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The inverse of the inverse, that is, the inverse of , is , 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 .
If a statement's inverse is false, then its converse is false (and vice versa). If a statement's negation is false, then the statement is true (and vice versa). If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a logical biconditional.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics .
Phrased another way, denying the antecedent occurs in the context of an indicative conditional statement and assumes that the negation of the antecedent implies the negation of the consequent. It is a type of mixed hypothetical syllogism that takes on the following form : [ 1 ]
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its converse, [2] unless the antecedent P and the consequent Q are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal."
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This is the inverse of the naturalistic fallacy. Naturalistic fallacy – inferring evaluative conclusions from purely factual premises [105] [106] in violation of fact-value distinction. Naturalistic fallacy (sometimes confused with appeal to nature) is the inverse of moralistic fallacy.