Material Conditional - Formal Properties

Formal Properties

When studying logic formally, the material conditional is distinguished from the semantic consequence relation, if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including classical logic. For example, the following principles hold:

• If then for some . (This is a particular form of the deduction theorem.)

• The converse of the above

• Both and are monotonic; i.e., if then, and if then for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do they hold in relevance logics.

Other properties of implication (following expressions are always true, for any logical values of variables):

• distributivity:

• transitivity:

• reflexivity:

• truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.

• commutativity of antecedents:

Note that is logically equivalent to ; this property is sometimes called currying. Because of these properties, it is convenient to adopt a right-associative notation for → where denotes .

Note also that comparison of truth table shows that is equivalent to, and it is sometimes convenient to replace one by the other in proofs. Such a replacement can be viewed as a rule of inference.