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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 ...
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 traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias . Illustrating a general tendency in applied logic, Aristotle 's law of noncontradiction states that "It is impossible that the same thing can at the same time both ...
So Plato's law of non-contradiction is the empirically derived necessary starting point for all else he has to say. [13] In contrast, Aristotle reverses Plato's order of derivation. Rather than starting with experience, Aristotle begins a priori with the law of non-contradiction as the fundamental axiom of an analytic philosophical system. [14]
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 .
John Locke claimed that the principles of identity and contradiction (i.e. the law of identity and the law of non-contradiction) were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is."
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 ...
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally ...