Relevance logic, classical logic, and disjunctive syllogism

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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7. Deducing

Philosophical Method: A Very Short Introduction ◽

10.1093/actrade/9780198810001.003.0007 ◽

2020 ◽

pp. 74-88

Author(s):

Timothy Williamson

Keyword(s):

Modal Logic ◽

Classical Logic ◽

Logical Inference ◽

Disjunctive Syllogism ◽

Philosophical Concept ◽

The Law ◽

The Difference ◽

Excluded Middle

Detective work is an important tool in philosophy. ‘Deducing’ explains the difference between valid and sound arguments. An argument is valid if its premises are true but is only sound if the conclusion is true. The Greek philosophers identified disjunctive syllogism—the idea that if something is not one thing, it must be another. This relates to another philosophical concept, the ‘law of the excluded middle’. An abduction is a form of logical inference which attempts to find the most likely explanation. Modal logic, an extension of classical logic, is a popular branch of logic for philosophical arguments.

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Relevance Logic as a Conservative Extension of Classical Logic

David Makinson on Classical Methods for Non-Classical Problems - Outstanding Contributions to Logic ◽

10.1007/978-94-007-7759-0_17 ◽

2013 ◽

pp. 383-398

Author(s):

David Makinson

Keyword(s):

Classical Logic ◽

Relevance Logic ◽

Conservative Extension

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Natural deduction and sequent calculus for intuitionistic relevant logic

Journal of Symbolic Logic ◽

10.1017/s0022481200029674 ◽

1987 ◽

Vol 52 (3) ◽

pp. 665-680 ◽

Cited By ~ 7

Author(s):

Neil Tennant

Keyword(s):

Natural Deduction ◽

Sequent Calculus ◽

Deductive System ◽

Relevant Logic ◽

Relevance Logic ◽

Minimal Logic ◽

Disjunctive Syllogism

Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A.Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism).But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.

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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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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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Relevantism, Material Detachment, and the Disjunctive Syllogism Argument

Canadian Journal of Philosophy ◽

10.1080/00455091.1984.10716376 ◽

1984 ◽

Vol 14 (2) ◽

pp. 167-188 ◽

Cited By ~ 3

Author(s):

R. Routley

Keyword(s):

Classical Logic ◽

Relevant Logic ◽

Disjunctive Syllogism

Relevantism, as a matter of definition, rejects classical logic as incorrect and adopts instead a relevant logic as encapsulating correct inference. It rejects classical logic on the grounds that the rule of Material Detachment, from A and not A or B to infer B, (that is, Disjunctive Syllogism considered as an inferential principle), sometimes leads from truth to falsity. Relevantism — although promoted by some relevant logicians (Routley and Routley), and an integral part of ultralogic (i.e. universal, all purpose, ultramodallogic; cf. [1], [8]) — has recently encountered heavy, but interesting, criticism from relevance logicians themselves (from Belnap, Dunn, and Meyer).

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