NEW RESULTS IN LOGIC OF FORMULAS WHICH LOSE SENSE

Transcription

1 Bulletin of the Section of Logic Volume 14/3 (1985), pp reedition 2007 [original edition, pp ] Adam Morawiec Krystyna Piróg-Rzepecka NEW RESULTS IN LOGIC OF FORMULAS WHICH LOSE SENSE Abstract This is an extended version of an abstract that appeared in the Proceedings of the National Conference Universal Algebra and its Applications. The Conference took place in Jarno ltówek, Poland, in May 1985, and was organized by the Mathematical Institute of College of Education, Opole. The aim of this abstract ie to present the latest results concerning a system of nonsense-logic, known as the system W. Thus, we recall only those of previous results which are indispensable for our present considerations. A more exhaustive survey and bibliography can be found in [1]; our notation and terminology are those of [1], too. Primitive symbols of W are,, and which stand for implication, conjunction, and negation, respectively. The system itself may be defined by the following three-valued matrix: M 3 = <{1, 0, 1 / 2 }, {1},,, > where the functions,, are defined as follows: / / / / / 2 1 / 2 1 / 2 1 / 2 1 / / 2 1 / 2

2 New Results in Logic of Formulas which Lose Sense 115 The values 1, 0, 1 / 2 we interpret as truth, falsity, and nonsense, respectively, and the functions,, in the usual way. [We will abuse the notation by using the same symbols for functors of W, functions of M 3, and, further, for primitive symbols of T W, but it will not provoke any misunderstandings. Moreover, we will not distinguish between an expression of a given formal language and its name.] For W it was shown that it is axiomatizable, L-decidable, 2-complete as well as structurally complete; a theory of consequence, T W, strongly adequate (see [2]) to the system itself, an extended algebra of sets (see [4]), and a suppositional system, each based upon W were also presented. New results refer to the theory T W and to a quantification theory built upon the system W. Primitive symbols of T W are: S, Cn,,,, S being a set, Cn a function from the power set of S, P (S), into itself, and,, names for the corresponding functors of W. Variables x, y, z,... will denote elements and X, Y, Z,... subsets of S. By an S-substitution of a formula of W we mean, somewhat unprecisely, the formula resulting from the given one, say ϕ, after all the sentential variables occurring in ϕ have been replaced, arbitrary but correctly, by some of x, y, z,.... To formulate axioms of the theory T W we need the following definition of an essential variable of formula of W : If a formula is a sentential variable, this variable is its only essential one; If the formula is of the form ϕ, then its all and only essential variables are those of ϕ. If the formula is of the form ϕ ψ, then its all and only essential variables are both those of ϕ and of ψ; There is no essential variable in ϕ ψ. A variable, x is an essential one of an S-substitution of ϕ iff the sentential variable replaced in ϕ by x was an essential one of ϕ. Besides all the axioms of Tarski s general methodology, we take the following sentences and schemata as the axioms of T W :

3 116 Adam Morawiec and Krystyna Piróg-Rzepecka AI x y CnX iff y Cn(X {x}), AII Cn{x, y} = Cn{x y}, AIII Cn{x, x} = S, AIV x y CnX iff Cn(X {x (y y)}) = S, AV if Cn{ϕ ψ} = S, then ϕ ψ CnΛ, AV I if Cn{χ, ϕ ψ} = S, then χ (ϕ ψ) CnΛ, where ϕ, ψ and χ are S-substitutions of W -formulas such that in AV every essential variable of ψ is an essential one of ϕ, and in AV I every essential variable of ψ is also an essential one of χ. Now, let T W be the theory obtained from T W by replacing AIII and AIV with the single axiom: AIII Cn(X {x x}) = S iff x CnX, and by omitting the schema AV I. Theorem 1. The theory T W is deductively equivalent to T W. The theorem is an immediate consequence of the following lemmas ( T stands for the predicate of provability in a given theory T ): Lemma 1. a. T W (Cn(X {x x}) = S iff x CnX, b. T W Cn{x, x} = S, c. T W (x y CnX iff Cn(X {x (y y)}) = S). Lemma 2. Schemata AV and AVI are equivalent to each other on the ground of the other axioms. Although simpler than T W, T W still requires for its formulation a notion external of T W itself the notion of an essential variable. It turns out, however, that we can define theory of deductive systems strongly adequate to W not using this notion. Just replace AV of T W by the sentences: AIV Cn{( x x) x, ( y y) y} = = Cn{( (x y) x y) x y}, AV Cn{ (x y) (x y)} = Cn{x y}, AV I Cn{ x} = Cn{x}.

4 New Results in Logic of Formulas which Lose Sense 117 Let T W be the resulting theory, i.e. its axioms consist of AI, AII, AIII, AIV, AV, AV I, and all the axioms of general methodology. Theorem 2. Both theories T W and T W are strongly adequate to the system W. To close this part, let us notice that property to be an essential variable can be espressed, in some sense, in W : Theorem 3. A sentential variable p is an essential one of ϕ iff W (( ϕ ϕ) ϕ) (( p p) p). Now, a quantification theory W built upon W will briefly be discussed. We enrich our symbolism by adding to the symbols of W an universal quantifier, and two countable sets of variables: individual x x 1, x 2, x 3,... and predicative ones, P 1, P 2, P 3,.... ϕ, ψ, χ,... will denote, from now on, formulas of this enriched language. For a given nonempty set V, we call a function f an interpretation in V if: 1. f(p) {1, 0, 1 / 2 }, for every sentential variable p; 2. f(x) V, for every, individual variable x; 3. f(p ) = < V P, F P >, for every predicative variable P, where V P, F P V and V P F P =. In [1] expressions Sp V,f (ϕ) = 1 and Sp V,f (ϕ) = 0, were defined for a system S; we extend that given inductive definition in the following way: a. Sp V,f ( x k ψ) = 1 iff for every a V Sp V,f a k (ψ) = 1, where f a k is such an interpretation in V that for every predicative variable P fk a(p ) = f(p ) and { fk a f(xj ), for j k, (x j ) = a, for j = k; b. Sp V,f ( ψ) = 0 iff there is a V such that Sp V,f a(ψ) = 0 and for k x k every a V Sp V,f a (ψ) = 1 or Sp V,f a (ψ) = 0. k k

5 118 Adam Morawiec and Krystyna Piróg-Rzepecka Definition 1. A formula ϕ is a tautology of W iff for every nonempty set V and every interpretation f in V Sp V,f (ϕ) = 1. It appears that W can be axiomatized, i.e. there is an axiomaticsystem W A such that for W and W A both the sets of all formulas and the sets of all acceptable formulas are equal to each other. Primitive symbols of W A are all those of W ; its axioms, besides all instants of some axioms for W, are the following schemata: A1. x k ϕ(x k ) ϕ(x j ); A2. x k (ψ ϕ(x k )) (ψ x k ϕ(x k )); A3. x k ϕ(x k ) (( ϕ(x j ) ϕ(x j )) ϕ(x j )); A4. x k (ψ (( ϕ(x k ) ϕ(x k )) ϕ(x k ))) (ψ (( x k ϕ(x k ) x k ϕ(x k )) x k ϕ(x k ))); provided that in A2 and A4 x k does not occur free in ψ. Modus ponens ϕ, ϕ ψ/ψ, and two kinds of generalization, ϕ(x k )/ x k ϕ(x k ) and ( ϕ(x k ) ϕ(x k )) ϕ(x k )/( x k ϕ(x k ) x k ϕ(x k )) x k ϕ(x k ) are the only rules of inferences of W A. Theorem 4. ϕ is a tautology of W iff ϕ is a theorem of W A. To the end let us notice that for formulas of W built of the quantifier, functors of W, and unary predicates of the same variable an effective decision method, analogous to that of the classical case (see [3]), was described. References [1] K. Piróg-Rzepecka, Systemy nonsense-logics, PWN, Warszawa- Wroc law, 1977 (in Polish). [2] W. A. Pogorzelski, Adekwatność teorii systemów dedukcyjnych wzglȩdem rachunków zdaniowych, Studia Logica, Vol. XIII, 1962 (in Polish).

6 New Results in Logic of Formulas which Lose Sense 119 [3] J. S lupecki, K. Ha lkowska and K. Piróg-Rzepecka, Logika matematyczna, PWN, Warszawa-Wroc law, 1976 (in Polish). [4] J. S lupecki and K. Piróg-Rzepecka, An extension of the algebra of sets, Studia Logica, Vol. XXXI, Institute of Mathematics College of Education Opole, Poland