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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. 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 }.

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 " 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 ...

5. Hume's principle - Wikipedia

   en.wikipedia.org/wiki/Hume's_principle

   (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 ...

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 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 ...