In his recent article, ``Self-Consciousness'’, George Bealer has set outa novel and interesting argument against functionalism in the philosophyof mind. I shall attempt to show, however, that Bealer's argument cannotbe sustained.In arguing for this conclusion, I shall be defending three main theses.The first is connected with the problem of defining theoreticalpredicates that occur in theories where the following two features arepresent: first, the theoretical predicate in question occurswithin both extensional and non-extensional contexts; secondly, thetheory in question asserts that the relevant (...) theoretical states enterinto causal relations. What I shall argue is that a Ramsey-styleapproach to the definition of such theoretical terms requires twodistinct quantifiers: one which ranges over concepts, and theother which ranges over properties in the world.My second thesis is a corollary: since the theories on whichBealer is focusing have both of the features just mentioned, and sincethe method that he employs to define theoretical terms in his argumentagainst functionalism does not involve both quantifiers that range overproperties and quantifiers that range over concepts, that method isunsound.My final thesis is that when a sound method is used, Bealer's argumentagainst functionalism no longer goes through.The structure of my discussion is as follows. I begin by setting out twoarguments – the one, a condensed version of Bealer's argument, andthe other, an argument that parallels Bealer's argument very closely.The parallel argument leads to a conclusion, however, that, rather thanbeing merely somewhat surprising, seems very implausible indeed. Forwhat the second argument establishes, if sound, is that there can betheoretical terms that apply to objects by virtue of their first-orderphysical properties, but whose meaning cannot be defined via aRamsey-style approach.Having set out the two parallel arguments, I then go on to focus uponthe second, to determine what is wrong with it. My diagnosis will bethat the problem with the argument arises from the fact that it involvesdefining a theoretical term that occurs both inside and outside ofopaque contexts, for the method employed fails to take into account thefact that the types of entities that are involved in the relevanttruthmakers are different when a sentence occurs within an extensionalcontext from those involved when a sentence occurs within anon-extensional context.I then go on to discuss how one should define a theoretical term thatoccurs within such theories, and I argue that in such a case one needstwo quantifiers, ranging over different types of entities – on theone hand, over properties and relations, and the other, over concepts. Ithen show that, when such an approach is followed, the argument inquestion collapses.I then turn to Bealer's argument against functionalism, and I show,first, that precisely the same method of defining theoretical terms canbe applied there, and, secondly, that, when this is done, it turns outthat that argument is also unsound.Next, I consider two responses that Bealer might make to my argument,and I argue that those responses would not succeed.Finally, I conclude by asking exactly where the problem lies in the caseof Bealer's argument. My answer will be that it is not simply the factthat one is dealing with a theoretical term that occurs in bothextensional and non-extensional contexts. It is rather the combinationof that feature together with the fact that the theory in questionasserts that the relevant type of theoretical state enters into causalrelations. For the first of these features means that the Ramseysentence for the theory must involve quantification over concepts, whilethe presence of the second feature means that the Ramsey sentence mustinvolve quantification over properties in the world, and so no attemptto offer a Ramsey-style account of the meaning of the relevanttheoretical term can succeed unless one employs both quantification overconcepts and quantification over properties. Bealer, however, in hisargument against functionalism, uses a method of defining theoreticalterms that does not involve both types of quantification, and it isprecisely because of this that his argument does not in the end succeed. (shrink)

In this paper the structuralist approach of theory reconstruction is applied to International Trade Theory. In the basic element the universal laws of the theory are stated and the general concepts are defined in terms of three sets and seven functions. Ricardo’s Theory of Comparative Advantage and the Factor Proportions Theory by Heckscher and Ohlin are reconstructed as specialisations of the basic element. Two intratheoretical constraints are formulated in order to ensure the consistency of the theory. A number of empirical (...) studies are shown to be paradigmatic, successful or questionable applications of International Trade Theory. The reconstruction allows the assessment of the theory from a metatheoretical angle and illuminates the logical interdependencies of its components. (shrink)

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a (...) BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper. (shrink)

The function of concepts must be taken seriously to understand the scientific practices of developing and working with concepts. Despite its significance, little philosophical attention has been paid to the function of concepts. A notable exception is Brigandt (2010), who suggests incorporating the epistemic goal pursued with the concept’s use as an additional semantic property along with the reference and inferential role. The suggestion, however, has at least two limitations. First, his proposal to introduce epistemic goals as the third component (...) of concepts lacks independent grounding, except to account for the rationality of semantic change (the Grounding Problem). Second, it is hardly justified to consider epistemic goals as a semantic property (the Misplacement Problem). To remedy these predicaments, we suggest a new perspective that takes concepts as cognitive entities with a 2-layered structure rather than as merely linguistic entities and develop an account of the function of concepts. We provide empirical evidence showing that functional information affects our cognitive processes. It is claimed that the function of concepts is not a semantic property but a type of meta-information regulating a body of concept-constitutive information. (shrink)

In this paper, I would like to point out some problems of the presently reigning functional concept of the logical form of sentences, which presents itself as the final answer to the question of true logical form of sentences and, with this, as the basic scheme of logic. I believe that the present conception of the logical form of sentences is a historical result, which in many ways surpasses all former concepts of logical form in (...) the history of logic, but which seems not the final concept of the logical form. It contains some immanent limitations which are, in my opinion, linked mainly to the ‘functional’ concept of elementary sentences, which is the foundation of all other logical structures of sentences. (shrink)

This paper begins the study of first-order functions, which are a generalization of truth-functions. The concepts of truth-table and systems of truth-functions, both introduced in propositional logic by Post, are also generalized and studied in the quantificational setting. The general facts about these concepts are given in the first five sections, and constitute a “general theory” of first-order functions. The central theme of this paper is the relation of definition among notions expressed by formulas of first-order logic. We emphasize that (...) logic is not concerned only with the consequence relation among notions expressed by formulas. It also attends to the relation of definition among notions, where a notion is defined from other notions. Sections 5 and 6 deal exclusively with the relation of definition among notions expressed by formulas of first-order logic. In these sections, we study the systems of first-order functions, which are the sets of first-order functions closed under definitions. Sections 7 and 8 are concerned with the relativization of first-order functions to a class of structures. The relativization to a class of structures is a fundamental operation which is used in order to relate the theory of first-order functions with set theory and first-order model theory, a subject which we have barely scratched the surface. The apparatus developed in this paper enables us to define what is a vehicle for the foundation of classical mathematics in set theory, and, in Sect. 8, we prove that first-order logic with one binary predicate variable is not a minimal vehicle for the foundation of classical mathematics in set theory. Sections 9 and 10 introduce further operations and ideals of first-order functions. Besides some results on the influence of the arguments of a first-order function, a result about definability is proved in Sect. 10.1. It is this theorem that provides necessary and sufficient conditions for a first-order function to be in a finitely generated ideal. In Sect. 11, this result is applied to the problem of predicate definability in classes of structures, the problem with which Beth’s theorem dealt in the case of elementary classes. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability (...) of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

The purpose of this note is to analyze the structure of concepts, treated as fuzzy subsets of a certain set. The concepts are assumed to be composed out of more elementary ones. Such an approach was developed mainly to get an empirical access to the values of membership functions of concepts.

Neuroconstructivism, Vol. 1: How the Brain Constructs Cognition implies that brain functioning depends on biofeedback and ecological trajectories. Using the building blocks of Boolean algebra known as logic gates and models of distributed control systems, I suggest that levels of regulatory states are responsible for optimal, pathological, and developmental processes. I include the impact of regulatory and nonregulatory functions on structural development.

This paper: (1) gives a schema of the logical structure of functional explanation in biology; (2) shows that it falls under the covering law model of explanation by proving that the explanandum follows from the explanans; and (3) supports the claim that it captures the logical structure underlying the biological usage by analyzing in detail two cases from biology.

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

This paper casts doubt on a recent criticism of Frege's theory of concepts and extensions by showing that it misses one of Frege's most important contributions: the derivation of the infinity of the natural numbers. We show how this result may be incorporated into the conceptual structure of Zermelo- Fraenkel Set Theory. The paper clarifies the bearing of the development of the notion of a real-valued function on Frege's theory of concepts; it concludes with a brief discussion of the claim (...) that the standard interpretation of second-order logic is necessary for the derivation of the Peano Postulates and the proof of their categoricity. (shrink)

The paper is devoted to the role (function) of reflection in cognition and its modes (types) as part of the cognitive ability. Along with logical and transcendental reflection, Kant's transcendental shift (turn) is discussed, as well as the role of reflection in Kant's schematism (the ability to judge) and the formation of schemas. Particular attention is paid to another mode of reflection – reflexive switching, which underlies not only the formation of pure rational concepts and schemes (Kant's concept of (...) epigenesis; metaphysical deduction of categories), but the epigenesis of a priori forms of sensibility of space and (especially) time, connected with the Kantian synthesis of apprehension in the intuition. The article proposes an original conception of time, the so-called onion model of time, which is based on a reflexive (iterative) apperceptive ladder. (shrink)

To explain the phenomena in the world of our experience, to answer the question “why?” rather than only the question “what?”, is one of the foremost objectives of all rational inquiry; and especially, scientific research in its various branches strives to go beyond a mere description of its subject matter by providing an explanation of the phenomena it investigates. While there is rather general agreement about this chief objective of science, there exists considerable difference of opinion as to the function (...) and the essential characteristics of scientific explanation. In the present essay, an attempt will be made to shed some light on these issues by means of an elementary survey of the basic pattern of scientific explanation and a subsequent more rigorous analysis of the concept of law and of the logical structure of explanatory arguments. (shrink)

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions (...) as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra). (shrink)

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth-conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the `logicality of language', accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter-examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is `blind' to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non-classical---indeed quite exotic---kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis-a-vis its open class terms and employs a deductive system that is basically classical. (shrink)

We develop a category theoretical scheme for the comprehension of the information structure associated with a complex system, in terms of families of partial or local information carriers. The scheme is based on the existence of a categorical adjunction, that provides a theoretical platform for the descriptive analysis of the complex system as a process of functorial information communication.

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth‐conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the ‘logicality of language’, accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter‐examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is ‘blind’ to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non‐classical—indeed quite exotic—kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis‐á‐vis its open class terms and employs a deductive system that is basically classical. (shrink)

R. M. HARE takes the following view of the task of moral philosophy, "The function of moral philosophy—or at any rate the hope with which I study it—is that of helping us think better about moral questions by exposing the logical structure of the language in which this thought is expressed." The purpose of this essay is to show that this restriction of ethics to the logical dimensions of moral discourse is grounded in an excessively narrow conception of (...) the resources of human reason. Specifically, I shall argue that a more thoroughgoing grasp of these resources shows that reason has the capacity to move beyond such a purely logical task and to function as a normative source of moral demands. Central to the argument is the contention that the capacities of reason are restricted neither to the discovery of facts, nor to the opening of human discourse to the demands of logic, but include also the capacity to function as a principle constitutive of human agency as capable of self-awareness. I shall try to show that this latter capacity enables reason to play a morally normative role in human life. The argument will be divided into three major components, beginning with a brief expository sketch of Hare’s position aimed at providing the problematic framework for my argument by stating the essentials of his case against naturalism and for prescriptivism. The second component will describe the reflective capacity of reason and will point out that this capacity is at work in Hare’s writings. Finally, the third component will be directed toward showing how reflectivity is capable of founding a normative ethics. (shrink)

This paper addresses the phenomenological experience of precarity and vulnerability in racialized gender-based violence from a structural perspective. Informed by Indigenous social theory and anti-colonial approaches to intergenerational trauma that link settler colonial violence to the modalities of stress-inducing social, institutional, and cultural violences in marginalized women’s lives, I argue that philosophical failures to understand trauma as a functional, organizational tool of settler colonial violence amplify the impact of traumatic experience on specific populations. It is trauma by design. I (...) explore this through the history of the concept of trauma and its connection to tragedy. I give a brief overview of prominent theories of trauma and contrast these with the work of Indigenous feminist scholar Dian Million (2013), who highlights functional complicity of settler colonial institutions in shaping accounts of trauma in the west. I begin the piece with an important illustration of the kinds of lives and experiences that call for a politicized understanding of trauma in anti-colonial feminist theory. I end by offering an expansive notion of structural trauma that is a methodological pivot for conducting trauma-based gender-based violence research in a decolonial context, which calls for an end to narratives of trauma that are severed from the settler colonial project of Native land dispossession and genocide. [Volume 23, no.1]. (shrink)

. What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what (...) way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. (shrink)

Deep Neural Networks (DNNs) are becoming increasingly important as scientific tools, as they excel in various scientific applications beyond what was considered possible. Yet from a certain vantage point, they are nothing but parametrized functions fθ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{f}_{\varvec{\theta }}(\varvec{x})$$\end{document} of some data vector x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x}$$\end{document}, and their ‘learning’ is nothing but an iterative, algorithmic fitting of the parameters to data. Hence, what could be special about (...) them as a scientific tool or model? I will here suggest an integrated perspective that mediates between extremes, by arguing that what makes DNNs in science special is their ability to develop functional concept proxies (FCPs): Substructures that occasionally provide them with abilities that correspond to those facilitated by concepts in human reasoning. Furthermore, I will argue that this introduces a problem that has so far barely been recognized by practitioners and philosophers alike: That DNNs may succeed on some vast and unwieldy data sets because they develop FCPs for features that are not transparent to human researchers. The resulting breach between scientific success and human understanding I call the ‘Actually Smart Hans Problem’. (shrink)

The concept of a trope (understood as an individual property and not as a figure of speech) plays an important role in contemporary analytical metaphysics. It is, however, often far from clear what the logic of this concept really is. Indeed, there are two equally important intuitions underlying the concept of trope, two intuitions that generate two quite different conceptual frameworks. According to the first intuition, a trope is a particularised property – a property taken as an individual aspect of (...) a particular object. A trope is thus the colour of this table or the shape of this sculpture. In the light of this conception tropes are, as it were, direct abstractions from concrete individuals. According to the second intuition a trope is still such an individual (and abstract) colour or such an individual (and abstract) shape, but now it is not only abstracted from the concrete individual of which it is a trope but also taken in abstraction from the very fact that it has been abstracted this way. We can imagine that the cognitive access to these tropes is obtained by a kind of a double abstraction. All this can sound puzzling at first blush, but in what follows I hope to make the matter a little clearer. Let us anticipate briefly. On the second view, tropes are construed not so much as abstractions from concrete individuals, but rather as primitive items of which the concrete individuals are composed. Tropes so construed are, as it were, metaphysically prior to concrete individuals. We will call tropes of this kind unstructured tropes, whereas tropes conceived as aspects of individual things will be called propositional tropes. This terminology is justified because tropes understood as aspects of things seem to involve something like propositional structure, a structure that tropes conceived as primitive “building blocks” of reality definitely lack. It is sometimes believed that these two concepts of trope are nearly equivalent, that the difference between them is for the most part verbal. Yet it will turn out that there is an important ontological difference here. Propositional tropes are non-trivially propositional and unstructured tropes are non-trivially deprived of such a structure (althought not necessarily deprived of any dependence-links with respect to other tropes). It will be argued that tropes are metaphysically interesting only on the second conception (i.e., as primitive, unstructured items), whereas only on the first conception (tropes as propositionally structured entities) can they function as semantically efficient truthmakers. The moral of the paper will be that our trust in tropes must be mitigated. A trope ontology could plausibly be viewed as a serious metaphysical alternative, but its semantical possibilities should not be exaggerated. (shrink)

[[[ (Here only the chapters 3 – 8, see *** ) First I argue that the prohibition of linguistic self-reference as a solution to the antinomy problem contains a pragmatic contradiction and is thus not only too restrictive, but just inconsistent (chap.1). Furthermore, the possibilities of non-restrictive strategies for antinomy avoidance are discussed, whereby the explicit inclusion of the – pragmatically presuposed – consistency requirement proves to be the optimal strategy (chap.2). ]]] The central question here is that about the (...) actual reason for antinomic structures. It turns out to be a form of negative self-conditioning (chap.3). This makes it necessary to clarify the status of negative concepts (chap.4). The generalization of these considerations (chap.5) leads to the actual analysis of the antinomic basic structure (chap.6): Decisively for the pragmatics of the concept is that it positively owns a meaning, so that positivity is always constituted by the concept qua concept. Thus a negative concept is characterized by a fundamental ambivalence: From a semantic point of view it has negative character, in its pragmatic status as a concept, however, it has positive character. If the meaning is especially that of a negative self-reference, the ambivalence leads to the antinomic constellation of a negative self-condition – that is the crux of the matter here! The concept thus possesses the property defined by it exactly when it does not possess it and vice versa. A closer analysis shows that the function of reflective structures for the occurrence of antinomies has to be judged much more differentiated than previous opinions suggest. Not only are four forms of reflectivity to be distinguished in this context – ontic, semantic, pragmatic reflectivity, and especially the form of negative self-condition; but it is also apparent that these are intertwined with each other in a way that is difficult to be understood. The astonishing variety of relationships associated with this makes the irritation that has always emanated from the antinomy problem appear more comprehensible. In the developed pragmatic perspective furtheron parallels to the structure of self-consciousness become visible (chap.7). I conclude with considerations about the significance of the antinomic structure for the problem of dialectics, especially for the synthesis formation of mutually exclusive terms (chap.8). (shrink)

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this (...) family one finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are. (shrink)

In recent years, academics and educators have begun to use software mapping tools for a number of education-related purposes. Typically, the tools are used to help impart critical and analytical skills to students, to enable students to see relationships between concepts, and also as a method of assessment. The common feature of all these tools is the use of diagrammatic relationships of various kinds in preference to written or verbal descriptions. Pictures and structured diagrams are thought to be more comprehensible (...) than just words, and a clearer way to illustrate understanding of complex topics. Variants of these tools are available under different names: “concept mapping”, “mind mapping” and “argument mapping”. Sometimes these terms are used synonymously. However, as this paper will demonstrate, there are clear differences in each of these mapping tools. This paper offers an outline of the various types of tool available and their advantages and disadvantages. It argues that the choice of mapping tool largely depends on the purpose or aim for which the tool is used and that the tools may well be converging to offer educators as yet unrealised and potentially complementary functions. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

Abstract - Evolutionary, ecological and ethical studies are, at the same time, specific scientific disciplines and, from an historical point of view, structurally linked domains of research. In a context of environmental crisis, the need is increasingly emerging for a connecting epistemological framework able to express a common or convergent tendency of thought and practice aimed at building, among other things, an environmental policy management respectful of the planet’s biodiversity and its evolutionary potential. -/- Evolutionary biology, ecology and ethics: at (...) first glance, three different objects of research, three different worldviews and three different scientific communities. In reality, there are both structural and historical links between these disciplines. First, some topics are obviously common across the board. Second, the emerging need for environmental policy management has gradually but radically changed the relationship between these disciplines. Over the last decades in particular, there has emerged a need for an interconnecting meta-paradigm that integrates more strictly evolutionary studies, biodiversity studies and the ethical frameworks that are most appropriate for allowing a lasting co-evolution between natural and social systems. Today such a need is more than a mere luxury, it is an epistemological and practical necessity. -/- In short, the authors of this volume address some of the foundational themes that interconnect evolutionary studies, ecology and ethics. Here they have chosen to analyze a topic using one of these specific disciplines as a kind of epistemological platform with specific links to topics from one or both of the remaining disciplines. Michael Ruse’s chapter, for instance, elucidates some of the structural links between Darwinismand ethics. Ruse analyzes the Evolutionism vs. Creationism debate, emphasizing the risks run by scientists when they ideologize the scientific content of their studies. In the case of the contributions of Jean Gayon and Jean-Marc Drouin, which respectively deal with the disciplines of evolutionary biology and ecology, some central connections have been developed between these two disciplines, while reserving the option to consider in detail their topic in order to discover essential features ormeanings. Gayon analyzes the multilayered meanings of “chance” in evolutionary studies and the methodological implications that accompany such disparatemeanings. Froma similar analytical perspective, Drouin’s contribution focuses on the identification and critical evaluation of the different conceptions of time in ecology. Chance and time, factors of evolution in species and ecological systems, play a very important function in both disciplines, and these chapters help to capture their polysemous structure and development. Bryan Norton’s chapter, on adaptive environmental management, is set within an epistemological context where the Darwinian paradigm, ecological knowledge and ethical frameworks meet to give rise to practical, conservationist policies. In his contribution, Patrick Blandin pleads for the necessity of an eco-evolutionary ethics capable of fully encompassing humanity’s responsibility in the future determination of the biosphere’s evolutionary paths. Our value systems must recognize the predominant place that humanity has taken in the evolutionary history of the planet, and integrate the ethical ramifications of scientific advances in evolutionary and ecological studies. The chapter by J. Baird Callicott introduces us to a metaphorical ecological reversion with direct consequences for our moral conduct. If ecology showed that ecosystems are not organisms, recognizing organisms as a kind of ecosystem could be the basis for a new post-modern ecological ethics that lays the foundation for a better moral integration of humans with the environment. The contributions of Robin Attfield and Tom Regan delve into some of the classical issues in environmental ethics, situating them within a broader ecological and evolutionary context. Attfield’s chapter tackles the confrontation between individualistic and ecologically holistic perspectives, their different approaches to the issue of intrinsic value, and their tangled relation to monism and pluralism. Regan’s contribution ponders the criteria that allow individual beings, human and non-human, to own moral rights, the role of the struggle for existence in the relationship between species, and the logical difficulties involved in attributing intrinsic value to collective entities (species, ecosystems). Catherine Larrère’s chapter discusses the opposition between two environmental and ethical worldviews with very different philosophical centers of gravity: nature and technology. These opposing perspectives have direct consequences not only for the perception of the problems at hand and for what entities are deemed morally significant, but also for the proposed solutions. -/- To set out some foundational events in the history of evolutionary biology, ecology and environmental ethics is a first necessary step towards a clarification of their major epistemological orientations. On the basis of this inevitably nonexhaustive history, it will be possible to better position the work of the different contributors, and to build a meta-paradigm, i.e. a connecting epistemological framework resulting from one common or convergent tendency of thought and practice shared by different disciplines. (shrink)

Sittlichkeit seduces: Hegel’s third category of Right, intended to synthesize impartially derived rights with a subjectively centered morality of the good, understandably piques the hopes of his modern readers. How could it not? Sittlichkeit, Ethical Life, holds out the prospect of so much that we still seek. It promises to reconcile welfare and autonomy while guaranteeing concrete content for their product. By combining legal duty with a place for conscience and freedom, it could solve a central problem of politics. Most (...) broadly, Sittlichkeit would graft Enlightenment individualism onto a more Aristotelian conception of social order, merging the benefits of ancient virtue and modern liberty. Who would not want to have their cake and eat it in this way? (shrink)

Sittlichkeit seduces: Hegel’s third category of Right, intended to synthesize impartially derived rights with a subjectively centered morality of the good, understandably piques the hopes of his modern readers. How could it not? Sittlichkeit, Ethical Life, holds out the prospect of so much that we still seek. It promises to reconcile welfare and autonomy while guaranteeing concrete content for their product. By combining legal duty with a place for conscience and freedom, it could solve a central problem of politics. Most (...) broadly, Sittlichkeit would graft Enlightenment individualism onto a more Aristotelian conception of social order, merging the benefits of ancient virtue and modern liberty. Who would not want to have their cake and eat it in this way? (shrink)

This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and (...) culture. Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces. Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecture-based or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge. (shrink)

In his 2010 paper ‘Grounding and Truth-Functions’, Fabrice Correia has developed the first and so far only proposal for a logic of ground based on a worldly conception of facts. In this paper, we show that the logic allows the derivation of implausible grounding claims. We then generalize these results and draw some conclusions concerning the structural features of ground and its associated notion of relevance, which has so far not received the attention it deserves.

This paper considers the role of constitutivity and normativity in Frege’s conception of logic. It outlines an historical interpretation with two goals. First, it traces these concepts back to their origins in Kant’s philosophy. Second, it considers some of the different ways in which the issue of normativity and its proper grounding was addressed in the neo-Kantian tradition and in early analytic philosophy. Some neo-Kantians worked out an epistemic-normative conception of objective judgment, according to which the objectivity of cognition is (...) constituted by distinctively logical norms. In Frege we find an original and sophisticated version of this line of thought. For Frege, the normative and constitutive roles of logic come to the fore in the articulation of scientific reason which follows the classical model of demonstrative science as cognitio ex principiis. Wittgenstein’s Tractatus then opens up a fresh Kantian perspective on the constitutivity of logic, one that grounds logic in structure rather than norms, and does so in conscious opposition to Frege and his normative science. Logic is transcendental, according to Wittgenstein, being the essence of the world and of all description. Hence, the normative function of logic becomes, in a way, superfluous. (shrink)

In this paper I examine a cluster of concepts relevant to the methodology of truth theories: 'informative definition', 'recursive method', 'semantic structure', 'logical form', 'compositionality', etc. The interrelations between these concepts, I will try to show, are more intricate and multi-dimensional than commonly assumed.

The functional approach to natural language (FANL) emerged in the late 1960s. It focused on the use and the sub-use of language expressions, taking into account role of the language context and the extra-linguistic situation of a given statements. This approach referred, both conceptually and methodologically, to the tradition of British analytical philosophy of language on the one hand, and to the achievements of the Lvov-Warsaw School on the other. It seems that despite the passage of more than half (...) a century since its inception, this approach has lost nothing of its relevance and its cognitive value, and may even provide inspiration for the exploration new research areas concerning the formulation of language utterances. The aim of this article is to show the potential applications of the FANL as a specific semiotic concept in the field of the philosophy of social sciences. The paper points out how the functional conception can be successfully used in methodological analyses of the logical methodology of social sciences. The case studies presented in the article refer to the research practice of social sciences. Examples of the use and sub-uses of terms are discussed in the light of the polyparadigmatic structure of social sciences and the differences in research patterns accepted by given scientific collectives. (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)

It has been noticed by several authors that the colloquial understanding of anonymity as mere unknown-ness is insufficient. This common-sense notion of anonymity does not recognize the role of the goal for which the anonymity is sought. Starting with the distinction between the intentional and unintentional anonymity (which are usually taken to be the same) and the general concept of the non-coordinatability of traits, we offer a logical analysis of anonymity and identification (understood as de-anonymization). In our enquiry, we (...) focus on the intentional aspect of anonymity and develop a metaphor of “anonymity game” between “perpetrator” and “detective”. Beginning from common-sense intuitions, we provide a formalized, critical notion of anonymity. (shrink)

This paper describes two collaborative projects that illustrate the value of learning symbolic logic and provide students (and instructors) a break from the routine work of learning new symbols or proof techniques. The first project has students work together to reconstruct the argument in Peter Singer’s “Famine, Affluence, and Morality”. This project has the benefit of showing students that what they are reading in college has an underlying logical structure and that their knowledge of conditionals, conjunctions, etc. functions in (...) real, argumentative discourse. The second project introduces students to four key concepts: self-reference, paradox, and metatheory, and then exposes them to key metatheoretic concepts (consistency and completeness) and to Gödel’s incompleteness proof. (shrink)

The work of Luciano Floridi lies at the interface of philosophy, information science and technology, and ethics, an intersection whose existence and significance he was one of the first to establish. His closely related concepts of a philosophy of information (PI), informational structural realism, information logic (IL), and information ethics (IE) provide a new ontological perspective from which moral concerns can be addressed, especially but not limited to those arising in connection with the new information and communication technologies. In this (...) paper, I relate Floridi's approach to another novel perspective, namely, that of an extension of logic to complex real processes, including those of information production and transfer. This non-propositional, non-truth-functional logic (logic in reality (LIR)) is grounded in the fundamental dualism (dynamic opposition) inherent in energy and accordingly present at all levels of reality. The LIR description of the dynamics of processes and their evolution is relevant to what Floridi refers to as the possible non-linguistic aspects of information. It suggests answers to some of Floridi's “outstanding problems” in PI related to the ontological status of information and how it is used in cognition. Floridi's IL retains the formal structure of the doxastic and epistemic logics from which he correctly distinguishes it and is the basis for his conceptual PI. However, LIR fulfills Floridi's implied requirement that logic be regarded as a natural phenomenon dealing with other natural phenomena, recovering its original philosophical function. LIR provides a logical foundation for discussion of ethical questions based on kinds of information that complements IL. Both are reconsiderations of logic that, as Marijuan suggests, may be necessary for the advancement of information technology in an ethical direction (cf. also Brenner). IE focuses on entities as constituted by information in an overall strategy that generalizes the concept of moral agents. LIR and its related ontology naturalize critical aspects of Floridi's theses, especially, the moral value of being as such and a non-separable joint responsibility of individuals and groups. I compare IE to other current approaches to ethics and information technology (e.g., phenomenological and social constructivist). Ethical information is defined “ecologically” in process terms as reality in a physical space (cf. Floridi), with an intentional “valence,” positive and negative, in the morally valued interaction between producer and receiver. LIR is neither topic-neutral nor context independent and can support an ethics involving apparently contradictory perspectives (e.g., internalist and externalist). Ethics involves practical reasoning, and unlike standard logics, LIR supports Magnani's approach to abductive reasoning in rational moral decision making. The basis of moral responsibility and the consequent behavior of individuals involved in information and communications technologies is the same logical–metaphysical principle of dynamic opposition instantiated at other levels of reality. The way moral responsibilities are actively accepted (or not) by individuals supervenes on their primitive psychological structure, which in turn reflects an evolutionary development grounded in the fundamental dualism of the physical world. The paper concludes with some suggestions of areas of philosophical research, such as causality, identity, and the ontological turn, where convergence of the Floridi and LIR approaches might be envisaged. Their overall motivation is the same, namely, the development of strategies for reinforcing and increasing ethical sensitivity wherever possible. The ethical information concept outlined in the paper supports the function of IE, assigned to it by Floridi, of potentially determining what is right and what is wrong. (shrink)

Recent developments in the semantics of natural language seem to lead to a genuine synthesis of ideas from linguistics and logic, producing novel concepts and questions of interest to both parent disciplines. This book is a collection of essays on such new topics, which have arisen over the past few years. Taking a broad view, developments in formal semantics over the past decade can be seen as follows. At the beginning stands Montague's pioneering work, showing how a rigorous semantics can (...) be given for complete fragments of natural language by creating a suitable fit between syntactic categories and semantic types. This very enterprise already dispelled entrenched prejudices concerning the separation of linguistics and logic. Having seen the light, however, there is no reason at all to stick to the letter of Montague's proposals, which are often debatable. Subsequently, then, many improvements have been made upon virtually every aspect of the enterprise. More sophisticated grammars have been inserted (lately, lexical-functional grammar and generalized phrase structure grammar), more sensitive model structures have been developed (lately, 'partial' rather than 'total' in their com position), and even the mechanism of interpretation itself may be fine-tuned more delicately, using various forms of 'representations' mediating between linguistic items and semantic reality. In addition to all these refinements of the semantic format, descriptive coverage has extended considerably. (shrink)

Fregean thoughts (i.e. the senses of assertoric sentences) are structured entities because they are composed of simpler senses that are somehow ordered and interconnected. The constituent senses form a unity because some of them are ?saturated? and some ?unsaturated?. This paper shows that Frege's explanation of the structure of thoughts, which is based on the ?saturated/unsaturated? distinction, is by no means sufficient because it permits what I call ?wild analyses?, which have certain unwelcome consequences. Wild analyses are made possible because (...) any ?unsaturated? sense that is a mode of presentation of a concept together with any ?saturated? sense forms a thought. The reason is that any concept can be applied to any object (which is presented by a ?saturated? sense). This stems from the fact that Frege was willing to admit only total functions. It is also briefly suggested what should be done to block wild analyses. (shrink)

Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept (...) of an effective procedure; mundane procedures and Turing machine procedures are different kinds of procedure. Moreover, the same sequence ofparticular physical action can realize both a mundane procedure and a Turing machine procedure; it is sequences of particular physical actions, not mundane procedures, which enter the world of mathematics. I conclude by discussing whether genuinely continuous physical processes can enter the world of real numbers and compute real-valued functions. I argue that the same kind of correspondence assumptions that are made between non-numerical structures and the natural numbers, in the case of Turing machines and personal computers, can be made in the case of genuinely continuous, physical processes and the real numbers. (shrink)

This exposition of ockham's theory of (common, Personal) supposition involves the logical form of the four descent/ascent conditions and the logical relations of these with the three main modes of supposition. Central theses: each condition is a one-Way entailment, Each mode is a truth-Functional combination of conditions, Two of the three modes are not even coextensive with the two-Way entailments commonly taken as their definitions. Ockham's idea of "the singulars" of a general proposition is vague and problematic (...) and the entailment used in the descent/ascent conditions probably cannot be taken to be logical consequence in any strict sense. (shrink)

In a nutshell, this dissertation builds a bridge between an informal psychological model of emotions on the one hand, and implementation of emotions in robots and virtual characters on the other hand. This is done by formalizing a psychological model of emotions. With that we1 mean translating the concepts used in the psy- chological model to a logical language with unambiguous semantics. The resulting formalization of emotions will give rise to theorems that can be proved to be true. These (...) theorems are properties of the formal model and can be checked against the original psychological model of emotions. If they match, this will mean that the formalization is truthful with respect to the psychological model. The description of the model of emotions in this formal language can then be used as a foundation for implementing emotions in robots and virtual characters. (shrink)

The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators form (...) the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators. (shrink)

In Process and Reality (1929) and subsequent writings, A.N. Whitehead builds on the success of the Frege-Russell generalization of the mathematical function and develops his philosophy on that basis. He holds that the proper generalization of the meaning of the function shows that it is primarily to be defined in terms of many-to-one mapping activity, which he terms 'creativity'. This allows him to generalize the range of the function, so that it constitutes a universal ontology of construction or 'process'. He (...) analyzes the concept of God in terms of functional mapping to structure, and he defines finite entities as iterative 'occasions' of mapping activity. He thus challenges the widespread logical-analytical view that the connectives and variables of a function in its different instantiations are merely numerically different, and he develops a fallibilist theory of activity as essentially serial in nature. (shrink)