Inverse (logic)
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In
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 premises ...
, an inverse is a type of
conditional sentence Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''cond ...
which is an
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", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ...
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 statemen ...
of the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
, 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 (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 In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
, 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".''


See also

*
Contraposition 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 contrapositive, proof by contraposition. The cont ...
*
Converse (logic) In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical propositi ...
*
Obversion In 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, ...
*
Transposition (logic) In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the tru ...


Notes

Immediate inference {{logic-stub