logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form

P \rightarrow Q

, the inverse refers to the sentence

\neg P \rightarrow \neg Q

. Since an inverse is the

contrapositive In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stat ...

of the converse, inverse and converse are logically equivalent to each other. For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition :"If it's raining, then Sam will meet Jack at the movies." would be :"If it's not raining, then Sam will not meet Jack at the movies." The inverse of the inverse, that is, the inverse of

\neg P \rightarrow \neg Q

, is

\neg \neg P \rightarrow \neg \neg Q

, and since the

double negation In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not ( ...

of any statement is equivalent to the original statement in classical logic, the inverse of the inverse is logically equivalent to the original conditional

P \rightarrow Q

. Thus it is permissible to say that

\neg P \rightarrow \neg Q

and

P \rightarrow Q

are inverses of each other. Likewise,

P \rightarrow \neg Q

and

\neg P \rightarrow Q

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 cannot be inferred from the conditional itself'' (e.g., the conditional might be true while its inverse might be false). For example, the sentence :"If it's not raining, Sam will not meet Jack at the movies" cannot be inferred from the sentence :"If it's raining, Sam will meet Jack at the movies" because in the case where it's not raining, additional conditions may still prompt Sam and Jack to meet at the movies, such as: :"If it's not raining and Jack is craving popcorn, Sam will meet Jack at the movies." In

traditional logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

, where there are four named types of categorical propositions, only forms A (i.e., "All ''S'' are ''P"'') and E ("All ''S'' are not ''P"'') have an inverse. To find the inverse of these categorical propositions, one must: replace the subject and the predicate of the inverted by their respective contradictories, and change the quantity from universal to particular.Toohey, John Joseph
An Elementary Handbook of Logic
Schwartz, Kirwin and Fauss, 1918 That is: *"All ''S'' are ''P"'' (''A'' form) becomes "Some non-''S'' are non-''P''". *"All ''S'' are not ''P"'' (''E'' form) becomes "Some non-''S'' are not non-''P".''

Notes

Immediate inference {{logic-stub

See also

Notes