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A (existential second-order) formula is one additionally having some existential quantifiers over second order variables, i.e. …, where is a first-order formula. The fragment of second-order logic consisting only of existential second-order formulas is called existential second-order logic and abbreviated as ESO, as , or even as ∃SO.
In addition to Fagin's 1974 paper, [1] the 1999 textbook by Immerman provides a detailed proof of the theorem. [4] It is straightforward to show that every existential second-order formula can be recognized in NP, by nondeterministically choosing the value of all existentially-qualified variables, so the main part of the proof is to show that every language in NP can be described by an ...
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. [1] It is particularly important in the logic of graphs , because of Courcelle's theorem , which provides algorithms for evaluating monadic second-order formulas over graphs ...
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions , where quantifiers may range either just over the Boolean truth values , or over the Boolean-valued truth functions .
However, with free second order variables, not every S2S formula can be expressed in second order arithmetic through just Π 1 1 transfinite recursion (see reverse mathematics). RCA 0 + (schema) {τ: τ is a true S2S sentence} is equivalent to (schema) {τ: τ is a Π 1 3 sentence provable in Π 1 2-CA 0}.
The system Π 1 1-comprehension is the system consisting of the basic axioms, plus the ordinary second-order induction axiom and the comprehension axiom for every (boldface [11]) Π 1 1 formula φ. This is equivalent to Σ 1 1 -comprehension (on the other hand, Δ 1 1 -comprehension, defined analogously to Δ 0 1 -comprehension, is weaker).
(I. III. I.) [2] Note Hume's use of the word number in the ancient sense, to mean a set or collection of things rather than the common modern notion of "positive integer". The ancient Greek notion of number (arithmos) is of a finite plurality composed of units. See Aristotle, Metaphysics, 1020a14 and Euclid, Elements, Book VII, Definition 1 and ...
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula.For instance, the universal quantifier in the first order formula () expresses that everything in the domain satisfies the property denoted by .
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