On the Interpretation of Conditional Conjunctions

Abstract

One may distinguish conditional sentences depending on whether they express a material implication or an equivalence. Conditional sentences are frequently represented in logic as if p then q where p and q are propositions: p is a sufficient condition for q, while q is a necessary condition for p. In terms of a truth table: p and q is true, p and not-q is false, not-p and q is true and not-p and not-q is true. In terms of syllogistic reasoning this means that given if p then q from p follows q and from not-q follows not-p; from not-p no conclusion can be drawn with respect to q or not-q and from q no conclusion can be drawn with respect to p or not-p. This interpretation of a conditional relation is commonly called material implication. If a condi­tional relation is interpreted as equivalence, then p and q is true, not-p and not-q is true, not-p and q is false, p and not-q is false. Given if p then q, from p follows q and vice versa; from not-p follows not-q and vice versa. This relation is also called a biconditional relation. The terms con­ditional and biconditional in this study will denote material implication and equivalence, respectively.