Search results
Results from the WOW.Com Content Network
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 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 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 }.
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).
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.
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 ...
It is known that the Löwenheim–Skolem number of second-order logic is larger than the first measurable cardinal, if there is a measurable cardinal. [2] (And the same holds for its Hanf number.) The Löwenheim number of the universal (fragment of) second-order logic however is less than the first supercompact cardinal (assuming it exists).