Relevance Logic as a Conservative Extension of Classical Logic

AbstractWe show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

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Relevance logic, classical logic, and disjunctive syllogism

Philosophical Studies ◽

10.1007/bf01236456 ◽

1975 ◽

Vol 27 (6) ◽

pp. 361-376 ◽

Cited By ~ 3

Author(s):

John A. Barker

Keyword(s):

Classical Logic ◽

Relevance Logic ◽

Disjunctive Syllogism

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Society semantics for four-valued Łukasiewicz logic

Logic Journal of IGPL ◽

10.1093/jigpal/jzy066 ◽

2018 ◽

Vol 28 (5) ◽

pp. 892-911

Author(s):

Edson Vinícius Bezerra

Keyword(s):

Computer Science ◽

Classical Logic ◽

Łukasiewicz Logic ◽

Conservative Extension ◽

Interesting Point ◽

Rational Agents ◽

Lukasiewicz Logic ◽

Closed Society

AbstractWe argue that many-valued logics (MVLs) can be useful in analysing informational conflicts by using society semantics (SSs). This work concentrates on four-valued Łukasiewicz logic. SSs were proposed by Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) to deal with conflicts of information involving rational agents that make judgements about propositions according to a given logic within a society, where a society is understood as a collection $\mathcal{A}$ of agents. The interesting point of such semantics is that a new logic can be obtained by combining the logic of the agents under some appropriate rules. Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) defined SSs for the three-valued logics $I^{1}$ and $P^{1}$. In this kind of semantics, all the agents reason according to classical logic (CL) and the molecular formulas behave in the same way as in CL (the non-classical character of these logics only appears at the propositional level). Marcos (unpublished data) provided SSs with classical agents for the three-valued Łukasiewicz logic Ł$_{3}$, but in this case, the molecular formulas do not behave classically. We prove here that one can characterize Ł$_{4}^{\prime}$, a conservative extension of Ł$_{4}$ obtained by adding a connective $\blacktriangledown$, by means of a closed society where the agents reason according to Ł$_{3}$. We shall emphasize the importance of recovery operators in the construction of this class of societies. Moreover, we shall relate this semantics to Suszko’s view on the ‘two-valuedness’ of logic.

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Rewriting the History of Connexive Logic

Journal of Philosophical Logic ◽

10.1007/s10992-021-09640-6 ◽

2022 ◽

Author(s):

Wolfgang Lenzen

Keyword(s):

Twentieth Century ◽

Classical Logic ◽

Paraconsistent Logic ◽

Historical Analysis ◽

Relevance Logic ◽

Overwhelming Majority ◽

Lewis Carroll ◽

Self Consistent ◽

History Of ◽

Connexive Logic

AbstractThe “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties (like the condition that no proposition implies its own negation) only apply to “normal” (e.g., self-consistent) antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning (“humbly”) connexive implication don’t give rise, however, to anything like a non-classical logic.

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THE RELEVANCE OF PREMISES TO CONCLUSIONS OF CORE PROOFS

The Review of Symbolic Logic ◽

10.1017/s1755020315000040 ◽

2015 ◽

Vol 8 (4) ◽

pp. 743-784 ◽

Cited By ~ 7

Author(s):

NEIL TENNANT

Keyword(s):

Intuitionistic Logic ◽

Classical Logic ◽

Relevance Logic ◽

Minimal Logic ◽

Classical Extension ◽

The One

AbstractThe rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logic R. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such as R and its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove that R or any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge.

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Computability logic: Giving Caesar what belongs to Caesar

Logical Investigations ◽

10.21146/2074-1472-2019-25-1-100-119 ◽

2019 ◽

Vol 25 (1) ◽

pp. 100-119

Author(s):

Giorgi Japaridze

Keyword(s):

Classical Logic ◽

Formal Theory ◽

Order Logic ◽

Conservative Extension ◽

Logical Operators ◽

First Order ◽

Game Semantics ◽

Wide Range ◽

Algorithmic Solvability

The present article is a brief informal survey o$\textit {computability logic}$ (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems – understood in their most general, interactive sense – are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics.

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Relevant entailment—semantics and formal systems

Journal of Symbolic Logic ◽

10.2307/2274169 ◽

1984 ◽

Vol 49 (2) ◽

pp. 334-342 ◽

Cited By ~ 21

Author(s):

Arnon Avron

Keyword(s):

Simple Form ◽

Point Of View ◽

Relevance Logic ◽

Conservative Extension ◽

Formal Systems ◽

Proper Extension

This work results from an attempt to give the vague notion of relevance a concrete semantical interpretation. The idea is that propositions may be divided into different “domains of relevance”. Each “domain” has its own “T” and “F” values, and propositions “belonging” to one domain can never entail propositions “belonging” to another, unconnected one.The semantics we have developed were found to correspond to an already known system, which we call here RMI⥲. Its axioms are the implication-negation axioms of the system RM ([1, Chapter 5]). However, as Meyer has shown, RM is not a conservative extension of RMI⥲, since RMI⥲ has the sharing-of-variables property ([5], and [1, pp. 148–149[), which the implication-negation fragment of RM has not.RMI⥲ has four advantages in comparison to its more famous sister R⥲ (the pure intentional fragment of the system R; see [1]):a) It has a very natural (from a relevance point of view) many-valued semantics, the simple form of which we describe here.b) RMI⥲, ⊢ A1 → [A2 → (… → (An → A) …)] iff there is a proof of A from the set {A1, …, An} that actually uses all the members of this set. In R⥲, this holds only if we talk about “sequences” instead of “sets”. This is somewhat less intuitive (see [1, pp. 394–395]).c) RMI⥲ is a maximal “natural” relevance logic, in the sense that every proper extension of it limits the number of “domains of relevance” (§III).

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Formal Logic for Informal Logicians

Informal Logic ◽

10.22329/il.v26i2.444 ◽

2008 ◽

Vol 26 (2) ◽

pp. 199 ◽

Cited By ~ 2

Author(s):

David Sherry

Keyword(s):

Modal Logic ◽

Classical Logic ◽

Propositional Logic ◽

Formal Logic ◽

Relevance Logic ◽

Valid Argument ◽

Insight Into

Classical logic yields counterintuitive results for numerous propositional argument forms. The usual alternatives (modal logic, relevance logic, etc.) generate counterintuitive results of their own. The counterintuitive results create problems—especially pedagogical problems—for informal logicians who wish to use formal logic to analyze ordinary argumentation. This paper presents a system, PL– (propositional logic minus the funny business), based on the idea that paradigmatic valid argument forms arise from justificatory or explanatory discourse. PL– avoids the pedagogical difficulties without sacrificing insight into argument.

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