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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 ...
A truth table is a mathematical table used in logic —specifically in connection with Boolean algebra, Boolean functions, and propositional calculus —which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. [ 1 ]
The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested.
In classical logic, particularly in propositional and first-order logic, a proposition is a contradiction if and only if. Since for contradictory φ {\displaystyle \varphi } it is true that ⊢ φ → ψ {\displaystyle \vdash \varphi \rightarrow \psi } for all ψ {\displaystyle \psi } (because ⊥ ⊢ ψ {\displaystyle \bot \vdash \psi } ), one ...
In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion[ a ][ b ] is the law according to which any statement can be proven from a contradiction. [ 1 ][ 2 ][ 3 ] That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion. [ 4 ][ 5 ] The ...
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.
Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other ...
In formal languages, truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, truth-functional connectives. For the rules which allow new well-formed formulas ...