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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 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 .
In logic, the law of non-contradiction (LNC; also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "the house is white" and "the house is not white" are mutually exclusive.
Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by Ernest Nagel and James R. Newman in their 1958 Gödel's Proof. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:
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 ...
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 ...
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.