.

We investigate an information based generalization of the incompatibility-frame treatment of logics with non-classical negation connectives. Our framework can be viewed as an alternative to the neighbourhood semantics for extensions of lattice logic by various negation connectives, investigated by Hartonas. We set out the basic semantic framework, along with some correspondence results for extensions. We describe three kinds of constructions of canonical models and show that double negation law is not canonical with respect to any of these constructions. We also compare our semantics to Hartonas’.

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra.

A conversation can be conceived as aiming to circumscribe a set of possibilities that are relevant to the goals of the conversation. This set of possibilities may be conceived as determined by the goals and objective circumstances of the interlocutors and not by their propositional attitudes. An indicative conditional can be conceived as circumscribing a set of possibilities that have a certain property: If the set of relevant possibilities is subsequently restricted to one in which the antecedent holds, then it will be restricted as well to one in which the consequent holds. We will identify a number of desiderata concerning the validity of arguments; we will develop a formally precise semantics for conditionals conceived in this way that satisfies the desiderata, and we will present a deductive calculus that is sound and complete with respect to the semantics. Finally, we will argue that the semantics compares well, both formally and foundationally, with two other semantic theories of indicative conditionals that satisfy the desiderata, namely, those of Gillies and Bledin.

This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is also discussed.

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The main result of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed

In this paper, we present Carnap’s modal logic C, which is one of the first attempts to use the concept of possible world in shaping the semantics for modalities. Some older technical results, which concern the logic C, are summarized, namely two different kinds of axiomatization of C, one unusual characterization of C as the only set of formulae having one special property, and semantical and syntactical relations of C to S5. The fact that C is not closed under the universal substitution is shortly discussed. Finally, the predicate version of C, which is not axiomatizable, is defined