Deduction with uncertain conditionals

Abstract

Uncertain conditional propositions or conditional events pose a special problem for deduction due to their three-valued nature – true, false or inapplicable. This three-valued-ness gives rise to several different kinds of deduction between conditionals depending upon the particular deductive relation being employed. There is a hierarchy of 13 deductive relations between conditionals built up from four elementary ones, which can be expressed in terms of Boolean relations on the components of the conditionals. These different, but interrelated deductive relations on conditionals in turn give rise to various deductively closed systems of conditionals. Theorems are proved relating the deductive relations with each other and with their associated deductively closed sets (DCSs). The principal DCS generated by a single conditional using any of the deductive relations is completely characterized. Except for two of the deductive relations, deduction with a finite number of conditionals or with additional conditional information is also completely characterized as the union of associated principal DCSs. Examples of finite sets of deductively closed conditionals are exhibited including many, unlike Boolean algebra, which are finite and yet non-principal, that is, not generated by any one conditional. An introductory section carefully provides motivations and a complete algebraic characterization of the four chosen operations on conditional events. Results are applied to the famous penguin problem.