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A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
However, because phrase structure trees and structurally simpler trees are always able to derive one another from each other and are both still used today, ICA is still relevant in many contemporary theories. An important aspect of ICA in phrase structure grammars is that each individual word is a constituent by definition.
(y 1 ∨ y 2 ∨ … ∨ y n); while the former is a disjunction of n conjunctions of 2 variables, the latter consists of 2 n clauses of n variables. However, with use of the Tseytin transformation , we may find an equisatisfiable conjunctive normal form formula with length linear in the size of the original propositional logic formula.
A truth table is a semantic proof method used to determine the truth value of a propositional logic expression in every possible scenario. [93] By exhaustively listing the truth values of its constituent atoms, a truth table can show whether a proposition is true, false, tautological, or contradictory. [94] See § Semantic proof via truth tables.
Syntactic theories based on phrase structure typically analyze subject–aux inversion using syntactic movement. In such theories, a sentence with subject–aux inversion has an underlying structure where the auxiliary is embedded deeper in the structure. When the movement rule applies, it moves the auxiliary to the beginning of the sentence. [5]
Another use is in the resolution method for first-order logic, where formulas are represented as sets of clauses understood to be universally quantified. (For an example see drinker paradox .) An important result in model theory is the Löwenheim–Skolem theorem , which can be proven via Skolemizing the theory and closing under the resulting ...
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In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains. In one of these, often the largest, the solution is accurately approximated by an asymptotic series [2] found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero).