Premise's negative {contradictory, logic} has opposite truth-value.
Changing subject to negative of predicate and predicate to negative of subject can make new statements {contrapositive}| {transposition, logic} {contraposition}. "If A then B" transposes to "If not B, then not A". If theorem is true, contrapositive is true. If contrapositive is true, theorem is true. "All A are B" transposes to "All not B are not A". "Some A are not B" transposes to "Some B are not A". "No A are B" transposes to "No not B are not A". "Some A are B" transposes to "Some not B are not A". For true statements, contrapositive of "All A are B" and "Some A are not B" are true. For true statements A and B, "No A are B" and "Some A are B" contrapositives are not true.
Exchanging subject and predicate can make new statements {converse, logic}|. For example, "No A is B" has converse "No B is A". "Some A is B" converts to "Some B is A". For categorical statements "No A is B" and "Some A is B", if statement is true, converse is true. All other categorical converse statements must have independent proof. Conversion {conversion per accidens} from "All A are B" to "Some B are A" is valid.
Changing connectives to opposite makes negative statements {denial} {negation}. NOT added to statements denies statement: NOT a or -a. Negation changes statement truth-value. Negative statements only entail similar qualities and are less specific. Negation can result in statements about ambiguous, non-existent, or arbitrary things. In common language, negative particles, word meanings, or inflections negate statements.
Negation can apply to sentences with quantities. NOT every a is b, so Every a is NOT b {equipollence, negation}|.
Changing subject to negative subject {inversion, logic} {inverse, logic} makes new statements. For example, "All A are B" inverts to "All not A are B". "All A are B" implies "Some not A are not B". If inverse is true, converse is true. If converse is true, inverse is true.
Changing predicate to complement or negative and negating the statement can make new statements {obverse}|. Obversions of the four categorical forms are valid. All A are B, so Not (All A are not B). Some A are B, so Not (Some A are not B). Some A are not B, so Not (Some A are B). No A are B, so Not (No A are not B). If converse is true, obverse is true.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225