Indefinites, Skolem Functions, and Arbitrary Objects

In this chapter I will look at the semantic analysis of indefinites in English, and their treatment in the framework of Dynamic Logic, choice functions (epsilon terms), and Kit Fine’s arbitrary objects. In the end I will make some comparative remarks about the latter and the account in terms of Skolem functions that Fine criticized, and propose an alternative framework (team semantics).

Comments on Gabriel Sandu’s “Indefinites, Skolem Functions, and Arbitrary Objects”

Gabriel Sandu has made important contributions to the development of independence-friendly logic and I am grateful to him for his searching and sympathetic critique of my own work on arbitrary objects in relation both to independence friendly logic and to other treatments of quantificational and anaphoric dependence....

Exponential-constructible functions in P-minimal structures

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of [Formula: see text]-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages.

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

Better Proof Output for Vampire

Vampire produces highly usable and informative proofs, but now they are even better and this paper explains how. It is important that the proofs produced by automated theorem provers are both understandable and machine checkable. Producing something that satisfies both of these goals is challenging, especially when dealing with complex steps performed by the solver. The main areas where proof output has been improved for understanding include (i) introduction of new symbols (such as Skolem functions) in preprocessing, (ii) representation of unifiers (for example, in resolution steps), and (iii) presentation of AVATAR proofs. These improvements will be illustrated via a number of examples. For checkable proofs Vampire provides a mode that outputs the proof as a number of individual (TPTP) problems that can be independently checked. This process is explained and illustrated with examples.

ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

A P-MINIMAL STRUCTURE WITHOUT DEFINABLE SKOLEM FUNCTIONS

We show there are intermediate P-minimal structures between the semialgebraic and subanalytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are P-minimal structures which do not admit classical cell decomposition.

Skolem Functions in Non-Classical Logics

This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A(x,y) determines the extension of a function and f is a Skolem function for A.