Nonstandard Functional Interpretations and Categorical Models

The twentieth-century mathematician Thoralf Skolem is known principally for two achievements. The first is the statement and proof of the Löwenheim-Skolem theorem. The second is his construction of countable nonstandard models of Peano arithmetic and set theory when these theories are expressed in first-order (predicate) logic. The Löwenheim-Skolem theorem, for which Skolem gave several proofs, showed that no first-order axiom system can characterise a unique infinite model. Skolem regarded this theorem as casting doubt on the belief that mathematics can be reliably founded on formal axiomatic systems alone. He discovered that first-order logics are necessarily incomplete descriptions of their intended models. There is a vivid contrast between the comprehensive syntactic expressive power of first-order theories and their very limited power to constrain their models to a priori desired models. The startling conclusion he derived from his theorem is known as ‘Skolem’s paradox’. It asserts that concepts such as cardinality must be interpreted relative to a given model and thus have no absolute meaning. Anticipating modern model theory, in a later paper he gave a direct construction of a nonstandard model of first-order arithmetic based on plus and times by a method which anticipated the ultraproduct construction of models of first-order theories. He realised that it might be possible to construct new models for set theory which demonstrate the consistency or the independence of the axiom of choice and the continuum hypothesis. The first goal was realised in the early 1930s by Kurt Gödel’s introduction of the notion of constructability for consistency proofs. The second goal was realised in the early 1960s by Paul Cohen’s introduction of the notion of forcing for independence proofs, for which he was awarded the Field’s medal in 1966.

Intuitionistic nonstandard bounded modified realisability and functional interpretation

Cartesian closed Dialectica categories

Data involving epistemic modals suggest that some classically valid argument forms, such as reductio, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a de re–de dicto collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that govern the use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic.

Ontology is an explicit representation of a particular domain of interest. This paper focuses on the inherent problem of intractability (data complexity) due to data capturing for heterogeneous sources. We proposed to follow the tradition of knowledge representation and reasoning processes to develop a Semantic Web. Building query systems is the ultimate goal for this study. Conjunctive queries were adopted over ontology methods to address the data complexity. The Conjunctive queries are the form of queries in first-order logic that deal with conjunction (Λ) and existential quantification (∃) in database atomic formulae. State of the art meteorological and oceanographic domain provides the motivation for this work. We found that the querying method facilitates the tractability in reasoning and logical systems.

Indefinites are notorious for permitting long distance (LD) interpretations, that is, for exhibiting the apparent ability to take scope out of syntactic islands. This article reviews a family of approaches to LD interpretations that appeal to the notion of a choice function . Within this family, we will distinguish two branches. The proposals in one branch take their cue from Discourse Representation Theory (; ), adopting the view that the existential import of indefinites can be separated from the semantics of the indefinite article. Proposals in the second branch capitalize on another striking feature of indefinites, namely the availability of so‐called functional interpretations (e.g. ; ). One main result reported is that LD interpretations come in two different kinds, corresponding to the two choice function approaches reviewed. Another result is that existential quantification over choice functions comes at a cost: to prevent the derivation of unattested interpretations, certain restrictions on the scope of this quantification must be stipulated.

We introduce a new typed combinatory calculus with a type constructor that, to each type $$\sigma $$ , associates the star type $$\sigma ^*$$ of the nonempty finite subsets of elements of type $$\sigma $$ . We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be calledHAIandHAIωrespectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

Full classical S5 in natural deduction with weak normalization

We give a Natural Deduction formulation of an adaptation of Gödel’s functional (Dialectica) interpretation to the extraction of (more) efficient programs from (classical) proofs. We adapt Jørgensen’s formulation of pure Dialectica translation by eliminating his “Contraction Lemma” and allowing free variables in the extracted terms (which is more suitable in a Natural Deduction setting). We also adapt Berger’s uniform existential and universal quantifiers to the Dialectica-extraction context. The use of such quantifiers without computational meaning permits the identification and isolation of contraction formulas which would otherwise be redundantly included in the pure-Dialectica extracted terms. In the end we sketch the possible combination of our refinement of Gödel’s Dialectica interpretation with its adaptation to the extraction of bounds due to Kohlenbach into a light monotone functional interpretation.KeywordsProgram extraction from (classical) proofsComplexity of extracted programsBerger’s uniform quantifiersGödel’s Functional interpretationProof-Carrying CodeProof Mining

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH wasHere (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this isIn [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because

The calculus of constructions as a framework for proof search with set variable instantiation

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragmenti?ź$$\varSigma _2$$ over infinite words. Here, $$\varSigma _2$$ consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is $$\mathbb {B}\varSigma _2$$. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an $$\omega $$-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun's recent decidability result for $$\mathbb {B}\varSigma _2$$ from finite to infinite words.

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one variable free. We investigate this fragment over words and trees, presenting a complete classification of the complexity of its satisfiability problem for various navigational signatures and comparing its expressive power with other important formalisms. These include the two-variable fragment with counting and the unary negation fragment.