Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics

Caleiro, Carlos; Marcos, João; Volpe, Marco

doi:10.1016/j.tcs.2015.07.016

Cited by 25 publications

(28 citation statements)

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“…It may be observed that: (i) agents may not be aware even of easy consequences of their assumptions and there is still a difference between implicit and explicit information, 11 (ii) even implicit information that can be feasibly extracted from the explicit one requires the consumption of resources. Both observations raise interesting questions that we do not address here.…”

Section: Among Others)mentioning

confidence: 99%

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An informational view of classical logic

D’Agostino

2015

Theoretical Computer Science

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We present an informational view of classical propositional logic that stems from a kind of informational semantics whereby the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent. In this view the inferential power of logical agents is naturally bounded by their limited capability of manipulating "virtual information", namely information that is not implicitly contained in the data. Although this informational semantics cannot be expressed by any finitely-valued matrix, it can be expressed by a non-deterministic 3-valued matrix that was first introduced by W.V.O. Quine, but ignored by the logical community. Within the general framework presented in [21] we provide an in-depth discussion of this informational semantics and a detailed analysis of a specific infinite hierarchy of tractable approximations to classical propositional logic that is based on it. This hierarchy can be used to model the inferential power of resource-bounded agents and admits of a uniform proof-theoretical characterization that is half-way between a classical version of Natural Deduction and the method of semantic tableaux.Keywords: Classical Propositional Logic, Informational Semantics, Non-deterministic matrices, Computational Complexity, Natural Deduction sep Semantic Tableaux.$ This paper elaborates on ideas and results arising from my collaboration with Marcelo Finger, Luciano Floridi and Dov Gabbay [22,21] and is based on an invited talk given at the 8th Workshop on Logical and Semantic Frameworks with Applications (LSFA) held in São Paulo on 2-3 September 2013. A preliminary and partial version was published in the proceedings of the workshop [20]. I wish to thank Maribel Fernandez and Marcelo Finger for inviting me to the workshop and all the participants for an interesting discussion. I also wish to thanks two anonymous referees for valuable comments and suggestions.

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Section: Among Others)mentioning

confidence: 99%

“…We shall 11 This issue is interestingly related to the vast literature on awareness (see, for example, [42,30,41,49]). 12 In fact, in this context, the signs "T " and "F " act as propositional attitudes and, in a multiagent setting, can be indexed by symbols "x, y, z, ..." standing for different agents.…”

Section: Among Others)mentioning

confidence: 99%

An informational view of classical logic

D’Agostino

2015

Theoretical Computer Science

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“…As a practical application of that idea, the Suszko Reduction, seen as the technical counterpart of Suszko's Thesis, has nowadays been given a fully algorithmic implementation that applies to any finite-valued logic, and this has been used to provide uniform classic-like analytic deductive counterparts to all such logics (cf. [5]).…”

Section: Introductionmentioning

confidence: 99%

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

2017

Self Cite

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Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko's Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued.An Inferentially Many-Valued Two-Dimensional Notion of Entailment 235 of a p-matrix [7,8]. A q-matrix (quasi matrix) is a structure A, D + , D − , where A is an algebra similar to a propositional language L, and where D + and D − are subsets of A, and D + ∩ D − = ∅. Usually, D + is referred to as the set of designated values and D − as the set of antidesignated values. A p-matrix (plausibility matrix) is a structure A, D + , D * , where A is an algebra similar to a propositional language L and D + ⊆ D * ⊆ A. The set D * is usually referred to as the set of plausible, non-antidesignated values.We adopt a compact notation that avoids superscripts and the barnotation for set-theoretic complementation, introducing the symbols Y, Y , N, and N to denote, respectively, the sets of designated, non-designated (V \ Y), antidesignated, and non-antidesignated (V \N) values. With a cognitive twist, they might be taken as representing acceptance, non-acceptance, rejection and non-rejection. 1

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“…We do not present actual Suszko reductions of specific logics for that matter, but mostly use the notion toward irreducibility arguments. In that regard, our perspective differs from the important work of Caleiro and Marcos (2012) and Caleiro et al (2015), who propose a constructive method to compute Suszko reductions from arbitrary semantics. Their approach is in a sense more general, because it is not restricted to logics for which the reduction preserves truth-functionality, and it is output-oriented.…”

Section: Ranks and Truth-functional Reductionmentioning

confidence: 90%

“…Suszko gave a precise answer to that problem for a large class of consequence relations. He proved that every Tarskian relation, namely every reflexive, monotonic, and transitive consequence relation, can be characterized by a class of two-valued models (Caleiro and Marcos, 2012;Caleiro et al, 2015;Suszko, 1977;Tsuji, 1998;Wansing and Shramko, 2008). Suszko's problem and his result are of fundamental importance to understand the role and nature of truth values in logic.…”

Section: Suszko's Problemmentioning

confidence: 99%

Suszko’s Problem: Mixed Consequence and Compositionality

Chemla

Égré

2019

The Review of Symbolic Logic

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Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.by Tarski and the Polish school (see Bloom et al., 1970;Tarski, 1930;Wójcicki, 1973Wójcicki, , 1988, namely: substitution-invariance, monotonicity, reflexivity and transitivity. Those properties play a central role in Suszko's reduction. We then articulate what we mean by a semantics for a logic, and state various semantic properties which we will prove to be counterparts of those structural properties in section 3. Syntactic notions LogicThroughout the paper, we work within a multiple-premise multiple-conclusion logic, following the tradition of Gentzen (1935), Scott (1974) and Shoesmith and Smiley (1978). One of the reasons for doing so is simplicity in that premises and conclusions are handled symmetrically. This choice is not neutral and some of our results are likely to differ in a multi-premise single-conclusion setting. We leave that issue aside, and define a logic as follows:Definition 2.1 (Logic). A logic is a triple L, C, , with L a language (set of formulae), C a (possibly empty) set of connectives, and a consequence relation, what we call a formula-relation (i.e. a subset of P (L) × P (L), also called a set of arguments, where ...

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Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics

Cited by 25 publications

References 25 publications

An informational view of classical logic

An informational view of classical logic

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

Suszko’s Problem: Mixed Consequence and Compositionality

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