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Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris ...
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.
Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's disjuncts.
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Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.. For example, the two-place truth function that always returns false is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are ...
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