Gently Paraconsistent Calculi

In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect the principle of gentle explosion.

New foundations for Lewis modal systems

10.2307/2964179 ◽

1957 ◽

Vol 22 (2) ◽

pp. 176-186 ◽

Cited By ~ 51

Author(s):

E. J. Lemmon

Keyword(s):

Modal Logic ◽

Propositional Calculus ◽

Modal System ◽

New Foundations ◽

Image Position ◽

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.

Classical Logic in the Quantum Context

Quantum Reports ◽

10.3390/quantum2040042 ◽

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 ◽

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

10.2307/2270049 ◽

1968 ◽

Vol 33 (1) ◽

pp. 27-38 ◽

Cited By ~ 34

Author(s):

R. A. Bull

Keyword(s):

Linear Time ◽

Propositional Calculus ◽

Tense Logic ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

Fourth Section ◽

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

10.2307/2274102 ◽

1984 ◽

Vol 49 (1) ◽

pp. 192-203 ◽

Cited By ~ 13

Author(s):

Nicolas D. Goodman

Keyword(s):

Propositional Calculus ◽

Cut Elimination ◽

Conservative Extension ◽

First Order ◽

Classical Mathematics ◽

Order Arithmetic ◽

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

10.2307/2274168 ◽

1984 ◽

Vol 49 (2) ◽

pp. 329-333 ◽

Cited By ~ 2

Author(s):

Branislav R. Boričić

Keyword(s):

Positive Solution ◽

Natural Deduction ◽

Propositional Calculus ◽

Conservative Extension ◽

Image Position ◽

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]).

Abacus logic: The lattice of quantum propositions as the poset of a theory

10.2307/2275403 ◽

1994 ◽

Vol 59 (2) ◽

pp. 501-515

Author(s):

Othman Qasim Malhas

Keyword(s):

Hilbert Space ◽

Quantum Logic ◽

Hidden Variables ◽

Propositional Calculus ◽

Real Numbers ◽

Quantum Observable ◽

Quantum Observables ◽

Classical Observable ◽

Graphic Interpretation

AbstractWith a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.

Algebraic proof of the separation theorem on classical propositional calculus

Proceedings of the Japan Academy ◽

10.3792/pja/1195522117 ◽

1966 ◽

Vol 42 (2) ◽

pp. 67-69 ◽

Cited By ~ 3

Author(s):

Tsutomu Hosoi

Keyword(s):

Propositional Calculus ◽

Separation Theorem ◽

Algebraic Proof ◽

Categorical proof theory of classical propositional calculus

Theoretical Computer Science ◽

10.1016/j.tcs.2006.08.002 ◽

2006 ◽

Vol 364 (2) ◽

pp. 146-165 ◽

Cited By ~ 10

Author(s):

Gianluigi Bellin ◽

Martin Hyland ◽

Edmund Robinson ◽

Christian Urban

Keyword(s):

Proof Theory ◽

Propositional Calculus ◽

Helena Rasiowa. Z dziedziny logiki matematycznej.—I. Klasyczny rachunek zdań. (From the domain of mathematical logic.—I. The classical propositional calculus.) Matemaíyka, vol. 2 (1949), pp. 3–8. - Stanisław Goląb. O kilku trudnościach zwiqzanych z prostymi kwestiami logicznymi (On some difficulties connected with simple logical questions). Matemaíyka, vol. 2 (1949), pp. 27–29. - Jerzy Słupecki. O własnościach okresu warunkowego (On properties of the conditional). Matemaíyka, vol. 2 (1949), pp. 32–35.

10.2307/2266987 ◽

1950 ◽

Vol 15 (2) ◽

pp. 139-140

Author(s):

Andrzej Mostowski

Keyword(s):

Mathematical Logic ◽

Propositional Calculus ◽

Quantum logic and the classical propositional calculus

10.1017/s0022481200029819 ◽

1987 ◽

Vol 52 (3) ◽

pp. 834-841 ◽

Cited By ~ 3

Author(s):

Othman Qasim Malhas

Keyword(s):

Quantum Logic ◽

Hyperbolic Geometry ◽

Euclidean Plane ◽

Propositional Calculus ◽

AbstractIn much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.