We investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω),≥*), in terms of classical cardinal invariants of the continuum.

We give characterizations for the sentences "Every $\Sigma^1_2$-set is measurable" and "Every $\Delta^1_2$-set is measurable" for various notions of measurability derived from well-known forcing partial orderings.

I present a formal theory of the logic and aboutness of imagination. Aboutness is understood as the relation between meaningful items and what they concern, as per Yablo and Fine’s works on the notion. Imagination is understood as per Chalmers’ positive conceivability: the intentional state of a subject who conceives that p by imagining a situation—a configuration of objects and properties—verifying p. So far aboutness theory has been developed mainly for linguistic representation, but it is natural to extend it to (...) intentional states. The proposed framework combines a modal semantics with a mereology of contents: imagination operators are understood as variably strict quantifiers over worlds with a content-preservation constraint. (shrink)

Let κ < λ be regular uncountable cardinals. Using a finite support iteration (in fact a matrix iteration) of ccc posets we obtain the consistency of b = a = κ < s = λ. If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detailed investigation of iterated Sacks indestructibility.

We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection. We prove the following results:Many classical examples of nice tall filters contain no towers.It is consistent that tall analytic P-filters contain towers of arbitrary regular height.It is consistent that all towers generate nonmeager filters, in particular Borel filters do not contain towers.The statement “Every ultrafilter contains towers.” is independent of ZFC.Furthermore, we study many possible logical implications between the existence of towers in filters, (...) inequalities between cardinal invariants of filters $,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left$), and the existence of Luzin type families, that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$. (shrink)

We show the consistency of ${\frak o} <{\frak d}$ where ${\frak o}$ is the size of the smallest off-branch family, and ${\frak d}$ is as usual the dominating number. We also prove the consistency of ${\frak b} < {\frak a}$ with large continuum. Here, ${\frak b}$ is the unbounding number, and ${\frak a}$ is the almost disjointness number.

It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

Brendle, J., H. Judah and S. Shelah, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic 59 185–199. Using a notion of rank for Hechler forcing we show: assuming ωV1 = ωL1, there is no real in V[d] which is eventually different from the reals in L[ d], where d is Hechler over V; adding one Hechler real makes the invariants on the left-hand side of Cichoń's diagram equal ω1 and those on the right-hand side equal 2ω (...) and produces a maximal almost disjoint family of subsets of ω of size ω1; there is no perfect set of random reals over V in V[ r][ d], where r is random over V and d Hechler over V[r], thus answering a question of the first and second authors. (shrink)

We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, the splitting number, and the existence of certain kinds of splitting families.

We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, (...) and it was proved that Σ21-Kσ-regularity implies Σ21-u-regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u-regularity uses a game argument it was clear that determinacy implies u-regularity of all sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u-regularity, namely it holds in Solovay's model. Finally we show that forcing with uniform trees is equivalent to Laver forcing. (shrink)

Unification was originally introduced in automated deduction and term rewriting, but has recently also found applications in other fields. In this article, we give a survey of the results on unification obtained in two closely related, yet different, application areas of unification: description logics and modal logics.

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the (...) matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into 10 different values. (shrink)

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal is large. We show that for uncountable cardinals κ≤λ and , (...) if all subsets of λ either contain, or are disjoint from, a member of , then has size at least etc. As an application, we solve the Gross space problem under by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. (shrink)

We consider several game versions of the cardinal invariants \, \ and \. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \ and \ are both relatively consistent with ZFC, where \ and \ are the principal game versions of \ and \, respectively. The corresponding question for \ remains open.

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichon's diagram are equal to κ while the others are equal to λ, where $\kappa < \lambda$ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when λ is singular. We also show that $\mathrm{cf}(\kappa_U(\mathscr{L})) < \kappa_A(\mathscr{M})$ is consistent with ZFC.

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichon's diagram are equal to $\kappa$ while the others are equal to $\lambda$, where $\kappa < \lambda$ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when $\lambda$ is singular. We also show that $\mathrm{cf}) < \kappa_A$ is consistent with ZFC.

We show that Σ1 4-Amoeba-absoluteness implies that $\forall a \in \mathbb{R}(\omega^{L\lbrack a \rbrack}_1 < \omega^V_1)$ and, hence, Σ1 3-measurability. This answers a question of Haim Judah (private communication).

Unlike the first English translation in 1974, this edition contains the text corresponding to Brentano's original 1874 edition.

We investigate cardinal invariants related to the structure of dense sets of rationals modulo the nowhere dense sets. We prove that , thus dualizing the already known [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 59–80, Theorem 3.6]. We also show the consistency of each of and . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. (...) 183 59–80, Questions 3.11]. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

First published in 1969. Routledge is an imprint of Taylor & Francis, an informa company.

We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA.

We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.

Die Idee zu der in diesem Band dokumentierten Tagung ist im Rahmen des disziplinübergreifenden Nachwuchswissenschaftsnetzwerkes THESIS entstanden. Der Dialog über die fachlichen und disziplinären Grenzen hinweg hat bei THESIS seit dessen Gründung im Jahre 1990 stets einen großen Raum eingenommen. Anderen Fächern Respekt und Interesse entgegenzubringen und sich nicht von stereotypen Vorurteilen leiten zu lassen, ist konstitutiver Bestandteil im Selbstverständnis des Netzwerkes. Selbstverständlich gibt es in einem solchen Verbund eine Reihe von Gelegenheiten (darunter auch den einen oder anderen entwicklungsfördernden Konflikt), (...) das eigene Bild vom Anderen und das andere Bild vom Eigenen zur Sprache zu bringen und zu hinterfragen. Auf der Basis dieser bestehenden Kommunikationsstrukturen trafen sich am 11. Oktober 2003 in Göttingen 12 junge Wissenschaftlerinnen und Wissenschaftler zwischen, über und auch auf den Grenzen ihrer eigenen Herkunftsdisziplinen, um den Trendbegriff [alpha]Transdisziplinarität" auf seine Substanz zu prüfen. (shrink)

The Asian Logic Conference (ALC) is a major international event in mathematical logic. It features the latest scientific developments in the fields of mathematical logic and its applications, logic in computer science, and philosophical logic. The ALC series also aims to promote mathematical logic in the Asia-Pacific region and to bring logicians together both from within Asia and elsewhere for an exchange of information and ideas. This combined proceedings volume represents works presented or arising from the 14th and 15th ALCs.

The Description Logic Handbook covers all aspects of the research in the field of knowledge representation. Written by some of the most prominent researchers in the field, and covering the basic technical material and implementational aspects, it is both a unique reference and a self-study guide.

Tractatus 4.024 inspired the dominant semantics of our time: truth-conditional semantics. Such semantics is focused on possible worlds: the content of p is the set of worlds where p is true. It has become increasingly clear that such an account is, at best, defective: we need an ‘independent factor in meaning, constrained but not determined by truth-conditions’ (Yablo 2014, p. 2), because sentences can be differently true at the same possible worlds. I suggest a missing comment which, had it been (...) included in the Tractatus, would have helped semantics get this right from the start. This is my 4.0241: ‘Knowing what is the case if a sentence is true is knowing its ways of being true’: knowing a sentence’s truth possibilities and what we now call its topic, or subject matter. I show that the famous ‘fundamental thought’ that ‘the “logical constants” do not represent’ (4.0312) can be understood in terms of ways-based views of meaning. Such views also help with puzzling claims like 5.122: ‘If p follows from q, the sense of “p” is contained in the that of “q”’, which are compatible with a conception of entailment combining truth-preservation with the preservation of topicality, or of ways of being true. (shrink)

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ1. This answers a question of Halbeisen.

We study some properties of the quotient forcing notions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{tr(I)} = \wp(2^{< \omega})/tr(i)}$$\end{document} and PI = B(2ω)/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2ω. We show that the remainder forcing RI = Qtr(I)/PI is σ-closed in these cases. We also study the cardinal invariant of the continuum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, the distributivity (...) number of the quotient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Dense(\mathbb{Q})/nwd}$$\end{document}, in order to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\wp(\mathbb{Q})/nwd}$$\end{document} collapses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59–80, 2004). (shrink)

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) (...) questions, and improves results, of Faber, Laver and McKenzie. (shrink)

Say that a function π:n<ω→n (henceforth called a predictor) k-constantly predicts a real xnω if for almost all intervals I of length k, there is iI such that x(i)=π(x↾i). We study the k-constant prediction number vnconst(k), that is, the size of the least family of predictors needed to k-constantly predict all reals, for different values of n and k, and investigate their relationship.

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, (...) with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $. (shrink)

Im Zentrum der 1874 erschienenen bewußtseinspsychologischen Abhandlungen Brentanos steht die Beschreibung und Analyse des Psychischen. Damit bahnte sich jene Abwendung von der sensualistischen Tradition der philosophischen Bewußtseinslehre an, die in der Phänomenologie Edmund Husserls vollendet wurde.

Various theories try to give accounts of how measures of this confidence do or ought to behave, both as far as the internal mental consistency of the agent as ...

We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.

We show that, consistently, there can be maximal subtrees of \\) and \ / {\mathrm {fin}}\) of arbitrary regular uncountable size below the size of the continuum \. We also show that there are no maximal subtrees of \ / {\mathrm {fin}}\) with countable levels. Our results answer several questions of Campero-Arena et al..

When one thinks—knows, believes, imagines—that something is the case, one’s thought has a topic: it is about something, towards which one’s mind is directed. What is the logic of thought, so understood? This book begins to explore the idea that, to answer the question, we should take topics seriously. It proposes a hyperintensional account of the propositional contents of thought, arguing that these are individuated not only by the set of possible worlds at which they are true, but also by (...) their topic: what they are about. The book then builds epistemic, doxastic, probabilistic, and conditional logics based on this view. It applies them to issues ranging from dogmatism, scepticism, and epistemic fallibilism, to imagination and suppositional reasoning, belief revision, framing effects, and the acceptability of indicative conditionals. (shrink)

Timothy Williamson has defended the claim that the semantics of the indicative ‘if’ is given by the material conditional. Putative counterexamples can be handled by better understanding the role played in our assessment of indicatives by a fallible cognitive heuristic, called the Suppositional Procedure. Williamson’s Suppositional Conjecture has it that the Suppositional Procedure is humans’ primary way of prospectively assessing conditionals. This paper raises some doubts on the Suppositional Procedure and Conjecture.

Description logics are a family of knowledge representation formalisms that are descended from semantic networks and frames via the system Kl-one. During the last decade, it has been shown that the important reasoning problems (like subsumption and satisfiability) in a great variety of description logics can be decided using tableau-like algorithms. This is not very surprising since description logics have turned out to be closely related to propositional modal logics and logics of programs (such as propositional dynamic logic), for which (...) tableau procedures have been quite successful.Nevertheless, due to different underlying institutions and applications, most description logics differ significantly from run-of-the-mill modal and program logics. Consequently, the research on tableau algorithms in description logics led to new techniques and results, which are, however, also of interest for modal logicians. In this article, we will focus on three features that play an important rôle in description logics (number restrictions, terminological axioms, and role constructors), and show how they can be taken into account by tableau algorithms. (shrink)