Propositional Calculus for Associatively Tied Implications

Recently, Morsi [28] has developed a complete syntax for the semantical domain of all adjointness algebras. In [1], Abdel-Hamid and Morsi enrich adjointness algebras with one more conjunction, this time a t-norm T (T need not be commutative) that ties an implication A in the following sense: A(T(a, b), z) = A(a, A(b, z)). In this paper, we develop a new complete syntax for quite a general multiple-valued logic whose semantics is based on this type of algebra. Such a formal system serves as a combined calculus for two, possibly different, types of uncertainty. 1 Tied Adjointness Algebras We here compile basics on implications and their adjoints that will be needed in this work. Throughout, P and L denote partially ordered sets (posets). P has a top element, denote it by 1, but L need not have a top element. We denote their two order relations by the symbols ≤P, ≤L,, respectively. Definition 1 [1] An implication triple on (P, L) (in the case P = L we say “on P” to mean “on (P, P)”), is an ordered triple (A, K, H) of binary operations, which satisfies the following four conditions: (i) A : P × L → L is antimonotone in the left argument and monotone in the right argument, and it has 1 ∈ P as a left identity element. We call A an implication on (P,L). (ii) K: P × L → L is monotone in both arguments and has 1 ∈ P as a left identity element. We call K a conjunction on (P,L). (iii) H: L × L → P is antimonotone in the left argument and monotone in the right argument, and it satisfies for each y, z ∈ L: H(y, z) = 1 iff y ≤L z. (1) We call H a forcing-implication on L. (iv) A, K and H are mutually related by the following Adjointness condition, for each a ∈ P and y, z ∈ L: y ≤L A(a, z) iff K(a, y) ≤L z iff a ≤P H(y, z). (2) Definition 2 [28] An adjointness algebra is a 7-tiple (L, ≤L, P, ≤P, A, K, H), in which (L, ≤L), (P, ≤P) are two posets with a top element for the latter, and (A, K, H) is an implication triple on (P, L). Notice that Definition 1 and Definition 2 differ from the standard ones of [1] and [28], respectively, in that L ≠ P. Also, most results related to the standard definition (i.e. P = L) (see [1] and [28]) remain valid, with virtually the same proofs, for our more general definition (L ≠ P). An adjointness algebra is called a complete adjointness lattice when P and L are complete lattices, and an adjointness chain when P and L are chains. The enrichment of adjointness algebras with bounded lattice structure is considered in [31] (in the case that L = P). The study of implications and conjunctions related by adjointness has recently been the subject of extensive research, becoming an important branch of multiple-valued logics and fuzzy logic. Interesting results are found in [3][7], [10]-[12], [1, 14, 15, 19, 28, 31] among other articles on this subject. On the one hand, the concept of Adjointness is the main tool in building a useful calculus for implications; by generating theorems for fuzzy logic, thus placing it on fertile mathematical grounds. This condition has precise descriptions in Lattice Theory in terms of Galois connections (see [171]), as well as in Category Theory (see Höhle [21] ) in terms of adjoint pairs of functors between posets (considered as categories). On the other hand, it is now known that the membership values (in fuzzy sets) have at least three different semantics, which frequently coexist in the same application [13]. It is natural to request a conjunction between truth values with same semantics to be both commutative and associative. It is equally natural to admit noncommutative, nonassociative conjunctions between truth values of differing semantics. Similar argument restricts the need for the forcing axiom (for each x, y in L: H(y, z) = 1 iff y ≤L z.)