In 2018, Fern & aacute;ndez-Bret & oacute;n proved that Hindman's theorem is only a countable phenomenon. In 2023, Carlucci and Fern & aacute;ndez-Bret & oacute;n established a positive example of an uncountable Hindman-type theorem called the Adjacent Hindman Theorem for uncountable groups. In this article, we investigate a kind of products called syndetic finite products, and we obtain an uncountable Hindman-type theorem in semigroups for these products. Moreover, we establish a result for adjacent finite products of a fixed number of elements under large cardinal assumptions, which can be viewed as the generalization of Rado's path decomposition theorem.
The paper defends the project of augmenting Stalnaker-Lewis semantics with impossible worlds to accommodate nonvacuous truth conditions for counter possible conditionals. A notorious consequence of the project is that hypothetical "if" is a hyperintensional operator that only sometimes permits the substitution of coreferential expressions. The present view attempts to explain this data, as well as other apparently epistemic features of counterfactual thought and talk, in terms of epistemic relations that order impossible worlds in context. The approach is unified: it neither hypothesizes two distinct counterfactuals, one circumstantial and one epistemic, nor foreshadows a need for distinct sets of cognitive processes for the assessment of counter possible versus noncounterpossible conditionals.
It is shown that Mansfield's completeness theorem for infinitary logic holds for vocabularies with relations of arbitrary arity. This answers a question of J. M. Santiago Su & aacute;rez and M. Viale.
This paper continues earlier work and extends it to an intuitionistic setting. Kripke frames are used to semantically define a family of intuitionisticlike logics for which the "local" part of the truth definition is supplied by manyvalued logics whose semantics are algebraically simple and natural. A uniform tableau system is given and soundness and completeness are proved. The tableau connection entails that the semantic family collectively determines just four logics, intuitionistic logic itself, and intuitionistic-like versions of FDE, K3, and LP. These, apparently, are new logics, and are of natural interest. For instance, all have the disjunction property, and standard double negation embeddings are applicable. In addition, intuitionistic analogues of ST (strict-tolerant logic) and TS (tolerant-strict) logics are defined, and shown to have the same relationships to intuitionistic logic that the usual ST and TS have to classical logic.
Common sense reasoning with truth involves both (i) the use of classical logic and (ii) the assumption of the transparency of truth (the equivalence between a sentence and the attribution of truth to it). The semantic paradoxes show that at least one of these must go, and different theorists make different choices. But whatever one's choice, it is valuable to carve out one or more domains where common sense reasoning ((i) and (ii) together) can be safely used; domains where everything is well-behaved. This paper explores adding a predicate of well-behavedness to various truth theories, both classical and nonclassical (including nonclassical theories with special conditionals). With such a predicate, one can reason more easily, and formulate important generalizations that are unavailable without such a predicate. I explore general model-theoretic techniques that can be applied to both classical and nonclassical theories of truth to get corresponding accounts of well-behaved truth, and some of the important generalizations that such model theories validate. There are incidental remarks about axiomatic theories that might be associated with the model theories, and their proof-theoretic strength.
We present two hierarchies of versions of Zorn's lemma that can be used directly in reverse mathematical analyses just as the original is used in standard mathematical arguments. We show that at the first two levels these versions are reverse mathematically equivalent (over RCA0) to II1k-CA0 for k D 1,2 and at higher levels to known choice axioms not provable in Z2. We give several examples of how they could be used in known proofs and a new reverse mathematical analysis of some theorems about injective choice functions (matchings) for countable families (of sets of numbers). These include a couple of unusual situations. One principle (MCSF) can be proven in II12-CA0 using our version of Zorn's lemma at II12. It is a II14 statement and perhaps might be equivalent to II12-CA0. Another (MRSF) is just a II13 statement and so cannot imply even A12-CA0 but its known proofs all use even more than II12-CA0 (II12-CAC0 ). Both of these principles are shown to imply II11-CA0. These results suggest several interesting reverse mathematical questions. We also briefly discuss some connections to similar work of Flood, Jura, Levin, and Markkanen on matchings in general graphs.
This article studies prime models in locally o-minimal theories with definable completeness, extending several classical results from the o-minimal context.
This short paper is a small contribution to the field of Boolean contact algebras. We analyze the nondefinability of the property of interior-connectedness, and we prove certain minimality conditions for algebras and spaces that can be used in demonstrating that the aforementioned property cannot be expressed by means of contact within regular closed algebras.
We explore several model-theoretic aspects of D-sets, which were studied in detail by Adeleke and Neumann. We characterize ultrahomogeneity in the class of colored D-sets and classify unbounded order-indiscernible sequences in such structures. We use these results to provide a characterization of distal colored D-sets and prove that all colored D-sets with quantifier elimination are dp-minimal.
In a recent paper, Krawczyk proved that there are continuum many axiomatic extensions of global consequence associated with the modal system E that do not admit the local deduction detachment theorem. In algebraic parlance, he showed that there are continuum many varieties of modal algebras lacking the congruence extension property. In this paper, we extend Krawczyk's results and construct a continuum of varieties of modal algebras that do not have the congruence extension property, but that do admit other, logically relevant properties, such as monotonicity, extensiveness, idempotency, normality, etc. This gives a continuum of axiomatic extensions of the corresponding modal systems not having the local deduction detachment theorem.
A well-ordering principle is a principle of the form: If X is well-ordered then F(X) is well-ordered, where F is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic of interest in Reverse Mathematics are known to be equivalent to well-ordering principles. We give a unified treatment for proving lower bounds on the logical strength of various Ramsey-theoretic principles relations using characterizations of the corresponding formal systems in terms of well-ordering principles. Our implications (over RCA_0) from combinatorial theorems to ACA_0 and ACA_0^+ also establish uniform computable reductions of the corresponding well-ordering principles to the corresponding Ramsey-type theorems.
We investigate properties of stationary tower forcings and give conditions on stationary towers which imply the universal Baireness of sets of reals in L(R).
In this paper, we build Fidel-structures valued models following the methodology developed for Heyting-valued models; recall that Fidel structures are not algebras in the universal algebra sense. Taking models that verify Leibniz law, we are able to prove that all set-theoretic axioms of ZF are valid over these models. The proof is strongly based on the existence of paraconsistent models of Leibniz law. In this setting, the difficulty of having algebraic paraconsistent models of law for formulas with negation using the standard interpretation map is discussed, showing that the existence of models of Leibniz law is essential to getting models for ZF.
We work in the language of rings augmented by a 1-ary predicate symbol Fin(x) with intended interpretation in a ring as "x is a finite union of atoms" in the Boolean algebra of idempotents of the ring. We exhibit a set of axioms in this language, and prove that any commutative unital ring R satisfying these axioms is elementarily equivalent to a restricted product, over the set of atoms e of R, of connected rings Re. Each connected ring Re is the localization of Rat the set of powers of e. This proves a Feferman-Vaught theorem for rings and a converse to the Feferman-Vaught theorem for restricted products of rings. The most important application is to axioms for rings closely resembling adele rings over number fields. Our axioms are inspired by the axioms of D'Aquino and Macintyre for products, and our results are an exact analogue of their results on products which have intriguing applications to nonstandard models of first-order Peano arithmetic.
This study introduces Gentzen-style natural deduction systems for logic. Gurevich logic is an extended constructive three-valued logic obtained from intuitionistic logic by adding strong negation, and Nelson logic is the intuitionistic-negation-less fragment of Gurevich logic. The proposed natural deduction systems are constructed in a modular manner based on primitive rules for negation, that is, rules of explosion, of negation introduction, and of excluded middle. Theorems for equivalence between these natural deduction systems and the corresponding previously proposed cut-free Gentzen-style sequent calculi are proved. Moreover, normalization theorems for the proposed natural deduction systems are proved.
In the literature, the question regarding how to axiomatize the transitive logic of false belief is thought of as hard and left as an open problem. In this paper, among other contributions, we deal with this problem. In more detail, although the standard doxastic operator is undefinable with the operator of false belief, the former is almost definable with the latter. On one hand, the involved almost definability schema guides us to find the desired core axioms for the transitive logic and the Euclidean logic of false belief. On the other hand, inspired by the schema and other considerations, we propose a suitable canonical relation, which can uniformly handle the completeness proof of various logics of false belief, including the transitive logic. We also extend the results to the logic of reliable belief, due to the interdefinability of the operators of false belief and reliable belief.
In earlier papers, we collected statements that are, in a weak formal context, equivalent to Brouwer's fan theorem. This time, we do the same for the principle of open induction on [0, 1] and the approximate-fan theorem. These principles follow from Brouwer's thesis on bars and imply the fan theorem.
Stratified formulas were introduced by Quine as an alternative way to attack Russell's paradox. Instead of limiting comprehension by size (as in ZF set theory, using its axiom scheme of separation), unlimited comprehension is given to formulas that are in some sense descended from formulas of typed set theory. By keeping variables in a stratified structure, the most common candidates for inconsistency such as {x j x ... x} are eliminated. Under the usual syntax of set theory, the set of stratified formulas form a formal language. We show that, unlike the full class of well-formed formulas of set theory, this language is not context-free, and extend the result to its complement. Therefore, much like the axioms of PA and ZF (under their usual axiomatizations), the theory NF (New Foundations) as a formal language is not context-free. We then introduce a nonstandard syntax of set theory and show that with this syntax there is a restricted class of formulas, the exo-stratified formulas, that is context-free and full (up to relabeling of variables).
The paper proves that for any low(2) A(2)(0)-degrees x(0) and x(1), there exists a low 2-c.e. degree a such that a not equal y(0) U y(1) for all degrees y(0 <=) x(0) and y(1) (<= )x(1). It follows that for all m > 1 and n = 1; 2, the posets of low(n) c.e. and low(n) m-c.e. degrees are not elementarily equivalent.
This paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behavior of the conditional in HYPE) and show its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behavior of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories.
