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For example, because is a tautology of propositional logic, ((=)) ((=)) is a tautology in first order logic. Similarly, in a first-order language with a unary relation symbols R , S , T , the following sentence is a tautology:
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.
In literary criticism and rhetoric, a tautology is a statement that repeats an idea using near-synonymous morphemes, words or phrases, effectively "saying the same thing twice". [1] [2] Tautology and pleonasm are not consistently differentiated in literature. [3] Like pleonasm, tautology is often considered a fault of style when
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 ...
For example, x 1 is a positive literal, ¬x 2 is a negative literal, and x 1 ∨ ¬x 2 is a clause. The formula ( x 1 ∨ ¬ x 2 ) ∧ (¬ x 1 ∨ x 2 ∨ x 3 ) ∧ ¬ x 1 is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not.
The "up" in "climb up" is not always redundant, as in the example "He climbed up and then fell down the mountain." Many other examples of pleonasm are redundant only if the speaker's knowledge is taken into account. For example, most English speakers would agree that "tuna fish" is redundant because tuna is a kind of fish.
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 ...