53 REND. SEM. MAT. UNI VERS. POLITECN. TORINO Vol. 39°, 1 (1981)

Salvatore Guccione — Roberto Tortora — Virginia Vaccaro DEDUCTION THEOREMS IN LUKASIEWICZ PROPOSITIONAL CALCULI

Summary: In this paper we give some results about the validity of a Deduction Theorem for the complete axiomatizations of Lukasiewicz many-valued propositional calculi. These results follow from detailed analysis of the rules of inference occurring in each system.

As it is well-known, Lukasiewicz developed in [2] the many-valued propositional calculi based on two primitive connectives, negation N and implication C, which were defined semantically. Complete axiomatizations followed later on. Among the works on this argument, we mention Wajsberg [9], Rosser and Turquette [8], Mc Naughton [3], Rose and Rosser [7], Meredith [4], Chang [1], Rose [5], [6]. It is also known that, if two formal systems have the same rules of inference and the same theorems, then the Deduction Theorem holds for the former if and only if it holds for the latter, independently from the axioms of the systems. The aim of the present work is to analyze the rules of inference occurring in the various axiomatizations of the Lukasiewicz propositional calculi, in order to discuss the validity of the Deduction Theorem for them. We emphasize that the Deduction Theorem here discussed is relative to the original Lukasiewicz connective C. The specification is necessary, since in the-

Classificazione per soggetto: 02C05

54 se calculi another implication can be defined, namely the function I (see, -g- [8]). A Deduction Theorem relative to I can be also formulated. We will briefly examine this point at the end of Section 2.

e

1. We denote by $ the set of truth values, ./must contain 0 and 1. If./ contains M > 2 elements, it consists of the rational numbers 1 M-2 ' M-l ' M-1 ' If J is infinite, it is generally identified with the set of rational numbers of the interval [0, 1]. A rational number s G [0, 1] is selected. Each element of «/ greater than s is called a designated truth value. If s G J>, it must be specified whether s is designated or not: if £ is finite, we suppose without loss of generality that s is designated, the distinction makes sense only in the case that J> is infinite. An assignment of truth values is a function v from the set of wellformed formulas (wffs) to the set«/. The functions N and C are defined semantically as usual: v(NP) = 1 - v(P) v(CPQ) = ww {171 ~ v(P) + *((?)}. C and N are the unique primitive function^. If J> is finite, we use the functions P arid A(P) (1 < k 0 OL>0 0i>\

EPNBOL-IP

BPP

and, for every a. > 0, Dc^jP can be defined in terms of C and N by virtue of a theorem of Mc Naughton [3] in such a way that: v0afiP)=

max {0, min{l, (0 + 1) v(P) - a } }

I Ma-i otP for KB^^^NP

M P

I MOL-WP

t

7*-iiu1

for V

^

nsfi a>2>

(Ma-i,otPr

^n ^

, .

m

2 < / 3 < a - l , (a,0) = l

For the functions Afa-0,aP can be proved that (1)

v(Ma^f0LP) = max{0, 0 - m i n d - v(P)t (a-

1) v(P) - 0 + 1}}

Functions G^aP are also used in [5]. We adopt for them the definition there suggested. G&aP for HmP(ci>2, l < 0 < a - l , (a, 0) = 1 ) where H 0 a P is defined by virtue of the above quoted Mc Naughton's theorem in such a way that: (2)

v(H&0LP) = max{0, min{l, a • v(P)-

0+1}}

The rather cumbersome definitions of FPQ, ,(P) and ^,-(P) can be found in [7], p. 20 and p. 52, respectively.

(1) The definition of (X)at for X a string of symbols, can be given, as usual, by induction on a.

56 A generalized summation 2 and a chain symbol T are defined, as in [7], by: if j3 = a then Sf stt P, denotes Ptt if 0 > a then $*ma P, denotes Afy Sfi^P,if 0 < a then T*ma P{Q denotes 0 if 0 > a then Tf=a P{Q denotes CP„ r?r& PfQ. Now, we list all the cases that arise when J> and s are specified. Case 1. ./finite (M = 3,4,....), s= 1 Case 2. ^/finite (Af= 3, 4,...), 0 < s < l Case J. . / infinite, s = 1 Case 4. J> infinite, 0 < s < l , s designated Case 5. ./infinite, 0 < s < l , s undesignated. For each of them, we collect here the relative complete axiomatizations. Let M > 3 . We denote byif^ , s v(NCQMa-pt aU) = v(Q) = max{0, 2s - l } < s . This proves that R l ' is not of type 5. In the case of R2', for each s = /?/o:, choose a formula P and an assignment v such that v(P) =