We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the $${\text {pcf}}$$ pcf -conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:TheoremIf ABSP holds for an ascending (...) sequence $$ \langle \aleph _{n_{m}} \rangle _{m}$$ ⟨ ℵ n m ⟩ m $$( n_{m} \in \omega )$$ ( n m ∈ ω ) then there is an inner model with measurables $$\kappa < \aleph _{\omega }$$ κ < ℵ ω of arbitrarily large Mitchell order below $$\aleph _{\omega }$$ ℵ ω, that is: $$\sup \left\{ \alpha \mid {\exists }\kappa < \aleph _{\omega } o ( \kappa ) \ge \alpha \right\} = \aleph _{\omega }$$ sup α ∣ ∃ κ < ℵ ω o ( κ ) ≥ α = ℵ ω. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales. (shrink)

We give a proof ofTheorem 1. Let κ be the smallest cardinal such that the free subset property Fr ω (κ,ω 1)holds. Assume κ is singular. Then there is an inner model with ω1 measurable cardinals.

We prove the independence of a strong partition relation on ℵ ω , answering a question of Erdos and Hajnal. We then give an almost complete answer to the free subset problem.

We give for ordinals α a lower bound for the least ordinal α such that Frordξ,β) and show that given enough measurable cardinals there are forcing extensions where the given bounds are sharp.

It is consistent that there exists a set mapping $F: \lbrack\omega_2\rbrack^2 \rightarrow \lbrack\omega_2\rbrack^{<\omega}$ such that $F(\alpha, \beta) \subseteq \alpha$ for $\alpha < \beta < \omega_2$ and there is no infinite free subset for F. This solves a problem of A. Hajnal and A. Mate.

Assuming the existence of ω compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on ω. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. thm2.1Suppose the consistency of “ZFC+GCH + there are infinitely many compact cardinals”. Then the following is consistent: ZFC+GCH + and for every family 0 (...) such that for all 0free subsets with at least one point in every infinite cardinal interval of ω. (shrink)

Free will, if such there be, involves free choosing: the ability to mentally choose an outcome, where the outcome is 'free' in being, in some substantive sense, up to the agent of the choice. As such, it is clear that the questions of how to understand free will and mental causation are connected, for events of seemingly free choosing are mental events that appear to be efficacious vis-a-vis other mental events as well as physical events. (...) Nonetheless, the free will and mental causation debates have proceeded largely independently of each other. Here we aim to make progress in determining how the free will and mental causation debates bear on one another. We first argue that the problems of free will and of mental causation can be seen as special cases of a more general problem, concerning whether and how mental events of a given type may be efficacious, qua the types of event they are---qualitative, intentional, freely deliberative---given their apparent causal irrelevancy for effects of the type in question; here we generalize what Horgan 1989 identifies as "the problem of mental quausation" (S1). We then build on this result to identify fruitful parallels between hard determinism and eliminative physicalism (S2) and soft determinism and non-reductive physicalism (S3). (shrink)

This essay begins by dividing the traditional problem of free will and determinism into a “correlation” problem and an “explanation” problem. I then focus on the explanation problem, and argue that a standard form of abductive (i.e. inference to the best-explanation) reasoning may be useful in solving it. To demonstrate the fruitfulness of the abductive approach, I apply it to three standard accounts of free will. While each account implies the same solution to the correlation problem, each implies (...) a unique solution to the explanation problem. For example, all libertarian-friendly accounts of free will imply that it is impossible to act freely when determinism is true. However, only a narrow subset of libertarians have the theoretical resources to defend the incompatibilist claim that deterministic laws (qua being deterministic) undermine free will, while other libertarians must reject this incompatibilist view. -/- [Version: Nov. 12, 2018]. (shrink)

This paper contributes to the semantic typology of dependent indefinites, by accounting for the distribution and interpretation of the Romanian indefinite vreun. It is shown that its occurrences are restricted to negative polarity and a subset of modal contexts. More specifically, the study of its behavior in intensional environments reveals that vreun is systematically incompatible with non-epistemic operators, a restriction we capture by proposing a novel empirical generalization (‘the epistemic constraint’). To account for the observed pattern, we adopt the unitary (...) approach to polarity in Chierchia (Logic in Grammar: polarity, free choice and intervention, 2013a; in: Fălăuş (ed.) Alternatives in semantics, 2013b) and derive the properties of vreun from its obligatory association with alternatives. Its distributional restrictions are argued to follow from the interplay between the types of alternatives it activates (scalar and subdomain alternatives), the way these alternatives are factored into meaning (via an independently motivated mechanism of exhaustification) and the lexical semantics of the operators in the context of occurrence. We propose that the epistemic constraint arises from the lexicalization of an inference (anti-total variation) that non-epistemic operators can never satisfy. (shrink)

Given a family of subsets of, we try to compute the least natural number n such that for every function there exists a bijection such that for all.

The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality (...) of a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

The claim that the free energy principle is somehow related to Hamilton’s principle in statistical mechanics is ubiquitous throughout the subject literature. However, the exact nature of this relationship remains unclear. According to some sources, the free energy principle is merely similar to Hamilton’s principle of stationary action; others claim that it is either analogous or equivalent to it, while yet another part of the literature espouses the claim that it is a version of Hamilton’s principle. In this (...) article, we aim to clarify the nature of the relationship between the two principles by investigating the two most likely interpretations of the claims that can be found in the subject literature. According to the strong interpretation, the two principles are equivalent and apply to the same subset of physical phenomena; according to the weak interpretation, the two principles are merely analogous to each other by virtue of their similar formal structures. As we show, adopting the stronger reading would lead to a dilemma that is untenable for the proponents of the free energy principle, thus supporting the adoption of the weaker reading for the relationship between the two constructs. (shrink)

Recall that a subset X of a Boolean algebra A is independent if for any two finite disjoint subsets F , G of X we have ∏ x∈F x ∏ y∈G −y≠0. The independence of a BA A , denoted by Ind, is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant i , well known in the (...) case A= P / fin . In this article we consider the collection of all cardinalities of infinite maximal independent subsets of a BA A ; we call this set the spectrum of infinite maximal independent subsets , denoted by Spind. Note that infinite maximal independent subsets exist in any BA which is not superatomic. The main result is that any set of infinite cardinals can occur as Spind for some infinite BA A . Beyond this we give results concerning the way that Spind changes under various algebraic operations. However, the basic components of most algebras that we deal with are free algebras. (shrink)

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2 |A| = 2 |B| . This implies in particular that B has 2 |B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to (...) be accessible as far as possible to a general audience, in particular we have confined the more technical material to a "black box" at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere. (shrink)

This paper investigates whether search engines and other new modes of online communication should be covered by free speech principles. It criticizes the analogical reason-ing that contemporary American courts and scholars have used to liken search engines to newspapers, and to extend free speech coverage to them based on that likeness. There are dissimilarities between search engines and newspapers that undermine the key analogy, and also rival analogies that can be drawn which don’t recommend free speech protection (...) for search engines. Partly on these bases, we argue that an analogical approach to questions of free speech coverage is of limited use in this context. Credible verdicts about how free speech principles should apply to new modes of online com-munication require us to re-excavate the normative foundations of free speech. This method for deciding free speech coverage suggests that only a subset of search engine outputs and similar online communication should receive special protection against government regulation. (shrink)

I will start by noting that Harry Frankfurt’s concept of wholeheartedness is in conflict with the intuition that free will should be efficacious in general, rather than pertain only to a small subset of decisions. To replace wholeheartedness I introduce a heuristic account for deliberation and decisions. I will show that introspective activity can lead to the individual having two types ‘introspective revelations’. By the onset of the introspective revelations a self-perpetuating loop is initiated. The loop consists of two (...) elements positively reinforcing each other. The two elements are introspective activity and introspective revelations. The effect is a propensity to ask oneself “what do I want?” whenever faced with a choice. This propensity, I submit, can give us what we want when we want free will. (shrink)

The purpose of this paper is to briefl y examine one of the fundamental assumptions made in contemporary liberal political philosophy, namely that persons are free and equal. Within the contemporary liberal political thought it would be considered very uncontroversial and even trivial to claim something of the following form: “persons are free and equal” or “people think of themselves as free and equal”. The widespread nature of this assumption raises the question what justifies this assumption, are (...) there good reasons for holding it? After establishing some methodological remarks, including a distinction between having freedom-equality and being free-equal and restricting the domain of discussion to include only a subset of all moral questions, namely the questions of political morality, the paper deals with some conceptual issues concerning this assumption of persons as free and equal, such as how do free-and-equal-making properties relate to person-making properties. It then moves on to examine three broad ways the free-and-equal-mak-ing properties could be established. First, necessary property approaches, which take some necessary feature of persons to be what makes them free and equal (e.g. possessing an immortal soul). Second, contingent property approaches, which take some contingent feature of persons to be what makes them free and equal (e.g heir practise of reasoning). Third, agreement based approaches, which take some agreement or contract among persons to be the basis for their being free and equal (e.g. evolutionary emergence of our treatment of others). Strengths and weaknesses of all approaches will be examined. (shrink)

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has (...) a limit point”. implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

Notice that this paper has not claimed that all natural languages are CFL's. What it has shown is that every published argument purporting to demonstrate the non-context-freeness of some natural language is invalid, either formally or empirically or both.18 Whether non-context-free characteristics can be found in the stringset of some natural language remains an open question, just as it was a quarter century ago.Whether the question is ultimately answered in the negative or the affirmative, there will be interesting further (...) questions to ask. If it turns out that natural languages are indeed always CFL's, it will be reasonable to ask whether this helps to explain why speakers apparently recognize so quickly whether a presented utterance corresponds to a grammatical sentence or not, and to associate structural and semantic details with it. It might also be reasonable to speculate about the explanation for the universally context-free character of the languages used by humans, and to wonder whether evolutionary biological factors are implicated in some way (Sampson (1979) could be read in this light). And naturally, it will be reasonable to pursue the program put forward by Gazdar (1981, in press) to see to what extent CFL-inducing grammatical devices can be exploited to yield insightful descriptions of natural languages that capture generalizations in revealing ways.If a human language that is not a CFL is proved to exist, on the other hand, a different question will be raised: give the non-context-free character of human languages in general, why has this property been so hard to demonstrate that it has taken over twenty-five years to bring it to light since the issue was first explicitly posed? If human languages do not have to be CFL's, why do so many (most?) of them come so close to having the property of context-freeness? And, since the CFL's certainly constitute a very broad class of mathematically natural and computationally tractable languages, what property of human beings or their communicative or cognitive needs is it that has caused some linguistic communities to reach beyond the boundaries of this class in the course of evolving a linguistic system?Either way, we shall be interested to see our initial question resolved, and further questions raised. One cautionary word should be said, however, about the implications (or lack of them) that the answer will have for grammatical studies. Chomsky has repeatedly stated that he does not see weak generative capacity as a theme of central importance in the theory of grammar, and we agree. It is very far from being the case that the recent resurgence of interest in exploring the potential of CF-PSG or equivalent systems will, or should, be halted dead in its tracks by the discovery (if it is ever forthcoming) that some natural language is not a CFL. In the area of parsing, for instance, it seems possible that natural languages are not only parsed on the basis of constituent structure such as a CF-PSG would assign, but are parsed as if they were finite state languages (see Langendoen (1975) and Church (1980) for discussion along these lines). That is, precisely those construction-types that figure in the various proofs that English is not an FSL appear to cause massive difficulty in the human processing system; the sentences crucial to the proofs are for the most part unprocessable unless they are extremely short (yet the arguments for English not being and FSL only go through if length is not an issue). This means that in practice properties of finite state grammars are still of great potential importance to linguistic theory despite the fact that they do not provide the framework for defining the total class of grammatical sentences. The same would almost certannly be true of CF-PSG's if they were shown to be inadequate in a similar sense. It is highly unlikely that the advances made so far in far in phrase structure description could be nullified by a discovery about weak generative capacity. Moreover, there are known to be numerous ways in which the power of CF-PSG's can be marginally enhanced to permit, for example, xx languages to be generated without allowing anything like the full class of recursively enumerable or even context-sensitive languages (see Hopcroft and Ullmann (1979), Chapter 14) for an introduction to this topic, noting especially Figure 14.7 on p. 393. The obvious thing to do if natural languages were ever shown not to be CFL's in the general case would be to start exploring such minimal enhancements of expressive power to determine exactly what natural languages call for in this regard and how it could be effectively but parsimoniously provided in a way that closely modelled human linguistic capacities.In the meantime, it seems reasonable to assume that the natural languages are a proper subset of the infinite-cardinality CFL's until such time as they are validly shown not to be. (shrink)

One broadly recognised characteristic feature of (a core subset of) the self-attributions constitutive of self-knowledge is that they are ‘immune to error through misidentification’ (hereafter IEM). In the last thirty years, Evans’s notion of “identification-freedom” (Evans 1982) has been central to most classical approaches to IEM. In the Evansian picture, it is not clear, however, whether there is room for a description of what may be the strongest and most interesting variant of IEM; namely what Pryor (1999) has first brought (...) to the fore under the name “Which-object IEM”, and which I’ll prefer calling existential IEM. I argue that recent development of relativist frameworks in semantics and pragmatics, particularly in Recanati (2007a-b), (2009), (2010), (2012a), may be precisely of a nature to address this limitation. The relativist theory of IEM, and in particular its suitability to cover existential IEM, may superficially seem to stem from a rejection of the core elements of the classical identification-freedom approach. However, I hope to show that, modulo a clarification and richer un-derstanding of the relevant notions of identification and identification-freedom, the relativist theory of IEM can be seen as both pushing further and complementing Evans’s intuitions, rather than conflicting with them. (shrink)

The following result is an approximation to the answer of the question of Kokorin about decidability of a quantifier-free theory of field of rational numbers. Let Q0 be a subset of the set of all rational numbers which contains integers 1 and −1. Let be a set containing Q0 and closed by the functions of addition, subtraction and multiplication. For example coincides with Q0 if Q0 is the set of all binary rational numbers or the set of all decimal (...) rational numbers. It is clear that , where Z is the set of all integers and Q is the set of all rational numbers. A negationless theory uses only conjunction and disjunction as logical connectives. Let T be a quantifier-free negationless elementary theory of signature , where Pol is the list of all polynomials with rational coefficients from Q0 in which exponent of the polynomials and rational coefficients are written in binary number system. Theorem 1. Theory T is NP-hard and if is everywhere dense then the theory T is decidable by an algorithm which belongs to EXPTIME. The set of all rational numbers or the set of all binary rational numbers may be taken instead of . The quantifier-free negationless theory of the field of rational numbers may be regarded as a base of constructive mathematics. (shrink)

The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As (...) a byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

In _The Philosophical Quarterly, 47 (1997), pp. 373-381, van Inwagen argues in a critical notice of my book _The Metaphysics of Free Will that the impression that Frankfurt-type examples show that moral responsibility need not require alternative possibilities results from insufficient analytical precision. He suggests various precise principles which imply that moral responsibility requires alternative possibilities. In reply, I seek to defend the conclusion I have drawn from Frankfurt-type examples: moral responsibility need not require alternative possibilities. I contend that (...) van Inwagen's principle the principle of possible prevention and the no-matter-what principle are invalid, and I suggest that their plausibility comes from thinking about a proper subset of the relevant cases. (shrink)

We improve the canonization theorems generalizing the Erdos-Rado theorem, and as a result complete the answer to "When does a Hausdorff space of cardinality χ necessarily have a discrete subspace of cardinality κ?" We also improve the results on existence of free subsets.

It is consistent that there is a set mapping from the four-tuples of ω n into the finite subsets with no free subsets of size t n for some natural number t n . For any $n it is consistent that there is a set mapping from the pairs of ω n into the finite subsets with no infinite free sets. For any $n it is consistent that there is a set mapping from the pairs (...) of ω n into ω n with no uncountable free sets. (shrink)

We survey the definition and fundamental properties of the family of short core models, which extend the core model K of Dodd and Jensen to include α-sequences of measurable cardinals . The theory is applied to various combinatorial principles to get lower bounds for their consistency strengths in terms of the existence of sequences of measurable cardinals. We consider instances of Chang's conjecture, ‘accessible’ Jónsson cardinals, the free subset property for small cardinals, a canonization property of ω ω , (...) and a non-closure property of elementary embeddings of the universe. In some cases, equiconsistencies are obtained. (shrink)

Results of Sierpiski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite (...) set. Such partitions or coverings can exist only when the sets forming the product are of limited size.This paper considers such coverings for products of infinitely many sets . In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering , then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on.One result given here is a relative consistency proof for a hypothesis used by S. Mrówka to produce a counterexample in the dimension theory of metric spaces. (shrink)

In a previous paper, I argued that neuroscience and psychology could in principle undermine libertarian free will by providing support for a subset of what I called “statements of local determination.” I also argued that Libet-style experiments have not so far supported statements of that sort. In a commentary to the paper, Adina Roskies and Eddy Nahmias accept the claim about Libet-style experiments, but reject the claim about the possibilities of neuroscience. Here, I explain why I still disagree with (...) their conclusion, despite being sympathetic to a lot of what they say in support of it. (shrink)

Summary To understand the present situation we must know something about its history. The ‘Rise of the West’, which grew out of the ‘European Miracle’, is a special case of cultural evolution. The development of science is an important element in this process. Cultural evolution went hand in hand with biological evolution. Evolutionary epistemology illuminates the achievements and the evolution of cognitive sensory apparatus of various species. Man's cognitive sensory apparatus is adapted to the ‘mesocosmos’, the world of medium-sized dimensions. (...) The biological structures constitute the hardware of the cognitive sensory apparatus, while certain expectations and theories, which are ontogenetically apriori, constitute the corresponding system software. A distinction is introduced between ‘primary theories’ (linked to the sensory apparatus) and ‘secondary theories’. The latter are the result of attempts to meet the demand for an explanation of phenomena that cannot be explained in terms of ‘primary theories’. Two subsets of ‘secondary theories’ are compared: spiritualistic-personalized theories and scientific theories. From the historical point of view the scientific secondary theories are but a special subset of the class of secondary theories. From the systematic point of view it is instructive to focus on a comparison of the two subsets: what do they have in common? in what respects do they differ? The rise of scientific thinking is closely linked to the ‘European Miracle’. How (and when and why) did the West grow rich? To answer this question we must first produce an explanation of the principle: theories about the consequences of institutional arrangements. Then we can give a historical explanation of why this development took place in Europe and only there. It is claimed that the secret of success, economic wealth and the first approximations to relatively free societies, was the taming of the state, the taming of cleptocracy (independent of the nature of the agency having cleptocratic appetites, be it princes or parliaments). The taming of the state is a pre-democratic achievement. Finally, the consequences of institutional arrangements for scientific progress and innovation are examined. Only if the system is market-like, will it link individual effort with reward and, through the competitive process, encourage the wide dissemination and use of new ideas. There is no tradeoff between freedom on the one hand and economic success and the growth of scientific knowledge on the other. Freedom and the ensuing flexibility is the key to the past and to the future. (shrink)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, (...) in the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we study the correspondences between (...) point-free and point-based models of space in terms of extended contact. More precisely, we prove new representation theorems for extended contact algebras. (shrink)

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)

During the mid-1970s the extraordinarily dangerous style of free solo climbing emerged in the collective practice of a small community of “Stonemaster” climbers actively developing new climbing routes and the new “free” style of roped climbing in what is now Joshua Tree National Park, California. While its emergence might be interpreted as an affectively-driven, macho embodied social semiotic or ethnomotricity, in actuality the evolution of free soloing in the case of Stonemaster-era climbing at Joshua Tree may be (...) more accurately understood as the logical consequence of an intensive regime of practice in which climbers developed a near absolute bodily familiarity with certain climbing routes. Eventually, the reasonableness of climbing with little and even no protection on a subset of these routes became self-evident and conventional. Free soloing, in the semiotic perspective of the later Peirce, manifested as a Normal Logical Interpretant in the Joshua Tree landscape. It embodied the living definition of self-controlled conduct, and the Symbols it cultivated testified persuasively to the growth of a creative corporeal intelligence actualizing near absolute degrees of free will. (shrink)

This article examines the ethical issues raised by the pricing of priceless goods. Priceless goods are defined as ones that are widely held to have some special non-market value that makes them unsuited for buying and selling. One subset of priceless goods isprescription drugs—particularly life-saving and life-enhancing ones. Drug makers are under pressure to price their medicines responsibly, which means to restrain their prices (and profits). However, this article argues that it is precisely because life-saving and life-enhancing medicines are priceless (...) that it is especially urgent to leave companies free to charge market prices for them. (shrink)

Given any subset A of ω1 there is a proper partial order which forces that the predicate xA and the predicate xω1A can be expressed by -provably incompatible Σ3 formulas over the structure Hω2,,NSω1. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable over Hω2,,NSω1 by a provably antisymmetric Σ3 formula with two free variables. The proofs of these results involve a technique for manipulating (...) the guessing properties of club-sequences defined on stationary subsets of ω1 at will in such a way that the Σ3 theory of Hω2,,NSω1 with countable ordinals as parameters is forced to code a prescribed subset of ω1. On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy. (shrink)

This paper serves three specific goals. First, it reports the development of an Indian Asian face set, to serve as a free resource for psychological research. Second, it examines whether the use of pre-tested U.S.-specific norms for stimulus selection or weighting may introduce experimental confounds in studies involving non-U.S. face stimuli and/or non-U.S. participants. Specifically, it examines whether subjective impressions of the face stimuli are culturally dependent, and the extent to which these impressions reflect social stereotypes and ingroup favoritism. (...) Third, the paper investigates whether differences in face familiarity impact accuracy in identifying face ethnicity. To this end, face images drawn from volunteers in India as well as a subset of Caucasian face images from the Chicago Face Database were presented to Indian and U.S. participants, and rated on a range of measures, such as perceived attractiveness, warmth, and social status. Results show significant differences in the overall valence of ratings of ingroup and outgroup faces. In addition, the impression ratings show minor differentiation along two basic stereotype dimensions, competence and trustworthiness, but not warmth. We also find participants to show significantly greater accuracy in correctly identifying the ethnicity of ingroup faces, relative to outgroup faces. This effect is found to be mediated by ingroup-outgroup differences in perceived group typicality of the target faces. Implications for research on intergroup relations in a cross-cultural context are discussed. (shrink)

We investigate the class of m-hypergraphs whose substructures with l elements have more than sm-element subsets that do not form a hyperedge. The class will have the free amalgamation property if s is small, but it does not if s is large. We find the boundary of s. Suppose the class has the free amalgamation property. In the case $$m \ge 3$$ m ≥ 3, we demonstrate that the random structure for the class has continuum-many automorphisms with (...) a single orbit. The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit. (shrink)

Given free information and unlimited processing power, should decision algorithms use as much information as possible? A formal model of the decision-making environment is developed to address this question and provide conditions under which informationally frugal algorithms, without any information or processing costs whatsoever, are optimal. One cause of compression that allows optimal algorithms to rationally ignore information is inverse movement of payoffs and probabilities (e.g., high payoffs occur with low probably and low payoffs occur with high probability). If (...) inversely related payoffs and probabilities cancel out, then predictors that correlate with payoffs and consequently condition the probabilities associated with different payoffs will drop out of the expected-payoff objective function, severing the link between information and optimal action rules. Stochastic payoff processes in which rational ignoring occurs are referred to as compressed environments, because optimal action depends on a reduced-dimension subset of the environmental parameters. This paper considers benefits and limitations of economic models versus other methods for studying links between environmental structure and the real-world success of simple decision procedures. Different methods converge on the normative proposition of ecological rationality, as opposed to axiomatic rationality based on informational efficiency and internal consistency axioms, as a superior framework for comparing the effectiveness of decision strategies and prescribing decision algorithms in application. (shrink)

Language employs various coordinators to connect propositions, a subset of which are “logical” in nature and thus analogous to the truth operators of formal logic. We here focus on two linguistic connectives and their negations: conjunction _and_ and (inclusive) disjunction _or_. Linguistic connectives exhibit a truth-conditional component as part of their meaning (their semantics), but their use in context can give rise to various implicatures and presuppositions (the domain of pragmatics) as well as to inferences that go beyond semantic/pragmatic properties (...) (the result of reasoning processes). We provide a comprehensive review of the role of the logical connectives in language and argue that three sets of factors—semantic, pragmatic, and those related to reasoning—are separate and separable, though some details may differ cross-linguistically. As a way to showcase the argument, we present two experiments in language comprehension in Spanish wherein pragmatic content was minimised and reasoning processes neutered, thus potentially highlighting what might be the default meanings of the connectives under study. In Experiment 1 we show that the conjunctive reading of inclusive disjunction is available in positive contexts other than in syntactically intricate cases such as downward entailing and free choice contexts, contrary to what has been claimed in the literature. In Experiment 2 we show that negated conjunctions and disjunctions in Spanish can easily receive the same interpretation when contrasted against the same context and, moreover, that these interpretations match those available in English, despite claims from the literature that linguistic connectives and local negation interact differently in English and Romance languages. (shrink)

For Kant and his followers, pure reason can be practical, and its substantive practical command is, broadly speaking, that we treat ourselves and others as worthy of respect as free and equal. If those who have defended the Kantian morality system are correct, this moral imperative will not be authoritative and inescapable simply because we don't know how to coherently reweave our practical commitments so as to leave it out, but because it is presupposed by the possibility of practical (...) reason. On this model, the rational authority of moral dictates would be understood as absolute and universal. ;But what if Kant and his followers have been wrong? What if the search for practical unconditionality turned out to be vain? Kant himself would most likely react to this news by pronouncing that, without a priori necessity, there could be no moral necessity, and without moral necessity, there could be no purity of motive, and without purity of motive, there could be no moral worth, and without moral worth, there could be no legitimate ascriptions of praise and blame. In short, for Kant and his orthodox followers, without unconditionality, the morality system would collapse. ;I will argue that the search for a priori necessity behind moral law is indeed hopeless, but that this does not entail the collapse of the morality system. It does however force us to conceive a new way of accounting for practical normativity in the absence of unconditionality. It also requires that, in order to preserve morality, we find a defensible place for it as a contingent subset of ethics broadly construed. My answer is to provide a new account of morality, and practical normativity in general, against the backdrop of neopragmatism. My hope is to provide an alternative story which preserves the Kantian insight that practical necessity is driven by rational authority, but which does not presuppose unconditionality to explain the latter's force and inescapability. In short I will attempt to show that ethical principles, when considered in linguistic space, need not be a priori necessary to bind individuals in a genuinely authoritative sense. (shrink)

Given that consciousness is poorly understood and vaguely defined, Paul Feyerabend's advice to "keep our options open" seems sound, but is frequently ignored in favour of an insistence that a scientific theory of consciousness must be reducible to current monist physics and biology. This book argues that such an insistence is historically unsupportable, theoretically incoherent and unnecessary. The author instead makes the case for emergent property pluralism. New concepts of emergent mental properties are needed because of the failure of mainstream (...) approaches satisfactorily to address issues like subjective volition, autonomy and creativity. Personal consciousness is active and classifiable as a subset of the wider problem of biological causation. The book is split into three sections. Part one builds an historical case for pluralism. Part two deconstructs insistent monism and mainstream models before addressing biological causation. Part three explores the consequences of such an alternative approach by examining specific phenomena like free will, the self and evolutionary emergence. (shrink)

In this paper we continue our work of Kalantari and Welch . There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the (...) literature. We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Slaves and the Generals of ArginusaePeter HuntIn the second half of 406 B.C. the Athenians made two shocking decisions. They freed the slaves who had fought in the battle of Arginusae and gave them citizenship, and they condemned to death their victorious generals. I suggest that these two events were related. Specifically, I would like to argue, first, that the competition for rowers to man the huge navies (...) of this period had extended even to slaves. The promise of freedom was intended to motivate the slaves: the grant of Athenian citizenship was an attempt to keep them rowing for Athens rather than deserting and ending up in the enemy's navy. Second, I would like to consider the trial of the generals in the context of this costly, divisive, and controversial policy. A priori considerations, historical parallels, and a few admittedly tenuous pieces of direct evidence suggest that their policy of rewarding the slaves contributed to the generals' unpopularity that made them so vulnerable to prosecution.Slaves to CitizensPassages in Aristophanes' Frogs and a scholiast's quotation of Hellanicus tell us that the slaves who fought in the battle of Arginusae were given freedom and citizenship. Most historians believe that the Athenians freed these slaves,1 but, given Athenian exclusiveness-manifest in Pericles' citizenship law-some doubt that they would have allowed a large group [End Page 359] of slaves to become citizens.2 The evidence, however, indicates that this was exactly what happened.Frogs won first prize at the Lenaean festival of 405-406, about six months after the Athenian naval victory at Arginusae. Early in the play, Dionysus and his slave Xanthias are traveling to Hades. When Dionysus mocks Xanthias for complaining about the load he is carrying, Xanthias exclaims:Dash it all, why wasn't I in that naval battle? Then I could really and truly tell you to go to blazes!(Ar. Ran. 33)3Fighting in the naval battle would have freed Xanthias from carrying Dionysus' baggage and allowed him to tell his master off. A bit later, Charon refuses to ferry Xanthias to Hades:I don't take slaves, unless they fought in the life-or-death naval battle.(190-91)4Again, slaves who have fought in a naval battle are treated differently from other slaves. From these passages alone, one might guess that the slaves who fought in some naval battle-Arginusae being the most recent and probable one-were given their freedom: they did not have to serve their masters and could ride with Charon rather than walking around the lake to get to Hades.Ian Worthington argues against this, the traditional reading of these lines. Xanthias' regret at not serving in the naval battle (33) is not that he would be then be free, but rather that then he would be dead and not forced to serve Dionysus.5 Aristophanes, however, was addressing an [End Page 360] audience many of whom had either fought at Arginusae or had relatives who had. We may associate Arginusae primarily with the deaths of the unrescued crews, but in 405 Arginusae was considered a successful relief mission and a great victory. Aristophanes was addressing the living victors, for whom "to fight at the naval battle" could not serve as shorthand for "to die." Worthington's reading of lines 190-91 is also less convincing than the traditional interpretation. He claims that those who "fought in the life-or-death naval battle" are the men whose drowning after the battle gave rise to the trial of the generals. The text gives no hint or suggestion that Charon is talking about this subset of the sailors who fought in the battle. Rather, Charon refers to another subset of those who fought in the battle, the slaves: "I don't take slaves, unless...." So we should stay with the traditional interpretation and the scholia to Frogs 33, which, although not an independent source, accords with our reading of Aristophanes: ("[I]n the previous year, in the archonship of Antigenes, the Athenians... won a naval battle around Arginusae and freed the slaves who had fought along with them").6Our first evidence for... (shrink)

Moral responsibility is centered on the idea that, given some conditions, people deserve blame or credit, punishment or reward. At least according to traditional readings, moral responsibility presupposes free will, understood as the ability to choose independently of previous events. The achievements of neuroscience in recent decades make a very good case for the hypothesis that the mind is a material entity, a subset of the electrochemical activity of the brain. However, if the mind is a material entity, then (...) it is subject to physical laws. According to some interpretations, this entails the affirmation of determinism and the denial of free will and moral responsibility. This article reviews some of those achievements and their likely impact on the attributions of moral responsibility. The first section presents evidence from lesion studies, imaging techniques and direct electrical stimulation of the brain and argues that the convergence of results from these methods supports the hypothesis that the mind is the activity of the brain. The next section concentrates on studies that are usually taken as directly threatening free will – Libet’s experiments and the apparent mental causation theory. The third section turns to the implications of these findings for moral responsibility, explaining determinism, epiphenomenalism, libertarianism, hard determinism and compatibilism. Finally, the last section defends that the requirement of free will as a precondition for moral responsibility should be replaced by the requirement of more naturalistic properties, such as behavioral flexibility, self-control and natural autonomy.Key words: moral responsibility, ethics, neuroscience, neuroethics, autonomy. (shrink)

For a free filter F on $$\omega $$, let $$N_F=\omega \cup \{p_F\}$$, where $$p_F\not \in \omega $$, be equipped with the following topology: every element of $$\omega $$ is isolated whereas all open neighborhoods of $$p_F$$ are of the form $$A\cup \{p_F\}$$ for $$A\in F$$. The aim of this paper is to study spaces of the form $$N_F$$ in the context of the Nikodym property of Boolean algebras. By $$\mathcal{A}\mathcal{N}$$ we denote the class of all those ideals $$\mathcal {I}$$ (...) on $$\omega $$ such that for the dual filter $$\mathcal {I}^*$$ the space $$N_{\mathcal {I}^*}$$ carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ of finitely supported signed measures such that $$\Vert \mu _n\Vert \rightarrow \infty $$ and $$\mu _n(A)\rightarrow 0$$ for every clopen subset $$A\subseteq N_{\mathcal {I}^*}$$. We prove that $$\mathcal {I}\in \mathcal{A}\mathcal{N}$$ if and only if there exists a density submeasure $$\varphi $$ on $$\omega $$ such that $$\varphi (\omega )=\infty $$ and $$\mathcal {I}$$ is contained in the exhaustive ideal $$\text{ Exh }(\varphi )$$. Consequently, we get that if $$\mathcal {I}\subseteq \text{ Exh }(\varphi )$$ for some density submeasure $$\varphi $$ on $$\omega $$ such that $$\varphi (\omega )=\infty $$ and $$N_{\mathcal {I}^*}$$ is homeomorphic to a subspace of the Stone space $$St(\mathcal {A})$$ of a given Boolean algebra $$\mathcal {A}$$, then $$\mathcal {A}$$ does not have the Nikodym property. We observe that each $$\mathcal {I}\in \mathcal{A}\mathcal{N}$$ is Katětov below the asymptotic density zero ideal $$\mathcal {Z}$$, and prove that the class $$\mathcal{A}\mathcal{N}$$ has a subset of size $$\mathfrak {d}$$ which is dominating with respect to the Katětov order $$\le _K$$, but $$\mathcal{A}\mathcal{N}$$ has no $$\le _K$$ -maximal element. We show that, when $$\mathcal {I}$$ is a density ideal, $$\mathcal {I}\not \in \mathcal{A}\mathcal{N}$$ holds if and only if $$\mathcal {I}$$ is totally bounded if and only if the Boolean algebra $$\mathcal {P}(\omega )/\mathcal {I}$$ contains a countable splitting family. Our results shed some new light on differences between the Nikodym property and the Grothendieck property of Boolean algebras. (shrink)