Showing papers in "Studia Logica in 2021"

Three ways of being non-material

Abstract: This paper develops a probabilistic analysis of conditionals which hinges on a quantitative measure of evidential support. In order to spell out the interpretation of ‘if’ suggested, we will compare it with two more familiar interpretations, the suppositional interpretation and the strict interpretation, within a formal framework which rests on fairly uncontroversial assumptions. As it will emerge, each of the three interpretations considered exhibits specific logical features that deserve separate consideration.

Containment logics: algebraic completeness and axiomatization

Abstract: The paper studies the containment companion (or, right variable inclusion companion) of a logic $$\vdash $$ . This consists of the consequence relation $$\vdash ^{r}$$ which satisfies all the inferences of $$\vdash $$ , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the Plonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.

Free Logics are Cut-Free

Abstract: The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is proved in a constructive way for sequent calculi adequate for all logics under consideration.

The Poset of All Logics III: Finitely Presentable Logics

Abstract: A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz and the Maltsev hierarchies.

Cut-free Sequent Calculus and Natural Deduction for the Tetravalent Modal Logic

Abstract: The tetravalent modal logic ( $${\mathcal {TML}}$$ ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic $${{\mathcal {TML}}}^N$$ ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for $${\mathcal {TML}}$$ with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.

The Entropy-Limit (Conjecture) for $$\Sigma _2$$ Σ 2 -Premisses

Abstract: The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: (i) applying it to finite sublanguages and taking a limit; (ii) comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some $$\Sigma _2$$ -premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture.

Axiomatization of Crisp Gödel Modal Logic

Abstract: In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Godel algebra $$[0,1]_G$$ . We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get $$\Diamond $$ as an abbreviation of $$\Box $$ , nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

Non-classical Models of ZF

Abstract: This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of $$\mathsf {ZF}$$ . Then, we build lattice-valued models of full $$\mathsf {ZF}$$ , whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from $$\mathsf {ZF}$$ .

Semi De Morgan Logic Properly Displayed

Abstract: In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi.

A Generalized Proof-Theoretic Approach to Logical Argumentation Based on Hypersequents

Abstract: In this paper we introduce hypersequent-based frameworks for the modelling of defeasible reasoning by means of logic-based argumentation and the induced entailment relations. These structures are an extension of sequent-based argumentation frameworks, in which arguments and the attack relations among them are expressed not only by Gentzen-style sequents, but by more general expressions, called hypersequents. This generalization allows us to overcome some of the known weaknesses of logical argumentation frameworks and to prove several desirable properties of the entailments that are induced by the extended (hypersequent-based) frameworks. It also allows us to incorporate as the deductive base of our formalism some well-known logics (like the intermediate logic LC, the modal logic S5, and the relevance logic RM), which lack cut-free sequent calculi, and so are not adequate for standard sequent-based argumentation. We show that hypersequent-based argumentation yields robust defeasible variants of these logics, with many desirable properties.

Relational Representation Theorems for Extended Contact Algebras

Abstract: In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we study the correspondences between point-free and point-based models of space in terms of extended contact. More precisely, we prove new representation theorems for extended contact algebras.

A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication

Abstract: We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

Incompleteness and the Halting Problem

Abstract: We present an abstract framework in which we give simple proofs for Godel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’, Chaitin’s and Kritchman-Raz’s Theorems.

Inquisitive Heyting Algebras

Abstract: 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 (prime elements represent declarative propositions, non-prime elements represent questions, join is a question-forming operation) 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.

Correction to: The Hahn Embedding Theorem for a Class of Residuated Semigroups

Abstract: Let be the class of odd involutive even the notion of partial lex products is not sufficiently general. One more tweak is needed, a slightly even more complex construction, called partial sublex product, introduced here.

A Binary Quantifier for Definite Descriptions for Cut Free Free Logics

Abstract: This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.

Pooling Modalities and Pointwise Intersection: Axiomatization and Decidability

Abstract: We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein (Pooling modalities and pointwise intersection: semantics, expressivity, and applications). The definition of our canonical models combines standard techniques with a so-called “puzzle piece construction”, which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique’s potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models.

Belnap–Dunn Modal Logic with Value Operators

Abstract: The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics for $${\mathscr {L}}_0$$ and its two expansions $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ with value operators. The local finitary consequence relation $$\models _4^k$$ in the language $${\mathscr {L}}_k$$ with respect to the class of all frames is axiomatized by a sequent system $$\mathsf {S}_k$$ where $$k=0, 1, 2$$ . We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language $${\mathscr {L}}_0$$ under the relational semantics for classical modal logic.

A Conservative Negation Extension of Positive Semilattice Logic Without the Finite Model Property

Abstract: In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

Sequent-Calculi for Metainferential Logics

Abstract: In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as $$\mathbf {ST}, \mathbf {LP}, \mathbf {K_3}$$ , and $$\mathbf {TS}$$ . What is distinctive of these metainferential logics is that they are mixed, i.e. the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics $$\mathbf {ST}, \mathbf {LP}, \mathbf {K_3}$$ and $$\mathbf {TS}$$ , and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level.

Transitive Logics of Finite Width with Respect to Proper-Successor-Equivalence

Abstract: This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The frame condition for each prefinite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points that have a finite lower bound of depth and have different proper successors. We will construct continuums of transitive logics of finite suc-eq-width but not of finite width, and continuums of those of prefinite suc-eq-width but not of prefinite width. This shows that our new completeness results cover uncountably many more logics than Fine’s theorem and Rybakov’s theorem respectively.

Recapturing Dynamic Logic of Relation Changers via Bounded Morphisms

Abstract: The present contribution shows that a Hilbert-style axiomatization for dynamic logic of relation changers is complete for the standard Kripke semantics not by a well-known rewriting technique but by the idea of an auxiliary semantics studied by van Benthem and Wang et al. A key insight of our auxiliary semantics for dynamic logic of relation changers can be described as: “relation changers are bounded morphisms.” Moreover, we demonstrate that this semantic insight can be used to provide a modular cut-free labelled sequent calculus for the logic in the sense that our calculus can be regarded as a natural expansion of a labelled sequent calculus of iteration-free propositional dynamic logic.

Residuated Structures and Orthomodular Lattices

Abstract: The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

Labelled Sequent Calculi for Lewis’ Non-normal Propositional Modal Logics

Abstract: C. I. Lewis’ systems were the first axiomatisations of modal logics. However some of those systems are non-normal modal logics, since they do not admit a full rule of necessitation, but only a restricted version thereof. We provide G3-style labelled sequent calculi for Lewis’ non-normal propositional systems. The calculi enjoy good structural properties, namely admissibility of structural rules and admissibility of cut. Furthermore they allow for straightforward proofs of admissibility of the restricted versions of the necessitation rule. We establish completeness of the calculi and we discuss also related systems.

Lattices of Finitely Alternative Normal Tense Logics

Abstract: A finitely alternative normal tense logic $$T_{n,m}$$ is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice $$\Lambda (T_{1,1})$$ is described. There are $$\aleph _0$$ logics in $$\Lambda (T_{1,1})$$ without the finite model property (FMP), and only one pretabular logic in $$\Lambda (T_{1,1})$$ . There are $$2^{\aleph _0}$$ logics in $$\Lambda (T_{1,1})$$ which are not finitely axiomatizable. For $$nm\ge 2$$ , there are $$2^{\aleph _0}$$ logics in $$\Lambda (T_{n,m})$$ without the FMP, and infinitely many pretabular extensions of $$T_{n,m}$$ .

Idempotent Variations on the Theme of Exclusive Disjunction

Abstract: An exclusive disjunction is true when exactly one of the disjuncts is true. In the case of the familiar binary exclusive disjunction, we have a formula occurring as the first disjunct and a formula occurring as the second disjunct, so, if what we have is two formula-tokens of the same formula-type—one formula occurring twice over, that is—the question arises as to whether, when that formula is true, to count the case as one in which exactly one of the disjuncts is true, counting by type, or as a case in which two disjuncts are true, counting by token. The latter is the standard answer: counting by tokens. James McCawley once suggested that, when the exclusively disjunctive construction in natural language (well, in English at least) is at issue, the construction should be treated as involving a multigrade connective whose semantic treatment is sensitive to the set of disjuncts rather than the corresponding multiset. Without any commitment as to whether there actually is such a construction (in English), and conceding that for obvious pragmatic reasons such ‘repeated disjunct’ cases would be at best highly marginal, we note that for the binary case, this requires a nonstandard answer—count by type rather than by token—to the earlier question, and thus, an idempotent exclusive disjunction connective. Section 2 explores that idea and Section 3, a further idempotent variant for which it is the propositions expressed by the disjuncts, rather than the disjuncts themselves, that get counted once only in the case of repetitions. Sections 1 and 4 respectively set the stage for these investigations and conclude the discussion (after noting an intimate connection between the logic of Section 3 and the modal logic S5). More detailed considerations of points arising from the discussion but otherwise in danger of interrupting the flow are deferred to a ‘Longer Notes’ appendix at the end (Section 5.)

Measuring Inconsistency in Some Logics with Modal Operators

Abstract: The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

Lambek Calculus with Conjugates

Abstract: We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

Sequent Calculi for the Propositional Logic of HYPE

Abstract: In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Endogenizing Epistemic Actions

Abstract: Through a series of examples, we illustrate some important drawbacks that the action model logic framework suffers from in its ability to represent the dynamics of information updates. We argue that these problems stem from the fact that the action model, a central construct designed to encode agents’ uncertainty about actions, is itself effectively common knowledge amongst the agents. In response to these difficulties, we motivate and propose an alternative semantics that avoids them by (roughly speaking) endogenizing the action model. We discuss the relationship between this new framework and action model logic, and provide a sound and complete axiomatization of several new logics that naturally arise.