# Implication

## Definition

**Implication** is a 1. (material implication) the combination in propositional logic of two formulae with the connective -> (*if ... then ...*), also called conditional. The implication of phi and psi, phi -> psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false:

(i) phi psi phi -> psi 1 1 1 1 0 0 0 1 1 0 0 1

2. (logical implication) the relation that exists between two sentences phi and psi if phi -> psi is a tautology. In other words, psi is the logical implication or *logical consequence* of phi if psi is true in every model in which phi is true.

## Example

That q is a logical implication of (p V q) can be demonstrated by merely setting up the truth table for the formula in (ii):

(ii) (p V q) -> q

This implication is true for every combination of truth values for p and q. A logical consequence of predicate logic is the consequence of ThereIs(x) [ P(x) ] from P(c).

## See also

## Link

## References

- Gamut, L.T.F. 1991.
*Logic, language, and meaning,*Univ. of Chicago Press, Chicago.