scholarly journals Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Marilynn Johnson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Variable Domain ◽  
Free Logic ◽  
Intuitionist Logic ◽  
Branching Rule ◽  
Branching Rules ◽  
Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

7. Deducing

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.

scholarly journals Modal logic and classical logic, Humanities press

1988 ◽  
Vol 67 (1) ◽  
pp. 141
Author(s):  
Gian-Carlo Rota
Keyword(s):  
Modal Logic ◽  
Classical Logic

A proof-theoretic study of the correspondence of classical logic and modal logic

2003 ◽  
Vol 68 (4) ◽  
pp. 1403-1414 ◽  
Author(s):  
H. Kushida ◽  
M. Okada
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Predicate Logic ◽  
Transformation Method ◽  
Modal Operator ◽  
Theoretic Method ◽  
Barcan Formula ◽  
Theoretic Proof ◽  
Classical Predicate

AbstractIt is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.

The extensions of the modal logic K5

1985 ◽  
Vol 50 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Michael C. Nagle ◽  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Abstract Characterization ◽  
Propositional Modal Logic ◽  
Normal Logic ◽  
Countably Infinite ◽  
Infinite Set

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

scholarly journals What Counts as Evidence for a Logical Theory?

2019 ◽  
Vol 16 (7) ◽  
pp. 250 ◽  
Author(s):  
Ole Thomassen Hjortland
Keyword(s):  
Classical Logic ◽  
A Priori ◽  
Scientific Theories ◽  
Alternative Account ◽  
Logical Theory ◽  
Crucial Question ◽  
Indispensability Arguments ◽  
Timothy Williamson ◽  
Graham Priest ◽  
Number Of Authors

Anti-exceptionalism about logic is the Quinean view that logical theories have no special epistemological status, in particular, they are not self-evident or justified a priori. Instead, logical theories are continuous with scientific theories, and knowledge about logic is as hard-earned as knowledge of physics, economics, and chemistry. Once we reject apriorism about logic, however, we need an alternative account of how logical theories are justified and revised. A number of authors have recently argued that logical theories are justified by abductive argument (e.g. Gillian Russell, Graham Priest, Timothy Williamson). This paper explores one crucial question about the abductive strategy: what counts as evidence for a logical theory? I develop three accounts of evidential confirmation that an anti-exceptionalist can accept: (1) intuitions about validity, (2) the Quine-Williamson account, and (3) indispensability arguments. I argue, against the received view, that none of the evidential sources support classical logic.

Robert Bull and Krister Segerberg. Basic modal logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 1–88. - John P. Burgess. Basic tense logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 89–133. - Richmond H. Thomason. Combinations of tense and modality. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 135–165. - Johan van Benthem. Correspondence theory. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 167–247. - James W. Garson. Quantification in modal logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 249–307. - Nino B. Cocchiarella. Philosophical perspectives on quantification in tense and modal logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 309–353.

1989 ◽  
Vol 54 (4) ◽  
pp. 1472-1477 ◽  
Author(s):  
Steven T. Kuhn
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Correspondence Theory ◽  
Tense Logic ◽  
Publishing Company ◽  
Reidel Publishing Company ◽  
Philosophical Logic

scholarly journals A Binary Quantifier for Definite Descriptions for Cut Free Free Logics

Studia Logica ◽  
2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Classical Logic ◽  
Final Section ◽  
Sequent Calculus ◽  
Definite Descriptions ◽  
Present Approach ◽  
Cut Elimination ◽  
Free Logic

AbstractThis paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.

MODAL LOGIC WITHOUT CONTRACTION IN A METATHEORY WITHOUT CONTRACTION

2019 ◽  
Vol 12 (4) ◽  
pp. 685-701
Author(s):  
PATRICK GIRARD ◽  
ZACH WEBER
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Kripke Semantics ◽  
Substructural Logic

AbstractStandard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using a nonclassical substructural logic as the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with a noncontractive logic in the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics.

scholarly journals Normalisation and subformula property for a system of classical logic with Tarski’s rule

2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Classical Logic ◽  
Similar Form ◽  
General Introduction ◽  
Intuitionist Logic ◽  
Introduction Rule ◽  
Definition Of ◽  
Subformula Property

AbstractThis paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic.

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