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Author: Admin | 2025-04-28
Symbols and and the axiom:.Then the elements satisfying are thought of as elements of the first sort, and elements satisfying as elements of the second sort. One can quantify over each sort by using the corresponding predicate symbol to limit the range of quantification. For example, to say there is an element of the first sort satisfying formula , one writes:.Additional quantifiers[edit]Additional quantifiers can be added to first-order logic.Sometimes it is useful to say that "P(x) holds for exactly one x", which can be expressed as ∃!x P(x). This notation, called uniqueness quantification, may be taken to abbreviate a formula such as ∃x (P(x) ∀y (P(y) → (x = y))).First-order logic with extra quantifiers has new quantifiers Qx,..., with meanings such as "there are many x such that ...". Also see branching quantifiers and the plural quantifiers of George Boolos and others.Bounded quantifiers are often used in the study of set theory or arithmetic.Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory.Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus, formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.The most commonly studied infinitary logics are denoted Lαβ, where α and β are each either cardinal numbers or the symbol ∞. In this notation, ordinary first-order logic is Lωω.In the logic L∞ω, arbitrary conjunctions or disjunctions are allowed when building formulas, and there is an unlimited supply of variables. More generally, the logic that permits conjunctions or
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