A geometric proof of the completeness of the Łukasiewicz calculus

1995 ◽  
Vol 60 (2) ◽  
pp. 563-578 ◽  
Author(s):  
Giovanni Panti
Keyword(s):  
Completeness Theorem ◽  
Geometric Proof ◽  
Sentential Calculus

AbstractWe give a self-contained geometric proof of the completeness theorem for the infinite-valued sentential calculus of Łukasiewicz.

Algebraic completeness results for R-mingle and its extensions

1970 ◽  
Vol 35 (1) ◽  
pp. 1-13 ◽  
Author(s):  
J. Michael Dunn
Keyword(s):  
Algebraic Logic ◽  
Completeness Theorem ◽  
Strong Completeness ◽  
Finite Characteristic ◽  
Sentential Calculus ◽  
Completeness Result

Schiller Joe Scroggs in [9] established remarkable facts concerning “normal” extensions of the modal sentential calculus S5, the most notable of these facts being that all such proper extensions have finite characteristic matrices. The major import of the present paper is that like facts hold for the relevant sentential calculus R-Mingle (RM). Robert K. Meyer in [6] has obtained an important completeness result for RM, which will play a central role in our results. However, in §2 we shall obtain a new proof of Meyer's result as a by-product of the algebraic logic that we develop in §1. Also in §2 we shall obtain the promised results for extensions of RM. In §3 we shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer.

On Statman's Finite Completeness Theorem

1992 ◽  
Author(s):  
Richard Statman ◽  
Gilles Dowek
Keyword(s):  
Completeness Theorem

Algorithmic Logic with Recursive Functions

1981 ◽  
Vol 4 (4) ◽  
pp. 975-995
Author(s):  
Andrzej Szałas
Keyword(s):  
Completeness Theorem ◽  
Inference Rules ◽  
Partial Correctness ◽  
Recursive Functions ◽  
Algorithmic Logic ◽  
Block Structured

A language is considered in which the reader can express such properties of block-structured programs with recursive functions as correctness and partial correctness. The semantics of this language is fully described by a set of schemes of axioms and inference rules. The completeness theorem and the soundness theorem for this axiomatization are proved.

PAL – Propositional Algorithmic Logic

1981 ◽  
Vol 4 (3) ◽  
pp. 675-760
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Data Structures ◽  
Completeness Theorem ◽  
Natural Numbers ◽  
First Order ◽  
Algorithmic Logic ◽  
Axiomatic Systems ◽  
Nondeterministic Program ◽  
Basic Sets

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

A Logically Justified Model of Computation I

1981 ◽  
Vol 4 (1) ◽  
pp. 151-172
Author(s):  
Pierangelo Miglioli ◽  
Mario Ornaghi
Keyword(s):  
Computer Science ◽  
Program Synthesis ◽  
Completeness Theorem ◽  
Point Of View ◽  
Program Execution ◽  
Computation Model ◽  
Clear Explanation ◽  
Direct Interpretation ◽  
Adequacy Condition ◽  
General Explanation

The aim of this paper is to provide a general explanation of the “algorithmic content” of proofs, according to a point of view adequate to computer science. Differently from the more usual attitude of program synthesis, where the “algorithmic content” is captured by translating proofs into standard algorithmic languages, here we propose a “direct” interpretation of “proofs as programs”. To do this, a clear explanation is needed of what is to be meant by “proof-execution”, a concept which must generalize the usual “program-execution”. In the first part of the paper we discuss the general conditions to be satisfied by the executions of proofs and consider, as a first example of proof-execution, Prawitz’s normalization. According to our analysis, simple normalization is not fully adequate to the goals of the theory of programs: so, in the second section we present an execution-procedure based on ideas more oriented to computer science than Prawitz’s. We provide a soundness theorem which states that our executions satisfy an appropriate adequacy condition, and discuss the sense according to which our “proof-algorithms” inherently involve parallelism and non determinism. The Properties of our computation model are analyzed and also a completeness theorem involving a notion of “uniform evaluation” of open formulas is stated. Finally, an “algorithmic completeness” theorem is given, which essentially states that every flow-chart program proved to be totally correct can be simulated by an appropriate “purely logical proof”.

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).

scholarly journals QUANTUM TEAM LOGIC AND BELL’S INEQUALITIES

2015 ◽  
Vol 8 (4) ◽  
pp. 722-742 ◽  
Author(s):  
TAPANI HYTTINEN ◽  
GIANLUCA PAOLINI ◽  
JOUKO VÄÄNÄNEN
Keyword(s):  
Quantum Mechanics ◽  
Completeness Theorem ◽  
Probability Logic ◽  
Bell’S Inequalities ◽  
Dependence Logic ◽  
Logical Approach ◽  
Team Semantics ◽  
Bell's Inequalities

AbstractA logical approach to Bell’s Inequalities of quantum mechanics has been introduced by Abramsky and Hardy (Abramsky & Hardy, 2012). We point out that the logical Bell’s Inequalities of Abramsky & Hardy (2012) are provable in the probability logic of Fagin, Halpern and Megiddo (Fagin et al., 1990). Since it is now considered empirically established that quantum mechanics violates Bell’s Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell’s Inequalities are not provable, and prove a Completeness theorem for this logic. For this end we generalise the team semantics of dependence logic (Väänänen, 2007) first to probabilistic team semantics, and then to what we call quantum team semantics.

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