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Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. [2] A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv ...
Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class K 1 or K 2 , the deduction violates the inheritance characteristic of tautology (i.e., the derivation ...
Not all tautologies of classical logic lift to Ł3 "as is". For example, the law of excluded middle, A ∨ ¬A, and the law of non-contradiction, ¬(A ∧ ¬A) are not tautologies in Ł3. However, using the operator I defined above, it is possible to state tautologies that are their analogues: A ∨ IA ∨ ¬A (law of excluded fourth)
Many tautologies in classical logic are not theorems in intuitionistic logic – in particular, as said above, one of intuitionistic logic's chief aims is to not affirm the law of the excluded middle so as to vitiate the use of non-constructive proof by contradiction, which can be used to furnish existence claims without providing explicit ...
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
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.
The three-term contingency (also known as the ABC contingency) is a psychological model describing operant conditioning in three terms consisting of a behavior, its consequence, and the environmental context, as applied in contingency management. The three-term contingency was first defined by B. F. Skinner in the early 1950s. [1]
The use of tautologies, however, is usually unintentional. For example, the phrases "mental telepathy", "planned conspiracies", and "small dwarfs" imply that there are such things as physical telepathy, spontaneous conspiracies, and giant dwarfs, which are oxymorons. [8] Parallelism is not tautology, but rather a particular stylistic device.