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- Unary Interpretability Logics for Sublogics of the Interpretability Logic
$$\textbf{IL}$$-
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Abstract: Abstract De Rijke introduced a unary interpretability logic \(\textbf{il}\) , and proved that \(\textbf{il}\) is the unary counterpart of the binary interpretability logic \(\textbf{IL}\) . In this paper, we find the unary counterparts of the sublogics of \(\textbf{IL}\) .
PubDate: 2023-09-20
- Semisimplicity and Congruence 3-Permutabilty for Quasivarieties with
Equationally Definable Principal Congruences-
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Abstract: Abstract We show that the properties of [relative] semisimplicity and congruence 3-permutability of a [quasi]variety with equationally definable [relative] principal congruences (EDP[R]C) can be characterized syntactically. We prove that a quasivariety with EDPRC is relatively semisimple if and only if it satisfies a finite set of quasi-identities that is effectively constructible from any conjunction of equations defining relative principal congruences in the quasivariety. This in turn allows us to obtain an ‘axiomatization’ of relatively filtral quasivarieties. We also show that a variety is 3-permutable and has EDPC if and only if there is a single pair of quaternary terms satisfying two simple equations, and whose equality defines principal congruences in the variety. Finally, we combine both results to obtain a neat characterization of semisimple, 3-permutable varieties with EDPC, which is applied to solve a problem posed by Blok and Pigozzi in the third paper of their series on varieties with EDPC.
PubDate: 2023-09-20
- Boolean Connexive Logic and Content Relationship
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Abstract: Abstract We present here some Boolean connexive logics (BCLs) that are intended to be connexive counterparts of selected Epstein’s content relationship logics (CRLs). The main motivation for analyzing such logics is to explain the notion of connexivity by means of the notion of content relationship. The article consists of two parts. In the first one, we focus on the syntactic analysis by means of axiomatic systems. The starting point for our syntactic considerations will be the smallest BCL and the smallest CRL. In the first part, we also identify axioms of Epstein’s logics that, together with the connexive principles, lead to contradiction. Moreover, we present some principles that will be equivalent to the connexive theses, but not to the content connexive theses we will propose. In the second part, we focus on the semantic analysis provided by relating- and set-assignment models. We define sound and complete relating semantics for all tested systems. We also indicate alternative relating models for the smallest BCL, which are not alternative models of the connexive counterparts of the considered CRLs. We provide a set-assignment semantics for some BCLs, giving thus a natural formalization of the content relationship understood either as content sharing or as content inclusion.
PubDate: 2023-09-20
- Decidability of Lattice Equations
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Abstract: Abstract We provide an alternative proof of the decidability of the equational theory of lattices. The proof presented here is quite short and elementary.
PubDate: 2023-09-15
- Intuitionistic Public Announcement Logic with Distributed Knowledge
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Abstract: Abstract We develop intuitionistic public announcement logic over intuitionistic \({\textbf{K}}\) , \({{\textbf{K}}}{{\textbf{T}}}\) , \({{\textbf{K}}}{{\textbf{4}}}\) , and \({{\textbf{S}}}{{\textbf{4}}}\) with distributed knowledge. We reveal that a recursion axiom for the distributed knowledge is not valid for a frame class discussed in [12] but valid for the restricted frame class introduced in [20, 26]. The semantic completeness of the static logics for this restricted frame class is established via the concept of pseudo-model.
PubDate: 2023-09-15
- Intuitionistic Modal Algebras
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Abstract: Abstract Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.
PubDate: 2023-09-15
- Correspondence Theory for Modal Fairtlough–Mendler Semantics of
Intuitionistic Modal Logic-
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Abstract: Abstract We study the correspondence theory of intuitionistic modal logic in modal Fairtlough–Mendler semantics (modal FM semantics) (Fairtlough and Mendler in Inf Comput 137(1):1–33, 1997), which is the intuitionistic modal version of possibility semantics (Holliday in UC Berkeley working paper in logic and the methodology of science, 2022. http://escholarship.org/uc/item/881757qn). We identify the fragment of inductive formulas (Goranko and Vakarelov in Ann Pure Appl Logic 141(1–2):180–217, 2006) in this language and give the algorithm \(\textsf{ALBA}\) (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012) in this semantic setting. There are two major features in the paper: one is that in the expanded modal language, the nominal variables, which are interpreted as atoms in perfect Boolean algebras, complete join-prime elements in perfect distributive lattices and complete join-irreducible elements in perfect lattices, are interpreted as the refined regular open closures of singletons in the present setting, similar to the possibility semantics for classical normal modal logic (Zhao in J Logic Comput 31(2):523–572, 2021); the other feature is that we do not use conominals or diamond, which restricts the fragment of inductive formulas significantly. We prove the soundness of \(\textsf{ALBA}\) with respect to modal FM-frames and show that \(\textsf{ALBA}\) succeeds on inductive formulas, similar to existing settings like (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012; Zhao 2021, in: Cia-battoni, Pimentel, Queiroz (eds) Logic, language, information, and computation, Springer International Publishing, Cham, 2022).
PubDate: 2023-08-29
- Hyperintensionality and Fine-Grained Semantics
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PubDate: 2023-08-10
- Systems for Non-Reflexive Consequence
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Abstract: Abstract Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.
PubDate: 2023-08-10
- From Belnap-Dunn Four-Valued Logic to Six-Valued Logics of Evidence and
Truth-
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Abstract: Abstract The main aim of this paper is to introduce the logics of evidence and truth \(LET_{K}^+\) and \(LET_{F}^+\) together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics \(LET_{K}\) and \(LET_{F}^-\) with rules of propagation of classicality, which are inferences that express how the classicality operator \({\circ }\) is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for \(LET_{K}\) is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for \(LET_{K}^+\) is then obtained by imposing restrictions on the semantics of \(LET_{K}\) . These restrictions correspond exactly to the rules of propagation of classicality that extend \(LET_{K}\) . The logic \(LET_{F}^+\) is obtained as the implication-free fragment of \(LET_{K}^+\) . We also show that the 6 values of \(LET_{K}^+\) and \(LET_{F}^+\) define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that \(LET_{K}^+\) is Blok-Pigozzi algebraizable and that its implication-free fragment \(LET_{F}^+\) coincides with the degree-preserving logic of the involutive Stone algebras.
PubDate: 2023-08-09
- Correction to: Group Representation for Even and Odd Involutive
Commutative Residuated Chains-
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PubDate: 2023-08-03
- The Elimination of Direct Self-reference
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Abstract: Abstract This paper provides a procedure which, from any Boolean system of sentences, outputs another Boolean system called the ‘m-cycle unwinding’ of the original Boolean system for any positive integer m. We prove that for all \(m>1\) , this procedure eliminates the direct self-reference in that the m-cycle unwinding of any Boolean system must be indirectly self-referential. More importantly, this procedure can preserve the primary periods of Boolean paradoxes: whenever m is relatively prime to all primary periods of a Boolean paradox, this paradox and its m-cycle unwinding have the same primary periods. In this way, we can produce an indirectly self-referential Boolean paradox with the same periodic characteristics as a known Boolean paradox.
PubDate: 2023-08-03
- A First-Order Expansion of Artemov and Protopopescu’s Intuitionistic
Epistemic Logic-
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Abstract: Abstract Intuitionistic epistemic logic by Artemov and Protopopescu (Rev Symb Log 9:266–298, 2016) accepts the axiom “if A, then A is known” (written \(A \supset K A\) ) in terms of the Brouwer–Heyting–Kolmogorov interpretation. There are two variants of intuitionistic epistemic logic: one with the axiom “ \(KA \supset \lnot \lnot A\) ” and one without it. The former is called \(\textbf{IEL}\) , and the latter is called \(\textbf{IEL}^{-}\) . The aim of this paper is to study first-order expansions (with equality and function symbols) of these two intuitionistic epistemic logics. We define Hilbert systems with additional axioms called geometric axioms and sequent calculi with the corresponding rules to geometric axioms and prove that they are sound and complete for the intended semantics. We also prove the cut-elimination theorems for both sequent calculi. As a consequence, the disjunction property and existence property are established for the sequent calculi without geometric implications. Finally, we establish that our sequent calculi can also be formulated with admissible structural rules (i.e., in terms of a G3-style sequent calculus).
PubDate: 2023-08-01
- Tense Operators on Distributive Lattices with Implication
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Abstract: Abstract Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters and tense centered deductive systems, respectively.
PubDate: 2023-08-01
- Williamson’s Abductive Case for the Material Conditional Account
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Abstract: Abstract In Suppose and Tell, Williamson makes a new and original attempt to defend the material conditional account of indicative conditionals. His overarching argument is that this account offers the best explanation of the data concerning how people evaluate and use such conditionals. We argue that Williamson overlooks several important alternative explanations, some of which appear to explain the relevant data at least as well as, or even better than, the material conditional account does. Along the way, we also show that Williamson errs at important junctures about what exactly the relevant data are.
PubDate: 2023-08-01
- Semantical Analysis of the Logic of Bunched Implications
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Abstract: Abstract We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a meta-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth’s clause for disjunction instead of Kripke’s). It is this complexity that motivates us to use BI as a case-study for this approach to semantics.
PubDate: 2023-08-01
- On Boolean Algebraic Structure of Proofs: Towards an Algebraic Semantics
for the Logic of Proofs-
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Abstract: Abstract We present algebraic semantics for the classical logic of proofs based on Boolean algebras. We also extend the language of the logic of proofs in order to have a Boolean structure on proof terms and equality predicate on terms. Moreover, the completeness theorem and certain generalizations of Stone’s representation theorem are obtained for all proposed algebras.
PubDate: 2023-08-01
- A Simple Way to Overcome Hyperconnexivity
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Abstract: Abstract The term ‘hyperconnexive logic’ (or ‘hyperconnexivity’ in general) in relation to a certain logical system was coined by Sylvan to indicate that not only do Boethius’ theses hold in such a system, but also their converses. The plausibility of the latter was questioned by some connexive logicians. Without going into the discussion regarding the plausibility of hyperconnexivity and the converses of Boethius’ theses, this paper proposes a quite simple way to escape the hyperconnexivity within the semantic framework of Wansing-style constructive connexive logics. In particular, we present a working method for escaping hyperconnexivity of constructive connexive logic \({{\textbf{C}}}\) , discuss the problem that creates an obstacle to using the same method in the case of logic \({{\textbf{C3}}}\) and provide a possible solution to this problem that allows us to construct a logical theory which is similar to \({{\textbf{C3}}}\) and free from hyperconnexivity. All new logics introduced in this paper are equipped with sound and complete Hilbert-style calculi, and their relationships with other well-known connexive logics are discussed.
PubDate: 2023-07-05
DOI: 10.1007/s11225-023-10056-3
- Executability and Connexivity in an Interpretation of Griss
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Abstract: Abstract Although the work of G.F.C. Griss is commonly understood as a program of negationless mathematics, close examination of Griss’s work suggests a more fundamental feature is its executability, a requirement that mental constructions are possible only if corresponding mental activity can be actively carried out. Emphasizing executability reveals that Griss’s arguments against negation leave open several types of negation—including D. Nelson’s strong negation—as compatible with Griss’s intuitionism. Reinterpreting Griss’s program as one of executable mathematics, we iteratively develop a pair of bilateral constructive logics and argue for their adequacy as accounts of the propositional basis of Griss’s work. We conclude by observing connexive features exhibited by the two bilateral logics and by investigating the difficulties connexive principles reveal for the development of executable mathematics.
PubDate: 2023-07-05
DOI: 10.1007/s11225-023-10055-4
- Petr Cintula, Carles Noguera, Logic and Implication. An Introduction to
the General Algebraic Study of Non-classical Logics, vol. 57 of Trends in
Logic, Springer, 2021, pp. 465+xxii; ISBN: 978-3-030-85674-8 (Hardcover)
117.69€, ISBN: 978-3-030- 85675-5 (eBook) 93.08 €.-
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PubDate: 2023-06-12
DOI: 10.1007/s11225-023-10050-9
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