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 Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 115 ◽  
Author(s):  
Joanna Golińska-Pilarek ◽  
Magdalena Welle
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Classical Propositional Logic ◽  
Tableau System ◽  
Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

Update Semantics in Communicated Information System

2011 ◽  
Vol 403-408 ◽  
pp. 1460-1465
Author(s):  
Guang Ming Chen ◽  
Xiao Wu Li
Keyword(s):  
Information Systems ◽  
Propositional Logic ◽  
General Purpose ◽  
Main Task ◽  
Information Communication ◽  
Classical Propositional Logic ◽  
Update Semantics ◽  
Essential Form ◽  
Soundness And Completeness ◽  
Multi Agents

An approach, which is called Communicated Information Systems, is introduced to describe the information available in a number of agents and specify the information communication among the agents. The systems are extensions of classical propositional logic in multi-agents context, providing with us a way by which not only the agent’s own information, but the information from other agents may be applied to agent’s reasoning as well. Communication rules, which are defined in the most essential form, can be regarded as the base to characterize some interesting cognitive proporties of agents. Since the corresponding communication rules can be chosen for different applications, the approach is general purpose one. The other main task is that the soundness and completeness of the Communicated Information Systems for the update semantics have been proved in the paper.

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 SUBFORMULA AND SEPARATION PROPERTIES IN NATURAL DEDUCTION VIA SMALL KRIPKE MODELS

2010 ◽  
Vol 3 (2) ◽  
pp. 175-227 ◽  
Author(s):  
PETER MILNE
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
Careful Attention ◽  
Classical Propositional Logic ◽  
First Order ◽  
Invalid Inference ◽  
Subformula Property

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck.

scholarly journals Parity, Relevance, and Gentle Explosiveness in the Context of Sylvan's Mate Function

2018 ◽  
Vol 15 (2) ◽  
pp. 381
Author(s):  
Thomas Macaulay Ferguson
Keyword(s):  
Infinite Number ◽  
Proof Theory ◽  
Possible Worlds ◽  
Consequence Relations ◽  
Deductive Systems ◽  
First Degree Entailment ◽  
Broad Field ◽  
Soundness And Completeness ◽  
Set Up ◽  
Hallmark Feature

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness.

scholarly journals Holistic Type Extension for Classical Logic via Toffoli Quantum Gate

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 636 ◽  
Author(s):  
Hector Freytes ◽  
Roberto Giuntini ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Quantum States ◽  
Quantum Gate ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Werner States ◽  
To Receive

A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called “quantum computation with mixed states”, where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of “holistic" contradiction.

PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC

2018 ◽  
Vol 13 (3) ◽  
pp. 509-540 ◽  
Author(s):  
MINGHUI MA ◽  
AHTI-VEIKKO PIETARINEN
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Theoretic Analysis ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System ◽  
Logical Algebra ◽  
Graphical System ◽  
Logical Calculi

AbstractThis article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system.

scholarly journals Logics for Knowability

2021 ◽  
pp. 1-42
Author(s):  
Mo Liu ◽  
Jie Fan ◽  
Hans Van Ditmarsch ◽  
Louwe B. Kuijer
Keyword(s):  
Propositional Logic ◽  
Single Agent ◽  
Classical Propositional Logic ◽  
Public Announcement ◽  
Public Announcement Logic ◽  
Multi Agent ◽  
Soundness And Completeness

In this paper, we propose three knowability logics LK, LK−, and LK=. In the single-agent case, LK is equally expressive as arbitrary public announcement logic APAL and public announcement logic PAL, whereas in the multi-agent case, LK is more expressive than PAL. In contrast, both LK− and LK= are equally expressive as classical propositional logic PL. We present the axiomatizations of the three knowability logics and show their soundness and completeness. We show that all three knowability logics possess the properties of Church-Rosser and McKinsey. Although LK is undecidable when at least three agents are involved, LK− and LK= are both decidable.

scholarly journals Sequent Systems without Improper Derivations

2021 ◽  
Author(s):  
Katsumi Sasaki
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Inference Rules ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Structural Rules ◽  
Sequent Systems ◽  
Natural Deduction System ◽  
Sequent System

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\).

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