Some calculi with strong negation primitive

1968 ◽  
Vol 33 (1) ◽  
pp. 97-100
Author(s):  
J. Jay Zeman
Keyword(s):  
Propositional Calculus ◽  
Strong Negation ◽  
Material Implication ◽  
Unary Operator ◽  
Modal Systems

We shall take “strong negation” to be a unary operator with some of the properties usually associated with negation, and such that the strong negation of a statement implies the “ordinary” negation ofthat statement, but not vice-versa. This paper will consider two varieties of strong negation: first, strong negation set in the context of the propositional calculus, specifically, the intuitionist PC, and secondly, strong negation in modal systems, that is, strong negation as impossibility. In the latter part of the paper, we shall present axiomatizations of several classical Lewis-modal systems having only material implication and impossibility primitive.

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.

Strong completeness of fragments of the propositional calculus

1951 ◽  
Vol 16 (3) ◽  
pp. 204-204 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Primitive Function ◽  
Strong Completeness ◽  
Substitution Rule ◽  
Material Implication ◽  
Rules Of Procedure

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of theTheorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.

Modal propositional logic on an orthomodular basis. I

1974 ◽  
Vol 39 (3) ◽  
pp. 478-488 ◽  
Author(s):  
L. Herman ◽  
R. Piziak
Keyword(s):  
Boolean Algebra ◽  
Algebraic Logic ◽  
Propositional Calculus ◽  
Natural Generalization ◽  
Point Of View ◽  
Material Implication ◽  
Transformation Rules ◽  
Orthocomplemented Lattice ◽  
Modal Propositional Logic

A common method of obtaining the classical modal logics, for example the Feys system T, the Lewis systems, the Brouwerian system etc., is to build on a basis for the propositional calculus by adjoining a new symbol L, specifying new axioms involving L and the symbols in the basis for PC, and imposing one or more additional transformation rules. In the jargon of algebraic logic, which is the point of view we shall adopt, the “necessity” symbol L may be interpreted as an operator on the Boolean algebra of propositions of PC. For example, the Lewis system S4 may be regarded as a Boolean algebra ℒ together with an operator L on ℒ having the properties: (1)Lp ≤ p for all p in ℒ, (2)L1 = 1, (3) L(p → q) ≤ Lp → Lq for all p, q in ℒ, and (4) Lp = L(Lp) for all p in ℒ. Here, of course, → denotes the material implication connective: p→q = p′ ∨ q. It is easy to verify that property (3) may be replaced by either (3′) L(p ∧ q) = Lp ∧ Lq for all p, q in ℒ, or by (3″) L(p → q) ∧ Lp ≤ Lq for all p, q in ℒ. In particular, it follows from (1) through (4) above that L is a decreasing, idempotent and isotone operator on ℒ. Such mappings are often called interior operators.In a previous paper [5], we considered the problem of introducing an implication connective into a quantum logic. This is greatly complicated by the fact that the quantal propositions band together to form an orthocomplemented lattice which is only “locally” distributive. Such lattices are called orthomodular. For definitions and further discussion, the reader is referred to that paper. In it, we argued that the Sasaki implication connective ⊃ defined by p ⊃ q = p′ ∨ (p ∧ q) is a natural generalization of material implication when the lattice of propositions is ortho-modular. Indeed, if unrestricted distributivity were permitted, p ⊃ q would reduce to the classical material implication p → q. For this reason, we choose ⊃ to play the role of material implication in an orthomodular lattice. Further properties of ⊃ are enumerated in Example 2.2(1) and Corollary 2.4 below.

Antitheses in systems of relevant implication

1983 ◽  
Vol 48 (1) ◽  
pp. 97-99 ◽  
Author(s):  
Jean Porte
Keyword(s):  
Propositional Calculus ◽  
Propositional Variable ◽  
Material Implication ◽  
Relevant Implication ◽  
Classical Negation ◽  
Classical Propositional Calculus ◽  
Propositional Calculi

There are several ways for a formula to be “the contrary” of a thesis. When there is a negation we could call a formula an “antithesis” if it is the negation of a thesis—or its negation is a thesis—both properties being equivalent when the negation is classical.When there is no negation or when the connective called “negation” is very different from classical negation, we are forced to look for a different notion.Whence:Definition 1. In a propositional calculus, a substitution antithesis—or, for short, an s-antithesis—is a formula no instance of which is a thesis.There is another kindred notion, but only for calculi for which a deducibility has been defined:Definition 2. A deduction antithesis—for short, a d-antithesis—is a formula from which every formula is deducible.Both notions have been used in the study of the deducibility of certain systems (see below, §4).In the classical propositional calculus, the s-antitheses are simply the formulas which are negations of theses (and whose negations are theses).In S5 and in all the weaker modal propositional calculi, there are s-antitheses which are not negations of theses (and whose negations are not theses); typical examples are ¬(Mp ⊃ Lp) and ¬(p ⊃ Lp) where p is a propositional variable, and ⊃, ¬, M and L are respectively material implication, negation, possibility and necessity (see [8]).

Fragments of the propositional calculus

1949 ◽  
Vol 14 (1) ◽  
pp. 42-48 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Propositional Calculus ◽  
Function Symbol ◽  
Material Implication ◽  
Complete Axiomatization ◽  
Truth Function ◽  
Conceptual Changes

Of the several methods for proving the completeness of sets of axioms for the prepositional calculus perhaps the simplest is due to Kalmár, although it does not appear to be widely known. In this paper we generalize Kalmár's method to indicate how to obtain a complete axiomatization of any fragment of the propositional calculus which includes material implication. We shall carry through the description and proofs for the case where, in addition to a symbol for implication, there is just one other primitive truth-function symbol. For systems in which there are no other function symbols, or more than one other such symbol, notational changes but not conceptual changes will be required.

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