Propositional calculus for adjointness lattices

Recently, Morsi has developed a complete syntax for the class of all adjointness algebras $\left( L,\leq ,A,K,H\right)$. There, $\left( L,\leq \right)$ is a partially ordered set with top element $1$, $K$ is a conjunction on $\left( L,\leq \right)$ for which $1$ is a left identity element, and the two implication-like binary operations $A$ and $H$ on $L$ are adjoints of $K$. In this paper, we extend that formal system to one for the class $ADJL$ of all 9-tuples $\left( L,\leq ,1,0,A,K,H,\wedge ,\vee \right)$, called \emph{% adjointness lattices}; in each of which $\left( L,\leq ,1,0,\wedge ,\vee \right)$ is a bounded lattice, and $\left( L,\leq ,A,K,H\right)$ is an adjointness algebra. We call it \emph{Propositional Calculus for Adjointness Lattices}, abbreviated $AdjLPC$. Our axiom scheme for $AdjLPC$ features four inference rules and thirteen axioms. We deduce enough theorems and inferences in $AdjLPC$ to establish its completeness for $ADJL$; by means of a quotient-algebra structure (a Lindenbaum type of algebra). We study two negation-like unary operations in an adjointness lattice, defined by means of $0$ together with $A$ and $H$. We end by developing complete syntax for all adjointness lattices whose implications are $S$-type implications

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