Taming Paraconsistent (and Other) Logics

Ciabattoni, Agata; Lahav, Ori; Spendier, Lara; Zamansky, Anna

doi:10.1145/2661636

Cited by 8 publications

(28 citation statements)

References 21 publications

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“…Nmatrices have been used already to give effective semantics to a big range of important non-classical logics [6,2,16,23]. These structures can for example be used to deal with paraconsistent behaviour [14], to model how a processor deals with information from multiple-sources [5], or for reasoning about computation errors [3]. The fact that ∧ ∨ can be seen as a defective ∧ or ∨ might indicate that (P)Nmatrices can be of value when reasoning about unreliable logic circuits [31,26].…”

Section: Discussionmentioning

confidence: 99%

An unexpected Boolean connective

Marcelino¹

2021

Preprint

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We consider a 2-valued non-deterministic connective ∧ ∨ defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it platypus. The value of ∧ ∨ is not completely determined by the input, contrasting with usual notion of Boolean connective. We call nondeterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. Unexpectedly, this very simple connective and the logic it defines, illustrate various key advantages in working with generalized notions of semantics (by incorporating non-determinism), calculi (by allowing multiple-conclusion rules) and even of logic (moving from Tarskian to Scottian consequence relations). We show that the associated logic cannot be characterized by any finite set of finite matrices, whereas with non-determinism two values suffice. Furthermore, this logic is not finitely axiomatizable using single-conclusion rules, however we provide a very simple analytical multiple-conclusion axiomatization using only two rules. Finally, deciding the associated multipleconclusion logic is coNP-complete, but deciding its single-conclusion fragment is in P.So they cut him to pieces, wrote a thesis A cranium of deceit, he's prone to lie and cheat; It's no wonder -a blunder from down under Duckbill, watermole, duckmole! Mr. Bungle, Platypus

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Section: Discussionmentioning

confidence: 99%

An unexpected Boolean connective

Marcelino¹

2021

Preprint

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“…Often, one works with a single-conclusion logic which can then be streghtened by the addition of new axioms [6,26,20] (and possibly also new syntax). Concretely, let Σ 1 , ⊢ 1 be a single-conclusion logic, Σ 2 be a signature, Ax ⊆ L Σ2 (P ) be a set of axiom schemata, and define Ax inst = {A σ : A ∈ Ax and σ : P → L Σ1∪Σ2 (P )}.…”

Section: Adding Axiomsmentioning

confidence: 99%

Modular many-valued semantics for combined logics

Caleiro¹,

Marcelino²

2022

Preprint

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We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial nondeterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics whereas, unsurprisingly, it may be lost in the context of single-conclusion logics.Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism.

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“…The added expressiveness allows for finite characterizations of logics that do not admit finite semantics based on logical matrices [23,24] and provide valuable insight about proof theoretical properties of said logics [2]. It also allows for general recipes for various practical problems in logic, including procedures to constructively updating semantics when imposing new axioms [9,12], including language extensions; or effectively combining the semantics of two logics capturing the effect of joining their axiomatizations [8,21]. Recently, PNmatrices also provided new interpretations of quantum states as valuations [15].…”

Section: Introductionmentioning

confidence: 99%

Computational Properties of Partial Non-deterministic Matrices and Their Logics

Marcelino

Caleiro

Filipe

2021

Logical Foundations of Computer Science

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Incorporating non-determinism and partiality in the traditional notion of logical matrix semantics has proven to be decisive in a myriad of recent compositionality results in logic. However, several important properties which are known to be computable for finite matrices have not been studied in this wider context of partial nondeterministic matrices (PNmatrices).This paper is dedicated to understanding how this generalization of the considered semantical structures affects the computational properties of basic problems regarding their induced logics, in particular their sets of theorems.We will show that the landscape is quite rich, as some problems keep their computational status, for others the complexity increases, and for a few decidability is lost. Namely, we show that checking if the logics defined by two finite PNmatrices have the same theorems is undecidable. This latter result is obtained by reduction from the undecidable problem of checking universality of term-DAG-automata.

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Taming Paraconsistent (and Other) Logics

Cited by 8 publications

References 21 publications

An unexpected Boolean connective

An unexpected Boolean connective

Modular many-valued semantics for combined logics

Computational Properties of Partial Non-deterministic Matrices and Their Logics

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