Inverse (logic)
Encyclopedia
In traditional logic, an inverse is a type of conditional sentence
which is an immediate inference
made from another conditional sentence. Any conditional sentence has an inverse: the contrapositive of the converse. The inverse of
is thus 
. For example, substituting propositions in natural language for logical variables, the inverse of the conditional proposition "If it's raining, then Sam will meet Jack at the movies." is "If it's not raining, then Sam will not meet Jack at the movies.".
The inverse of the inverse, that is, the inverse of
, is 
. Since a double negation has no logical effect, the inverse of the inverse is logically equivalent to the original conditional 
. Thus it is permissible to say that 
and 
are inverses of each other. Likewise, we may say that 
and 
are inverses of each other.
The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. But the inverse of a conditional is not inferable from the conditional. For example, "If it's not raining, then Sam will not meet Jack at the movies." cannot be inferred from "If it's raining, then Sam will meet Jack at the movies.". It could easily be the case that Sam and Jack are attending the movies no matter the weather.
Conditional sentence
In grammar, conditional sentences are sentences discussing factual implications or hypothetical situations and their consequences. Languages use a variety of conditional constructions and verb forms to form such sentences....
which is an immediate inference
Immediate inference
An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green." we can make the immediate inference that "No toads are not green." This new statement is known as the contrapositive of the original statement...
made from another conditional sentence. Any conditional sentence has an inverse: the contrapositive of the converse. The inverse of
is thus . For example, substituting propositions in natural language for logical variables, the inverse of the conditional proposition "If it's raining, then Sam will meet Jack at the movies." is "If it's not raining, then Sam will not meet Jack at the movies.".The inverse of the inverse, that is, the inverse of
, is . Since a double negation has no logical effect, the inverse of the inverse is logically equivalent to the original conditional . Thus it is permissible to say that and are inverses of each other. Likewise, we may say that and are inverses of each other.The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. But the inverse of a conditional is not inferable from the conditional. For example, "If it's not raining, then Sam will not meet Jack at the movies." cannot be inferred from "If it's raining, then Sam will meet Jack at the movies.". It could easily be the case that Sam and Jack are attending the movies no matter the weather.
See also
- Conversion (logic)
- ObversionObversionIn traditional logic, obversion is a "type of immediate inference in which from a given proposition another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original...
- Transposition (logic)Transposition (logic)In the methods of deductive reasoning in classical logic, transposition is the rule of inference that permits one to infer from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. Its symbolic expression is:...
- ContrapositionContrapositionIn traditional logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality . For its symbolic expression in modern logic see the rule...