Arithmetical interpretations of dynamic logic

1983 ◽  
Vol 48 (3) ◽  
pp. 704-713 ◽  
Author(s):  
Petr Hájek
Keyword(s):  
Propositional Logic ◽  
Dynamic Logic ◽  
Completeness Theorem ◽  
Free Variable ◽  
Peano Arithmetic ◽  
Logical Connectives ◽  
Arithmetical Interpretation

AbstractAn arithmetical interpretation of dynamic propositional logic (DPL) is a mapping ƒ satisfying the following: (1) ƒ associates with each formula A of DPL a sentence ƒ(A) of Peano arithmetic (PA) and with each program α a formula ƒ(α) of PA with one free variable describing formally a supertheory of PA; (2) ƒ commutes with logical connectives; (3) ƒ([α]A) is the sentence saying that ƒ(A) is provable in the theory ƒ(α); (4) for each axiom A of DPL, ƒ(A) is provable in PA (and consequently, for each A provable in DPL, ƒ(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation ƒ, ƒ(A) is provable in PA. Various modifications of this result are considered.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Vaughan R. Pratt. Semantical considerations on Floyd–Hoare logic. 17th Annual Symposium on Foundations of Computer Science, Institute of Electrical and Electronics Engineers, New York1976, pp. 109–121. - Michael J. Fischer and Richard E. Ladner. Propositional dynamic logic of regular programs. Journal of computer and system sciences, vol. 18 (1979), pp. 194–211. - Krister Segerberg. A completeness theorem in the modal logic of programs. Universal algebra and applications. Papers presented at Stefan Banach International Mathematical Center at the semester “Universal algebra and applications” held February 15–June 9, 1978, edited by Tadeuz Traczyk, Banach Center Publications, vol. 9, PWN—Polish Scientific Publishers, Warsaw1982, pp. 31–46. - Rohit Parikh. The completeness of propositional dynamic logic. Mathematical foundations of computer science 1978, Proceedings, 7th symposium, Zakopane, Poland, September 4–8, 1978, edited by J. Winkowski, Lecture notes in computer science, vol. 64, Springer-Verlag, Berlin, Heidelberg, and New York, 1978, pp. 403–415. - Dexter Kozen and Rohit Parikh. An elementary proof of the completeness of PDL. Theoretical computer science, vol. 14(1981), pp. 113–118.

1986 ◽  
Vol 51 (1) ◽  
pp. 225-227 ◽  
Author(s):  
Robert Goldblatt
Keyword(s):  
New York ◽  
Computer Science ◽  
Universal Algebra ◽  
Dynamic Logic ◽  
Completeness Theorem ◽  
Theoretical Computer Science ◽  
Propositional Dynamic Logic ◽  
Theoretical Computer ◽  
Mathematical Foundations ◽  
Scientific Publishers

scholarly journals Performability of Actions

2021 ◽  
Author(s):  
Janusz Czelakowski
Keyword(s):  
Action Theory ◽  
Dynamic Logic ◽  
Completeness Theorem ◽  
Axiom System ◽  
Formal Ontology ◽  
Strong Completeness ◽  
Practical Applications ◽  
System Equation ◽  
Central Result

AbstractAction theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical applications, has not been investigated in the literature in a systematic way thus far. This work, being a companion to the book as reported (Czelakowski in Freedom and enforcement in action. Elements of formal action theory, Springer 2015), elaborates the theory of performability of actions based on relational models and formal constructs borrowed from formal lingusistics. The discussion of performability of actions is encapsulated in the form of a strict logical system "Equation missing". This system is semantically defined in terms of its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since due to the nature of compound actions the system "Equation missing" is not finitary, other semantic variants of "Equation missing" are defined. The focus in on the system "Equation missing" of performability of finite compound actions. An adequate axiom system for "Equation missing" is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is highlighted. The relationship between performability of actions and dynamic logic is also discussed.

Proof theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Completeness Theorem ◽  
Formal Proof ◽  
Order Logic ◽  
Truth Table ◽  
First Order Logic ◽  
Formal Proofs ◽  
Systematic Method ◽  
First Order

As with any logic, the semantics of first-order logic yield rules for deducing the truth of one sentence from that of another. In this chapter, we develop both formal proofs and resolution for first-order logic. As in propositional logic, each of these provides a systematic method for proving that one sentence is a consequence of another. Recall the Consequence problem for propositional logic. Given formulas F and G, the problemis to decide whether or not G is a consequence of F. From Chapter 1, we have three approaches to this problem: • We could compute the truth table for the formula F → G. If the truth values are all 1s then we conclude that F → G is a tautology and G is a consequence of F. Otherwise, G is not a consequence of F. • Using Tables 1.5 and 1.6, we could try to formally derive G from {F}. By the Completeness Theorem for propositional logic, G is a consequence of F if and only if {F} ├ G. • We could use resolution. By Theorem1.76, G is a consequence of F if and only if ∅ ∈ Res(H) where H is a formula in CNF equivalent to (F ∧￢G). Using these methods not only can we determine whether one formula is a consequence of another, but also we can determine whether a given formula is a tautology or a contradiction. A formula F is a tautology if and only if F is a consequence of (A∨￢A) if and only if ￢F is a contradiction. In this chapter, we consider the analogous problems for first-order logic. Given formulas φ and ψ, how can we determine whether ψ is a consequence of φ? Equivalently, how can we determine whether a given formula is a tautology or a contradiction? We present three methods for answering these questions. • In Section 3.1, we define a notion of formal proof for first-order logic by extending Table 1.5. • In Section 3.3, we “reduce” formulas of first-order logic to sets of formulas of propositional logic where we use resolution as defined in Chapter 1.

scholarly journals Subalgebras, direct products and associated lattices of MV-algebras

1992 ◽  
Vol 34 (3) ◽  
pp. 301-307 ◽  
Author(s):  
L. P. Belluce ◽  
A. Di Nola ◽  
A. Lettieri
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Propositional Logic ◽  
Abelian Groups ◽  
Completeness Theorem ◽  
Direct Products ◽  
Algebraic Proof ◽  
Mv Algebras

MV-algebras were introduced by C. C. Chang [3] in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory [1].

scholarly journals Tense θ-valued Moisil propositional logic

2010 ◽  
Vol 5 (5) ◽  
pp. 642 ◽  
Author(s):  
Carmen Chiriţă
Keyword(s):  
Representation Theorem ◽  
Propositional Logic ◽  
Propositional Calculus ◽  
Completeness Theorem ◽  
Logical System ◽  
Tense Operators

In this paper we study the tense θ-valued Moisil propositional calculus, a logical system obtained from the θ-valued Moisil propositional logic by adding two tense operators. The main result is a completeness theorem for tense θ-valued Moisil propositional logic. The proof of this theorem is based on the representation theorem of tense θ-valued Łukasiewicz-Moisil algebras, developed in a previous paper.

Prerequisites

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Logical Validity ◽  
First Order ◽  
Gödel’S Completeness Theorem ◽  
Logical Connectives ◽  
Incompleteness Theorems ◽  
Individual Variables

As we remarked in the preface, although this volume is a sequel to our earlier volume G.I.T. (Gödel’s Incompleteness Theorems), it can be read independently by those readers familiar with at least one proof of Gödel’s first incompleteness theorem. In this chapter we give the notation, terminology and main results of G.I.T. that are needed for this volume. Readers familiar with G.I.T. can skip this chapter or perhaps glance through it briefly as a refresher. §0. Preliminaries. we assume the reader to be familiar with the basic notions of first-order logic—the logical connectives, quantifiers, terms, formulas, free and bound occurrences of variables, the notion of interpretations (or models), truth under an interpretation, logical validity (truth under all interpretations), provability (in some complete system of first-order logic with identity) and its equivalence to logical validity (Gödel’s completeness theorem). we let S be a system (theory) couched in the language of first-order logic with identity and with predicate and/or function symbols and with names for the natural numbers. A system S is usually presented by taking some standard axiomatization of first-order logic with identity and adding other axioms called the non-logical axioms of S.we associate with each natural number n an expression n̅ of S called the numeral designating n (or the name of n).we could, for example, take 0̅,1̅,2̅, . . . ,to be the expressions 0,0', 0",..., as we did in G.I.T. we have our individual variables arranged in some fixed infinite sequence v1, v2,..., vn , . . . . By F(v1, ..., vn) we mean any formula whose free variables are all among v1,... ,vn, and for any (natural) numbers k1,...,kn by F(к̅1 ,... к̅n), we mean the result of substituting the numerals к̅1 ,... к̅n, for all free occurrences of v1,... ,vn in F respectively.

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