An interpolation theorem for denumerably long formulas

López‐Escobar, Esteban

doi:10.4064/fm-57-3-253-272

Cited by 126 publications

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“…The proof theoretic argument for the interpolation property of L ωω , proceeding via the Cut Elimination Theorem of a suitable Gentzen system, works also in L ω 1 ω , as was demonstrated by Lopez-Escobar (1965). It works also in the countable admissible fragments of L ω 1 ω , as demonstrated by Barwise (1969).…”

Section: Positive Resultsmentioning

confidence: 90%

The Craig Interpolation Theorem in abstract model theory

Väänánen

2008

Synthese

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The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying "natural," robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

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Section: Positive Resultsmentioning

confidence: 90%

The Craig Interpolation Theorem in abstract model theory

Väänánen

2008

Synthese

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“…And so they do have a complete notion of proof (Lopez-Escobar, 1965). Unlike the notion of proof for classical logic, this cannot show the consequence relation to be any simpler than what's given by its a priori specification.…”

Section: Countability and Categoricitymentioning

confidence: 99%

LOGIC IN THE TRACTATUS

Weiss¹

2017

The Review of Symbolic Logic

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Abstract. I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein's "form-series" device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is 1 1 -complete. But third, it is only granted the assumption of countability that the class of tautologies is 1 -definable in set theory.Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.We have by now a quite systematic and rigorous grasp of the logical work of two of Wittgenstein's teachers, Frege and Russell. This is thanks in part to decades of flourishing scholarship, and thanks also to Frege and Russell themselves. In contrast, despite comparably voluminous commentary there is still no received understanding of anything describable as the logical system of the Tractatus (Wittgenstein, 1921). It is hard to resist the conclusion that the Tractatus did not, despite its professed program and its large reputation, offer any systematic alternative conception of the nature of logic.But the conclusion is mistaken. To the contrary, there is a system, or a class of similar systems, which can be understood to explicate the development of logic in the Tractatus. They differ rather sharply from those of Frege or Russell, as well as from classical first-or second-order logic. Nonetheless they can be exactly described and investigated metamathematically, for epistemological and metaphysical evaluation. In this paper, I will present one such system, and investigate some of its properties. These turn out to be of surprising and indeed independent interest.In seeking to understand what logic is supposed to be according to the early Wittgenstein, we may distinguish two kinds of evidence. First, there are the contours of his own construction, famously, for example, in the truth-functionality thesis and its enactment through iterated joint denial. Second, there are in the Tractatus apparent declarations of epistemological constraints on the nature of logic: some of these, for example, have been taken to suggest that according to Wittgenstein, logic must be decidable.I wish to separate these two strands. In the early decades of the 20th century, it was entirely possible for a proficient researcher to develop a computationally intractable system

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“…This is an easy consequence, due to Scott, of the interpolation theorem for L ω 1 ω [4]: If ϕ is an L ω 1 ω -sentence in the signature σ and ψ is an L ω 1 ω -sentence in the signature τ such that ϕ → ψ holds in all countable models, then there is an L ω 1 ω -interpolant θ in the signature σ∩τ such that ϕ → θ and θ → ψ hold in all countable models. To derive the Lopez-Escobar theorem from interpolation, note that every Borel set is defined by an L ω 1 ω -sentence from a sequence of parameters n i ∈ N. If we use constants c i and d i for each n i , the assumption that the Borel set is isomorphism invariant implies that there is an L ω 1 ω -interpolant (without parameters) defining the set.…”

Section: Introductionmentioning

confidence: 99%

Non-isomorphism invariant Borel quantifiers

Engström

Schlicht

2011

Proc. Amer. Math. Soc.

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Abstract. Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in L ω 1 ω by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group Sym(N) for families of relations and non-isomorphism invariant generalized quantifiers on the natural numbers such as "for all even numbers". Moreover we produce a binary quantifier Q for every closed subgroup of Sym(N) such that the Borel sets of structures invariant under the subgroup action are exactly the sets of structures definable in L ω 1 ω (Q).

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An interpolation theorem for denumerably long formulas

Cited by 126 publications

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The Craig Interpolation Theorem in abstract model theory

The Craig Interpolation Theorem in abstract model theory

LOGIC IN THE TRACTATUS

Non-isomorphism invariant Borel quantifiers

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