logical conditional

Material Conditional, or material implication has a truth table is the following:

T	T	T
T	F	F
F	T	T
F	F	T
The notation for conditional is , and alternative notations and is also used.

Definition via Negation and Disjunction

We can define material conditional with negation and disjunction. is logical equivalent to .

Relation with Natural Language

See: logic and natural language For a sentence “if A then B”, we can translate it as . For “A if B,” we translate it as . Interestingly, for “A only if B”, we translate it as .

However, there are some semantic difference between natural languages and conditional. In particular, is automatically true if the antecedent is false or the consequent is true. This can be surprising if we understand conditional as “if … then …” in English.