The role and precise significance of Intelligible Matter in the philosophy of Plotinus has been neglected or dismissed with many questions unanswered. In view of the fact that, unless this role can be properly understood, the whole doctrine of the procession of the Second Hypostasis must remain mysterious, this paper is intended to shed light on two important aspects of that Hypostasis: the nature of Intelligible Matter itself and the relation of that Matter to the Forms.

The following hopes to bring Plato’s unwritten doctrines into proximity with Irigaray’s concept of the Two as found in works like To Be Two or I love to you. By focusing on the the indefinite Dyad, Plato's reported co-archai with the One, it will be evidenced that Platonism begins and ends with a One which is not One (a kind of Two). Further, in this Dyad's failure to be One, it ultimately comes to possess its own productive (...) and destructive power or, to use Irigaray's words, to love toward the Other in such a way as to produce or give birth to other others, real difference. Finally, Irigaray’s conception of the Two has dramatic social implications for her as a feminist, grounding a transformation of the political order of things. Due to this political import, the final section shifts to questioning whether Plato’s reported protological principles, the One and the indefinite Dyad, could perform a similar work. (shrink)

The aim of this paper is to show how some passages in the treatise VI 6 of the Enneads could be considered a Plotinian attempt to achieve a sort of mediation between Plato’s dialogues and the Academic models concerning the generation of numbers and ideas by the two principles of the One and the Indefinite Dyad. Through a complex exegetical contamination, Plotinus gets to identify numbers, ideas and the Dyad with the hypostasis of the Intellect, saving for (...) the One the condition of the sole principle of every reality. This new reading modifies the ontological role played by ideal numbers : they gain an ontological priority on any other being, thanks to their being conceived as the a priori condition of the display of multiplicity. The so-called Intermediates, on the other hand, have a reshaped status, as a result of the fact that calculation is unable to grasp the essence of a given reality. (shrink)

In this essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the (...) One and the Indefinite Dyad provides, in the later Plato, a unitary theoretical formalism accounting, by means of an iterated mixing without synthesis, for the structural origin and genesis of both supersensible Ideas and the sensible particulars which participate in them. As these commentators also argue, this duality furthermore provides a maximally general answer to the problem of temporal becoming that runs through Plato’s corpus: that of the relationship of the flux of sensory experiences to the fixity and order of what is thinkable in itself. Additionally, it provides a basis for understanding some of the famously puzzling claims about forms, numbers, and the principled genesis of both attributed to Plato by Aristotle in the Metaphysics, and plausibly underlies the late Plato’s deep considerations of the structural paradoxes of temporal change and becoming in the Parmenides, the Sophist, and the Philebus. After extracting this structure of duality and developing some of its formal, ontological, and metalogical features, we consider some of its specific implications for a thinking of time and ideality that follows Deleuze in a formally unitary genetic understanding of structural difference. These implications of Plato’s duality include not only those of the constitution of specific theoretical domains and problematics, but also implicate the reflexive problematic of the ideal determinants of the form of a unitary theory as such. We argue that the consequences of the underlying duality on the level of content are ultimately such as to raise, on the level of form, the broader reflexive problem of the basis for its own formal or meta-theoretical employment. We conclude by arguing for the decisive and substantive presence of a proper “Platonism” of the Idea in Deleuze, and weighing the potential for a substantive recuperation of Plato’s duality in the context of a dialectical affirmation of what Deleuze recognizes as the “only” ontological proposition that has ever been uttered. This is the proposition of the univocity of Being, whereby “being is said in the same sense, everywhere and always,” but is said (both problematically and decisively) of difference itself. (shrink)

This chapter analyzes the relationship between number and substance in the intelligible realm. Plotinus formulates three hypotheses about the existence of number in the intelligible: 1) number is posterior to the Forms; 2) number is simultaneous with the Forms; and 3) number is anterior to the Forms. He proves that the last hypothesis is true. Based on the distinction between substance as ontological actualization of beings and quantity as the countability of individual units, he defines two kinds of number: Substantial (...) Number, which is the activity of substance and a power of being; and Monadic Number, which enumerates individual things. As an activity of substance, the former enacts the limiting role of the Monad, while, as a power of substance, it enacts the role of the Indefinite Dyad as potentiality, limited into existence by the Monad. While substantial number actualizes the existence of that which has separated from the One in the intelligible, monadic number expresses quantitatively that which has been already defined by substantial number. Plotinus also pioneers the view of the substantial number as henad and, in turn, shapes the development of later Neoplatonic theory of number. (shrink)

An article by John Dillon ardues that the Platonism that Plotinus inherits – setting aside Ammonius Saccas, of whom we know all too little – is by the later second century distinctly dualist in tendency, and is able, especially in the case of Plutarch, to quote Plato to its purpose. Plato himself, though, as the author maintains, is, despite appearances to the contrary, what one might term a ‘modified monist’. That is to say, he fully recognizes the degree of imperfection (...) and evil in the world, and holds it to be ineradicable, but he does not in the last resort believe in a positive countervailing force to the Good or the One. What we have is simply a negative force, whether Indefinite Dyad, disorderly World-Soul, or Receptacle, which is an inevitable condition of their being a world at all, but which, as a side-effect of introducing diversity, generates various sorts of imperfection. It is this scenario that justifies his follower Hermodorus in declaring that Plato recognizes only a single first principle, and it to this sort of monism – if anything, in a more pronounced form – that Plotinus returns. (shrink)

In his Commentary on the Timaeus Proclus says that in some occasions Plato speaks of a model (from which the world is created) that is identical to the Demiurge while in other occasions he suggests that the model is distinct from the Demiurge. Here, identity and difference refer to the similarity with or dissimilarity from the intelligible One, identified with eternity (stability; fixedness). However, Plato also speaks in the Timaeus that the Cosmos is pretty and its Constructor (the Demiurge) is (...) good inasmuch as He fixed his sight in the perpetual model. If we ask what is produced when the Demiurge fixed his sight in a model that it is not identical to Himself, we find the reply in the Timaeus: envy, defects, disorder, limitation regarding intelligence and the soul, incompleteness, ugliness, and imperfection. Plotinus, on his turn, mentions (Enneads, II.4) the existence of two types of matter, intelligible and sensible. The intelligible or divine matter is equivalent to the "indefinite Dyad", identified with the procession of the Noûs. For Plotinus, though the intelligible matter is eternal and invariant, its archê is constituted by difference and movement - conditions that prepare the naturation of sensible matter. For Proclus, the Soul, an unfolding of the Noûs, mediates between the Intellect, intelligible matter and sensible matter. Many interpreters seem to have understood inteligible matter as the matter of the Soul, and to explain the interaction between the intelligible and the sensible many theories have appeared regarding the constitution of intelligible matter and about the nature of the "vehicle" or "vehicles" that would be associated with the Soul's "descent" and incarnation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:A Less Familiar Plato: From Phaedo to Philebus by Kevin CorriganKristian SheeleyCORRIGAN, Kevin. A Less Familiar Plato: From Phaedo to Philebus. New York: Cambridge University Press, 2023. xi + 306 pp. Cloth, $110.00Corrigan makes a substantial contribution to the body of Plato scholarship that offers rigorous and textually supported corrections to [End Page 711] superficial (yet all too common) readings of Plato’s dialogues. The book covers a range (...) of core Platonic topics such as embodiment and selfhood, participation in the divine, the critique of artworks in the Republic, the form of the good, love, friendship, desire, and the relationship between mythos and logos. Corrigan devotes his primary attention to the so-called middle dialogues and the Philebus, but to varying degrees he engages with many more Platonic dialogues as well as other philosophers in the Platonic tradition.Chapter 1 combats common interpretations of the Phaedo claiming that Plato wholly denigrates the body and idealizes the state of disembodiment that can only be fully reached in death. Corrigan’s opponents claim that Plato’s negative comments about the body encourage an unhealthy attitude toward the body and the emotions. Corrigan responds by thoroughly showing how the passages about body–soul separation, when read in view of the dialogue as a whole, instead reveal a “model of positive embodiment” that is “full of conscientious protest, good hopes, thoughts, arguments, refutations, and friendship.” Next, Corrigan turns his focus to passages from both Plato and Aristotle that shed light on classic perplexities surrounding the key concept of participation. He argues that “participation” is not the best option for translating methexis or its related terms, and he suggests “having of” or “sharing” as alternative translations. If they are correctly understood, Plato’s discussions of the forms and their relationship to particulars are not vulnerable to criticisms concerning infinite regress or self-predication issues according to Corrigan’s analysis. Chapters 3 and 4 confront Plato’s well-known criticisms of artworks in the Republic and beyond. Here Corrigan makes a highly intriguing point about the role of what he calls “the intelligible painter” in Republic 6 and 7, through which Socrates both sketches and exemplifies “positive philosophical art” as an alternative to the traditional art he criticizes in books 2, 3, and 10. The intelligible painter uses the good as his “perfect and sufficient model” to cultivate himself and others through education, just as Socrates fashions the images of the sun-good, the cave, and other images for the sake of benefitting his interlocutors.In chapter 5 Corrigan draws inspiration from Plotinus’s nuanced understanding of the famous statement in the Republic that the form of the good is “beyond being.” According to the view Corrigan works out, the form of the good is the ground of all intelligibility, and so its relationship to everything that is intelligible can be understood, but it is not in itself intelligible. Chapter 6 takes a close look at the Philebus in light of the Republic’s discussions of the forms and dialectic, the One and the indefinite Dyad of the “unwritten doctrines,” and the work of Speusippus and Xenocrates. Corrigan then shows in chapter 7 how Plato reveals the wide variety of ways in which myths can be used, ranging from the misleading tales of unwise poets to Plato’s own use of stories to aid philosophical logos. Finally, chapters 8 and 9 focus on Plato’s dialogues concerning love and friendship: Symposium, Phaedrus, Lysis, and First Alcibiades, which Corrigan regards as authentic. Corrigan’s main [End Page 712] objective is to show that these dialogues collectively provide a “guide for the perplexed” in matters of love that, contrary to modern criticisms of Plato, is both sensitive and relevant to real, complex, and concrete relationships between humans.As mentioned above, Corrigan sometimes uses terminology and conceptual resources from authors throughout the various ancient and medieval Platonic traditions, including Aristotle and his inheritors. Corrigan’s interpretive method often aims to show that the dialogues harmonize with one another as well as with Plato’s “unwritten teachings,” and he argues that some Neoplatonic interpretations of Plato are correct. This interpretive approach might be seen as... (shrink)

This chapter analyzes Plotinus’ refutation of the Aristotle’s criticism of Plato’s view of number in the Parmenides. By rejecting any quantitative value of number in the intelligible realm, Plotinus specifically focuses on Aristotle’s inability to understand the Monad and the Indefinite Dyad as the principles of creation and order of the intelligible. The chapter shows that Plotinus not only follows the steps of his Platonic and Neopythagorean predecessors in defence of Plato’s position, but cleverly uses Aristotle’s own ideas (...) in arguing that number in the intelligible is activity and a property of primary substance. The result is an original and ontologically elaborate theory of substantial number which offers a new and more successful defence of Plato’s “true numbers” against Aristotle’s criticism and explains the relationship between substantial non-quantitative number and monadic quantitative number as that between intelligible paradigm and its material copy. (shrink)

The argument of this dissertation is that despite the intellectual gendered burden of the problem of disembodiment I define, it can be employed from within the limitations of a gendered account in feminist philosophy of the continental-realist type. I formulate the problem of disembodiment as rooted in the notion of the boundless (apeiron) associated with femininity. Both boundlessness and disembodiment are subject to radicalization in Plato (chōra) and Plotinus (to hen). Read as a dyad, they culminate in a tendency (...) towards gendered disembodiment, mediated by Plato’s soul-body dualism. The dissertation seeks to compare the gendered dimension of disembodiment in the work of Plato and Plotinus and that of the non- philosophers François Laruelle and Katerina Kolozova. “Part I. The Problem of Boundlessness: Radicalizing Disembodiment” is divided in three chapters, which present an intellectual history of the problem of boundlessness as femininity. I survey the problem of boundlessness as drafting relations between elements and principles and femininity in Greek mythology (Chapter 1), Plato’s cosmology (Chapter 2), and Plotinus’ metaphysics (Chapter 3). I argue that the relation between death and the female was ambivalent by the time of the Anaximandrean apeiron and that it became a subject of radicalization via Plato’s chōra and Plotinus’ One, mediated by the notion of the Indefinite Dyad and the doctrine of divided matter. The problem of boundlessness was subject to conceptual radicalization that led to hierarchical metaphysics and deepened the division between body and soul via the association of femininity, reproductivity and matter. “Part II. The Problem of Disembodiment: Revising Boundlessness” is divided in two chapters focusing on the contemporary relevance and importance of the problem of disembodiment as a way of revising boundlessness. I present and explain the legacy of the Platonic chōra and the Plotinian One and what they as a dyad entail for contemporary continental philosophy. I offer (Chapter 4) a trajectory for a continental feminist philosophy interpretation of disembodiment by combining continental feminist philosophy, non- philosophy and new realism. With the aid of Laruelle’s non-philosophy, I explain how and why chōra and the One can be used for/from continental feminist philosophy, followed by a presentation of how chōra and the One are revised in continental philosophy from non- philosophical and new realist perspectives. I then develop (Chapter 5) a continental feminist philosophical interpretation of and approach to the problem of disembodiment from a realist perspective by problematizing continental and feminist philosophical anti-realism. The approach presented is itself an argument in defence of a feminist engagement with disembodiment and the dissertation’s contribution: a non-philosophical contribution to the problem of disembodiment via a continental feminist-realist philosophical approach. The approach is offered through the intersection of continental feminist realism and non- philosophy, and partially new realism. My conclusion is that an affirmative project and consideration of disembodiment for continental feminist philosophy is possible via a non- philosophical and new realist reconsideration of the One. (shrink)

The Treatise On Numbers develops a typical Plotinian inquiry about oneness and intelligible plurality, in which the analysis is based on the problem of the infinite inside the intelligible realm, as it seems to be presented in the socalled generation of numbers in Plato's Parmenides, where numbers are linked to Being, identified by Plotinus with the hypostasis of Intellect, and produce a process of unlimited division. In order to avoid this interpretation, Plotinus finds in other Plato's dialogues the proof that (...) number is an eidetic nature, encompassing all the intelligible entities, so that the infinite has to be interpreted as what is produced by calculus. Other arguments are rather depending on the model of the Old Academy, concerning the relationship between forms and numbers: Plotinus wonders if numbers come ontologically before forms and indeed he states right this position. Other aspects of Plotinus' investigation, which can be traced back to the Old Academy, are the distinction between two kinds of numbers and the doctrine of the unified number. This model of number, quite ambiguous, may be identified with the indefinite dyad, even though the former has to be conceived, differently from that of Old Academy, within a mono-polar framework, according to which the sole principle is the One. (shrink)

The following paper argues that Blaise Pascal, in spite of his famous opposition between the God of the Philosophers and the God of “Abraham, Isaac and Jacob” has significant affinities with the tradition of Renaissance Platonism and is in fact a Platonist in his overall outlook. This is shown in three ways. Firstly, it is argued that Pascal’s skeptical fideism has roots in the notion of faith developed in post-Plotinian neo-Platonism. Secondly, it is argued that Pascal makes considerable use of (...) the Platonic notion of an indefinite dyadic principle. Thirdly, it is argued that Pascal’s religious psychology gives a centrality to the body that brings it close to the theurgical standpoint of figures like Iamblichus. Pascal is then contrasted to figures like Cusanus and Pico in that a dyadic principle of opposition is more prominent in his work than a triadic logic of mediation. (shrink)

In a well-known passage Aristotle ascribes to Plato, or as some think to his followers, the dictum, γρ ριθμóς στιν κ νòς κα τς ορίστον , ‘Number is from the one and the undetermined dyad ’, but what this apparently simple statement means has remained a mystery until modern times. In other passages Aristotle expands it to explain that the indefinite duality is a duality of the great and small, e.g., ς μν ον λην τò μγα κα τò (...) μικρòν εναι ρχς, ς δ' οσίαν τò ν. ξ κείνων γρ κατ μθεξιν το νòς εναι τος ριθμος . ‘As the matter he posits the great and small for principles, as substance the one; for by the mixture of the one with them he says numbers .’ This identification of the dyad with the great and small, elsewhere called τò νισον and τò πειρον , gives a first clue to its nature. In a notable article in Mind 35 , 419–40, continued in vol. 36, , 12–33, and amplified by D' Arcy Wentworth Thompson in vol. 38 , 43–55, A. E. Taylor first suggested a connexion between the indefinite duality and the modern theory of continued fractions. In the light of subsequent research in the history of Greek mathematics it may now be asserted with a high degree of confidence that his conjecture was almost certainly correct; but it was then no more than a conjecture, and when he looked for confirmation he looked in the wrong direction. (shrink)

Across languages, we find indefinites that trigger modal inferences. This article contributes to a semantic typology of these items by contrasting Spanish algún with indefinites like German irgendein or Italian uno qualsiasi. While irgendein-type indefinites trigger a Free Choice effect (Kratzer and Shimoyama 2002; Chierchia 2006), algún simply signals that at least two individuals in its domain are possibilities. Additionally, algún, but not irgendein, can convey that the speaker does not know how many individuals satisfy the existential claim in the (...) world of evaluation. We contend that the two types of indefinites impose different constraints on their domain of quantification: irgendein and its kin are domain wideners (Kratzer and Shimoyama 2002), whereas algún is an ‘anti-singleton’ indefinite (its domain cannot be restricted to a singleton). This, together with the fact that algún does not require uniqueness, allows us to derive the contrast between irgendein and algún by using the pragmatic reasoning presented by Kratzer and Shimoyama. (shrink)

In this article, I consider Descartes’ enigmatic claim that we must assert that the material world is indefinite rather than infinite. The focus here is on the discussion of this claim in Descartes’ late correspondence with More. One puzzle that emerges from this correspondence is that Descartes insists to More that we are not in a position to deny the indefinite universe has limits, while at the same time indicating that we conceive a contradiction in the notion that (...) the universe has a limit. I reject one attempt to resolve this apparent conflict which appeals to Descartes’ admission to More that divine omnipotence requires that God can create a vacuum in nature, and focus instead on his response to More’s claim that this omnipotence requires the possibility of a completion of the division of matter that results in atoms. Finally, I distinguish Descartes’ indefinite from two other kinds of infinity with which it might well be confused. The first is an essentially incomplete ‘potential infinity’ that Descartes discusses in the Third Meditation, and the second is the sort of quantitative infinity that Leibniz – contrary to Descartes – denies can constitute a completed whole. (shrink)

Across languages, we find indefinites that overtly mark a speaker’s lack of knowledge: they signal that the speaker is unable to give any further information about who or what satisfies her existential claim (Becker, 1999; Haspelmath, 1997). From now on, we will refer to the marking of the speaker’s lack of knowledge as an epistemic effect and to the indefinites that induce epistemic effects as epistemic indefinites.

In the war on terrorism, the imprisonment of suspected terrorists by the United States has raised a host of issues,1 among them that of indefinite detention. Over the years, there has been a great...

It is argued that an indefinite statement as introduced by Aristotle in De Int. 7 refers to a universal which may partly partake in contradictory accidental predicates together. This fact is mirrored on the semantic level by ascribing truth to some degree to both parts of a contradiction. Accordingly, Aristotle should be interpreted as saying in De Int. 9 that the statement that a certain individual object will be F at some time in its contingent future is to be (...) taken to be true to some degree. This is because an individual object cannot yet, with respect to its contingent future, be regarded as factual but only as - time-independently - exemplifying a universal. In this context, fundamental connections become apparent between indefinite statements on the one hand and Aristotelian modal logic, statistics and theory of science on the other. (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

This paper investigates the truth conditions of sentences containing indefinite noun phrases, focusing on occurrences in attitude reports, and, in particular, a puzzle case due to Walter Edelberg. It is argued that indefinites semantically contribute the (thought-)object they denote, in a manner analogous to attributive definite descriptions. While there is an existential reading of attitude reports containing indefinites, it is argued that the existential quantifier is contributed by the de re interpretation of the indefinite (as the de re (...) reading adds existential quantification to the interpretation of definites on Kaplan’s analysis). (shrink)

I argue that indefinite descriptions are referring terms. This is not the ambiguity thesis: that sometimes they are referring terms and sometimes something else, such as quantifiers . No. On my view they are always referring terms; and never quantifiers. I defend this thesis by modifying the standard conception of what a referring term is: a modification that needs to be made anyway, irrespective of the treatment of indefinites. I derive this approach from my speech-act theoretic semantics . The (...) basic thought is that referring terms have as their meanings speech-acts of a certain kind called proto-referring acts. These are acts in which speakers advertise or present intentions to denote, where denotation is a word-world relation, and advertising an intention is acting as if one has intentions, where it is open whether one has them or not, or whether the referring term used denotes or not. I show how this works for proper names. The meaning of a proper name is the speech-act proto-referring act type defined by a certain referential tree. This gives us the basis for an account of proper name meaning irrespective of denotation: a uniform treatment of full and empty names. Applied to indefinites, we can capture cases where speakers perform proto-acts—in which they advertise an intention to denote something—where they intend to denote, but others where they do not, but they still perform the proto-act: advertising an intention to denote. Two cases are ‘Fred saw a hippo’ and ‘Jane did not see a hippo’. In both cases U performs the same proto-act, represent it as Rpro, but possesses the advertised intention in the first case, but not in the second. In the first case, Rpro gains referential content—its descriptive content is expanded to include seen by Fred, whereas in its second token use it has no content augmentation or sentential determination of reference. In the second case ‘a hippo’ is an empty referring term, just as ‘Pegasus’ is empty in ‘Fred did not see Pegasus; he does not exist’. But in both sentences ‘a hippo’ functions in the same basic way: in both cases the basic proto-referring act Rpro is performed. Using this approach, I show how definite descriptions can be construed as indefinites with added meaning. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in (...) requiring nonstandard assumptions about mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)

In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...) paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how. (shrink)

One of the big questions about indefinites is why they can escape scope islands. In the recent approach of Brasoveanu and Farkas :1–55, 2011) scopal relations with syntactically dominating quantifiers are hard wired into the semantic definition of the existential quantifier, which immediately explains why the semantic scope of indefinites may exceed their syntactic scope. In this paper, I argue for the revival of an alternative approach which places the explanatory burden on the idea that indefinites are essentially referential expressions, (...) similar to definites, and not plain existential quantifiers. I propose one fully explicit variant of such theories and argue that it comes with a number of conceptual and empirical advantages over competing theories. (shrink)

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

This paper stems from previous works of the authors in which a new measure of the simplicity of an explanation based on its degree of arbitrariness is proposed: The more the explanation is arbitray, the less appealing it is, with explanations having no arbitrariness – called constrained – being the preferred ones. In previous works, as commonly done in the literature of abductive logic programming, a set of hypotheses is not an explanation, unless it is definite, i.e. it explains all (...) the data belonging to the observation. In this paper, we follow a different perspective and define the concept of indefinite constrained explanations, i.e. constrained explanations that are not definite, but admit some indefiniteness. An indefinite constrained explanation captures the intuition of the existence of an explanation (indefinite explanation) that would best explain the given evidence, while not making arbitrary choices (constrained explanation). The main contribution of the paper is the study of abduction in indefinite deductive theories: specifically, the paper studies the framework of abductive logic programming extended with integrity constraints in the setting in which both the initial knowledge base and the abductive explanations are indefinite (may contain occurrences of null values) and the domain is possibly infinite. Furthermore, the paper discusses the complexity of problems concerning indefinite (constrained) explanations. (shrink)

The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefinite extensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefinite extensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefinite extensibility of the concept set, plausible principles permit the derivation of modal collapse.

The paper proposes a novel solution to the problem of scope posed by natural language indefinites that captures both the difference in scopal freedom between indefinites and bona fide quantifiers and the syntactic sensitivity that the scope of indefinites does nevertheless exhibit. Following the main insight of choice functional approaches, we connect the special scopal properties of indefinites to the fact that their semantics can be stated in terms of choosing a suitable witness. This is in contrast to bona fide (...) quantifiers, the semantics of which crucially involves relations between sets of entities. We provide empirical arguments that this insight should not be captured by adding choice/Skolem functions to classical first-order logic, but in a semantics that follows Independence-Friendly Logic, in which scopal relations involving existentials are part of the recursive definition of truth and satisfaction. These scopal relations are resolved automatically as part of the interpretation of existentials. Additional support for this approach is provided by dependent indefinites, a cross-linguistically common class of special indefinites that can be straightforwardly analyzed in our semantic framework. (shrink)

Working on the assumption that ideas are embedded in socio-technical arrangements which actualize them, this essay sheds light on the way the Open Method of Co-ordination (OMC) achieves the Lisbon strategic goal: to become the most competitive and dynamic knowledge-based economy in the world . Rather than framing the issue in utilitarian terms, it focuses attention on quantified indicators, comparable statistics and common targets resulting from the increasing practice of intergovernmental benchmarking, in order to tackle the following questions: how does (...) the OMC go about co-ordinating Member States through the benchmarking of national policies? And to what extent does this managerial device impact the path of European construction? Beyond the ideological and discursive construction of the competitive imperative, this technology of government transforms it into an indefinite discipline (Foucault) which constantly urges decision-makers to hit the top of the charts. This contribution thus argues that the practice of intergovernmental benchmarking is far from being neutral in purpose and effect. On the contrary, it lays the foundation for building a competitive Europe which unites Member States through competition. (shrink)

In his thought-provoking, carefully crafted, and impressively researched article, Don Scheid makes a compelling argument for an indefinite detention paradigm for “mega-terrorists.” Scheid's proposa...

Cognitive Science, Volume 46, Issue 5, May 2022.

I argue that alternative-denoting expressions interact with their semantic context by taking scope. With an empirical focus on indefinites in English, I show how this approach improves on standard alternative-semantic architectures that use point-wise composition to subvert islands, as well as on in situ approaches to indefinites more generally. Unlike grammars based on point-wise composition, scope-based alternative management is thoroughly categorematic, doesn’t under-generate readings when multiple sources of alternatives occur on an island, and is compatible with standard treatments of binding. (...) Unlike all in situ treatments of indefinites, relying on a true scope mechanism prevents over-generation when an operator binds into an indefinite. My account relies only on function application, some mechanism for scope-taking, and two freely-applying type-shifters: the first is Karttunen’s :3–44, 1977. https://doi.org/10.1007/BF00351935) proto-question operator, aka Partee’s Studies in discourse representation theory and the theory of generalized quantifiers, Foris, Dordrecht, 1986) IDENT, and the second can be factored out of extant approaches to the semantics of questions in the tradition of Karttunen. These type-shifters form a decomposition of LIFT, the familiar function mapping values into scope-takers. Exceptional scope of alternative-generating expressions arises via scopal pied-piping: indefinites take scope over their island, which then itself takes scope. (shrink)

Linguistic predictions may be generated from and evaluated against a representation of events and referents described in the discourse. Compatible with this idea, recent work shows that predictions about novel noun phrases include their definiteness. In the current follow-up study, we ask whether people engage similar prediction-related processes for definite and indefinite referents. This question is relevant for linguistic theories that imply a processing difference between definite and indefinite noun phrases, typically because definiteness is thought to require a (...) uniquely identifiable referent in the discourse. We addressed this question in an event-related potential (ERP) study (N = 48) with preregistration of data acquisition, preprocessing, and Bayesian analysis. Participants read Dutch mini-stories with a definite or indefinite novel noun phrase (e.g., “het/een huis,” the/a house), wherein (in)definiteness of the article was either expected or unexpected and the noun was always strongly expected. Unexpected articles elicited enhanced N400s, but unexpectedly indefinite articles also elicited a positive ERP effect at frontal channels compared to expectedly indefinite articles. We tentatively link this effect to an antiuniqueness violation, which may force people to introduce a new referent over and above the already anticipated one. Interestingly, expectedly definite nouns elicited larger N400s than unexpectedly definite nouns (replicating a previous surprising finding) and indefinite nouns. Although the exact nature of these noun effects remains unknown, expectedly definite nouns may have triggered the strongest semantic activation because they alone refer to specific and concrete referents. In sum, results from both the articles and nouns clearly demonstrate that definiteness marking has a rapid effect on processing, counter to recent claims regarding definiteness processing. (shrink)

We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.

A temporal counterpart to Bell nonlocality would intuitively refer to the presence of non-classical correlations between timelike-separated events. The hypothesis of temporal nonlocality has received recent support in the literature, and its existence would likely influence the future development of physical theories. This paper shows how Adlam's principle of temporal locality can be violated within a protocol involving indefinite causal orders. While the derivations of Leggett-Garg inequalities or the temporal CHSH inequality are said to involve problematic assumptions preventing a (...) targeted probing of a well-defined notion of temporal nonlocality, the present test is free from such worries. However, it is shown that the test, in its current formulation, fails to be fully model-independent. We provide several considerations regarding the physicality of ICOs that could help alleviate this drawback. In the present work, a specific physical interpretation of ICOs in terms of retrocausal influences would explain the presence of temporally nonlocal correlations. It is argued that, as the physical underpinnings of temporal nonlocality might also account for standard Bell nonlocality, focusing on the former as a consequence of ICOs might support under-explored strategies to make sense of the latter. (shrink)

The paper rst lays out a non-congurational approach to scope ambiguities in which scope dependencies are treated as dependencies between evaluation indices of variables. The notions of dependent and domain variables are dened naturally in this framework. These concepts are then used to account for the distribution and interpretation of determiner reduplication in Hungarian, a phenomenon that has not received much attention so far.1 1. Introduction This paper contributes to the study of the semantics of indenites in natural language by (...) introducing on the scene a new type of indenite, called dependent. We meet it in Hungarian, where one may reduplicate certain determiners, as illustrated in [1]-[3]. (shrink)

Quantum mechanics has recently indicated that, at the fundamental level, temporal order is not fixed. This phenomenon, termed Indefinite Causal Order, is yet to receive metaphysical or theological engagement. We examine Indefinite Causal Order, particularly as it emerges in a 2018 photonic experiment. In this experiment, two operations A and B were shown to be in a superposition with regard to their causal order. Essentially, time, intuitively understood as fixed, flowing, and fundamental, becomes fuzzy. We argue that if (...) Indefinite Causal Order is true, this is good evidence in favor of a B‐theory of time, though such a B‐theory requires modification. We then turn to theology, suggesting that a B‐theoretic temporal ontology invites serious reconsideration of the doctrine of salvation. This paper concludes that the best explanation for salvation given a B‐theory is mind‐dependent salvific becoming, a type of psychological soteriological change that occurs through downward causation. (shrink)

In this paper I argue against grounding being necessarily well-founded, and provide some reasons to think it's actually not well-founded.

Every member of the club was convinced that if a (particular) friend of his from Texas had died in the fire, he would have inherited a fortune.

A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second (...) is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). -/- Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences? (shrink)