Studia Logica最新文献

In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (LFI) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such LFIs. Here, we intend to make a first step in this direction. On the other hand, the logic Ciore was developed to provide new logical systems in the study of inconsistent databases from the point of view of LFIs. An interesting fact about Ciore is that it has a strong consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of Ciore, namely QCiore, was defined preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both Ciore and QCiore respectively. In first place, we introduce a two-sided sequent system for Ciore. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique.

In this paper, we discuss semantical properties of the logic (textbf{GL}) of provability. The logic (textbf{GL}) is a normal modal logic which is axiomatized by the the Löb formula ( Box (Box psupset p)supset Box p ), but it is known that (textbf{GL}) can also be axiomatized by an axiom (Box psupset Box Box p) and an (omega )-rule ((Diamond ^{*})) which takes countably many premises (phi supset Diamond ^{n}top ) ((nin omega )) and returns a conclusion (phi supset bot ). We show that the class of transitive Kripke frames which validates ((Diamond ^{*})) and the class of transitive Kripke frames which strongly validates ((Diamond ^{*})) are equal, and that the following three classes of transitive Kripke frames, the class which validates ((Diamond ^{*})), the class which weakly validates ((Diamond ^{*})), and the class which is defined by the Löb formula, are mutually different, while all of them characterize (textbf{GL}). This gives an example of a proof system P and a class C of Kripke frames such that P is sound and complete with respect to C but the soundness cannot be proved by simple induction on the height of the derivations in P. We also show Kripke completeness of the proof system with ((Diamond ^{*})) in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations (Box xle Box Box x) and (bigwedge _{nin omega }Diamond ^{n}1=0) is not a variety.

Abstract

Recent work by Busaniche, Galatos and Marcos introduced a very general twist construction, based on the notion of conucleus, which subsumes most existing approaches. In the present paper we extend this framework one step further, so as to allow us to construct and represent algebras which possess a negation that is not necessarily involutive. Our aim is to capture the main properties of the largest class that admits such a representation, as well as to be able to recover the well-known cases—such as (quasi-)Nelson algebras and (quasi-)N4-lattices—as particular instances of the general construction. We pursue two approaches, one that directly generalizes the classical Rasiowa construction for Nelson algebras, and an alternative one that allows us to study twist-algebras within the theory of residuated lattices.

Abstract

This paper studies a combined system of intuitionistic and classical propositional logic from proof-theoretic viewpoints. Based on the semantic treatment of Humberstone (J Philos Log 8:171–196, 1979) and del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996), a sequent calculus (textsf{G}(textbf{C}+textbf{J})) is proposed. An approximate idea of obtaining (textsf{G}(textbf{C}+textbf{J})) is adding rules for classical implication on top of the intuitionistic multi-succedent sequent calculus by Maehara (Nagoya Math J 7:45–64, 1954). However, in the semantic treatment, some formulas do not satisfy heredity, which leads to the necessity of a restriction on the right rule for intuitionistic implication to keep the soundness of the calculus. The calculus (textsf{G}(textbf{C}+textbf{J})) enjoys cut elimination and Craig interpolation, whose detailed proofs are described in this paper. Cut elimination enables us to show the decidability of this combination both directly and syntactically. This paper also employs a canonical model argument to establish the strong completeness of Hilbert system (mathbf {C+J}) proposed by del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996).

In this paper we propose and defend the Synonymy account, a novel account of metaphysical equivalence which draws on the idea (Rayo in The Construction of Logical Space, Oxford University Press, Oxford, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous—and so, in our view, equivalent—just in case (i) they take the same propositions to stand in the same entailment relations, and (ii) they are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the Synonymy account fares better than some of its competitors, specifically, McSweeney’s (Philosophical Perspectives 30(1):270–293, 2016) epistemic account and Miller’s (Philosophical Quarterly 67(269):772–793, 2017) hyperintensional account.

Over the past ten years, the community researching connexive logics is rapidly growing and a number of papers have been published. However, when it comes to the terminology used in connexive logic, it seems to be not without problems. In this introduction, we aim at making a contribution towards both unifying and reducing the terminology. We hope that this can help making it easier to survey and access the field from outside the community of connexive logicians. Along the way, we will make clear the context to which the papers in this special issue on Frontiers of Connexive Logic belong and contribute.

We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form (langle H, mu rangle ) that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) (mu : H longrightarrow [0,1]_{mathbb {Q}}) satisfies the following condition: if (alpha ), (beta ), (alpha wedge beta ), (alpha vee beta in H), then (mu (alpha vee beta ) = mu (alpha ) + mu (beta ) - mu (alpha wedge beta )). Since the range of (mu ) is the set ([0,1]_{mathbb {Q}}) of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability.

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper (Chajda, Länger and Paseka, in: Proceeding of 2022 IEEE 52th International Symposium on Multiple-Valued Logic, Springer, 2022), we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset ({textbf{A}}), namely its Dedekind-MacNeille completion ({{,mathrm{textbf{DM}},}}({textbf{A}})) and a completion (G({textbf{A}})) which coincides with ({{,mathrm{textbf{DM}},}}({textbf{A}})) provided ({textbf{A}}) is finite. In particular we prove that if ({textbf{A}}) is a Kleene poset then its extension (G({textbf{A}})) is also a Kleene lattice. If the subset X of principal order ideals of ({textbf{A}}) is involution-closed and doubly dense in (G({textbf{A}})) then it generates (G({textbf{A}})) and it is isomorphic to ({textbf{A}}) itself.

We exhibit infinitely many semisimple varieties of semilinear De Morgan monoids (and likewise relevant algebras) that are not tabular, but which have only tabular proper subvarieties. Thus, the extension of relevance logic by the axiom ((prightarrow q)vee (qrightarrow p)) has infinitely many pretabular axiomatic extensions, regardless of the presence or absence of Ackermann constants.