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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 ...
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.
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 ...
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 ...
The logic of questions, including the study of the forms and principles of questions and their relationships to answers. Eubulides paradox A paradox presented by Eubulides of Miletus, including the liar paradox, which involves a statement declaring itself to be false, creating a contradiction.
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:
0. The null assumption, i.e., we are proving a tautology 1. Our first subproof: we assume the l.h.s. to show the r.h.s. follows 2. A subsubproof: we are free to assume what we want. Here we aim for a reductio ad absurdum 3. We now have a contradiction 4. We are allowed to prefix the statement that "caused" the contradiction with a not 5.