Abstract: Publication date: Available online 9 January 2019Source: Annals of Pure and Applied LogicAuthor(s): M. Malliaris, S. Shelah We investigate the interpretability ordering ⊴⁎ using generalized Ehrenfeucht–Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.

Abstract: Publication date: Available online 28 December 2018Source: Annals of Pure and Applied LogicAuthor(s): Sebastian Enqvist, Fatemeh Seifan, Yde Venema We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.Besides these main applications, our result covers the Kozen-Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfers to the level of the full coalgebraic modal mu-calculus.

Abstract: Publication date: Available online 28 December 2018Source: Annals of Pure and Applied LogicAuthor(s): Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan, Jan van Mill For a topological space X, let L(X) be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [1] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grzn (n≥1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [1, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or S4.Grzn for some n≥1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.

Abstract: Publication date: Available online 21 December 2018Source: Annals of Pure and Applied LogicAuthor(s): Yo Matsubara, Hiroshi Sakai, Toshimichi Usuba Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of Pκλ and showed that the existence of skinny stationary subsets of Pκλ is related to large cardinal properties of ideals over Pκλ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

Abstract: Publication date: Available online 17 December 2018Source: Annals of Pure and Applied LogicAuthor(s): Liu Yong There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called (m,n)-cupping degree and show that there exists an isolated (2,ω)-cupping degree, and there exists a proper (2,1)-cupping degree. It helps understanding various degree structures in the Ershov Hierarchy.

Abstract: Publication date: Available online 13 December 2018Source: Annals of Pure and Applied LogicAuthor(s): S.N. Popova, M.E. Zhukovskii In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ϕ such that P(Gn⊨ϕ) does not converge as n→∞ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices {1,…,n}). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.

Abstract: Publication date: Available online 22 November 2018Source: Annals of Pure and Applied LogicAuthor(s): Michael Shulman We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or “structural” set theories in the spirit of Lawvere's “Elementary Theory of the Category of Sets”. We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or “material” set theories, using a version of the classical construction based on internal well-founded relations.

Abstract: Publication date: Available online 13 November 2018Source: Annals of Pure and Applied LogicAuthor(s): Fabio Pasquali We characterize categories whose internal logic is Hilbert's ε-calculus as those categories which have a proper factorization system satisfying the axiom of choice and in which every non-initial object is injective. We provide an example of such a category where the law of excluded middle is not valid.

Abstract: Publication date: Available online 6 November 2018Source: Annals of Pure and Applied LogicAuthor(s): Sam Sanders Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic, have been proposed. We single out one particular system by Dinis-Gaspar, which is categorised by the authors as being part of intuitionistic nonstandard arithmetic. Their system is indeed inconsistent with the Transfer axiom of Nonstandard Analysis, and the latter axiom is classical in nature as it implies (higher-order) comprehension. Inspired by this observation, the main aim of this paper is to provide answers to the following questions:(Q1)In the spirit of Reverse Mathematics, what is the minimal fragment of Transfer that is inconsistent with the Dinis-Gaspar system'(Q2)What other axioms are inconsistent with the Dinis-Gaspar system' Our answer to the first question suggests that the aforementioned inconsistency actually derives from the axiom of extensionality relative to the standard world, and that other (much stronger) consequences of Transfer are actually harmless. Perhaps surprisingly, our answer to the second question shows that the Dinis-Gaspar system is inconsistent with a number of (non-classical) continuity theorems which one would -in our opinion- categorise as intuitionistic in the sense of Brouwer. Finally, we show that the Dinis-Gaspar system involves a standard part map, suggesting this system also pushes the boundary of what still counts as ‘Nonstandard Analysis’ or ‘internal set theory’.

Abstract: Publication date: Available online 29 October 2018Source: Annals of Pure and Applied LogicAuthor(s): Dominique Lecomte We provide, for each non-self dual Borel class Γ, a concrete finite antichain basis for the class of non-potentially Γ Borel relations whose closure has an acyclic symmetrization, considering the quasi-order of injective continuous reducibility. Along similar lines, we provide a sufficient condition for reducing the oriented graph G0 involved in the Kechris-Solecki-Todorčević dichotomy. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.

Abstract: Publication date: Available online 27 October 2018Source: Annals of Pure and Applied LogicAuthor(s): John Krueger We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ω1-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal I[ω2] does not have a maximal set modulo clubs.

Abstract: Publication date: Available online 9 October 2018Source: Annals of Pure and Applied LogicAuthor(s): Steffen Lewitzka In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.

Abstract: Publication date: Available online 16 October 2018Source: Annals of Pure and Applied LogicAuthor(s): Benno van den Berg, Sam Sanders Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.

Abstract: Publication date: Available online 15 October 2018Source: Annals of Pure and Applied LogicAuthor(s): Peter Holy, Philipp Lücke, Ana Njegomir We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.

Abstract: Publication date: Available online 21 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Sohei Iwata, Taishi Kurahashi In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs LP0. Moreover, we prove a version of the uniform arithmetical completeness theorem of LP0.

Abstract: Publication date: Available online 21 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Yatir Halevi We give a geometric description of the pair (V,p), where V is an algebraic variety over a non-trivially valued algebraically closed field K with valuation ring OK and p is a Zariski dense generically stable type concentrated on V, by defining a fully faithful functor to the category of schemes over OK with residual dominant morphisms over OK.Under this functor, the pair (an algebraic group, a generically stable generic type of a subgroup) gets sent to a group scheme over OK. This returns a geometric description of the subgroup as the set of OK-points of the group scheme, generalizing a previous result in the affine case.We also study a maximum modulus principle on schemes over OK and show that the schemes obtained by this functor enjoy it.

Abstract: Publication date: Available online 10 September 2018Source: Annals of Pure and Applied LogicAuthor(s): Christian Espíndola 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 and Joyal. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.