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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 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 ]
[citation needed] For example, consider the following English sentences: "I know you're coming." "I know that you're coming." In this construction, the conjunction that is optional when joining a sentence to a verb phrase with know. Both sentences are grammatically correct, but the word that is pleonastic in this case.
For example, If P, then Q. P. ∴ Q. In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent. The second premise "affirms" the antecedent. The conclusion, that the consequent must be true, is deductively valid.
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 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 ...
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 .
The negation of a tautology is a contradiction, a sentence that is false regardless of the truth values of its propositional variables, and the negation of a contradiction is a tautology. A sentence that is neither a tautology nor a contradiction is logically contingent. Such a sentence can be made either true or false by choosing an ...