Showing papers in "Mathematical Logic Quarterly in 1979"

On Fuzzy Logic III. Semantical completeness of some many‐valued propositional calculi

Abstract: Now we come to speak of the first non-trivial question which, in a way, puts to the test all the definitions presented so far, viz. the question of balance of syntax against semantics. Every choice of an enriched complete residuated lattice & = ((L, 8, +), S ) of a certain type ( A T : A + N,, Ex: A + N*), plus a set P of propositional variables, poses the problem of axiomatizability of the 8-valued propositional calculus over P. According to [3], M. 25, the question actually goes like this: (Q) Do there exist an L-fuzzy set of logical axioms A : F(P, L, A ) --f L and a set W of L-valued rules of inference in F(P, L, A ) such that for any L-fuzzy theory X: F(P, L, A ) + L over P and any formula e) E F(P, L, A ) the degree ( g y ( p , 8 ) X ) e), to which q~ follows from X in the L-semantical system ( F ( P , L , A ) , Y ( P , a)), equals exactly the degree ( q A , @ X ) q ~ , to which q~ is provable from X in the L-syntactical system ( F ( P , L, A ) , A, W ) ? l ) In this paper we investigate the case when the underlying lattice of B is a chain. an abstract set of formulas ([3]);

Automorphisms of the Lattice of Recursively Enumerable Vector Spaces

Abstract: Recursively enumerable (r.e,) algebraic structures have received attention because of their recursion theoretic depth and richness. Although the development of such theories is somewhat analoguous to the development of the theory of r.e. sets, the former is not reducible to or a corollary of the latter. I n fact, on more than one occasion the development of such theories has increased our insight into the theory of r.e. sets. The initial works in this area are due to FROHLICH and SHEPHERDSON [6] and Rasm [14]. The more recent works on vector space structure are due to DEKKER [4, 51, CROSSLEY and NERODE [3], METAKIDES and NERODE [13], REMMEL [El , RETZLAFF [19], and the author [8]. An excellent motivational reference is METAKIDES and NERODE [El. I n this paper we confine ourselves with vector space structure and study the lattice of r.e. vector spaces and ith automorphisms. Let V , denote a countably infinite dimensional, f d y effwtive vector space over a countable recursive field 3. By fully effective we mean that 8, under a fixed Godel numbering has the following properties : 0) operations of vector addition and scalar multiplication on V, are presented by partial recursive functions on the Godel numbers of 8,. (ii) V , has a depedence algorithm, i.e., there is a uniform effective procedure which applied to any TL vectors of V , determines whether or not they are linearly independent.

The Completeness of Presupposition‐Free Tense Logic

Abstract: In this paper, we continue the work begun by MCARTHUR and LEBLANC~) on the strong completeness of a minimal quantificational tense logic2) We establish the strong completeness and soundness of QK: = , the presupposition-free (with equality) version of MCARTHUR and LEBLANC’S QK, .3) The tense-counterparts VsGA 2 GVXA and VxHL4 3 HVzA of the Barcan Formula are not provable in QKf =, nor are the counterparts of the converse of the Barcan Formula G V d 3 VxGA and HVxA 3 3 V . r H A ; a term may designate a t one time but not a t another. Like MCARTHUR and LERLANC, we employ a Henkin-style proof with MAKTNSON attendants4) and truth-value semantics. In order to handle the presupposition-free character of QB: = , our construction of maximally consistent and omega-complete sets differs from others to be found in the literature.

Some Classes of Kripke Frames Characteristic for the Intuitionistic Logic

Abstract: It was claimed by MCKAY [l] that the set of all finite binary trees is a characteristic set of Kripke frames for the intuitionistic propositional logic (I). This claim, however, was shown to be false by GABBAY and DEJONGH [2] who exhibited an I-invalid formula which holds in any structure based on a frame consisting of a finite binary tree. It is shown in this note that the infinite binary tree is characteristic for I. By a Kripke frame is understood a pair 9 = (A, R) consisting of a non-empty set A and a reflexive transitive relation R on A. A Kripke structure is a Kripke frame together with a valuation w which assigns the value 0 or 1 to each pair consisting of a point in A and a propositional formula in accordance with the following conditions:

Entscheidbarkeit der Theorie der Linearen Ordnung in Logiken mit Mächtigkeitsquantoren bzw. mit Chang-Quantor

Abstract: In [l] und [2 ] ist die Entscheidbarkeit der Theorie der linearen Ordnung in dcr Logik LQ,, mit dem zusatzlichen Quantor ,,es gibt N, viele\" gezeigt worden, wobei N, als regular vorausgesetzt wurde. Fur x = 0 folgt dieses Resultat schon aus den Ergebnissen von M. 0. RABIN [6]. Fur x = 1 hat H. P. TUSCHIK [7] einen anderen Beweis angegeben. Er zeigte weiterhin in [8], da13 die in Frage stehende Theorie entscheidbar bleibt, wenn man sie in einer Logik mit endlich vielen Quantoren Q,, n < 0 , betrachtet. Die vorliegende Arbeit ist eine Weiterfuhrung von [l] und [2 ] fur den Pall, daB K, singular ist, woraus dann schlieSlich die Entscheidbarkeit der Theorie der linearen Ordnung in LQ* fur beliebiges x und in LQ, folgt, falls QC der Chang-Quantor i&. Bhnlich wie in [a] wird mit Hilfe gewisser Erzeugungsregeln eine rekursiv aufzahlbare Menge M , von Ordnungstypen definiert, deren LQw-Theorien rekursiv aufzahlbar sind. Die hier benotigten Operationen sind jedoch komplizierter als die in [2]. Unter Verwendung von spieltheoretischen Methoden wird gezeigt, da13 M , in der Klasse aller geordneten Mengen bezuglich LQ_ dicht liegt! Aus der Vollstandigkeit von LQo> folgt schlieBlich die Behauptung.