We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus. From the Tractatus’s Picture theory of language we can strictly infer the two contradictory propositions simultaneously: the world and the language are equinumerous; the world and the language are not equinumerous. I call this antinomy the world-language paradox. (...) Based on a rigorous analysis of the Tractatus, with the help of the technical resources of Cantor’s naive set theory and Zermelo-Fraenkel set theory with the axiom of choice, I outline the world-language paradox and assess the unique possible solution plan, i.e., the mathematical plan utilizing ‘infinity’. I conclude that Wittgenstein cannot solve the hidden set-theoretical paradox of the Tractatus successfully unless he gives up his finitism. (shrink)

I defend the “settist” view that set theory can be done consistently without any form of distinction between sets and “classes” (by whatever name), if we think clearly about belief and the expression of belief—and this, furthermore, entirely within classical logic. Standard arguments against settism in classical logic are seen to fail because they assume, falsely, that expressing commitment to a set theory is something that must be done in a meaningful language, the semantics of which requires, on pain of (...) Russellian paradox, a more powerful set theory. I explore the consequences of this response to the standard argument against “classical logic settism” for our notion of belief, and argue that what is revealed is that representationalist theories of belief cannot be right as long as it is possible to believe that every set is self-identical. (shrink)

We present an order-theoretic analysis of set-theoretic paradoxes. This analysis will show that a large variety of purely set-theoretic paradoxes (including the various Russell paradoxes as well as all the familiar implementations of the paradoxes of Mirimanoff and Burali-Forti) are all instances of a single limitative phenomenon.

Is there a key for ‘translating' some set-theoretical paradoxes into counterpart semantical paradoxes and vice-versa? There is, and this encourages the hope of a unified solution. The solution turns not on inventing new axioms that do not entail contradiction, but on imposing a completely intuitive restriction on the comprehension axiom of naive set theory in order to avoid illegitimate (circular) stipulation.

In his paper "Are There Set Theoretic Possible Worlds?", Selmer Bringsjord argued that the set theoretic definition of possible worlds proffered by, among others, Robert Adams and Alvin Plantinga is incoherent. It is the purpose of this note to evaluate that argument. The upshot: these set theoretic accounts can be preserved, but only by abandoning the power set axiom.

It is a platitude among decision theorists that agents should choose their actions so as to maximize expected value. But exactly how to define expected value is contentious. Evidential decision theory (henceforth EDT), causal decision theory (henceforth CDT), and a theory proposed by Ralph Wedgwood that this essay will call benchmark theory (BT) all advise agents to maximize different types of expected value. Consequently, their verdicts sometimes conflict. In certain famous cases of conflict—medical Newcomb problems—CDT and BT seem to get (...) things right, while EDT seems to get things wrong. In other cases of conflict, including some recent examples suggested by Andy Egan, EDT and BT seem to get things right, while CDT seems to get things wrong. In still other cases, EDT and CDT seems to get things right, while BT gets things wrong. It's no accident, this essay claims, that all three decision theories are subject to counterexamples. Decision rules can be reinterpreted as voting rules, where the voters are the agent's possible future selves. The problematic examples have the structure of voting paradoxes. Just as voting paradoxes show that no voting rule can do everything we want, decision-theoretic paradoxes show that no decision rule can do everything we want. Luckily, the so-called “tickle defense” establishes that EDT, CDT, and BT will do everything we want in a wide range of situations. Most decision situations, this essay argues, are analogues of voting situations in which the voters unanimously adopt the same set of preferences. In such situations, all plausible voting rules and all plausible decision rules agree. (shrink)

Emmanuel Levinas sets up alterity as a fundamental ontological category, irreducible to being and nothingess. There are two difficulties in understanding this ontological alterity. On the one hand, Levinas formulates it with negative terms - infinition, abstraction, ab-solutenes, trace of a past that has never been present. On the other hand, Levinas invokes the notions of the superlative, the Good, and God. These notions are very difficult to separate from the notion of a redoubling of the positivity by which the (...) things of the world are posited in their own subsistent being. The quasi-concepts with which Levinas has thematized the alterity of the others who face us in an ontological sphere of infinition, absoluteness, and abstractness have determined negatively what he envisions as beyond negativity and positivity. They reduce the determinateness of the wants and needs of another, and reduce the otherness of one other from other others. In addition, the positivism with which Levinas eliminates the imperatives with which the elements summon us and the material imperatives with which the things put demands on us effaces the phenomenality of the other whose want and needs are inscribed on the susceptibility and vulnerability of his surfaces of skin. (shrink)

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for by (...) introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes (...) do exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

This paper is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc. While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rest on an intuition providing an unshakeable foundation,'each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom (...) systems for non-well-founded universes, and an attempt is made to motivate such axiom systems on the basis of an old and respected ‘algebraic’ intuition. (shrink)

A close-knit family of conceptual structures underlies the range of philosophical phenomena from Descartes' Cogito through semantic and set-theoretical paradoxes to some of the major limitative results of twentieth-century logic. At issue are questions of indexicals, the nature of semantics, free will and determinism, and contemporary debates regarding the nature of consciousness. The conceptual structures that underlie all of these are variations on a single theme: the theme of reflexivity. OUr attempt here is to characterize reflexive conceptual structures more (...) thoroughly and more precisely than has been done before, making explicit the structure of paradox and the clear connections to major logical results. (shrink)

Paradoxes are arguments that lead from apparently true premises, via apparently uncontroversial reasoning, to a false or even contradictory conclusion. Paradoxes threaten our basic understanding of central concepts such as space, time, motion, infinity, truth, knowledge, and belief. In this volume Roy T Cook provides a sophisticated, yet accessible and entertaining, introduction to the study of paradoxes, one that includes a detailed examination of a wide variety of paradoxes. The book is organized around four important types of paradox: the (...) semantic paradoxes involving truth, the set-theoretic paradoxes involving arbitrary collections of objects, the Soritical paradoxes involving vague concepts, and the epistemic paradoxes involving knowledge and belief. In each of these cases, Cook frames the discussion in terms of four different approaches one might take towards solving such paradoxes. Each chapter concludes with a number of exercises that illustrate the philosophical arguments and logical concepts involved in the paradoxes. _Paradoxes_ is the ideal introduction to the topic and will be a valuable resource for scholars and students in a wide variety of disciplines who wish to understand the important role that paradoxes have played, and continue to play, in contemporary philosophy. (shrink)

A paradox about sets of properties is presented. The paradox, which invokes an impredicatively defined property, is formalized in a free third-order logic with lambda-abstraction, through a classically proof-theoretically valid deduction of a contradiction from a single premise to the effect that every property has a unit set. Something like a model is offered to establish that the premise is, although classically inconsistent, nevertheless consistent, so that the paradox discredits the logic employed. A resolution through the ramified (...) theory of types is considered. Finally, a general scheme that generates a family of analogous paradoxes and a generally applicable resolution are proposed. (shrink)

Our target is collectivities--all types of collectivities, beyond formal treatment in terms of sets alone. Collectivities are collections that can have members under all modalities: actual and potential members, definite and indefinite members, past and future members, members identifiable or unknown. The null collectivity aside, collectivities will indeed have members, but their membership need not be enumerable individual by individual or identifiable with precision. Collectivities are pluralities we generally access in terms of qualifying features and modalities rather than lists of (...) identifiable members. Our attempt is to do justice to collectivities in their generality and indeterminacy, with plena as a specific focus: large-scale macro-totalities on the order of all facts, all truths, and all things...a collectivity well beyond the technical machinery of sets. (shrink)

The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is circular, Beall (...) concludes that Yablo's paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (1) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of the description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular. (shrink)

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the (...) logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches. (shrink)

As the story goes, the source of the paradoxes of naive set theory lies in a conflation of two distinct conceptions of set: the so-called iterative, or mathematical, conception, and the Fregean, or logical, conception. While the latter conception is provably inconsistent, the former, as Godel notes, "has never led to any antinomy whatsoever". More important, the iterative conception explains the paradoxes by showing precisely where the Fregean conception goes wrong by enabling us to distinguish between sets and proper classes, (...) collections that are "too big" to be sets. While I agree wholeheartedly with this distinction, in this paper I argue first that the iterative conception does not provide an explanation of all of the set theoretic paradoxes. I then argue that we need to reconsider the distinction between sets and proper classes rather more carefully. The result will be that ZFC does not capture the iterative conception in its full generality. I close by offering a more general theory that, arguably, does. (shrink)

Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of naive set theory (...) inevitably leads one into a severe scepticism with regard to the feasibility of giving a systematic semantics for set theory. It is further argued that this is not just a problem for philosophers of mathematics. Semantic scepticism in set theory will almost inevitably spill over into total pessimism regarding the prospects for an explanatory theory of language and meaning in general. The conclusion is that those who wish to avoid such intellectual defeatism need to look seriously at the possibility that it is the logic used in the derivation of the paradoxes, and not the naive set theory itself, which is at fault. (shrink)

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

What gave rise to Ernst Zermelo's axiomatization of set theory in 1908? According to the usual interpretation, Zermelo was motivated by the set-theoretic paradoxes. This paper argues that Zermelo was primarily motivated, not by the paradoxes, but by the controversy surrounding his 1904 proof that every set can be wellordered, and especially by a desire to preserve his Axiom of Choice from its numerous critics. Here Zermelo's concern for the foundations of mathematics diverged from Bertrand Russell's on the one hand (...) and from Felix Hausdorff's on the other. (shrink)

DEFINING OUR TERMS A “paradox" is an argumentation that appears to deduce a conclusion believed to be false from premises believed to be true. An “inconsistency proof for a theory" is an argumentation that actually deduces a negation of a theorem of the theory from premises that are all theorems of the theory. An “indirect proof of the negation of a hypothesis" is an argumentation that actually deduces a conclusion known to be false from the hypothesis alone or, more (...) commonly, from the hypothesis augmented by a set of premises known to be true. A “direct proof of a hypothesis" is an argumentation that actually deduces the hypothesis itself from premises known to be true. Since `appears', `believes' and `knows' all make elliptical reference to a participant, it is clear that `paradox', `indirect proof' and `direct proof' are all participant-relative. PARTICIPANT RELATIVITY In normal mathematical writing the participant is presumed to be “the community of mathematicians" or some more or less well-defined subcommunity and, therefore, omission of explicit reference to the participant is often warranted. However, in historical, critical, or philosophical writing focused on emerging branches of mathematics such omission often invites confusion. One and the same argumentation has been a paradox for one mathematician, an inconsistency proof for another, and an indirect proof to a third. One and the same argumentation-text can appear to one mathematician to express an indirect proof while appearing to another mathematician to express a direct proof. WHAT IS A PARADOX’S SOLUTION? Of the above four sorts of argumentation only the paradox invites “solution" or “resolution", and ordinarily this is to be accomplished either by discovering a logical fallacy in the “reasoning" of the argumentation or by discovering that the conclusion is not really false or by discovering that one of the premises is not really true. Resolution of a paradox by a participant amounts to reclassifying a formerly paradoxical argumentation either as a “fallacy", as a direct proof of its conclusion, as an indirect proof of the negation of one of its premises, as an inconsistency proof, or as something else depending on the participant's state of knowledge or belief. This illustrates why an argumentation which is a paradox to a given mathematician at a given time may well not be a paradox to the same mathematician at a later time. -/- The present article considers several set-theoretic argumentations that appeared in the period 1903-1908. The year 1903 saw the publication of B. Russell's Principles of mathematics, [Cambridge Univ. Press, Cambridge, 1903; Jbuch 34, 62]. The year 1908 saw the publication of Russell's article on type theory as well as Ernst Zermelo's two watershed articles on the axiom of choice and the foundations of set theory. The argumentations discussed concern “the largest cardinal", “the largest ordinal", the well-ordering principle, “the well-ordering of the continuum", denumerability of ordinals and denumerability of reals. The article shows that these argumentations were variously classified by various mathematicians and that the surrounding atmosphere was one of confusion and misunderstanding, partly as a result of failure to make or to heed distinctions similar to those made above. The article implies that historians have made the situation worse by not observing or not analysing the nature of the confusion. -/- RECOMMENDATION This well-written and well-documented article exemplifies the fact that clarification of history can be achieved through articulation of distinctions that had not been articulated (or were not being heeded) at the time. The article presupposes extensive knowledge of the history of mathematics, of mathematics itself (especially set theory) and of philosophy. It is therefore not to be recommended for casual reading. AFTERWORD: This review was written at the same time Corcoran was writing his signature “Argumentations and logic”[249] that covers much of the same ground in much more detail. https://www.academia.edu/14089432/Argumentations_and_Logic . (shrink)

The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view that the truth-paradoxes and (...) the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible (...) principles. Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them. (shrink)

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)

A growing body of empirical evidence suggests the existence of a Paradox of Choice, whereby a larger choice set leads to a lower expected payoff to the decisionmaker. These empirical findings contradict traditional choice-theoretic results in microeconomics and even social psychology, suggesting profound yet unexplained aspects of choice settings/behavior. The Paradox remains largely as a theoretically-rootless empirical phenomenon. We neither understand the mechanisms and conditions that generate it, nor whether the Paradox stems from choice behavior, choice setting, (...) or both. We present a general search model of consumer choice under incomplete information to examine the theoretical existence and conditions for the Paradox. The model departs from traditional microeconomic choice theory in assuming no “cognitive free lunches.” Search and evaluative efforts require time, focus, and opportunity cost in the model, as in real life. The model shows that conditions for Paradox readily exist, diminishing returns to potential value of the choice in choice set size is a necessary condition for a Paradox outcome, that a Paradox outcome can occur with or without a “choice overload” response by the decisionmaker, and that avoidance of a Paradox outcome is an impossibility in some choice settings (primitive to setting regardless of choice behavior). Lastly, we develop a pareto-improving choice set reduction algorithm that eliminates weakly dominated choice set elements and presents the restricted choice set to the decisionmaker. As such, it is shown that the algorithm acts to treat or mitigate the Paradox of Choice, as well as to pareto-improve choice-supporting evaluation more generally. Like established AI recommender systems, the novel algorithm filters choices. In the present case, the filtering is based upon identification of weakly dominated choice set elements. (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)

There are two paradoxes of satisfaction, and they are of different kinds. The classic satisfaction paradox is a version of Grelling’s: does ‘does not satisfy itself’ satisfy itself? The Unsatisfied paradox finds a predicate, P, such that Px if and only if x does not satisfy that predicate: paradox results for any x. The two are intuitively different as their predicates have different paradoxical extensions. Analysis reduces each paradoxical argument to differing rule sets, wherein their respective pathologies (...) lie. Having different pathologies, they are paradoxes of different kinds. Furthermore, each of these satisfaction paradoxes has an analogue with the same pathology in set theory. Therefore, these analogues are respectively of the same two kinds. This level of abstraction is significant in that it tracks two related but different pathologies. Thus, not all paradoxes of semantics and set theory share the same pathology: there are at least two kinds of paradox cutting across the semantic and set-theoretic distinction. (shrink)

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are (...) tightly related, and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics. (shrink)

In this paper, I present an interpretation of Kant’s view that reason’s hypostasis of the idea of a sum-total of reality is dogmatic and illegitimate. In the section on the ‘Transcendental Ideal’, the second section of the Ideal of Pure Reason chapter in the Critique of Pure Reason, Kant starts by describing reason’s procedure from the affirmation of the principle of thoroughgoing determination to the hypostasis in question. According to the interpretation I defend, the argument for hypostasis deployed in this (...) section constitutes an improvement upon an argument defended by the pre-critical Kant himself in his 1673 essay “The Only Possible Argument in Support of a Demonstration of the Existence of God”. By making reference to the concept of omnitudo realitatis, the argument in the ‘Transcendental Ideal’ section presents a much more radical and convincing interpretation of the thesis that ‘possibility presupposes actuality’. Second, I present transcendental idealism and its related distinction between objects of sense and objects in general as the main dissuasive argument of the critical philosopher against hypostasis. Finally, I consider an argument against hypostasis that is independent of transcendental idealism: the threat of set-theoretical paradoxes if we hypostatize the relevant idea, intended as the concept of an absolutely comprehensive totality. (shrink)

Thought: A Journal of Philosophy, EarlyView.

This paper presents and comments the content of a note by Beppo Levi on logical paradoxes. Though the existence of this contribution is known, very little analysis of it is available in the literature. I put the emphasis on Levi’s usage of “elementation procedures” for solving the set-theoretical paradoxes, which is the most original part of Levi’s approach to the topic.

Graham Priest 2002 argues that all logical paradoxes that include set-theoretic paradoxes and semantic paradoxes share a common structure, the Inclosure Schema, so they should be treated as one family. Through a discussion of Berry's Paradox and the semantic notion ?definable?, I argue that (i) the Inclosure Schema is not fine-grained enough to capture the essential features of semantic paradoxes, and (ii) the traditional separation of the two groups of logical paradoxes should be retained.

A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered here (...) revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes. (shrink)

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that the idea of classical (...) recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

Frege's definition of the natural number n in terms of the set of n-membered sets has been treated rudely by history. It has suffered not one but two crippling blows. The discovery of Russell's Paradox revealed a fatal flaw in the ‘naive’ conception of set. In spite of its intuitive appeal, Frege's Basic Law V turned out to be impermissible, leaving us only with the etiolated concept of set that survives in the axiomatic treatments initiated by Zermelo. The independence (...) results, however, of Godel and Cohen, concerning Cantor's Continuum Hypothesis, have left us adrift in choosing between Cantorian and non-Cantorian set theories, which has induced in some logicians a skepticism in regard to the very idea of set-theoretic platonism. (shrink)

What one can say about the past, present and future of set theory depends on what one expects or at least hopes set theory will accomplish. In order to gauge the early expectations, I begin with a quote from the inaugural lecture in 1903 of my mathematical grandfather, the internationally known Finnish mathematician Ernst Lindelöf. The subject of his lecture was – guess what – Cantor’s set theory. In his conclusion, Lindelöf says of Cantor’s results: For mathematics they have lent (...) new tools and opened up new fields of research, they have thrown entirely new light on the foundations of analysis and brought clarity and order where there was only disorder and contradictions. Thus they have greatly contributed to the harmony that is the essence of mathematics, a harmony a grasp of which is the reward of mathematical research. (Quoted in Olli Lehto, Tieteen aatelia, Otava, Helsinki, 2008, p. 263) We can all agree with the compliments Lindelöf pays to set theory as an impressive specimen of mathematical research, including the theory of infinite cardinals and ordinals. But as far as the foundational role of set theory is concerned, in the perspective of the subsequent century his words read as an example of supreme historical irony. Far from bringing harmony into the foundations of mathematics, problems arising from set theory led to a schism between different schools of thought. Few mathematicians think of set theory as a tool for reaching new results outside set theory.. On the contrary, an interesting rich tradition called reverse mathematics takes significant mathematical results and asks what set-theoretical assumptions are needed to prove them. Set-theoretical paradoxes have greatly increased mathematicians’ concerns about contradictions instead of assuaging them. Many foundationalists would blandly deny that we have even now, more than a hundred years later, reached “clarity and order” about the foundations of analysis. What Lindelöf took to be the results of set theory thus were in reality so many hopes that set theory was expected to fulfill.. (shrink)

Alain Badiou is a highly original, indeed decidedly iconoclastic thinker whose work has ranged widely over areas of equal concern to philosophers in the ‘continental’ and mainstream analytic traditions. These areas include ontology, epistemology, ethics, politics, and – above all – philosophy of mathematics. It is unfortunate, and symptomatic of prevailing attitudes, that his work has so far receivedminimal attention from commentators in the analytic line of descent. Here I try to help the process of reception along by describing Badiou’s (...) remarkably ambitious approach to issues of mathematical (more specifically: of set-theoretical) ontology, and by explaining just where his project stands in relation to some major issues within current analytic debate. Chief among them are: the issue between realists and anti-realists – along with various avowed middle-ground or compromise solutions – and those oddly tenacious problems-from-Wittgenstein (e.g., concerning what it means to follow a rule) that have so preoccupied philosophers over the past decade. In particular I stress the unusual, indeed unique combination in his thought of high formal rigour and conceptual clarity allied to a speculative scope and inventiveness which tend to make those other discussions appear somewhat self-absorbed and parochial. Most importantly, Badiou engages these issues at a level of creative as well as of technical or analytic grasp, which puts his thinking closely in touch with the way that set theory has itself evolved through a constant process of – in Badiou’s phrase – ‘turning paradoxes into concepts.’ I also discuss his strong and principled rejection of the ‘linguistic turn’ in its manifold (analytic and continental) variants, and his idea of the ‘event’ as that which inherently eludes or surpasses the conceptual resources of any received ontology, whether in mathematics and the natural sciences or in the history of genuinely epochal changes in politics and ethics. All in all, I put the case for Badiou as a thinker of the first importance not only for the impressive range, depth and originality of his work, but also because it points to an escape-route from some of the more cramped or windowless quarters of present-day philosophic thought. (shrink)

The principle _determinatio est negatio_—that determination is achieved through negation—has philosophical roots extending back to Plato and Aristotle, and it later influenced early modern thinkers such as Francisco Suárez and Spinoza. This paper has two aims. The first demonstrates how the principle of negation functions as a tool for conceptual determination across various philosophical frameworks, and the second demonstrates that the principle plays a key role in the analysis and resolution of the Burali-Forti paradox within the context of the (...) naïve and axiomatic set theories. In the first section, the analysis focuses on the evolution of the principle from ancient philosophy to early modern metaphysics, examining Plato’s dialectics, Aristotle’s metaphysical distinctions, Suárez’s scholastic theories, and Spinoza’s monist metaphysics. The second section shifts to mathematics, where _determinatio est negatio_ plays a key role in resolving set-theoretical paradoxes, particularly the Burali-Forti paradox. By exploring Cantor’s solution and its reliance on the distinction between consistent and inconsistent sets, this study demonstrates how this principle is essential for avoiding self-referential inconsistencies. The contributions of Zermelo and von Neumann, who developed axiomatic frameworks to address these paradoxes, further elaborate the philosophical foundations of set theory. Ultimately, the study reveals a deep connection between metaphysical principles and the mathematical treatment of paradoxes, emphasizing the ongoing relevance of _determinatio est negatio_ in both domains. (shrink)

There is a paradox, or self-defeating supposition in the core of constructionism, for it would appear that any attempt to resolve a dispute in historical interpretation within a convention of self-contained criteria of confirmation by appealing to justificatory criteria outside the convention -to wit, the theory of constructionism -is self-defeating. Through the theoretical consideration of historians isolated in a vat, following Hilary Putnam's metaphor, it becomes clear that the vat language of the historians does not have the possibility (...) of referring, intrinsically or extrinsically, to anything external. The implications of the vat metaphor for an understanding of historical inquiry are: 1) we need to recognize and insist upon different levels of abstraction both in historical writing and in the justification of its claims; 2) applied to history, the set paradox reveals the need to recognize that at the most abstract level of consideration we encounter inescapable incoherence; and 3) we need to recognize that in some important sense reference must be completed in the world, regardless of the problems in characterizing this sense philosophically. (shrink)

The starting point of this paper is a claim defended most famously by Graham Priest: that given certain observed similarities between the set-theoretic and the semantic paradoxes, we should be looking for a ‘uniform solution’ to the members of both families. Despite its indisputable surface attractiveness, I argue that this claim hinges on a problematic reasoning move. This is seen most clearly, I suggest, when the claim and its underlying assumptions are examined by the lights of a novel, quite general (...) and, I contend, promising take on inter-theoretic analogy. The ensuing discussion is intended to serve as both a possible case study and a first step towards the broader aim of the paper: namely, to initiate a wider conversation on the methodology of paradox-solving on the one hand, and the use of inter-theoretic analogies on the other. (shrink)

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) in chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turing thesis) related to the possible “solution of supertasks,” and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for physical applications (...) are discussed: Canlorian “naive” (i.e., nonaxiomatic) set theory, contructivism, and operationalism. In the author's opinion, an attitude of “suspended attention” (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same time, physicists should be open to “bizarre” or “mindboggling” new formalisms, which need not be operationalizable or testable at the lime of their creation, but which may successfully lead to novel fields of phenomenology and technology. (shrink)

Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a (...) specific reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)