Propositional Logic: Deductive Systems

2012 ◽  
pp. 49-73
Author(s):  
Mordechai Ben-Ari
Keyword(s):  
Propositional Logic ◽  
Deductive Systems

scholarly journals THE UBIQUITY OF CONSERVATIVE TRANSLATIONS

2012 ◽  
Vol 5 (4) ◽  
pp. 666-678 ◽  
Author(s):  
EMIL JEŘÁBEK
Keyword(s):  
Propositional Logic ◽  
Broad Class ◽  
Paraconsistent Logic ◽  
Lambek Calculus ◽  
Classical Propositional Logic ◽  
Consequence Relation ◽  
Deductive Systems ◽  
Nonclassical Logics ◽  
Conservative Translations

AbstractWe study the notion of conservative translation between logics introduced by (Feitosa & D’Ottaviano2001). We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.

scholarly journals Hybrid Deduction–Refutation Systems

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Valentin Goranko
Keyword(s):  
Propositional Logic ◽  
Proof Theory ◽  
Natural Deduction ◽  
Basic Theory ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Deductive Systems ◽  
Refutation Systems ◽  
Soundness And Completeness

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which I show soundness and completeness for both deductions and refutations.

scholarly journals Proving infinitary formulas

2016 ◽  
Vol 16 (5-6) ◽  
pp. 787-799 ◽  
Author(s):  
AMELIA HARRISON ◽  
VLADIMIR LIFSCHITZ ◽  
JULIAN MICHAEL
Keyword(s):  
Propositional Logic ◽  
Formal System ◽  
Answer Set Programming ◽  
Logic Programs ◽  
Deductive Systems ◽  
Answer Set

AbstractThe infinitary propositional logic of here-and-there is important for the theory of answer set programming in view of its relation to strongly equivalent transformations of logic programs. We know a formal system axiomatizing this logic exists, but a proof in that system may include infinitely many formulas. In this note we describe a relationship between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. This relationship allows us to use finite proofs to justify the validity of infinitary formulas.

Hilbert-style proof Calculus for Propositional Logic in ABC notation

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
Modus Ponens ◽  
Axiomatic System ◽  
First Contact

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.

Metatheories of deductive systems

2004 ◽  
Author(s):  
Sergei V. Babyonyshev
Keyword(s):  
Deductive Systems

Reducing Separation Formulas to Propositional Logic

2003 ◽  
Author(s):  
Ofer Strichman ◽  
Sanjit A. Seshia ◽  
Randal E. Bryant
Keyword(s):  
Propositional Logic

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

scholarly journals ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1456
Author(s):  
Stefka Fidanova ◽  
Krassimir Todorov Atanassov
Keyword(s):  
Decision Making ◽  
Knapsack Problem ◽  
Propositional Logic ◽  
Optimization Problems ◽  
Real Life ◽  
Combinatorial Optimization Problems ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Multiple Constraint

Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO.

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