Propositional Logic - Transformations of the Conditional

Contrapositive

The contrapositive of a conditional statement is equivalent to the conditional statement.

A contrapositive takes the conditional P→Q, then derives ¬Q→¬P.

The best way to think about this is via modus tollens. However, you can also prove P→Q≡¬Q→¬P:

P

Q

¬P

¬Q

P→Q

¬Q→¬P

T

 T F F T T

T

 F F T F F

F

 T T F T T

F

 F T T T T

Table 13

Another example of showing equivalency can be via natural language. Let us say the conditional proposition states that, “If I study hard, then I will pass.” The contrapositive would state that, “If I do not pass, then I have not studied hard.” This is somewhat common sensical.

Converse

A converse is not equivalent to the conditional.

A converse takes the conditional P→Q, then derives Q→P.

An example in natural language is to take the conditional, “If it’s a dog, then it’s a mammal” and deriving, “If it’s a mammal, then it’s a dog.” This is obviously incorrect since there could be a non-dog mammal.

Inverse

An inverse is not equivalent to the conditional.

An inverse takes the conditional , then derives .

An example in natural language is to take the conditional, “If it’s a dog, then it’s a mammal” and deriving, “If it’s not a dog, then it’s a mammal.” This is obviously incorrect since there could be a non-dog animal.

It is also neat to note that: Q→P≡¬P→¬Q

P

Q

P→Q

¬P→¬Q

T

 T T T

T

 F F T

F

 T T F

F

 F T T

Table 14

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