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Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. [2] A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv ...
Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class K 1 or K 2 , the deduction violates the inheritance characteristic of tautology (i.e., the derivation ...
However, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. " every ", " some ", and "is"), a truth-functional tautology is true because of the logical terms it ...
Starting from these eight tautologies and a tacit use of the "rule" of substitution, PM then derives over a hundred different formulas, among which are the Law of Excluded Middle 1.71, and the Law of Contradiction 3.24 (this latter requiring a definition of logical AND symbolized by the modern ⋀: (p ⋀ q) = def ~(~p ⋁ ~q).
Formally a pps is a polynomial-time function P whose range is the set of all propositional tautologies (denoted TAUT). [1] If A is a formula, then any x such that P(x) = A is called a P-proof of A. The condition defining pps can be broken up as follows: Completeness: every propositional tautology has a P-proof,
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A pair of an affirmative statement and its negation is, he calls, a 'contradiction' (in medieval Latin, contradictio). Examples of contradictories are 'every man is white' and 'not every man is white' (also read as 'some men are not white'), 'no man is white' and 'some man is white'.