A new axiomatization of Belnap's conditional assertion
A conditional assertion, as distinguished from the assertion of a conditional, asserts just its consequent, provided that the antecedent condition of assertion is met. If that condition is not met, there is no assertion at all. Nonassertive subformulas make no contribution to truth- functional compounds of which they are a part. This feature enables conditional assertions, in conjunction with standard, unrestricted quantifiers, to effect restricted quantification. However, it also means that classically valid schemata will have non-true and even falsifiable instances. This paper provides an axiomatization of the never-false formulas of the language (and, by a result of \textit{J. M. Dunn} [J. Philos. Logic 4, 383-397 (1975; Zbl 0329.02006)], also of the always-true formulas). The key technical innovation is the definition of a function from formulas to formulas, such that the latter are true just when the former are assertive (true or false), and false otherwise. This makes it possible to specify a decidable axiom set using ''impure'', conditioned axiom schemata - there is no single logical form that all instances of a schemata share - or, alternatively, to provide actual axioms with a conditioned, i.e. restrictive, rule of substitution.
scientific article; zbMATH DE number 3950492