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  2. Second-order logic - Wikipedia

    en.wikipedia.org/wiki/Second-order_logic

    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.

  3. Monadic second-order logic - Wikipedia

    en.wikipedia.org/wiki/Monadic_second-order_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 ...

  4. Second-order arithmetic - Wikipedia

    en.wikipedia.org/wiki/Second-order_arithmetic

    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 "" expressing the fact that n is a member of X (X being a free set variable): in this case, the induction axiom for φ is

  5. Hume's principle - Wikipedia

    en.wikipedia.org/wiki/Hume's_principle

    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.

  6. Courcelle's theorem - Wikipedia

    en.wikipedia.org/wiki/Courcelle's_theorem

    The satisfiability problem for a formula of monadic second-order logic is the problem of determining whether there exists at least one graph (possibly within a restricted family of graphs) for which the formula is true. For arbitrary graph families, and arbitrary formulas, this problem is undecidable.

  7. S2S (mathematics) - Wikipedia

    en.wikipedia.org/wiki/S2S_(mathematics)

    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}.

  8. Spectrum of a sentence - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_sentence

    Let ψ be a sentence in first-order logic.The spectrum of ψ is the set of natural numbers n such that there is a finite model for ψ with n elements.. If the vocabulary for ψ consists only of relational symbols, then ψ can be regarded as a sentence in existential second-order logic (ESOL) quantified over the relations, over the empty vocabulary.

  9. Gödel's completeness theorem - Wikipedia

    en.wikipedia.org/wiki/Gödel's_completeness_theorem

    Second-order logic, for example, does not have a completeness theorem for its standard semantics (though does have the completeness property for Henkin semantics), and the set of logically valid formulas in second-order logic is not recursively enumerable. The same is true of all higher-order logics.