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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.
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 .
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 ...
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 ...
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. [1] A sentence is said to be a logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must ...
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.
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
[2] Classical propositional logic is a truth-functional logic, [3] in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. [4] On the other hand, modal logic is non-truth-functional.