Arnon Avron and Anna Zamansky Non-deterministic Semantics for Logical Systems

The principle of truth-functionality (or compositionality) is a basic principle in many-valued logic in general, and in classical logic in particular. According to this principle, the truth-value of a complex formula is uniquely determined by the truth-values of its subformulas. However, real-world information is inescapably incomplete, uncertain, vague, imprecise or inconsistent, and these phenomena are in an obvious conflict with the principle of truth-functionality. One possible solution to this problem is to relax this principle by borrowing from automata and computability theory the idea of non-deterministic computations, and apply it in evaluations of truth-values of formulas. This leads to the introduction of non-deterministic matrices (Nmatrices) — a natural generalization of ordinary multi-valued matrices, in which the truth-value of a complex formula can be chosen nondeterministically out of some non-empty set of options. There are many natural motivations for introducing non-determinism into the truth-tables of logical connectives. We discuss some of them below. They give rise to two different ways in which non-determinism can be incorporated: the dynamic and the static. In both the value v(¦(ψ1, . . . , ψn)) assigned to the formula ¦(ψ1, ..., ψn) is selected from a set ¦̃(v(ψ1), . . . , v(ψn)) (where ¦̃ is the interpretation of ¦). In the dynamic approach this selection is made separately and independently for each tuple 〈ψ1, . . . , ψn〉. Thus the choice of one of the possible values is made at the lowest possible (local) level of computation, or on-line, and v(ψ1), . . . , v(ψn) do not uniquely determine v(¦(ψ1, . . . , ψn)). In contrast, in the static semantics this choice is made globally, system-wide, and the interpretation of ¦ is a function, which is selected before any computation begins. This function is a “determinisation” of the non-deterministic interpretation ¦̃, to be applied in computing the value of any formula under the given valuation. This limits non-determinism, but still leaves the freedom of choosing the above function among all those that are compatible with the non-deterministic interpretation ¦̃ of ¦.