First-order Goedel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Goedel logics G_V (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that G_V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Goedel logics are also characterized.
The prenex fragments of first-order infinite-valued Goedel logics are classified. It is shown that the prenex Goedel logics characterized by finite and by uncountable subsets of [0, 1] are axiomatizable, and that the prenex fragments of all countably infinite Goedel logics are not axiomatizable.
Some common fallacies about fundamental themes of Logic are exposed: the First and Second incompleteness Theorem interpretations, Chaitin's various superficialities and the usual classification of the axiomatic Theories in function of its language order.
It is shown that G-up, the quantified propositional Goedel-Dummett logic based on the truth-values set V-up = {1 - 1/n : n >= 1} u {1}, is decidable. This result is obtained by reduction to Buechi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of G-up as the intersection of all finite-valued quantified propositional Goedel logics.
A Henkin-style proof of completeness of first-order classical logic is given with respect to a very small set (notably missing cut rule) of Genzten deduction rules for intuitionistic sequents. Insisting on sparing on derivation rules, satisfiability theorem is seen to need weaker assumptions than completeness theorem, the missing request being exactly the rule ~ p --> p, which gives a hint of intuitionism's motivations from a classical point of view. A bare treatment of standard, basic first-order syntax somehow more algebraic-flavoured than usual is also given.
Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Goedel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Goedel logics by Avron. It is shown that the system is sound and complete, and allows cut-elimination. A question by Takano regarding the eliminability of the Ta…
The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a lan…
We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.
We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.
We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a continuous version of conceptual completeness showing that $T^\eq$ is the maximal conservative expansion of $T$, and 3. define the concept of a metric pre-topos.
We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
In this paper we present a tableau proof system for first order logic of proofs FOLP. We show that the tableau system is sound and complete with respect to Mkrtychev models of FOLP.
We extend the main result of (G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11--30 (2020)) to the first-order intuitionistic logic (with and without equality), showing that it is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.
This paper introduces the logic $QLET_{F}$, a quantified extension of the logic of evidence and truth $LET_{F}$, together with a corresponding sound and complete first-order non-deterministic valuation semantics. $LET_{F}$ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment ($FDE$) with a classicality operator ${\circ}$ and a non-classicality operator $\bullet$, dual to each other: while ${\circ} A$ entails that $A$ behaves classically, ${\bullet} A$ follows from $A$'s violating some classically valid inferences. The semantics of $QLET_{F}$ c…
We study the first-order primal infon logic. It is the core of the policy language DKAL. We provide Gentzen-style calculi for two versions of this logic that are not equivalent. For both versions we investigate the semantics: one of them is a generalization of the so-called quasi-boolean semantics, the other one is a Krypke-style semantics. We prove the completeness results and the disjunction property for both logics.
I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist appro…
We study three kinds of compactness in some variants of Gödel logic: compactness, entailment compactness, and approximate entailment compactness. For countable first-order underlying language we use the Henkin construction to prove the compactness property of extensions of first-order Gödel logic enriched by nullary connective or the Baaz's projection connective. In the case of uncountable first-order language we use the ultraproduct method to derive the compactness theorem
We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.
We investigate the logical structure of intuitionistic Kripke-Platek set theory IKP, and show that the first-order logic of IKP is intuitionistic first-order logic IQC.
In this paper we prove a Robinson consistency theorem for a class of many-sorted hybrid logics as a consequence of an Omitting Types Theorem. An important corollary of this result is an interpolation theorem.
It is known that the set of tautologies of second order intuitionistic propositional logic, $\mathrm{IPC} 2$, is undecidable. Here, we prove that the sets of formulas of $\mathrm{IPC} 2$ which are true in the algebra of open subsets of reals or rationals are decidable.
In this paper we propose a semantics in which the truth value of a formula is a pair of elements in a complete Boolean algebra. Through the semantics we can unify largely two proofs of cut-eliminability (Hauptsatz) in classical second order logic calculus, one is due to Takahashi-Prawitz and the other by Maehara.
This paper is devoted to systematic studies of some extensions of first-order Gödel logic. The first extension is the first-order rational Gödel logic which is an extension of first-order Gödel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational Gödel logic has the completeness property, that is any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. …
Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic using the notion of a first-order hyperdoctrine from categorical logic to a structure which we will call the Medvedev hyperdoctrine. We also study the intermediate logic that the Medvedev…
We propose an extension of the Ehrenfeucht-Fraisse game able to deal with logics augmented with Lindstrom quantifiers. We describe three different games with varying balance between simplicity and ease of use.
Product update is an operation on models introduced into epistemic logic in order to represent a broad class of informational events. If adding modalities representing product update to a language does not alter its expressive power then we say that the language is "closed for product update." The basic modal language is known to be closed for product update. We establish that monadic second order logic is closed for product update (Theorem 5). Our technique is to pass via an intermediate language with what we call "action nominals." We obtain as corollaries that propositionally quantified mod…
One of the nice properties of the first-order logic is the compactness of satisfiability. It state that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in many-valued logics, poses various kind of the compactness in these logics. One of this issues is the compactness of $K$-satisfiability. Here, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in many-valued logic based on continuous t-norms (basic logic), we extend the results around this topic. To this end, we consider a reverse semantic…
I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser's basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.
Using a recently introduced algebraic framework for the classification of fragments of first-order logic, we study the complexity of the satisfiability problem for several ordered fragments of first-order logic, which are obtained from the ordered logic and the fluted logic by modifying some of their syntactical restrictions.
Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first-order consequences in inclusion logic.
Intuitionistic dependence logic was introduced by Abramsky and Vaananen (2009) as a variant of dependence logic under a general construction of Hodges' (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.
In this article we study linear temporal logics with team semantics (TeamLTL) that are novel logics for defining hyperproperties. We define Kamp-type translations of these logics into fragments of first-order team logic and second-order logic. We also characterize the expressive power and the complexity of model-checking and satisfiability of team logic and second-order logic by relating them to second- and third-order arithmetic. Our results set in a larger context the recent results of Lück showing that the extension of TeamLTL by the Boolean negation is highly undecidable under the so-calle…
Dependence logic, introduced in [8], cannot be axiomatized. However, first-order consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem.
Independence logic cannot be effectively axiomatized. However, first-order consequences of independence logic sentences can be axiomatized. In this article we give an explicit axiomatization and prove that it is complete in this sense. The proof is a generalization of the similar result for dependence logic.
We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player (duplicator) does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a nonelementary succinctn…
Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possi…
We show a model construction for a system of higher-order illative combinatory logic $\mathcal{I}_ω$, thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system $\mathcal{I}_0$ of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.
We present a unified categorical treatment of completeness theorems for several classical and intuitionistic infinitary logics with a proposed axiomatization. This provides new completeness theorems and subsumes previous ones by Gödel, Kripke, Beth, Karp, Joyal, Makkai and Fourman/Grayson. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.
We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two closely related but different logics. We provide a natural deduction system that is complete for validities for both of these logics, and we also investigate a range of related basic decision problems. For example, the validity problems of the two-variable fragments of the logics are shown coNexpTime-complete, which is in stark contrast with the high undecidabili…
We define the notion of a model of higher-order modal logic in an arbitrary elementary topos $\mathcal{E}$. In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier $Ω_{\mathcal{E}}$, but rather by a suitable complete Heyting algebra $H$. The canonical map relating $H$ and $Ω_{\mathcal{E}}$ both serves to interpret equality and provides a modal operator on $H$ in the form of a comonad. Examples of such structures arise from surjective geometric morphisms $f : \mathcal{F} \to \mathcal{E}$, where $H = …
The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setti…
In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k]. We also introduce several useful operators that can be expressed in INEX[k]. In particular, we de…
University of Torino（都灵大学）；Freie Universität Berlin（柏林自由大学）；King’s College London（伦敦国王学院）；University of Luxembourg（卢森堡大学）
A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics.
This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable assignments. Unlike in the propositional case, both logics are shown to have the same expressive power and their proof calculi to have the same deductive power. Both calculi are also mutually relatively complete. In the presence of differential equations, corollaries obtain usable and complete translations between differential game logic, a logic for the deductive verification of hybrid ga…
