Philosophy:Inverse (logic)
In logic, 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 [math]\displaystyle{ P \rightarrow Q }[/math], the inverse refers to the sentence [math]\displaystyle{ \neg P \rightarrow \neg Q }[/math]. Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other.[1] 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 [math]\displaystyle{ \neg P \rightarrow \neg Q }[/math], is [math]\displaystyle{ \neg \neg P \rightarrow \neg \neg Q }[/math], and since the double negation of any statement is equivalent to the original statement in classical logic, the inverse of the inverse is logically equivalent to the original conditional [math]\displaystyle{ P \rightarrow Q }[/math]. Thus it is permissible to say that [math]\displaystyle{ \neg P \rightarrow \neg Q }[/math] and [math]\displaystyle{ P \rightarrow Q }[/math] are inverses of each other. Likewise, [math]\displaystyle{ P \rightarrow \neg Q }[/math] and [math]\displaystyle{ \neg P \rightarrow Q }[/math] 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.[1] 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[2]). 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, 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.[3] 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
Notes
- ↑ 1.0 1.1 Taylor, Courtney K.. "What Are the Converse, Contrapositive, and Inverse?" (in en). https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458.
- ↑ "Mathwords: Inverse of a Conditional". https://www.mathwords.com/i/inverse_conditional.htm.
- ↑ Toohey, John Joseph. An Elementary Handbook of Logic. Schwartz, Kirwin and Fauss, 1918
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