Description: If ph and ps are wff's, so is ( ph -> ps ) or " ph implies ps ". Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when ph is true and ps is false; it is true otherwise. Think of the truth table for an OR gate with input ph connected through an inverter. After we state the axioms of propositional calculus ( ax-1 , ax-2 , ax-3 , and ax-mp ) and define the biconditional ( df-bi ), the constant true T. ( df-tru ), and the constant false F. ( df-fal ), we will be able to prove these truth table values: ( ( T. -> T. ) <-> T. ) ( truimtru ), ( ( T. -> F. ) <-> F. ) ( truimfal ), ( ( F. -> T. ) <-> T. ) ( falimtru ), and ( ( F. -> F. ) <-> T. ) ( falimfal ). These have straightforward meanings, for example, ( ( T. -> T. ) <-> T. ) just means "the value of ( T. -> T. ) is T. ".
The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of ( ph -> ( ps -> ch ) ) , the middle ps may be informally called either an antecedent or part of the consequent depending on context. Contrast with <-> ( df-bi ), /\ ( df-an ), and \/ ( df-or ).
This is called "material implication" and the arrow is usually read as "implies". However, material implication is not identical to the meaning of "implies" in natural language. For example, the word "implies" may suggest a causal relationship in natural language. Material implication does not require any causal relationship. Also, note that in material implication, if the consequent is true then the wff is always true (even if the antecedent is false). Thus, if "implies" means material implication, it is true that "if the moon is made of green cheese that implies that 5=5" (because 5=5). Similarly, if the antecedent is false, the wff is always true. Thus, it is true that, "if the moon is made of green cheese that implies that 5=7" (because the moon is not actually made of green cheese). A contradiction implies anything ( pm2.21i ). In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication".
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wi | wff ( 𝜑 → 𝜓 ) |