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A formula of propositional logic is a tautology if the formula itself is always true, regardless of which valuation is used for the propositional variables. There are infinitely many tautologies. In many of the following examples A represents the statement "object X is bound", B represents "object X is a book", and C represents "object X is on ...
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 .
Sometimes the following alternative definition is considered: a pps is given as a proof-verification algorithm P(A,x) with two inputs. If P accepts the pair (A,x) we say that x is a P-proof of A. P is required to run in polynomial time, and moreover, it must hold that A has a P-proof if and only if it is a tautology.
A less trivial example of a redundancy is the classical equivalence between and . Therefore, a classical-based logical system does not need the conditional operator " → {\displaystyle \to } " if " ¬ {\displaystyle \neg } " (not) and " ∨ {\displaystyle \vee } " (or) are already in use, or may use the " → {\displaystyle \to } " only as a ...
The tee (⊤, \top in LaTeX), also called down tack (as opposed to the up tack) or verum, [1] is a symbol used to represent: . The top element in lattice theory.; The truth value of being true in logic, or a sentence (e.g., formula in propositional calculus) which is unconditionally true.
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 graphical representation of a partially built propositional tableau. In proof theory, the semantic tableau [1] (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux), also called an analytic tableau, [2] truth tree, [1] or simply tree, [2] is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. [1]
Tautology may refer to: Tautology (language) , a redundant statement in literature and rhetoric Tautology (logic) , in formal logic, a statement that is true in every possible interpretation