A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS

1983 ◽  
Vol 48 (1) ◽  
pp. 1-20 ◽  
Author(s):  
H. C. M. de Swart
Keyword(s):  
Decision Procedure ◽  
Type System ◽  
Propositional Calculus ◽  
Axiom System ◽  
Completeness Proof ◽  
Counterfactual Conditional ◽  
System A ◽  
Counterfactual Logic ◽  
Classical Propositional Calculus

In [1] and [2] D. Lewis formulates his counterfactual logic VC as follows. The language contains the connectives ∧, ∨, ⊃, ¬ and the binary connective ≤. A ≤ B is read as “A is at least as possible as B”. The following connectives are defined in terms of ≤.A < B: = ¬(B ≤ A) (it is more possible that A than that B).◊ A ≔ ¬(⊥ ≤ A) (⊥ is the false formula; A is possible).□ A ≔ ⊥ ≤ ¬A (A is necessary). (if A were the case, then B would be the case). (if A were the case, then B might be the case). and are two counterfactual conditional operators. (AB) iff ¬(A ¬B).The following axiom system VC is presented by D. Lewis in [1] and [2]: V: (1) Truthfunctional classical propositional calculus.

An algebraic study of Diodorean modal systems

1965 ◽  
Vol 30 (1) ◽  
pp. 58-64 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Discrete Time ◽  
Decision Procedure ◽  
John Locke ◽  
Original Model ◽  
Completeness Proof ◽  
Semantic Tableaux ◽  
Algebraic Treatment ◽  
Algebraic Result ◽  
Modal Systems

Attention was directed to modal systems in which ‘necessarily α’ is interpreted as ‘α. is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 withALCLpLqLCLqLp,is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 withALNLpLCLCLCpLpLpLp,is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski.A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 andCLCLCpLpLpCMLpLp.While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.

Effect of residual Doppler averaging on the probe absorption in cascade type system: A comparative study

2018 ◽  
Vol 27 (9) ◽  
pp. 094204 ◽  
Author(s):  
Suman Mondal ◽  
Arindam Ghosh ◽  
Khairul Islam ◽  
Dipankar Bhattacharyya ◽  
Amitava Bandyopadhyay
Keyword(s):  
Comparative Study ◽  
Type System ◽  
System A ◽  
Probe Absorption ◽  
Cascade Type

New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

scholarly journals Classical Logic in the Quantum Context

2020 ◽  
Vol 2 (4) ◽  
pp. 600-616
Author(s):  
Andrea Oldofredi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Classical Logic ◽  
Theory Of Measurement ◽  
Propositional Calculus ◽  
Bohmian Mechanics ◽  
Physical Content ◽  
Present Essay ◽  
Classical Propositional Calculus

It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.

An algebraic study of tense logics with linear time

1968 ◽  
Vol 33 (1) ◽  
pp. 27-38 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Linear Time ◽  
Propositional Calculus ◽  
Tense Logic ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Fourth Section ◽  
Classical Propositional Calculus ◽  
Axiom Systems ◽  
Time Linear

In [2] Prior puts forward a tense logic, GH1, which is intended to axiomatise tense logic with time linear and rational; he also contemplates the tense logic with time linear and real. The purpose of this paper is to give completeness proofs for three axiom systems, GH1, GHlr, GHli, with respect to tense logic with time linear and rational, real, and integral, respectively.1 In a fourth section I show that GH1 and GHlr have the finite model property, but that GHli lacks it.GH1 has the operators of the classical propositional calculus, together with operators P, H, F, G for ‘It has been the case that’, ‘It has always been the case that’, ‘It will be the case that’, ‘It will always be the case that’, respectively.

Epistemic arithmetic is a conservative extension of intuitionistic arithmetic

1984 ◽  
Vol 49 (1) ◽  
pp. 192-203 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Propositional Calculus ◽  
Cut Elimination ◽  
Conservative Extension ◽  
First Order ◽  
Classical Mathematics ◽  
Order Arithmetic ◽  
Classical Propositional Calculus ◽  
Epistemic Arithmetic ◽  
Classical Arithmetic ◽  
Intuitionistic Arithmetic

Questions about the constructive or effective character of particular arguments arise in several areas of classical mathematics, such as in the theory of recursive functions and in numerical analysis. Some philosophers have advocated Lewis's S4 as the proper logic in which to formalize such epistemic notions. (The fundamental work on this is Hintikka [4].) Recently there have been studies of mathematical theories formalized with S4 as the underlying logic so that these epistemic notions can be expressed. (See Shapiro [7], Myhill [5], and Goodman [2]. The motivation for this work is discussed in Goodman [3].) The present paper is a contribution to the study of the simplest of these theories, namely first-order arithmetic as formalized in S4. Following Shapiro, we call this theory epistemic arithmetic (EA). More specifically, we show that EA is a conservative extension of Hey ting's arithmetic HA (ordinary first-order intuitionistic arithmetic). The question of whether EA is conservative over HA was raised but left open in Shapiro [7].The idea of our proof is as follows. We interpret EA in an infinitary propositional S4, pretty much as Tait, for example, interprets classical arithmetic in his infinitary classical propositional calculus in [8]. We then prove a cut-elimination theorem for this infinitary propositional S4. A suitable version of the cut-elimination theorem can be formalized in HA. For cut-free infinitary proofs, there is a reflection principle provable in HA. That is, we can prove in HA that if there is a cut-free proof of the interpretation of a sentence ϕ then ϕ is true. Combining these results shows that if the interpretation of ϕ is provable in EA, then ϕ is provable in HA.

A note on some intermediate propositional calculi

1984 ◽  
Vol 49 (2) ◽  
pp. 329-333 ◽  
Author(s):  
Branislav R. Boričić
Keyword(s):  
Positive Solution ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Conservative Extension ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Propositional Calculi

This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.Let Cn(n ≥ 1) be the string of the following sequents:Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

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