According to act theories, propositions are structured cognitive act‐types. Act theories appear to make propositions inherently representational and truth‐evaluable, and to provide solutions to familiar problems with alternative theories, including Frege’s and Russell’s problems, and the third‐realm and unity problems. Act theories have critical problems of their own, though: acts as opposed to their objects are not truth evaluable, not structured in the right way, not expressed by sentences, and not the objects of propositional attitudes. I show how identifying propositions (...) with other cognitive event‐types, namely thoughts, has the perceived virtues of act theories without the defects. (shrink)

Cet article s'ajoute à la liste déjà longue de ceux qui traitent des rapports entre la théorie russellienne des descriptions définies et la théorie ramifiée des types. Seule la prétention d'aborder cette question dans une perspective nouvelle justifie ici sa présence: d'une part, la théorie des descriptions définies sera resituée dans le contexte initial et souvent méconnu de la théorie des expressions dénotantes et d'autre part, c'est à la logique intensionnelle de Russell, objet d'une méconnaissance au moins égale, que nous (...) nous intéresserons. De plus, c'est du point de vue de la théorie de la signification et non pas du point de vue ontologique, génético-historique ou logico-mathematique que nous nous proposons de traiter de ces rapports. (shrink)

It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...) hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)

There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...) is hard to see how one might argue, on epistemological grounds, for moral antirealism while maintaining commitment to mathematical realism. But it may be possible to do the opposite. (shrink)

We cannot know something unless it is true. The things that we know, therefore, must be logically consistent. Moreover, we cannot know something unless we are justified in believing it. But it does not obviously follow that the things that we are justified in believing must be consistent with each other. For we can be justified in believing something that turns out to be false. Knowledge entails truth and hence consistency. Rationally justified belief does not entail truth and it may (...) not entail consistency.Knowledge, however, requires especially good justification. We can have reason to believe something that happens to be true, even good reason, without our belief being so well-grounded that we know it to be true. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

I consider here several versions of finitism or conceptions that try to work around postulating sets of infinite size. Restricting oneself to the so-called potential infinite seems to rest either on temporal readings of infinity (or infinite series) or on anti-realistic background assumptions. Both these motivations may be considered problematic. Quine’s virtual set theory points out where strong assumptions of infinity enter into number theory, but is implicitly committed to infinity anyway. The approaches centring on the indefinitely large and the (...) use of schemata would provide a work-around to circumvent usage of actual infinities if we had a clear understanding of how schemata work and where to draw the conceptual line between the indefinitely large and the infinite. Neither of this seems to be clear enough. Versions of strict finitism in contrast provide a clear picture of a (realistic) finite number theory. One can recapture standard arithmetic without being committed to actual infinities. The major problem of them is their usage of a paraconsistent logic with an accompanying theory of inconsistent objects. If we are, however, already using a paraconsistent approach for other reasons (in semantics, epistemology or set theory), we get finitism for free. This strengthens the case for paraconsistency. (shrink)

L'informatique se définissant comme le traitement rationnel de l'information par machine automatique et l'intelligence se caractérisant par une même capacité de traitement rationnel, il était inévitable que l'on songe à associer l'intelligence au traitement automatique de l'information. C'est ce qu'a fait John McCarthy en forgeant le terme d'intelligence artificielle. Par «intelligence artificielle» on peut vouloir exprimer l'ambition de1. Recréer, transformer ou développer l'intelligence artificiellement2. Simuler l'intelligence en la reconstituant dans des modéles imitant certains aspects de notre intelligence dite naturelle.

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...) equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

Quine’s “New Foundations” (NF) was first presented in Quine [1937] and later on in Quine [1963]. Ernst Specker [1958, 1962], building upon a previous result of Ehrenfeucht and Mostowski [1956], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of a such a model remains an open problem.

The ecological approach to perception-action is unlike the standard approach in several respects. It takes the animal-in-its-environment as the proper scale for the theory and analysis of perception-action, it eschews symbol based accounts of perception-action, it promotes self-organization as the theory-constitutive metaphor for perception-action, and it employs self-referring, non-predicative definitions in explaining perception-action. The present article details the complexity issues confronted by the ecological approach in terms suggested by Rosen and introduces non-well-founded set theory as a potentially useful tool for (...) expressing them. The issues and the tool are brought to focus in the concept of affordance that is the basis for explanation of prospective control of action in the ecological approach. (shrink)

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

During the past few decades, a radical shift has occurred in how philosophers conceive of the relation between science and philosophy. A great number of analytic philosophers have adopted what is commonly called a ‘naturalistic’ approach, arguing that their inquiries ought to be in some sense continuous with science. Where early analytic philosophers often relied on a sharp distinction between science and philosophy—the former an empirical discipline concerned with fact, the latter an a priori discipline concerned with meaning—philosophers today largely (...) follow Willard Van Orman Quine (1908-2000) in his seminal rejection of this distinction. -/- This book offers a comprehensive study of Quine’s naturalism. Building on Quine’s published corpus as well as thousands of unpublished letters, notes, lectures, papers, proposals, and annotations from the Quine archives, this book aims to reconstruct both the nature (chapters 2-4) and the development (chapter 5-7) of his naturalism. As such, this book aims to contribute to the rapidly developing historiography of analytic philosophy, and to provide a better, historically informed, understanding of what is philosophically at stake in the contemporary naturalistic turn. (shrink)

I construe the question Are species sets? as a question about whether species can be conceived of as sets, as the term set is understood by contemporary logicians. The question is distinct from the question Are species classes?: The conception of classes invoked by Hull and others differs from the logician's conception of a set. I argue that species can be conceived of as sets, insofar as one could identify a set with any given species and that identification would satisfy (...) three desiderata: the set would be a set of organisms, the identification would be apposite, and the identification would permit the formulation of statements about species in set-theoretic terms. One cannot, however, identify a species with any given set. Understanding the claim that species are sets in this way enables one to understand better the dispute between some who accept the claim (e.g., Kitcher 1984, 1987) and some who apparently reject it (e.g., Mayr 1987). (shrink)

The following considerations are to be seen in the framework of nontraditional theory of predication (NTP), which stems in its basic features from Sinowjew (cf. Sinowjew 1970, Sinowjew/Wessel 1975), and which is described in detail in Wessel 1989.

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S...

We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink the commonly (...) accepted methods for dealing with the logical paradoxes. (shrink)

In , Frank Ramsey separates paradoxes into two groups, now taken to be the logical and the semantical. But he also revises the logical system developed in Whitehead and Russellthe intensional paradoxess interest in these problems seriously, then the intensional paradoxes deserve more widespread attention than they have historically received.

It’s generally taken to be established that no local hidden-variable theory is possible. That conclusion applies if our world is a _thread_, where a thread is a world where particles follow trajectories, as in Pilot-Wave theory. But if our world is taken to be a _set_ of threads locality can be recovered. Our world can be described by a _many-threads_ theory, as defined by Jeffrey Barrett in the opening quote. Particles don’t follow trajectories because a particle in our world is (...) a set of _elemental_ particles following different trajectories, each in a thread. The “elements” of a superposition are construed as subsets in such a way that a particle in our world only has definite position if all its set-theoretic elements are at corresponding positions in each thread. Wavefunction becomes a 3D density distribution of particles’ subset measures, the stuff of an electron’s “probability cloud”. Current Pilot-Wave theory provides a non-relativistic dynamics for the elemental particles (approximated by Many Interacting Worlds theory). EPR-Bell nonlocality doesn’t apply because the relevant measurement outcomes in the absolute elsewhere of an observer are always in superposition. (shrink)

Objective probability in quantum mechanics is often thought to involve a stochastic process whereby an actual future is selected from a range of possibilities. Everett’s seminal idea is that all possible definite futures on the pointer basis exist as components of a macroscopic linear superposition. I demonstrate that these two conceptions of what is involved in quantum processes are linked via two alternative interpretations of the mind-body relation. This leads to a fission, rather than divergence, interpretation of Everettian theory and (...) to a novel explanation of why a principle of indifference does not apply to self-location uncertainty for a post-measurement, pre-observation subject, just as Sebens and Carroll claim. Their Epistemic Separability Principle is shown to arise out of this explanation and the derivation of the Born rule for Everettian theory is thereby put on a firmer footing. (shrink)

The contemporary popularity of semantic externalism has arisen from so-called Twin Earth thought experiments which suggest that the representational content of a natural kind term cannot be wholly determined by processes within a speaker's body. Such arguments depend on the intuition that the extensions of natural kind terms cannot have changed as the result of the scientific investigation of natural kinds' constitutions. I demonstrate that this externalist intuition depends on an assumption about the mentality of isomorphic doppelgangers which has never (...) been questioned but which is nonetheless arguably false. I develop an alternative view of the instantiation of mind which entails a revision of our understanding of the constitution of environmental objects. The picture seems to be fully coherent despite its oddity and I can find no good reason to reject it. The conclusion must be that the case for semantic externalism is thus less compelling than is often supposed. (shrink)

The mereological predicate ‘is part of’ can be used to define the predicate ‘is identical with’. I argue that this entails that mereological theories can be ideologically simpler than nihilistic theories that do not use the notion of parthood—contrary to what has been argued by Ted Sider. Moreover, if one accepts an extensional mereology, there are good philosophical reasons apart from ideological simplicity to give a mereological definition of identity.

The semantic rules that govern ordinary property discourse appear to give rise to a version of Russell's antinomy. Do we therefore have an inconsistent conception of properties? This paper firstly develops a consistent conception of properties and secondly argues that we may indeed interpret ordinary property discourse as expressing the consistent conception rather than an inconsistent one.

Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

Formal semantics is an approach to SEMANTICS1, the study of meaning, with roots in logic, the philosophy of language, and linguistics, and since the 1980’s a core area of linguistic theory. Characteristics of formal semantics to be treated in this article include the following: Formal semanticists treat meaning as mind-independent (though abstract), contrasting with the view of meanings as concepts “in the head” (see I-LANGUAGE AND E-LANGUAGE and MEANING EXTERNALISM AND INTERNALISM); formal semanticists distinguish semantics from knowledge of semantics (Lewis (...) 1975, Cresswell 1978), which has consequences for the notion of semantic COMPETENCE. A central part of the meaning of a sentence on this approach is its TRUTH CONDITIONS, and most although not all formal semantics is model-theoretic, relating linguistic expressions to model-theoretically constructed semantic values cast in terms of truth, REFERENCE, and possible worlds. This sets formal semantics apart from approaches which view semantics as relating a sentence just to a representation on another linguistic “level” (LOGICAL FORM) or a representation in an innate LANGUAGE OF THOUGHT. The formal semanticist could accept such representations as an aspect of semantics but would insist on asking what the model-theoretic semantic interpretation of the given representationlanguage is (Lewis 1970). Formal semantics is centrally concerned with COMPOSITIONALITY at the SYNTAX-SEMANTICS INTERFACE, how the meanings of larger constituents are built up from the meanings of their parts on the basis of their syntactic structure, and with the relation between compositional SENTENCE MEANING and meaning in discourse. (shrink)

Putting individual constants and quantifiers into the same syntactic category within first-order language promises to have far-reaching consequences: a syntax of this kind can reveal the potential of any such language, allowing us to realize that a vast class of noun phrases, including non-denoting terms, can be accommodated in the new syntax as expressions suited to being subjects of sentences. In the light of this, a formal system that is an extension of classical first-order logic is developed here, and is (...) equipped with an appropriate semantics. An ontological interpretation of the new logic is then also provided, with several categories of object and notions of existence being distinguished. Last but not least, a modal version of the logic, with some interesting formal features, is proposed. (shrink)

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘4’ or the descriptive ‘the inhabitants of London’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously (...) if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. (shrink)

In his “Epistemology Naturalized,” Quine embraces a form of Humeanism. In this paper, I try to work out the significance of this Humeanism. In particular, I argue that it represents an anti-metaphysical position that Quine shares with Carnap. Crucial to my account is that contrary to much contemporary thinking on metaphysics, Carnap, and Quine following him, recognize both an ontological and an epistemological sense of metaphysics. As commentators have frequently acknowledged, Carnap and Quine disagree over rejecting metaphysics in the ontological (...) sense. I will argue, though, that they agree in rejecting metaphysics in their shared epistemological sense of the term. (shrink)

I consider the well-known criticism of Quine's characterization of first-order logical truth that it expands the class of logical truths beyond what is sanctioned by the model-theoretic account. Briefly, I argue that at best the criticism is shallow and can be answered with slight alterations in Quine's account. At worse the criticism is defective because, in part, it is based on a misrepresentation of Quine. This serves not only to clarify Quine's position, but also to crystallize what is and what (...) is not at issue in choosing the model-theoretic account of first-order logical truth over one in terms of substitutions. I conclude by highlighting the need for justifying the belief that the definition of first-order logical truth in terms of models is superior to its definition in terms of substitutions. (shrink)

La relation d'identité est-elle univoque, ou est-elle au contraire relative au concept sous lequel le jugement est énoncé? Depuis quelques années, parmi les philosophes anglo-américains, partisans et adversaires de la thèse que l'on a appelée « la relativité de l'identité », se sont affrontés sur ce point.

I compare Russell’s theory of mathematical functions, the “descriptive functions” from Principia Mathematica ∗30, with Frege’s well known account of functions as “unsaturated” entities. Russell analyses functional terms with propositional functions and the theory of definite descriptions. This is the primary technical role of the theory of descriptions in P M . In Principles of Mathematics and some unpublished writings from before 1905, Russell offered explicit criticisms of Frege’s account of functions. Consequenly, the theory of descriptions in “On Denoting” can (...) be seen as a crucial part of Russell’s larger logicist reduction of mathematics,aswellasanexcursionintothetheoryof reference.
. (shrink)

The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern (...) setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see [3]. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)