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15, true, Tautology. Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.
Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq". The tee symbol ⊤ {\displaystyle \top } is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ {\displaystyle \bot } ( falsum ) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value " true ", as ...
In propositional logic, tautology is either of two commonly used rules of replacement. [ 1 ] [ 2 ] [ 3 ] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs .
Tautological consequence can also be defined as ∧ ∧ ... ∧ → is a substitution instance of a tautology, with the same effect. [2]It follows from the definition that if a proposition p is a contradiction then p tautologically implies every proposition, because there is no truth valuation that causes p to be true and so the definition of tautological implication is trivially satisfied.
A stronger form of reductio ad absurdum, [56] where instead of only deriving from showing that leads to a contradiction, one can also derive from showing that leads to a contradiction. coextensive Having the same scope or range, especially referring to two terms or concepts that apply to the same set of objects.
Formulas and are logically equivalent if and only if the statement of their material equivalence is a tautology. [ 2 ] The material equivalence of p {\displaystyle p} and q {\displaystyle q} (often written as p ↔ q {\displaystyle p\leftrightarrow q} ) is itself another statement in the same object language as p {\displaystyle p} and q ...
This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that:
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 ...