What is a Non-truth-functional Logic?

Marcos, João

doi:10.1007/s11225-009-9196-z

Cited by 24 publications

(19 citation statements)

References 30 publications

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“…If C is the set of connectives of L, then A, D can be presented as a tuple V, D, {f c | c ∈ C} , where f c is a function on V with the same arity as c. With a view towards obtaining a semantics for L, an entailment relation is associated to a given matrix in a certain canonical way. For that purpose, a class of valuations is fixed, and often, in order to obtain a 'truth-functional semantics', the class Hom(L, A) of all homomorphisms of L into A is considered (see [22]). If M = A, D is a matrix, the single-conclusion entailment relation |= M ⊆ 2 L × L induced by M is defined as follows:…”

Section: Introductionmentioning

confidence: 99%

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

2017

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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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Section: Introductionmentioning

confidence: 99%

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

2017

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“…In this section, we start out by reviewing Suszko's proposal: we basically explain the sort of reduction that both Scott and Suszko used in order to reduce truth values to a minimum 'logical' rank (Scott, 1974;Suszko, 1977). As observed by several authors and first by Suszko himself (Béziau, 2001;Caleiro, Carnielli, Coniglio, & Marcos, 2003;Caleiro & Marcos, 2012;Font, 2009;Marcos, 2009;Shramko & Wansing, 2011;Suszko, 1977), this kind of reduction on the number of algebraic truth values meets weak demands, in particular regarding compositionality (see Appendix A for a more nuanced view). Although the reduction does not respect truth-compositionality in general, we show that a modification of it does so for compact logics operating with what we call regular connectives.…”

Section: Monotonic + Reflexive + Transitive = Intersection Of Pure Rementioning

confidence: 99%

“…Suszko's focus was on the notion of a logical truth value, and only concerns the existence of a sound and complete semantics for a logic. However, one may demand more from a semantics, such as valuationality, compositionality and truth-functionality (see Font, 2009;Marcos, 2009). Such properties may be seen as irrelevant in relation to logical truth values, insofar as they concern connectives more than consequence relations, but they surely are relevant in relation to the notion of an algebraic truth value.…”

Section: Stronger Notions Of Rankmentioning

confidence: 99%

“…Thirdly, we give a specific attention to the issue of semantic compositionality. Suszko (1977) pointed out that his reduction technique generally ends up in a semantics that fails compositionality, understood as truth-functionality (see Béziau, 2001;Caleiro & Marcos, 2012;Caleiro et al, 2015;Font, 2009;Marcos, 2009 for discussions, and our refinements of the notion of compositionality below). This failure, as stressed by Font (2009), makes it "difficult to call it a semantics, which makes the meaningfulness of his Reduction hard to accept".…”

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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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“…These situations pose a threat to the application of logical systems obeying the principle of truth-functionality in those problems. As is well known, the TFP fails in many logical systems of interest (see examples in [2]).…”

Section: Introductionmentioning

confidence: 95%

Non-Deterministic Semantics for Quantum States

Jorge

Holik

2020

Entropy

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In this work, we discuss the failure of the principle of truth functionality in the quantum formalism. By exploiting this failure, we import the formalism of N-matrix theory and non-deterministic semantics to the foundations of quantum mechanics. This is done by describing quantum states as particular valuations associated with infinite non-deterministic truth tables. This allows us to introduce a natural interpretation of quantum states in terms of a non-deterministic semantics. We also provide a similar construction for arbitrary probabilistic theories based in orthomodular lattices, allowing to study post-quantum models using logical techniques.

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What is a Non-truth-functional Logic?

Cited by 24 publications

References 30 publications

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

An Inferentially Many-Valued Two-Dimensional Notion of Entailment

Suszko’s Problem: Mixed Consequence and Compositionality

Non-Deterministic Semantics for Quantum States

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