Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception (...) and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). (...) A modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...) the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (shrink)

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the (...) axiom AFA is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

Surveying and criticising attitudes towards the role and strength of the iterative conception of set—widely seen as the justificatory basis of Zermelo-Fraenkel set theory with Choice—this paper highlights a tension in both contemporary and historic accounts of the iterative conception’s justificatory role: on the one hand its advocates wish to claim that it justifies ZFC, but on the other hand they abstain from stating whether the preconditions for such justification exists. Expanding the number of axioms that the (...) class='Hi'>conception is standardly charged with failing to justify, in the forms of the Emptyset and Powerset, this paper aims to extend the critique that the iterative conception does not justify ZFC to its subsystems. Exploring the historic and contemporary relationship between the iterative conception and set theory, the paper then attempts to defuse strategies that avoid the problems of intrinsic justification by weakening the iterative conception’s role to the ‘motivational’ or ‘heuristic’ by showing that what the iterative conception has been seen to motivate has changed over time. It is suggested that the conjunction of these arguments seriously weakens the programme of reasoning on an ‘atheoretic’ notion of set. (shrink)

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set (...) theory NF. (shrink)

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set (...) is a good one, for all the usual reasons. However, the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean – in the terminology of modern computer science at least – is the recursive conception of sets. Nevertheless, having got that quibble off my chest, I shall continue to write of the iterative conception like everyone else.). (shrink)

It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This article advances reasons (...) internal to Lear’s account to show that his argument for intuitionistic set theory is unsuccessful. (shrink)

The paper critically examines whether the open-endedness of the set concepts mandates the use of intuitionistic logic in set theory, as some philosophers think. [The sequel to this paper is ‘The Open-Endedness of the Set Concept and the Semantics of Set Theory' published in Synthese in 2003.].

ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

Although a large body of literature exists propounding the importance of space in aged care and care of the older person with dementia, there is, however, only limited exploration of the ‘acute care space’ as a particular type of space with archetypal constraints that maybe unfavourable to older people with cognitive impairment and nurses wanting to provide care that is person‐centred. In this article, we explore concepts of space and examine the implications of these for the delivery of care to (...) older people who are cognitively impaired. Our exploration is grounded in theorisations of space offered by key geographers and phenomenologists, but also draws on how space has been constructed within the nursing literature that refers specifically to acute care. We argue that space, once created, can be created and that nursing has a significant role to play in the process of its recreation in the pursuit of care that is person‐centred. We conclude by introducing an alternative logic of space aimed at promoting the creation of more salutogenic spaces that invokes a sense of sanctuary, safeness, and inclusion, all of which are essential if the care provided to the older person with cognitive impairment is apposite to their needs. The concept of ‘person‐centred space’ helps to crystallize the relationship between space and person‐centred care and implies more intentional manipulation of space that is more conducive to caring and healing. Significantly, it marks a return to Nightingale's wisdom, that is, to put the person in the best possible conditions for nature to act upon them. (shrink)

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate "ist" that expresses (...) the relation that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing "ist" to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example. (shrink)

This is a survey paper of approaches to the concept of logical implication. Roughly stated the main motivation of these approaches is to provide a necessary and sufficient condition for a set of propositions to logically imply a single proposition. In regard to their affinities these approaches are grouped into two: the transformational conception and the informational conception. Some approaches in each conception are philosophical and some are mathematical in character, their common assumption being that they (...) reflect a previous intuition – in some cases – already at work in classical mathematics. The first part of the paper is devoted to approaches within the transformational conception of logical implication emphasizing the contributions of Bolzano, Russell, Tarski, Quine and current model – theory. The second part of the paper discusses approaches within the informational conception of logical implication. Carnap and Bar Hillel’s extrinsic approach and Corcoran’s intrinsic approach developed in his information-theoretic logic are respectively compared and some of their philosophical underpinnings are brought to light. The paper concludes with some final remarks concerning the ontic question of the nature of logical implication and the epistemic question of the human access to this relation in both conceptions of logical implication. The article also attempts to connect the views of these representative thinkers with the current philosophical debate on the nature of logical consequence and the role of set-theoretic model theory in philosophy of logic. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

Tarski's general conception of logic placed it at the center of all rational thought, and he took its aim to be the creation of a unified conceptual apparatus. In pursuit of this conviction, from his base at the University of California in Berkeley in the post-war years he campaigned vigorously on behalf of logic, locally, nationally and internationally. Though Tarski was ecumenical in his efforts to establish the importance of logic in these various ways, in his own work—even that (...) part of it which is considered to be philosophically significant—he was first and foremost a mathematician, and, even more so, a set-theoretical one. The paper traces this particular conception of logic back to the influence of his teachers in set theory and topology in Warsaw, and then details how it affected his explication of the notions of satisfaction, truth, logical consequence and logical term. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

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

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove (...) ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃ x) [ x is a strongly inaccessible cardinal]. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical (...) terms, since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

Some philosophers find the following thesis attractive: for every logical constant C there is a set of logical rules of inference R such that a subject knows the meaning of C if and only if she accepts the rules in R. I point out some obvious but, apparently, easily forgotten difficulties concerning this thesis.

The article is devoted to identifying the concept of the meaning of life in the critical philosophy of A.I. Vvedensky, where special attention is paid to the methodological foundations and the historical and philosophical context. The formulation of the question about the meaning of life is one of the ultimate questions in philosophy, the answer to which makes it possible to determine the motives of human activity. In Vvedenskyʼs philosophy, the problem of goal-setting in life is revealed in the prism (...) of a key research intention – the construction of a scientifically revised worldview based on the principles of criticism. As a result, the question of life and its purpose is reduced to the formulation of the question of the logical permissibility and boundaries of questioning. The article attempts to reconstruct the philosopherʼs reasoning and identify the most significant definitions, which include morality, mental life, and the purpose of life. The disclosure of the concept of life in Vvedenskyʼs philosophy invariably requires consideration of the context in which the development of the philosopherʼs ideas takes place. Based on a comprehensive analysis, it is concluded that the concept of life is an extension of the main goal of Vvedenskyʼs research – the disclosure of the conditions for the formation of a reliable worldview, focused on the principles of scientific knowledge in accordance with moral values. Another important consequence is the provision on the priority of morality in relation to scientific and philosophical knowledge. The foundations of morality, to which the meaning of life belongs, can be presented only in a special form of applicability of critical reason – critical faith. Thus, the question of the specifics of critical metaphysics and the importance of differentiation of types and forms of faith in the work of the Russian neo-Kantian are clarified. (shrink)

From the editors of Voprosy filosofii:From August 17 to 22, the Eighth International Congress on the Logic. Methodology, and Philosophy of Science will convene in Moscow. The theme of this congress is "Man, Science, Humanism."The work of the congress will be organized in the following sections: 1. Foundations of mathematical reasoning. 2. The theory of models. 3. Foundations of calculability and recursion theory. 4. The theory of sets. 5. General logic. 6. The general methodology of science. 7. Foundations of probability (...) and statistical inference. 8. Foundations of the physical sciences. 9. Foundations of the biological sciences. 10. Foundations of psychology and the cognitive sciences. 11. Foundations of the social sciences. 12. Foundations and philosophy of linguistics. 13. The history of logic, methodology, and the philosophy of science.Two symposia are planned for the congress: "Science and Ethics" and "The New Style of Scientific Thinking in Natural Science." In addition to the main sections, several round tables will be organized at the congress on the following topics: J. The methodology of solving global problems of the modern epoch: a systems approach. 2. Methodological and social problems of computerization. 3. Factors in the development of modern scientific knowledge. 4. Methodological problems in the all-around development of man.The journal Voprosy filosofii has published a number of articles related to these problems over the past year . Soviet scholars are taking an active part in the preparation and organization of this congress. The journal plans in subsequent issues to publish materials for the congress under the most varied rubrics. In this issue, with D. P. Gorskii's article "The Concept of Real and Ideal Types," we are opening a special section in which we shall publish basic materials devoted to the Eighth International Congress on the Logic, Methodology, and Philosophy of Science. (shrink)

I argue that the medieval form of dialectical disputation known as obligationes can be viewed as a logical game of consistency maintenance. The game has two participants, Opponent and Respondent. Opponent puts forward a proposition P; Respondent must concede, deny or doubt, on the basis of inferential relations between P and previously accepted or denied propositions, or, in case there is none, on the basis of the common set of beliefs. Respondent loses the game if he concedes a contradictory (...) set of propositions. Opponent loses the game if Respondent is able to maintain consistency during the stipulated period of time. The obligational rules are here formalised by means of familiar notational devices, and the application of some game-theoretical concepts, such as (winning) strategy, moves, motivation, allows for an analysis of some crucial properties of the game. In particular, the primacy of inferential (syntactic) relations over semantic aspects and the dynamic character of obligations are outlined. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s (...) Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

Far from being of interest only to argumentation theorists, conceptions of speech acts play an important role in practitioners’ self-reflection on their own activities. After a brief review of work by Houtlosser, Jackson and Kauffeld on the ways that speech acts provide normative frameworks for argumentative interactions, this essay examines an ongoing debate among scientists in natural resource fields as to the appropriateness of the speech act of advocating in policy settings. Scientists’ reflections on advocacy align well with current scholarship, (...) and the scholarship in turn can provide a deeper understanding of how to manage the communication challenges scientists face. (shrink)

The goal of this paper is a philosophical explication and logical rectification of the notion of concept. We take into account only those contexts that are relevant from the logical point of view. It means that we are not interested in contexts characteristic of cognitive sciences, particularly of psychology, where concepts are conceived of as some kind of mental objects or representations. After a brief recapitulation of various theories of concept, in particular Frege’s and Church’s ones, we propose (...) our own theory based on procedural semantics of Transparent Intensional Logic (TIL) and explicate concept in terms of the key notion of TIL, namely construction viewed as an abstract, algorithmically structured procedure. (shrink)

The concept of molecular structure is one of the most important concepts of chemistry. In fact, molecular structure is closely related to the concept of chemical substance and its set of properties, and it is the main factor in the explanation of reactivity. In fact, much of the behavior of substances is explained in terms of the structure of their component molecules. This may explain why people tend to take the notion of molecular structure for granted. However, the problem begins (...) already when it comes to specifying the concept, since there is no clear definition of ‘molecular structure’ valid for all chemistry. The purpose of the present article is to show that the term ‘molecular structure’ subsumes very different notions, which depend on how the molecule and its components are conceived, and each of which brings its own difficulties. In particular, we will focus on how the molecule and its structure are devised from a classical view and from a quantum–mechanical view, and will discuss the different problems related with molecular structure that arise in each case. (shrink)

We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we investigate the case where this is done using the wellfoundedness property. Our hypothesis is that non-wellfounded sets (so-called hypersets) give rise to a different quality of virtual reality than do familiar wellfounded sets. To elaborate this hypothesis, we describe virtual reality through Sommerhoff’s categories of first- and second-order self-awareness; introduced as necessary conditions for (...) consciousness in terms of higher cognitive functions. We then propose a representation of first- and second-order self-awareness through sets, and assume that these sets, which we call events, originally form a collection of wellfounded sets. Strong virtual reality characterizes virtual reality environments which have the limited capacity to create only events associated with wellfounded sets. In contrast, the logically weaker and more general concept of weak virtual reality characterizes collections of virtual reality mediated events altogether forming an entirety larger than any collection of wellfounded sets. By giving reference to Aczel’s hyperset theory we indicate that this definition is not empty because hypersets encompass wellfounded sets already. Moreover, we argue that weak virtual reality could be realized in human history through continued progress in computer technology. Finally, within a more general framework, we use Baltag’s structural theory of sets (STS) to show that within this hyperset theory Sommerhoff’s first- and second-order self-awareness as well as both concepts of virtual reality admit a consistent mathematical representation. To illustrate our ideas, several examples and heuristic arguments are discussed. (shrink)

Though it enjoys widespread support, the claim that perceptual experiences possess nonconceptual content has been vigorously disputed in the recent literature by those who argue that the content of perceptual experience must be conceptual content. Nonconceptualism and conceptualism are often assumed to be well-defined theoretical approaches that each constitute unitary claims about the contents of experience. In this paper I try to show that this implicit assumption is mistaken, and what consequences this has for the debate about perceptual experience. I (...) distinguish between two different ways that nonconceptualist proposals about perceptual content can be understood: as claims about the constituents that compose perceptual contents or as claims about whether a subject’s undergoing experiences with those contents requires them to possess the concepts that characterize those contents. I maintain that these ways of understanding conceptualism and nonconceptualism are orthogonal to one another. This is revealed by the conceptual coherence of positions in which the contents of experiences have both conceptual and nonconceptual features; positions which possess their own distinctive sources of philosophical motivation. I argue that the fact that there is a place in conceptual space for such positions, and that there may be good reason for theorists to adopt them, creates difficulties for both the central argument for nonconceptualism and the central argument for conceptualism. I set out each of these arguments; the Argument from Possession-Independence and the Epistemically-Driven Argument. I then try to show how the existence of mixed positions about perceptual content derived from a clear distinction between compositional and possessional considerations constitutes a significant obstacle for those arguments as they stand. The takehome message of the paper is that unless one clearly acknowledges the distinction between issues about the composition of perceptual content and issues about how subject’s capacities to undergo certain experiences relates to their possession of concepts one runs the risk of embracing unsatisfying philosophical arguments in which conclusions relevant to one conception of nonconceptual and conceptual content are grounded on arguments that concern only the other; arguments that cannot, in themselves, sustain them. (shrink)

In this paper, I examine several commentaries and glosses on the prolegomenon of Najm al-Dīn al-Kātibī’s (d. 675/1276–77) Shamsiyya that relate to debates on the Aristotelian and Ibn Sīnān theory of science in the postclassical period. Chief among the commentaries of the Shamsiyya is Quṭb al-Dīn al-Rāzī’s (d. 766/1365) Taḥrīr al-qawāʿid al-manṭiqiyya. This commentary, rather than the base text of the Shamsiyya, set the stage for later interpretations by Mirak al-Bukhārī (fl. 733/1332), Naṣīr al-Dīn al-Qāshānī (d. 755/1354), Saʿd al-Dīn al-Taftāzānī (...) (d. 792/1390), al-Sayyid al-Sharīf al-Jurjānī (d. 816/1413), Dāwūd al-Khwāfī (fl. 839/1465) and ʿIṣām al-Dīn Isfarāyinī (d. 945/1538). I focus on three issues that were raised in these interpreta­tions of the Shamsiyya’s prolegomenon: (1) the place of the elements of sciences in logical corpus, (2) the notion of the prolegomenon and its content, and (3) the real essence of a science. I attend to the particular debates and contentions on these issues to reveal the general idea of science at that time. (shrink)

This textbook is a second edition of the successful, Mathematical Logic: On Numbers, Sets, Structures, and Symmetry. It retains the original two parts found in the first edition, while presenting new material in the form of an added third part to the textbook. The textbook offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Part I, Logic Sets, and Numbers, shows how mathematical logic is used to (...) develop the number structures of classical mathematics. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments. The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. The added Part III to the book is closer to what one finds in standard introductory mathematical textbooks. Definitions, theorems, and proofs that are introduced are still preceded by remarks that motivate the material, but the exposition is more formal, and includes more advanced topics. The focus is on the notion of countable categoricity, which analyzed in detail using examples from the first two parts of the book. This textbook is suitable for graduate students in mathematical logic and set theory and will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background. (shrink)

I provide a survey of the contents of the works belonging to Aristotle's Organon in order to define their nature, in the light of his declared intentions and of other indications (mainly internal ones) about his purposes. No unifying conception of logic can be found in them, such as the traditional one, suggested by the very title Organon, of logic as a methodology of demonstration. Logic for him can also be formal logic (represented in the main by the De (...) Interpretatione), axiomatized syllogistic (represented in the main by the Prior Analytics) and a methodology of dialectical and rhetorical discussion. The consequent lack of unity presented by those works does not exclude that both the set of works called Analytics and the set of works concerning dialectic (Topics and Sophistici Elenchi) form a unity, and that a certain priority is attributed to the analytics with respect to dialectic. (shrink)

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven to be (...) extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students. (shrink)

The concept of formal ontology was first developed by Husserl. It concerns problems relating to the notions of object, substance, property, part, whole, predication, nominalization, etc. The idea of formal ontology is present in many of Husserl?s works, with minor changes. This paper provides a reconstruction of such an idea. Husserl?s proposal is faced with contemporary logical orthodoxy and it is presented also an interpretative hypothesis, namely that the original difference between the general perspective of usual model theory and (...) formal ontology is grounded in the fact that this latter starts from an intended interpretation and not from the set of all the possible interpretations. (shrink)

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two (...) concepts as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely. (shrink)

Spinoza teaches that the nature of things expresses itself in two ways: in motion and in the world of bodies, on the one hand, and in the intellect with its world of ideas, on the other. Physical bodies are in perpetual motion – arising, changing, disappearing; since the human body also participates in these processes, they are perceived by senses. Spinoza calls the knowledge of sensual properties and duration of the existence of bodies “imagination”. The human mind processes images of (...) bodies with the help of a special kind of abstractions, the “beings of reason”, and then it forms, on the basis of “general concepts”, adequate ideas about the “inner essences” of things, their “order and connec­tion”, i.e. about the causes for the existence of things and their causal relations. Scientific definitions, formulae and models represent these causes and relations in a perfectly pure form – “under a species of eternity”. The article shows how Spinoza delineates the limits of human cognition: they are set by the nature of mind as the idea of the body (man con­sists of two modes of substance – the living body and its idea/mind in the infinite intel­lect). The wider the scope of the body’s activity, the more our mind knows about the world and about itself. The last section of the article describes the relationship of rea­son to virtue, logic to ethics, and characterises the way of life “according to the guidance of reason”. Social life appears to Spinoza as a battle of “passions”, the passive affects of solidarity and enmity. Society cannot live according to the guidance of reason, but our minds can find a form of government that minimises the damage caused by the “blind af­fects” of the rulers and the crowd: Spinoza considered democracy to be the “completely absolute state”. (shrink)

This paper sets out to analyze how causation works by focusing on biology, as represented by epidemiology and by scientific information on how the body works (“physiology”). It starts by exploring the specificity of evolved physiological systems, in which evolutionary, developmental and proximal causes all fit together, and the concept of function is meaningful; in contrast, this structure does not apply in epidemiology (or outside biology). Using these two contrasting branches of biology, I examine the role both of mechanism and (...) of difference making in causation. I find that causation necessarily involves both mechanism and difference making, and that they play complementary roles. Both are seen as ontologically necessary, even if the evidence is not always available for both. Influential monist accounts that focus on one of these, at the expense of ignoring the other, are found to be inadequate on these and on other grounds. Recent attempts to combine them are reviewed, notably that of Russo and Williamson (Int Stud Philos Sci 21:157–170, 2007), and it is argued that their epistemic view requires there to be a source of the different types of evidence that a rational agent would consider, and that this source must be ontic. I then analyze how causal relationships work in evolved physiological systems and in those studied by epidemiology, with a particular focus on how mechanism interacts with input. Finally, I consider this concept of causation from the perspective of everyday language, and of its possible generalisability outside biology. (shrink)

The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove (...) new theorems in. (shrink)

In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further (...) defends her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences. (shrink)

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after (...) all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory. (shrink)

In everyday speech, expressions of the type "that thing exists" are frequently employed. What do they mean? They must be dealt with at the logical level where we seek greater precision. Also at the philosophical level, the predicate "exists" stands in need of analysis, inasmuch as its meanings are associated in one way or another with the meanings of the term "reality." It might also be stated that every entity, to the degree that it is "real" in one sense (...) or another, exists in a manner distinctive to it. In this sense, one might say that this is a problem of cognition of the history of an object, and its results are set forth in various existential judgments. Nevertheless, analysis of "existence" precisely as a predicate is necessary. It bears a specifically logical character and, moreover, is related in a lesser degree to the purely linguistic side of things than in the case of analysis of the meanings of the predicate "to be.". (shrink)