scholarly journals Classical Logic with n Truth Values as a Symmetric Many-Valued Logic

2020 ◽  
Author(s):  
A. Salibra ◽  
A. Bucciarelli ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Propositional Calculus ◽  
Boolean Algebras ◽  
Truth Values ◽  
First Order ◽  
Rewriting System ◽  
Central Elements ◽  
Algebraic Framework ◽  
Classical Propositional Calculus

Abstract We introduce Boolean-like algebras of dimension n ($$n{\mathrm {BA}}$$ n BA s) having n constants $${{{\mathsf {e}}}}_1,\ldots ,{{{\mathsf {e}}}}_n$$ e 1 , … , e n , and an $$(n+1)$$ ( n + 1 ) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of $$n{\mathrm {BA}}$$ n BA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The $$n{\mathrm {BA}}$$ n BA s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, $$n{\mathrm {CL}}$$ n CL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in $$n{\mathrm {CL}}$$ n CL , and, via the embeddings, in all the finite tabular logics.

scholarly journals Sobre a lógica deôntica não-clássica

1987 ◽  
Vol 19 (55) ◽  
pp. 19-37
Author(s):  
Leila Z. Puga ◽  
Newton C.A. Da Costa
Keyword(s):  
Propositional Logic ◽  
Ethical Theory ◽  
Classical System ◽  
Propositional Calculus ◽  
Moral Dilemmas ◽  
Prima Facie ◽  
First Order ◽  
Starting Point ◽  
Expository Paper ◽  
Classical Propositional Calculus

Our starting point, in this basically expository paper, is the study of a classical system of deontic propositional logic, classical in the sense that it constitutes an extension of the classical propositional calculus. It is noted, then, that the system excludes ab initio the possibility of the existence of real moral dilemmas (contradictory obligations and prohibitions), and also can not cope smoothly with the so-called prima facie moral dilemmas. So, we develop a non-classical, paraconsistent system of propositional deontic logic which is compatible with such dilemmas, real or prima facie. In our paraconsistent system one can handle them neatly, in particular one can directly investigate their force, operational meaning, and the most important consequences of their acceptance as not uncommon moral facts. Of course, we are conscious that other procedures for dealing with them are at hand, for example by the weakening of the specific deontic axioms. It is not argued that our procedure is the best, at least as regards the present state of the issue. We think only that owing, among other reasons, to the circumstance that the basic ethical concepts are intrinsically vague, it seems quite difficult to get rid of moral dilemmas and of moral deadlocks in general. Apparently this speaks in favour of a paraconsistent approach to ethics. At any rate, a final appraisal of the possible solutions to the problem of dilemmas and deadlocks, if there is one, constitutes a matter of ethical theory and not only of logic. On the other hand, the paraconsistency stance looks likely to be relevant also in the field of legal logic. It is shown, in outline, that the systems considered are sound and complete, relative to a natural semantics. All results of this paper can be extended to first-order and to higher-order logics. Such extensions give rise to the question of the transparency (or oppacity) of the deontic contexts. As we shall argue in forthcoming articles, they normally are transparent. [L.Z.P., N.C.A. da C.] (PDF en portugués)

scholarly journals On an interpretation of second order quantification in first order intuitionistic propositional logic

1992 ◽  
Vol 57 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Andrew M. Pitts
Keyword(s):  
Propositional Logic ◽  
Heyting Algebra ◽  
Propositional Calculus ◽  
Interpolation Theorem ◽  
Second Order ◽  
Heyting Algebras ◽  
Intuitionistic Propositional Logic ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

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.

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.

Natural deduction based set theories: a new resolution of the old paradoxes

1986 ◽  
Vol 51 (2) ◽  
pp. 393-411 ◽  
Author(s):  
Paul C. Gilmore
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Truth Values ◽  
First Order ◽  
Axiom Scheme ◽  
Universe Of Sets ◽  
Comprehension Axiom

AbstractThe comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

Free L-algebras

1969 ◽  
Vol 34 (3) ◽  
pp. 475-480 ◽  
Author(s):  
Alfred Horn
Keyword(s):  
Heyting Algebra ◽  
Propositional Calculus ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Classical Propositional Calculus

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.

The metatheory of the classical propositional calculus is not axiomatizable

1985 ◽  
Vol 50 (2) ◽  
pp. 451-457 ◽  
Author(s):  
Ian Mason
Keyword(s):  
Propositional Logic ◽  
Propositional Calculus ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Property ◽  
Classical Propositional Logic ◽  
Open Problems ◽  
First Order

In this paper we investigate the first order metatheory of the classical propositional logic. In the first section we prove that the first order metatheory of the classical propositional logic is undecidable. Thus as a mathematical object even the simplest of logics is, from a logical standpoint, quite complex. In fact it is of the same complexity as true first order number theory.This result answers negatively a question of J. F. A. K. van Benthem (see [van Benthem and Doets 1983]) as to whether the interpolation theorem in some sense completes the metatheory of the calculus. Let us begin by motivating the question that we answer. In [van Benthem and Doets 1983] it is claimed that a folklore prejudice has it that interpolation was the final elementary property of first order logic to be discovered. Even though other properties of the propositional calculus have been discovered since Craig's orginal paper [Craig 1957] (see for example [Reznikoff 1965]) there is a lot of evidence for the fundamental nature of the property. In abstract model theory for example one finds that very few logics have the interpolation property. There are two well-known open problems in this area. These are1. Is there a logic satisfying the full compactness theorem as well as the interpolation theorem that is not equivalent to first order logic even for finite models?2. Is there a logic stronger than L(Q), the logic with the quantifierthere exist uncountably many, that is countably compact and has the interpolation property?

A LOGICAL FORMULATION OF ROUGH SET DEFINABILITY

2011 ◽  
Vol 19 (01) ◽  
pp. 27-37 ◽  
Author(s):  
TUAN-FANG FAN ◽  
CHURN-JUNG LIAU ◽  
DUEN-REN LIU
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Classical Logic ◽  
Propositional Logic ◽  
Rough Set Theory ◽  
First Order ◽  
Decision Logic ◽  
Data Tables ◽  
Attribute Value

In data mining problems, data is usually provided in the form of data tables. To represent knowledge discovered from data tables, a decision logic (DL) is proposed in rough set theory. DL is an instance of propositional logic, but we can use other logical formalisms to describe data tables. In this paper, we propose two descriptions of data tables based on first-order data logic (FODL) and attribute value-sorted logic (AVSL) respectively. In the context of FODL, we show that explicit definability and implicit definability in classical logic implies the notion of definability in rough set theory. We also show that AVSL is particularly useful for the representation of properties of many-valued data tables.

scholarly journals Extensions in graph normal form

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.

On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

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