125 research outputs found
A non-transitive relevant implication corresponding to classical logic consequence
In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence. Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to thesequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word. By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible
Grounding axioms for (relevant) implication
Most of the logics of grounding that have so far been proposed contain grounding axioms, or grounding rules, for the connectives of conjunction, disjunction and negation, but little attention has been dedicated to the implication connective. The present paper aims at repairing this situation by proposing adequate grounding axioms for relevant implication. Because of the interaction between negation and implication, new grounding axioms concerning negation will also arise
Theorems of Alternatives for Substructural Logics
A theorem of alternatives provides a reduction of validity in a substructural
logic to validity in its multiplicative fragment. Notable examples include a
theorem of Arnon Avron that reduces the validity of a disjunction of
multiplicative formulas in the R-mingle logic RM to the validity of a linear
combination of these formulas, and Gordan's theorem for solutions of linear
systems over the real numbers, that yields an analogous reduction for validity
in Abelian logic A. In this paper, general conditions are provided for
axiomatic extensions of involutive uninorm logic without additive constants to
admit a theorem of alternatives. It is also shown that a theorem of
alternatives for a logic can be used to establish (uniform) deductive
interpolation and completeness with respect to a class of dense totally ordered
residuated lattices
Relevance via decomposition: A project, some results, an open question
We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the ocedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition
Relevance via decomposition
We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition
The Third Way on Objective Probability: A Skeptic's Guide to Objective Chance
The goal of this paper is to sketch and defend a new interpretation or theory of objective chance, one that lets us be sure such chances exist and shows how they can play the roles we traditionally grant them. The subtitle obviously emulates the title of Lewis seminal 1980 paper A Subjectivist s Guide to Objective Chance while indicating an important difference in perspective. The view developed below shares two major tenets with Lewis last (1994) account of objective chance: (1) The Principal Principle tells us most of what we know about objective chance; (2) Objective chances are not primitive modal facts, propensities, or powers, but rather facts entailed by the overall pattern of events and processes in the actual world. But it differs from Lewisâ account in most other respects. Another subtitle I considered was A Humean Guide ... But while the account of chance below is compatible with any stripe of Humeanism (Lewis , Hume s, and others ), it presupposes no general Humean philosophy. Only a skeptical attitude about probability itself is presupposed (as in point (2) above); what we should say about causality, laws, modality and so on is left a separate question. Still, I will label the account to be developed âHumean objective chanceâ
Substructural Logics and Pragmatic Enrichment
In this dissertation, we argue for a Pragmatic Logical Pluralism, a pluralist thesis about logic which endorses Classical, Relevant, Linear, and Ordered logic. We justify that the formal languages of these four logics are legitimate codifications of the logical vocabulary and capture legitimate senses of logical consequence. This will be justified given a particular interpretation of the four formal languages: logical consequence and conditional, disjunction, and conjunction of the four different logics codify different and legitimate senses of âfollows fromâ, âif...thenâ, âorâ and âandâ which diverge in their different pragmatic enrichments. The dissertation is twofold. First, we will explore the effect that the lack of structural rules has on logical connectives, in four substructural logics, and its connection with certain pragmatic enrichments. Second, we will defend a pluralist thesis according to which pragmatics has an important role for capturing the inferential role of logical vocabulary, both of the notions of âfollows fromâ and the logical constants, although classical logic preserves truth and captures their lit- eral meaning. In sum, we defend a version of logical pluralism based on the plurality of legitimate translations from natural language to formal languages, arguing that more than one translation is legitimate for logical vocabulary, which makes it possible to adopt more than one logic.En aquesta tesi presentem el Pluralisme Lògic PragmĂ tic, una tesi pluralista sobre la lògica que accepta les lògiques ClĂ ssica, Rellevant, Lineal i Ordenada. Justifiquem que els llenguatges formals dâaquestes quatre lògiques sĂłn codificacions legĂtimes del vocabulari lògic i capturen sentits legĂtims de la conseqßència lògica. Això es justificarĂ donant una interpretaciĂł particular dels quatre llenguatges formals: la conseqßència lògica i el condicional, la disjunciĂł i la conjunciĂł de les quatre lògiques acceptades codifiquen diferents i legĂtims sentits de âsi...llavorsâ, âoâ i âiâ, que es distingeixen pels diferents enriquiments pragmĂ tics que codifiquen. La tesi tĂŠ dos vessants. Primer, explorem lâefecte que la falta de regles estructurals tĂŠ en les connectives lògiques de les quatre lògiques presentades, i la seva connexiĂł amb certs enriquiments pragmĂ tics. Segon, defensem una visiĂł pluralista segons la qual la pragmĂ tica juga un rol important a lâhora de capturar el rol inferencial del vocabulari lògic, tant per la nociĂł de conseqßència lògica com per les connectives, tot i que la lògica clĂ ssica preserva la veritat i captura el seu significat literal. En resum, defensem una versiĂł del pluralisme lògic basat en la pluralitat de traduccions legĂtimes del llenguatge natural al llenguatge formal, argumentant que mĂŠs dâuna traducciĂł ĂŠs legĂtima pel vocabulari lògic, la qual cosa ens permet adoptar mĂŠs dâuna lògica
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