Soundness and Completeness of Fuzzy Propositional Logic with Three Kinds of Negation

Tableau System ◽

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.403-408.1460 ◽

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 ◽

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.

Hybrid Deduction–Refutation Systems

Axioms ◽

10.3390/axioms8040118 ◽

2019 ◽

Vol 8 (4) ◽

pp. 118 ◽

Cited By ~ 1

Author(s):

Valentin Goranko

Keyword(s):

Propositional Logic ◽

Proof Theory ◽

Natural Deduction ◽

Basic Theory ◽

Logical System ◽

Deductive Systems ◽

Refutation Systems ◽

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.

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning ◽

A Preference-Based Approach to Defeasible Deontic Inference

10.24963/kr.2020/33 ◽

2020 ◽

Author(s):

James Delgrande

Keyword(s):

Propositional Logic ◽

Polynomial Hierarchy ◽

Deontic Reasoning ◽

Desirable Outcome ◽

Interesting Difference ◽

Completeness Result ◽

Syntactic Approach ◽

In this paper we present an approach to defeasible deontic inference. Given a set of rules R expressing conditional obligations and a formula A giving contingent information, the goal is to determine the most desirable outcome with respect to this information. Semantically, the rules R induce a partial preorder on the set of models, giving the relative desirability of each model. Then the set of minimal A models characterises the best that can be attained given that A holds. A syntactic approach is also given, in terms of maximal subsets of material counterparts of rules in R, and that yields a formula that expresses the best outcome possible given that A holds. These approaches are shown to coincide, providing an analogue to a soundness and completeness result. Complexity is not unreasonable, being at the second level of the polynomial hierarchy when the underlying logic is propositional logic. The approach yields desirable and intuitive results, including for the various “paradoxes” of deontic reasoning. The approach also highlights an interesting difference in how specificity is dealt with in nonmonotonic and deontic reasoning.

Logics for Knowability

Logic and Logical Philosophy ◽

10.12775/llp.2021.018 ◽

2021 ◽

pp. 1-42

Author(s):

Mo Liu ◽

Jie Fan ◽

Hans Van Ditmarsch ◽

Louwe B. Kuijer

Keyword(s):

Propositional Logic ◽

Single Agent ◽

Public Announcement ◽

Public Announcement Logic ◽

Multi Agent ◽

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.

Hilbert-style proof Calculus for Propositional Logic in ABC notation

10.31219/osf.io/jd3gp ◽

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.

Reducing Separation Formulas to Propositional Logic

10.21236/ada461197 ◽

2003 ◽

Cited By ~ 1

Author(s):

Ofer Strichman ◽

Sanjit A. Seshia ◽

Randal E. Bryant

Keyword(s):

Propositional Logic

Intuitionistic Propositional Logic

Cardinalities of proper ideals in some lattices of strengthenings of the intuitionistic propositional logic

Studia Logica ◽

10.1007/bf01063837 ◽

1983 ◽

Vol 42 (2-3) ◽

pp. 173-177

Author(s):

Wies?aw Dziobiak

Keyword(s):

Propositional Logic ◽

Kripke Semantics for Intersection Formulas

ACM Transactions on Computational Logic ◽

10.1145/3453481 ◽

2021 ◽

Vol 22 (3) ◽

pp. 1-16

Author(s):

Andrej Dudenhefner ◽

Paweł Urzyczyn

Keyword(s):

Kripke Semantics ◽

Type Assignment ◽

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

Extensions in graph normal form

Logic Journal of IGPL ◽

10.1093/jigpal/jzaa054 ◽

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.