Abstract Logic in a Lattice

Let S be a nonempty set and denote by P(S) the class of all subsets of S. Then a closure operator in S is any map J : P(S) → P(S) such that, for X, Y ∈ P(S), $$X \subseteq Y \Rightarrow J(x) \subseteq J(Y);X \subseteq J(X);J(J(X)) = J(X)$$ (see, e.g., Cohn [1965]). The theory of closure operators is a powerful and elegant tool for (monotone) logics. Indeed, given any deduction apparatus whose set of well-formed formulas is &1D53D;, we can consider the related deduction operator D: P(F) → P(F), i.e., the operator defined by setting, for any X ∈ P(F), D(X) equal to the set of formulas we can derive from X. Then D is a closure operator. This led several authors to propose a general approach in which an abstract logic is a pair (F, D) where F is the set of formulas in a given language and D a closure operator in F (see Tarski [1956], Brown and Suszko [1973], Wójcicki [1988]). The extension of such an approach to fuzzy logic is straightforward. It is enough to substitute the lattice of all subsets of F with the lattice of all fuzzy subsets of F. This is in accordance with the definition of deduction operator given in Pavelka [1979].