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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.
The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas. One particularly important instance of the induction scheme is when φ is the formula " n ∈ X {\displaystyle n\in X} " expressing the fact that n is a member of X ( X being a free set variable): in this case, the induction axiom ...
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 ...
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 }.
Second order approximation, an approximation that includes quadratic terms; Second-order arithmetic, an axiomatization allowing quantification of sets of numbers; Second-order differential equation, a differential equation in which the highest derivative is the second; Second-order logic, an extension of predicate logic
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 ...
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 .
Hume's principle or HP says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs. HP can be stated formally in systems of second-order logic.
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