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Logical equality (also known as biconditional or exclusive nor) is an operation on two logical values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true. The truth table for p XNOR q (also written as p ↔ q, Epq, p = q, or p ≡ q) is as follows:
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 ...
Some systems of classical logic include dedicated symbols for false (0 or ), while others instead rely upon formulas such as p ∧ ¬ p and ¬(p → p). In both Boolean logic and Classical logic systems, true and false are opposite with respect to negation; the negation of false gives true, and the negation of true gives false.
We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is in fact true.
It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument. The history of the inference rule modus tollens goes back to antiquity. [4]
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 ...
True or False Questions About Disney. 86. Cars was Pixar’s first movie. Answer: False – it was Toy Story. 87. Disney’s first full-color animated film was Snow White and the Seven Dwarfs.
Law of noncontradiction – no proposition can be both true and not true; Paraconsistent logic – a family of logics used to address contradictions; Paradox of entailment – a seeming paradox derived from the principle of explosion; Reductio ad absurdum – concluding that a proposition is false because it produces a contradiction