Quantification and Dependence Logics

Abstract

Knowing the truth-values of the components of an arbitrary conditional is not enough for evaluating it: this is emphasized by material implication’s paradoxes. So lots of systems, rejecting these paradoxes, obey a relevance principle that says that a conditional can be true only if there is some connection between its antecedent and its consequent. The connections involved are either based on the notion of deductibility or on the notion of containment:

• In the first approach, the logical systems satisfy a principle which requires that A be actually used in some deduction of B for A → B to be valid. Among such systems are J. Myhill’s real implication system and the logics of relevance and necessity, R and E, defined by Anderson and Belnap.

• The second approach links connection and containment. For example, in the system of Analytic Implication of Parry, in order to have A imply B, every concept occuring in B must also occur in A. Similarly, in Anderson and Belnap’s systems, antecedent and consequent of valid conditionals must share variables.