The Family of Ambiguity Logics

Abstract We investigate propositional logics which enrich classical logic with a binary connective ‘ $$\Vert $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>‖</mml:mo> </mml:math> ’ representing ambiguity. Some of these logics have been established in the literature. We briefly present and compare the most interesting of these. We generalize all existing approaches, defining the family of full ambiguity logics by certain basic requirements they have to meet. We introduce further examples of ambiguity logics, investigate the structure of the family, and show how conceptual properties correspond to formal properties. The most important notion for ambiguity logics is what we call trust: either we trust that ambiguous terms are used consistently in one sense in an argument, or we do not. Formally, every reasonable ambiguity logic is either closed under uniform substitution (this corresponds to trust), or it is closed under substitution of equivalents – but closure under both results in (a specific type of) triviality. This correlation between conceptual and mathematical properties is not straightforward, but our results show that it is well-founded.