In this sense the use of the term " contraposition " is usually referred to by " transposition " when applied to hypothetical propositions or material implications.
12.
In recognition of this problem, the sign ?! of formal implication in the propositional calculus is called material implication to distinguish it from the everyday, intuitive implication.
13.
These problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions.
14.
The obvious suggestion that q is relevant to p if q is implied by p breaks down because under standard definitions of material implication, a false proposition implies all other propositions.
15.
The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the tautology is implied by anything whatsoever, since a tautology is always true.
16.
The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication ( a material conditional is equivalent to the disjunction of the negation of its material equivalence.
17.
The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the tautology is implied by anything whatsoever, since a tautology is always true.
18.
To sum up, although it is deceptively similar to what we mean by " logically follows " in ordinary usage, material implication does not capture the meaning of " if . . . then ".
19.
(The implication arrow denotes material implication in the metalanguage . ) The " minimal conditional logic " "'M "'is characterized by the first six properties, and stronger conditional logics include some of the other ones.
20.
Hence " if I'm a donkey, then two and two is four " is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance.
