Reasoning with Ambiguity

We treat the problem of reasoning with ambiguous propositions. Even though ambiguity is obviously problematic for reasoning, it is no less obvious that ambiguous propositions entail other propositions (both ambiguous and unambiguous), and are entailed by other propositions. This article gives a formal analysis of the underlying mechanisms, both from an algebraic and a logical point of view. The main result can be summarized as follows: sound (and complete) reasoning with ambiguity requires a distinction between equivalence on the one and congruence on the other side: the fact that $$\alpha $$ α entails $$\beta $$ β does not imply $$\beta $$ β can be substituted for $$\alpha $$ α in all contexts preserving truth. Without this distinction, we will always run into paradoxical results. We present the (cut-free) sequent calculus $$\mathsf {AL}^{\textit{cf}}$$ AL cf , which we conjecture implements sound and complete propositional reasoning with ambiguity, and provide it with a language-theoretic semantics, where letters represent unambiguous meanings and concatenation represents ambiguity.