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The paper investigates the class of probability functions defined on sentences of a predicate language containing binary and possibly also unary predicates which satisfy the Principle of Binary Exchangeability (a strengthening of the Principle of Exchangeability). Following a survey of relevant properties of such functions we prove the main theorems of this paper showing that under some mild conditions any such probability function can, by conditioning on suitable sentences, closely approximate any other probability function from this class. These results on conditioning extend earlier results obtained for purely unary languages. [ABSTRACT FROM AUTHOR]

In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure M = (N , S , ∈) of the natural numbers and collection of sets of natural numbers S has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of M governed by the axioms of Kripke-Platek set theory, KP. In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory; see [2]. However, by contrast, the admissible extension of T is usually not a conservative extension of T. Owing to the interplay of T and KP , either theory's axioms may force new sets of naturals to exist which in turn may engender yet new sets of naturals on account of the axioms of the other. The paper discerns a general pattern though. It turns out that for many familiar theories T , the second order part of the admissible cover of T equates to T augmented by transfinite induction over all initial segments of the Bachmann-Howard ordinal. Technically, the paper uses a novel type of ordinal analysis, expanding that for KP to the higher set-theoretic universe while at the same time treating the world of subsets of N as an unanalyzed class-sized urelement structure. Among the systems of reverse mathematics, for which we determine the admissible extension, are Π 1 1 - CA 0 and ATR 0 as well as the theory of bar induction, BI. [ABSTRACT FROM AUTHOR]

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of ZF to new algebra-valued models. This paper presents, for the first time, non-trivial paraconsistent models of full ZF. Moreover, due to the validity of Leibniz's law in these structures, we will show how to construct proper models of set theory by quotienting these algebra-valued models with respect to equality, modulo the filter of the designated truth-values. [ABSTRACT FROM AUTHOR]

In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP 2 can be witnessed by a formula with a tree of tuples holding 'arbitrary homogeneous inconsistency' (e.g., weak k -TP 1 conditions or other possible inconsistency configurations). And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP 1 , and by using this, we investigate the problem of (in)equality of SOP 1 and SOP 2. Assuming the existence of a formula having SOP 1 such that no finite conjunction of it has SOP 2 , we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see Definition 4.1). We show that ATP implies SOP 1 and TP 2 , but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP 1 theories and the class of NTP 2 theories. At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP 2. So this example shows that SOP 1 and SOP 2 are not the same at the level of formulas, i.e., there is a formula having SOP 1 , while any finite conjunction of it does not witness SOP 2 (but a variation of the formula still has SOP 2). [ABSTRACT FROM AUTHOR]

In this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property R 1 , ℵ 1 . They are stronger than the countable chain condition. Some of their typical examples are forcing notions about indestructible gaps introduced by Kunen. Both chain conditions are similar and have common examples, however, no distinction between them is known so far. In this paper, we propose a difference of two chain conditions from the view point of Todorčević's fragments of Martin's Axiom. More precisely, it is proved that some Todorčević's fragment of Martin's Axiom for forcing notions with property R 1 , ℵ 1 is consistent with the existence of a non-special Aronszajn tree, and that some Todorčević's fragment of Martin's Axiom for forcing notions with the rectangle refining property in some weak sense implies that every Aronszajn tree is special. [ABSTRACT FROM AUTHOR]

In 1967 the author introduced a pre-ordering of all first order complete theories where T is lower than U if it is easier for an ultrapower of a model of T than an ultrapower of a model of U to be saturated. In a long series of recent papers, Malliaris and Shelah showed that this pre-ordering is very rich and gives a useful way of classifying simple theories. In this paper we investigate the analogous pre-ordering in continuous model theory. [ABSTRACT FROM AUTHOR]

The present paper investigates proof-theoretical and algebraic properties for the probability logic FP (Ł , Ł) , meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP (Ł , Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly to probabilistic reasoning. More precisely, leveraging on such translation map, we will show proof-theoretical properties for FP (Ł , Ł) and introduce a class of algebras with respect to which FP (Ł , Ł) will be proved to be locally sound and complete. Finally, we will apply these previous results to investigate what we called "probabilistic unification problem". In this respect, we will prove that Ghilardi's algebraic view on unification can be extended to our case and, on par with the Łukasiewicz propositional case, we show that probabilistic unification is of nullary type. [ABSTRACT FROM AUTHOR]

This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely 'qualitative' comparative language to a highly 'quantitative' language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive reasoning from those that encode additive and multiplicative reasoning. The latter includes not only languages with explicit multiplication but also languages expressing notions of dependence and conditionality. We show that the distinction tracks a divide in computational complexity: additive systems remain complete for NP , while multiplicative systems are robustly complete for ∃ R. We also address axiomatic questions, offering several new completeness results as well as a proof of non-finite-axiomatizability for comparative probability. Repercussions of our results for conceptual and empirical questions are addressed, and open problems are discussed. [ABSTRACT FROM AUTHOR]

Causal Modeling Semantics (CMS, e.g., [6,22,12]) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas. Our main idea is to assign a probability to a counterfactual ▪ at a causal model M as a weighted average of the probability of C in those submodels that truthmake A ∨ B [1,3,4]. The weights of the submodels are given by the inverse distance to the original model M , based on a distance metric proposed by Eva et al. [2]. Apart from solving a major problem in the epistemology of counterfactuals, our paper shows how work in semantics, causal inference and formal epistemology can be fruitfully combined. [ABSTRACT FROM AUTHOR]

In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the effect of large cardinal assumptions on this comparison. Using the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem in [5] , where he proves the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree T such that if U ⊂ T is a Kurepa tree with the inherited order from T , then U has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: If MA ω 2 holds and ω 2 is not a Mahlo cardinal in ▪ then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's ρ function which might be useful in other contexts. [ABSTRACT FROM AUTHOR]

By combining tree representation of sets with the method introduced in the previous three papers I–III [39,35,37] in the series, we give a new Π 2 1 -preserving interpretation of KP ω r + (Π n + 2 - Found) + θ (Kripke–Platek set theory with the foundation schema restricted to Π n + 2 , and augmented by θ) in Σ 1 1 - AC 0 + (Π n + 2 1 - TI) + θ for any Π 2 1 sentence θ , where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any Π 2 1 theorem of BI ≡ Σ 1 1 - AC 0 + (Π ∞ 1 - TI) does not increase the consistency strength of KP ω. Among such Π 2 1 theorems are several fixed point principles for positive arithmetical operators and ω -model reflection (the cofinal existence of coded ω -models) for theorems of BI. The reader's familiarity to the previous works I–III in the series might help, but is not necessary. [ABSTRACT FROM AUTHOR]

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring F q [ t ] over a finite field F q that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p > 0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and p > 0 for some prime p. [ABSTRACT FROM AUTHOR]

This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics. [ABSTRACT FROM AUTHOR]

Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system S , which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of S without deductively independent sets of axioms. [ABSTRACT FROM AUTHOR]

Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function x + y + 1 2. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε 0. They also investigated a recursively defined function M : R → R. They showed that the set of points of discontinuity of M is a subset of F of order type ε 0. They also showed that, although M is a total function on R , the fact that the restriction of M to Q is total is not provable in first-order Peano arithmetic PA. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets F of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function g : R n → R. The most straightforward generalization of x + y + 1 2 to an n -ary function is the function x 1 + ⋯ + x n + 1 n. We show that this function generates a set F n whose order type is just φ n − 1 (0). For this, we develop recursively defined functions M n : R → R naturally generalizing the function M. Furthermore, we prove that for any linear function g : R n → R , the order type of the resulting F is at most φ n − 1 (0). Finally, we show that there do exist continuous functions g : R n → R for which the order types of the resulting sets F approach the small Veblen ordinal. [ABSTRACT FROM AUTHOR]

The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type □ < κ 2 λ , where we prove that for a regular uncountable cardinal κ , if □ < κ 2 λ is paracompact for every λ ≥ κ , then κ is at least inaccessible. The case of the products of the type □ < κ X λ for κ singular has not been studied much in the literature and we offer various results. The question if □ < κ 2 λ can be paracompact for all λ when κ is singular has been partially answered but remains open in general. [ABSTRACT FROM AUTHOR]

Kunen refuted the existence of an elementary embedding from the universe of sets to itself assuming the Axiom of Choice. This paper concerns the ramifications of this hypothesis when the Axiom of Choice is not assumed. For example, the existence of such an embedding implies that there is a proper class of cardinals λ such that λ + is measurable. [ABSTRACT FROM AUTHOR]

A paper on ordinal partitions by Erdős and Milner [7] has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo–Fraenkel set theory. The work is part of a project on formalising the partition calculus. The chosen material is particularly appropriate in view of the substantial corrections [8] later published by its authors, illustrating the potential value of formal verification. [ABSTRACT FROM AUTHOR]