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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. 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 }.

  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. Second-order propositional logic - Wikipedia

    en.wikipedia.org/wiki/Second-order_propositional...

    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 .

  7. Monadic predicate calculus - Wikipedia

    en.wikipedia.org/wiki/Monadic_predicate_calculus

    Allowing monadic function symbols changes the logic only superficially [citation needed] [clarification needed], whereas admitting even a single binary function symbol results in an undecidable logic. Monadic second-order logic allows predicates of higher arity in formulas, but restricts second-order quantification to unary [clarification ...

  8. Beth definability - Wikipedia

    en.wikipedia.org/wiki/Beth_definability

    For first-order logic, the theorem states that, given a theory T in the language L' ⊇ L and a formula φ in L', then the following are equivalent: . for any two models A and B of T such that A|L = B|L (where A|L is the reduct of A to L), it is the case that A ⊨ φ[a] if and only if B ⊨ φ[a] (for all tuples a of A);

  9. Axiom schema - Wikipedia

    en.wikipedia.org/wiki/Axiom_schema

    Schematic variables in first-order logic are usually trivially eliminable in second-order logic, because a schematic variable is often a placeholder for any property or relation over the individuals of the theory. This is the case with the schemata of Induction and Replacement mentioned above. Higher-order logic allows quantified variables to ...

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