We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the (...) meager ideal was established by Bartoszyński and Kada. (shrink)

In an ideal democracy, people from all walks of life would come together to talk meaningfully and respectfully about politics. But we do not live in an ideal democracy. In contemporary democracies, which are marked by deep social divisions, different groups for the most part avoid talking to each other. And when they do talk to each other, their speech often seems to be little more than a vehicle for rage, hatred, and deception. -/- Democratic Speech in Divided (...) Times argues that we should nevertheless not give up on the ideal of democratic public speech. Drawing on the resources of political theory, epistemology, and philosophy of language, this book develops a sustained account of the norms that should govern public discourse in deeply divided circumstances. Should we try to find common ground when we talk to our political opponents, even though they seem unreasonable? Should we refrain from expressing anger, if we want to get things done? How can we use our speech to fight hate speech and disinformation? And is it even possible to speak to 'the other side,' in settings where different groups dislike one another, live apart from one another, and don't know much about one another? By tackling these questions, Lepoutre demonstrates that, when governed by the right set of norms, public speech can be a powerful force for good even amidst profound social divisions. (shrink)

This article defends the democratic ideal of inclusive public discourse, as articulated in Democratic Speech in Divided Times, against the critiques offered by Billingham, Fraser, and Hannon. Specifically, it considers and responds to three core challenges. The first challenge argues, notably, that the “shared reasons” constraint should either apply everywhere or not at all, and that, if this constraint is to apply in divided circumstances, its justificatory constituency must be idealized. The second challenge contends that the resistance of hate (...) speech and misinformation to counterspeech cannot adequately be explained by considerations of salience, and therefore cannot adequately by countered (as I suggest) by “positive” forms of counterspeech. Finally, the last challenge objects that the ideal of inclusive public discourse I defend remains, as pessimists allege, excessively idealistic. (shrink)

An influential critique of anger holds that anger comes at an important epistemic cost. In particular, feeling angry typically makes risk less visible to us. This is anger’s ‘red mist.’ These epistemic costs, critics suggest, arguably outweigh the epistemic benefits commonly ascribed to anger. This essay argues that the epistemic critique of anger is importantly misleading. This is not because it underestimates anger’s epistemic benefits, but rather because it overlooks the fact that anger’s red mist performs a crucial moral function. (...) By concealing risk, the red mist helps protect the dignity and self-respect of those who live under severe and unchosen risk. This function is vitally important in non-ideal circumstances, where many are unjustly subjected to dignity-impairing risk. And it is irreplaceable, in that it cannot readily be performed by other political emotions, such as hope. Beyond providing a novel defence of anger, this argument has a broader upshot. Anger’s red mist reveals that ignorance is not always problematic in politics. Accordingly, it should not be treated—as has standardly been the case—as an unmitigated political evil. In the first instance, our efforts are better spent ascertaining when ignorance is a problem, and when, instead, it performs a felicitous function. (shrink)

Public opinion research has shown that voters accept many falsehoods about politics. This observation is widely considered troubling for democracy—and especially participatory ideals of democracy. I argue that this influential narrative is nevertheless flawed, because it misunderstands the nature of political understanding. Drawing on philosophical examinations of scientific modelling, I demonstrate that accepting falsehoods within one’s model of political reality is compatible with—and indeed can positively enhance—one’s understanding of that reality. Thus, the observation that voters accept many political falsehoods does (...) not necessarily establish that they lack political understanding. I then address three worries: that voters cannot generally engage in such political modelling; that political modelling obscures facts that are crucial to political understanding; and that successful political modelling would require knowing that one’s model contains falsehoods. My responses reveal how, going forward, we should measure political ignorance; and they highlight the standing importance of participatory democracy. (shrink)

Counterspeech is communication that tries to counteract potential harm brought about by other speech. Theoretical interest in counterspeech partly derives from a libertarian ideal – as captured in the claim that the solution to bad speech is more speech – and partly from a recognition that well-meaning attempts to counteract harm through speech can easily misfire or backfire. Here we survey recent work on the question of what makes counterspeech effective at remedying or preventing harm, in those cases where (...) it is effective, as well as work investigating when and why there is a duty to engage in counterspeech. We suggest that the most fruitful area for philosophical inquiry on this topic, currently, relates to the questions about efficacy. Specifically, we argue that there is a need for better frameworks for conceptualizing the efficacy of counterspeech. Philosophers have collaborative work to do, alongside social scientists, in developing these frameworks. (shrink)

We investigate the cofinality of the partial order N κ of functions from a regular cardinal κ into the ideal N of Lebesgue measure zero subsets of R. We show that when add(N) = κ and the covering lemma holds with respect to an inner model of GCH, then cf(N κ ) = max {cf(κ κ ), cf([ cf(N)] κ )}. We also give an example to show that the covering assumption cannot be removed.

The essay contrasts two recurring phenomena of European culture: renaissance and avant-garde. The author discusses the paradigmatic Renaissance of 15th and 16th centuries and the paradigmatic Avant-Garde of early 20th century from the point of view of a practicing artist, interested in philosophical, social, religious, and political involvements of artists and their creation. The author shows the artistic and social history of 20th century as a struggle between the Avant-Garde and the Renaissance ideals, which, as he points out, found a (...) fertile ground in in the 20 years that followed immediately the Second World War. (shrink)

The essay contrasts two recurring phenomena of European culture: renaissance and avant-garde. The author discusses the paradigmatic Renaissance of 15th and 16th centuries and the paradigmatic Avant-Garde of early 20th century from the point of view of a practicing artist, interested in philosophical, social, religious, and political involvements of artists and their creation. The author shows the artistic and social history of 20th century as a struggle between the Avant-Garde and the Renaissance ideals, which, as he points out, found a (...) fertile ground in in the 20 years that followed immediately the Second World War. (shrink)

We give a proof of Hechler's theorem that any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\aleph_1$\end{document}-directed partial order can be embedded via a ccc forcing notion cofinally into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega^\omega$\end{document} ordered by eventual dominance. The proof relies on the standard forcing relation rather than the variant introduced by Hechler.

This paper assesses James M. Buchanan's claim of following a positive approach in stark contrast to the normative approach to public finance of Richard A. Musgrave. The goal of this paper is to shed light on the foundations of modern American public finance by analysing one aspect of the methodology of its two most prominent fathers. I show (1) that it is difficult to distinguish Musgrave's and Buchanan's theories of public goods along the positive/normative dividing line and (2) that Buchanan's (...) theory can also be considered normative. In the first three parts, I follow the Weberian methodological tradition in looking for value judgements in the theories, and by reflecting on the nature of ideal types. In the last part, I propose a broader interpretation of Buchanan's methodological stance within the academic context of the 1960s. (shrink)

On doit à l’économiste américain d’origine allemande, Richard A. Musgrave, la formulation des critères d’Équité Horizontale (ÉH) et d’Équité Verticale (ÉV). Étant donnée une base fiscale, l’ÉH veut que les égaux soient traités également, alors que l’ÉV exige que les inégaux soient traités inégalement (par exemple que les plus riches payent proportionnellement plus d’impôts). Musgrave opère en 1959 une redescription créative de considérations d’équité qui ont une longue histoire de Hobbes à Buchanan, en passant notamment par la fameuse maxime de (...) Adam Smith. La reconstruction généalogique montre que l’ÉV et l’ÉH relèvent de la justice distributive au sens aristotélicien du terme. Un critère purement formel, l’équité en matière d’impôt signifie l’égalité proportionnelle dans la répartition des charges communes à tous les individus membres d’une communauté politique. À défaut d’un consensus théorique sur la manière de conceptualiser les contribuables, l’ÉH conserve tout de même une valeur de principe régulateur dans le cadre non idéal de débats de réforme fiscale.Codes JEL : H20, B11, B12, B20, B40, D63. (shrink)

In the present note we give a direct deduction of the Axiom of Choice from the Maximal Ideal Theorem for commutative rings with unit.

We generalize the Bolzano-Weierstrass theorem on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.

It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal (...) of nonstationary subsets ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+ −I), then anI-partitionofAis a maximal collectionW⊆,P ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y.. (shrink)

We obtain an improvement of some coloring theorems from Eisworth :1216–1243, 2010), Eisworth and Shelah :1287–1309, 2009) for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an “idealized” version of the combinatorial principle Pr1) that maximizes the indecomposability of the associated ideal.

We look at a hierarchical arrangement of ideals in an MV -algebra. The principal classes of ideals studied are the maximals, the primes, the local and perfect ideals and the semi-locals. Beyond these special classes of ideals are the general ideals. Herein we study some relationships among these classes and, more specifically, the products of ideals of these classes. Among the results obtained are the square of a prime ideal is a local ideal, the finite product of prime (...) ideals is a semilocal ideal. We also show that the set of local ideals shares many of the properties of the class of prime ideals, as the Lying Over Theorem and the Going Up Theorem. (shrink)

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2math image and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.

In 1996, Krivine applied Friedman's A-translation in order to get an intuitionistic version of Gödel completeness result for first-order classical logic and countable languages and models. Such a result is known to be intuitionistically underivable 559), but Krivine was able to derive intuitionistically a weak form of it, namely, he proved that every consistent classical theory has a model. In this paper, we want to analyze the ideas Krivine's remarkable result relies on, ideas which where somehow hidden by the heavy (...) formal machinery used in the original proof. We show that the ideas in Krivine's proof can be used to intuitionistically derive some crucial mathematical results, which were supposed to be purely classical up to now: the Ultrafilter Theorem for countable Boolean algebras, and the maximal ideal theorem for countable rings. (shrink)

It is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel-Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a `generic' extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in (...) the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure. (shrink)

Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion (...) Scott-style entailment relation over a single-conclusion one. (shrink)

We establish that, in ZF, the statementRLT: Given a setIand a non-empty setF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}of non-empty elementary closed subsets of 2Isatisfying the fip, ifF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}has a choice function, then⋂F≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcap\mathcal{F} \ne \emptyset}$$\end{document},which was introduced in Morillon :739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem. The result provides, on one hand, an affirmative answer to (...) Morillon’s corresponding question in Morillon and, on the other hand, a negative answer—in the setting of ZFA —to the question in Morillon of whether RLT is equivalent to Rado’s selection lemma. (shrink)

We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)+ and (rTT)++; each is equivalent to (rTT) in ZF. The variation (rTT)++ applies (...) rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado. (shrink)

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0.

In this paper, we study some new examples of ideals on $$\omega $$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal (...) above all other analytic P-ideals in the weak Rudin–Keisler order. (shrink)

We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch :269–282, 2017).

We prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal J (coded in the ground model) a cofinal subset of (J, ⊆*) is order isomorphic to (Q, ≤).

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that (...) this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory. (shrink)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) should reject these claims, and lay the foundations of decision theory on firmer ground. (shrink)

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the (...) proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity. (shrink)

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

The notion that natural selection is a process of fitness maximization gets a bad press in population genetics, yet in other areas of biology the view that organisms behave as if attempting to maximize their fitness remains widespread. Here I critically appraise the prospects for reconciliation. I first distinguish four varieties of fitness maximization. I then examine two recent developments that may appear to vindicate at least one of these varieties. The first is the ‘new’ interpretation of Fisher's fundamental (...) class='Hi'>theorem of natural selection, on which the theorem is exactly true for any evolving population that satisfies some minimal assumptions. The second is the Formal Darwinism project, which forges links between gene frequency change and optimal strategy choice. In both cases, I argue that the results fail to establish a biologically significant maximization principle. I conclude that it may be a mistake to look for universal maximization principles justified by theory alone. A more promising approach may be to find maximization principles that apply conditionally and to show that the conditions were satisfied in the evolution of particular traits. (shrink)