Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (or bodies). (...) Here I focus on what is arguably the first thorough proposal of this sort, Whitehead’s theory of “extensive abstraction”, offering a critical reconstruction of the theory through its successive installments: from the purely mereological version of ‘La théorie relationniste de l’espace’ (1916) to the refined versions presented in An Enquiry Concerning the Principles of Natural Knowledge (1919) and in The Concept of Nature (1920) to the last, mereotopological version of Process and Reality (1929). (shrink)

The death of Alfred North Whitehead late in 1947 was a double loss. Those who knew Whitehead, even slightly, feel keenly the loss of a warm and stimulating personality, which in the last years of his retirement had mellowed to a benign radiance. The wider circle of students and teachers of philosophy who knew him through his writing alone regret the passing of the man who, many thought, was the most capable cosmologist of our time.

Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$. On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok (...) and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as,, or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$, with,, and. (shrink)

This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters (...) over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics. (shrink)

Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting in (...) which they can still be used. (shrink)

Rudolph Carnap’s Aufbau was one of the more ambitious philosophical programs of the twentieth century. His proposal was to begin with elementarerlebnisse —cross sections of one total stream of experience temporally limited by the least perceivable segment of time—and an undefined primitive relation, recollection of similarity, holding between the elementary experiences. Without any further non-logical terms, the goal was to utilize a logic, such as that of Principia Mathematica, and actually to construct logically, or to define formally, all the kinds (...) of entities, phenomenal, physical, psychological and cultural, which we usually take to make up our world. Unfortunately, the program is not completed. Only what Carnap refers to as the lower levels is actually formally constructed—that is, what might be called the phenomenal level. The remainder of the constructional program is only suggested in outline. (shrink)

We describe a tool that allows the merging of extensions of argumentation frameworks, following the approach defined by 33–42). The tool is implemented in Java, and is highly modular thanks to Object Oriented Programming principles. We describe a short experimental study that assesses the scalability of the approach, as well as the impact on runtime of using an integrity constraint.

De 1905 à 1922, l uvre d'Alfred North Whitehead a pour but principal de montrer comment les objets fondamentaux de la géométrie, de la physique et de la perception sont abstraits à partir d'un seul et unique type d' entités définies comme les éléments ultimes de l'expérience sensible : les événements. WHitehead développe dès lors la méthode de l'abstraction extensive : un modèle logico-mathématique qui permet d'exprimer ces différents types d'objets dans les termes mêmes des événements et de (...) leurs relations. L'élaboration d'un concept événementiel de nature constitue l'une des principales lignes de force des recherches whiteheadiennes : l'identité doit être définie dans les seuls termes des événements. (shrink)

The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)

Abstract Objections to life extension often focus on its effects for individual well-being. Prominent amongst these concerns is the possibility that life extending technologies will extend lifespan without preventing the ageing of the mind. Writers on the subject express the fear that life extending drugs will keep us physically youthful whilst our minds decay, succumbing to dementia, boredom, and loneliness. Generally these fears remain speculative, in part due to the absence of genuine life extending technologies. In this paper, however, I (...) examine the implications of an existing life extension technology. Caloric restriction (CR) and drugs that mimic its effects, such as rapamycin, metformin and resveratrol have been shown to increase average and maximum lifespan in a wide variety of organisms, and seem likely to do so in humans. Moreover, some CR mimetic drugs (CRMs) are already widely used. This means that they present a pressing test case for fears about mental ageing in an extended life. Misgivings about mental ageing can be divided into biomedical factors such as the likelihood of brain ageing, and psychological factors such as loss of meaning and boredom. I argue that studies of CR suggest that brain ageing will be beneficially slowed. However, it is less clear that deleterious aspects of psychological ageing can be similarly retarded. I argue that this reduces the desirability of life extension unless major social changes can be made. (shrink)

In this article, I argue for the redundancy of Whitehead’s Platonic notion of the extensive continuum, counterposing it to his related notion of an atomic “ether of events.” I argue that Whitehead’s atomic ether is more compatible with orthodox general relativity than generally supposed and remarkably close to the contemporary idea of a discrete manifold in the causal set theory of quantum gravity. I argue that the method of extensive abstraction complements Whitehead’s atomic hypothesis by demonstrating the (...) ultimately fictive nature of any continuum. (shrink)

In his pioneering work on Abstract Argumentation, P.M. Dung set a wide scenario by connecting stable models in Logic and Game Theory to simple Abstract Argumentation Frameworks ( AAF), which are essentially directed graphs in which arguments are represented as nodes, and the attack relation is represented by arrows. From such abstraction and simplicity, it is possible to capture important properties in many different fields. The Stable Marriage ( SM) problem is exactly one of such representable problems. Given two (...) sets of individuals partitioned into men and women, a matching is stable when there does not exist any man-woman match by which both man and woman would be individually better off than they are with the person to which they are currently matched. Stable matchings correspond to stable extensions if an AAF is correctly generated from the given SM problem. In this paper we elaborate on the original formulation by Dung, by also encoding SM problems with ties and incomplete preference lists into AAFs. Moreover, we use Weighted Abstract Argumentation to represent optimality criteria in the optimal extension of SM problems, where some matchings are better than others: criteria may consider only the preference of either men, or women, or a more global view obtained by combining the preferences of both. (shrink)

Carneades is a recently proposed formalism for structured argumentation with varying proof standards, inspired by legal reasoning, but more generally applicable. Its distinctive feature is that each statement can be given its own proof standard, which is claimed to allow a more natural account of reasoning under burden of proof than existing formalisms for structured argumentation, in which proof standards are defined globally. In this article, the two formalisms are formally related by translating Carneades into the ASPIC+ framework for structured (...) argumentation. Since ASPIC+ is defined to generate Dung-style abstract argumentation frameworks, this in effect translates Carneades graphs into abstract argumentation frameworks. For this translation, we prove a formal correspondence and show that certain rationality postulates hold. It is furthermore proved that Carneades always induces a unique Dung extension, which is the same in all of Dung's semantics, allowing us to generalise Carneades to cycle-containing structures. (shrink)

In his pioneering work on Abstract Argumentation, P.M. Dung set a wide scenario by connecting stable models in Logic and Game Theory to simple Abstract Argumentation Frameworks ( AAF), which are essentially directed graphs in which arguments are represented as nodes, and the attack relation is represented by arrows. From such abstraction and simplicity, it is possible to capture important properties in many different fields. The Stable Marriage ( SM) problem is exactly one of such representable problems. Given two (...) sets of individuals partitioned into men and women, a matching is stable when there does not exist any man-woman match by which both man and woman would be individually better off than they are with the person to which they are currently matched. Stable matchings correspond to stable extensions if an AAF is correctly generated from the given SM problem. In this paper we elaborate on the original formulation by Dung, by also encoding SM problems with ties and incomplete preference lists into AAFs. Moreover, we use Weighted Abstract Argumentation to represent optimality criteria in the optimal extension of SM problems, where some matchings are better than others: criteria may consider only the preference of either men, or women, or a more global view obtained by combining the preferences of both. (shrink)

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties (...) confronting the most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

The present paper aims to contribute to the substantivalism versus relationalism debate and to defend general relativity (GR) against pseudoscientific attacks in a novel, especially inclusive way. This work was initially motivated by the desire to establish the incompatibility of any ether theories with accelerated cosmic expansion and inflation (motto: where would a hypothetical medium supposedly come from so fast?). The failure of this program is of interest for emergent GR concepts in high energy particle physics. However, it becomes increasingly (...) important to guard scientific results against their misrepresentation by fundamentally anti-scientific agendas. We therefore argue that although it is not known whether the perceived space-time is fundamental (rather than a condensed state or a particular membrane), in a fundamental theory, space-time must be abstract relational: fundamental space-time is the consistent spatial-temporal arrangement of events. To pursue its own goals, this work should be accessible to a wide audience: physicists, philosophers of science, those being tempted by anti-relativity theories, but also those who vigorously defend some orthodox relativity that is not actually supported by GR. It must thus be extensive in order to satisfy the different parties’ desire to have included and understood their respective positions. (shrink)

Dung’s abstract argumentation provides us with a general framework to deal with argumentation, non-monotonic reasoning and logic programming. For the extension-based semantics, one of the basic principles is I-maximality which is in particular related with the notion of skeptical justification. Another one is directionality which can be employed for the study of dynamics of argumentation. In this paper, we introduce two new extension-based semantics into Dung’s abstract argumentation, called grounded-like semantics and initial semantics which satisfy the I-maximality and directionality principles. (...) The initial semantics has many good properties and can be expected to play a central role in studying other extension-based semantics, such as admissible, complete and preferred semantics. (shrink)

RESUMEN El presente artículo está dirigido a sistematizar una concepción teórica y metodológica que sustente el proceso de extensión universitaria en la carrera de Medicina en Cuba. Entre los resultados se destaca el lugar y papel de la extensión universitaria en el sistema de la formación integral del profesional a la que se asigna una connotación especial, de marcado contenido axiológico, coherente con las necesidades y proyecciones sociales que facilita la formación del educando y fortalece la relación institución-comunidad. Es factible (...) de ser aplicado pues puede devenir en una importante contribución a elevar el impacto social de la docencia médica en tanto atiendan aspectos como la preparación de los implicados en las diferentes instancias; la colaboración de los consejos populares y los factores de la comunidad que se manifiestan en el modo de actuación del profesional. ABSTRACT The present article is directed to systematize a theoretical and methodological conception that sustains the process of university extension in the career of Cuban medicine. Among the results he/she stands out the place and paper of the university extension in the system of the integral formation from the professional to which a special connotation is assigned, of marked axiological contained, coherent with the necessities and social projections that it facilitates the formation of the educating and it strengthens the relationship institution-community. It is feasible of being applied then it can become in an important contribution to elevate the social impact of the medical teaching as long as they assist aspects like the preparation of those implied in the different instances; the collaboration of the popular advice and the factors of the community that are manifested in the way of the professional's performance. (shrink)

Words in Semitic texts often consist of a concatenation of word segments, each corresponding to a Part-of-Speech (POS) category. Semitic words may be ambiguous with regard to their segmentation as well as to the POS tags assigned to each segment. When designing POS taggers for Semitic languages, a major architectural decision concerns the choice of the atomic input tokens (terminal symbols). If the tokenization is at the word level the output tags must be complex, and represent both the segmentation of (...) the word and the POS tag assigned to each word segment. If the tokenization is at the segment level, the input itself must encode the different alternative segmentations of the words, while the output consists of standard POS tags. Comparing these two alternatives is not trivial, as the choice between them may have global effects on the grammatical model. Moreover, intermediate levels of tokenization between these two extremes are conceivable, and, as we will aim to show, beneficial. To the best of our knowledge, the problem of tokenization for POS tagging of Semitic languages has not been addressed before in full generality. In this paper, we study this problem for the purpose of POS tagging of Modern Hebrew texts. After extensive error analysis of the two simple tokenization models, we propose a novel, linguistically-motivated, intermediate tokenization model that gives better performance for Hebrew over the two initial architectures. Our study is based on the well-known Hidden Markov Models (HMMs). We start out from a manually devised morphological analyzer and a very small annotated corpus, and describe how to adapt an HMM-based POS tagger for both tokenization architectures. We present an effective technique for smoothing the lexical probabilities using an untagged corpus, and a novel transform for casting the segment-level tagger in terms of a standard, word-level, HMM implementation. The results obtained using our model are on par with the best published results on Modern Standard Arabic, despite the much smaller annotated corpus available for Modern Hebrew.. (shrink)

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to Łoś and Suszko. However, it was recently observed by Cintula and Noguera that both of (...) these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the Łoś–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality \ may have more than one conservative extension of cardinality \ by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana. (shrink)

Gilead et al. offer a thoughtful and much-needed treatment of abstraction. However, it fails to build on an extensive literature on abstraction, representational diversity, neurocognition, and psychopathology that provides important constraints and alternative evidence-based conceptions. We draw on conceptions in software engineering, socio-technical systems engineering, and a neurocognitive theory with abstract representations of gist at its core, fuzzy-trace theory.

Research on temporal comparison has shown that people dissociate themselves from their past to attain a positive self view. Social comparison research has demonstrated that the distinctness of contextually activated information determines whether a recalled self exerts assimilation or contrast effects on the current self. However, hardly any study addressed individual differences. Also, very little is known about whether the ease or difficulty to date past events and experiences influences current self-judgments. We present a new scale capturing the degree of (...) the current self time extension. Three studies support the notion that temporal self-extension determines how past selves are accessed and processed, regarding both the abstractness with which self-knowledge is retrieved and the experienced temporal distance to the past. These findings have important implications for temporal as well as social comparison processes. (shrink)

The need for quantitative measurement represents a unifying bond that links all the physical, biological, and social sciences. Measurements of such disparate phenomena as subatomic masses, uncertainty, information, and human values share common features whose explication is central to the achievement of foundational work in any particular mathematical science as well as for the development of a coherent philosophy of science. This book presents a theory of measurement, one that is "abstract" in that it is concerned with highly general axiomatizations (...) of empirical and qualitative settings and how these can be represented quantitatively. It was inspired by, and represents a generalization and extension of, the last major research work in this field, Foundations of Measurement Vol. I, by Krantz, Luce, Suppes, and Tversky published in 1971. (shrink)

When teaching a novel mathematical concept, should we present learners with abstract or concrete examples? In this experiment, we conduct a critical replication and extension of a well‐known study that argued for the general advantage of abstract examples (Kaminski, Sloutsky, & Heckler, 2008a). We demonstrate that theoretically motivated yet minor modifications of the learning design put this argument in question. A key finding from this study is that participants who trained with improved concrete examples performed as well as, or better (...) than, participants who trained with abstract examples. We argue that the previously reported “advantage of abstract examples” manifested not because abstract examples are advantageous in general, but because the concrete condition employed suboptimal examples. (shrink)

Would be fairer to call Peirce’s philosophy of language “extensionalist” or “intensionalist”? The extensionalisms of Carnap and Quine are examined, and Peirce’s view is found to be prima facie similar, except for his commitment to the importance of “hypostatic abstraction”. Rather than dismissing this form of abstraction (famously derided by Molière) as useless scholasticism, Peirce argues that it represents a crucial (though largely unnoticed) step in much working inference. This, it is argued, allows Peirce to transcend the extensionalist-intensionalist (...) dichotomy itself, through his unique triadic analysis of reference and meaning, by transcending the distinction between (as Quine put it) “things” and “attributes”. (shrink)

ABSTRACT Joseph Halpern and Judea Pearl draw upon structural equation models to develop an attractive analysis of ‘actual cause’. Their analysis is designed for the case of deterministic causation. I show that their account can be naturally extended to provide an elegant treatment of probabilistic causation. 1Introduction 2Preemption 3Structural Equation Models 4The Halpern and Pearl Definition of ‘Actual Cause’ 5Preemption Again 6The Probabilistic Case 7Probabilistic Causal Models 8A Proposed Probabilistic Extension of Halpern and Pearl’s Definition 9Twardy and Korb’s Account 10Probabilistic (...) Fizzling 11Conclusion. (shrink)

Le Corbusier still believed that architecture’s fundamental measure should be man. With the exception of Hilberseimer, who more perspicaciously believed in the advent of an abstract, conceptual and computational form of production, architecture has been late in grasping the interlinkage of science, industry and capitalism. Even if it means a questioning of its own foundations, it can no longer feign ignorance of a production process that relies increasingly on linguistic constructions, on an « ambient factory » of interconnected PCs.

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem1.Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0= 1.Contrary to an impression which exists in (...) some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary1.Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4],Section27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification oftamesimpleK*-groupsof finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption oftameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results fromK-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness. (shrink)

Contents Acknowledgements iii I. Royce and the Interpretation of His Contemporaries Introduction 1 1. Royce and Lotze 25 Introduction to Lotze. Lotze's ontology. Royce's response to Lotze. Successfull metaphysics renders experience broadly consistent without denying types of human experience. Logically testing metaphysical assumptions offers a promising methodology. The individual is not immediately given but realized through a process. The individual is not immediately given but realized through a process. Conclusion. 2. Royce and James 66 The friendship of James and Royce. (...) James's position. James summarized. Royce's response. The ‘abstract’ ideal has its pragmatic aspect. Logic can supplement pragmatism. Experience itself drives the unifying efforts of speculative philosophy. A more consistent pragmatism hypothesizes broader experience and allows an active and constructive process of response to that experience. Conclusion. 3. Royce and Bradley 108 Bradley's destruction of appearances. Bradley's construction? Royce's response - non-technical. Metaphysics is interpretively applicable to special experiential problems. In active willful attention the ethical complements the theoretical, rendering the relation of finite and absolute interpretable. Royce's response - technical development. Conclusion. 4. Royce and Whitehead 163 Statement of Whitehead's three primary presuppositions. Scheme of Whitehead's philosophy: the role of extended events. The building blocks of Whitehead's metaphysics. Process as an action within and upon experience. The process of concrescence. Consciousness and propositions in Whitehead's philosophy. Royce in comparison with Whitehead. Royce's cosmology in outline. Explicit contrast and comparison of Royce's and Whitehead's philosophies. Principles suggested by the contrast of Royce and Whitehead. II. An Extension of Royce’s Philosophy of Interpretation Preamble: The Need for a Re-Envisioned Philosophy of Interpretation 212 5. The Roycean Hypothetical Method and Its Prospects 226 The interpretation of Christianity. Metaphysical foundations of community. The synthesis and promise of Royce's insights. 6. Hyperthematics 269 Working definition. Some methodological considerations. Some general considerations. Some definitions. The foundational hypertheme and the derivatives. Hyperthematic expansion. The hyperthematic structure of the game. The hyperthematic structure of narrative. The hyperthematic structure of dance. The hyperthematic structure of music. Retrospect and prospect. 7. Hyperthematics Applied: The Corporation As a Hyperthematic Structure 339 Introduction. Restatement of the hyperthemes. The problematic of the corporation. Some exemplifications of the historical development of the corporation. Consideration of several prominent views regarding the purpose of commercial corporations. Consideration of consistency of the corporation as a hyperthematic structure. Hyperthematic reconstruction of the corporation as a community of epsionomic interpretation. Bibliography 412. (shrink)

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects (...) simpler, proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

A critical re-examination of the history of the concepts of space (including spacetime of general relativity and relativistic quantum field theory) reveals a basic ontological elusiveness of spatial extension, while, at the same time, highlighting the fact that its epistemic primacy seems to be unavoidably imposed on us (as stated by A.Einstein “giving up the extensional continuum … is like to breathe in airless space”). On the other hand, Planck’s discovery of the atomization of action leads to the fundamental recognition (...) of an ontology of non-spatial, abstract entities (Quine) for the quantum level of reality (QT), as distinguished from the necessarily spatio-temporal, experimental revelations (measurements). The elementary quantum act (measured by Planck’s constant) has neither duration nor extension, and any genuinely quantum process literally does not belong in the Raum and time of our experience. As Heisenberg stresses: “Während also die klassische Physik ein objectives Geschehen in Raum and Zeit zum Gegenstand hat, für dessen Existenz seine Beobachtung völlig irrelevant war, behandelt die Quantentheorie Vorgänge, die sozusagen nur in den Momenten der Beobachtung als raumzeitliche Phänomene aufleuchten, und über die in der zwischenzeit anschaulische physikalische Aussagen sinloss sind”. An admittedly speculative, hazardous conjecture is then advanced concerning the relation of such quantum ontology with the role of the pre-phenomenal continuum (Husserl) in the perception of macroscopically distinguishable objects in the Raum and time of our experience. Although rather venturesome, it brings together important philosophical issues. Coherently with recent general results in works on the foundations of QT, it is assumed that the linearity of quantum dynamical evolution does not apply to the central nervous system of living beings at a certain level of the evolutionary ramification and at the pre-conscious stage of subjectivity. Accordingly, corresponding to the onset of a non-linear dynamic evolution, a ‘primary spatial’ reduction is ‘continually’ taking place, thereby constituting the neural precondition for the experience of distinguishable macroscopic objects in the continuous spatial extension. While preventing the theoretically possible quantum superpositions of macroscopic objects from being perceivable by living beings, the ‘primary reduction’ has no effect on the standard processes concerning quantum level entities involved in laboratory man-made experiments. In this connection, an experimental check which might falsify the conjecture is briefly discussed. The approach suggested here, if sound, leads to a naturalization of that part of Kant’s Transcendental Aesthetics than can survive the Euclidean catastrophe. According to such naturalized transcendentalism, “space can well be transcendental without the axioms being so”, in agreement with a well-known statement by Boltzman. Finally, as far as QT is concerned, the conjecture entails that a scheme for quantum measurement of the von Neumann type cannot even ‘leave the ground’, vindicating Bohr’s viewpoint. A quantum theory of measurement, in a proper sense, turns out to be unnecessary and in fact impossible. (shrink)

ion is arguably one of the most important methods in modern science in analysing and understanding complex phenomena. In his book The Philosophy of Information, Floridi (The philosophy of information. Oxford University Press, Oxford, 2011) presents the method of levels of abstraction as the main method of the Philosophy of Information. His discussion of abstraction as a method seems inspired by the formal methods and frameworks of computer science, in which abstraction is operationalised extensively in programming languages (...) and design methodologies. Is it really clear what we should understand by levels of abstraction? How should they be specified? We will argue that levels of abstraction should be augmented with annotations, in order to express semantic information for them and reconcile the method of level of abstraction (LoA’s) with other approaches. We discuss the extended method when applied e.g. to the analysis of abstract machines. This will lead to an example in which the number of LoA’s is unbounded. (shrink)

Many formalisms for reasoning about knowing commit an agent to be logically omniscient. Logical omniscience is an unrealistic principle for us to use to build a real-world agent, since it commits the agent to knowing infinitely many things. A number of formalizations of knowledge have been developed that do not ascribe logical omniscience to agents. With few exceptions, these approaches are modifications of the possible-worlds semantics. In this paper we use a combination of several general techniques for building non-omniscient reasoners. (...) First we provide for the explicit representation of notions such as problems, solutions, and problem solving activities, notions which are usually left implicit in the discussions of autonomous agents. A second technique is to take explicitly into account the notion of resource when we formalize reasoning principles. We use the notion of resource to describe interesting principles of reasoning that are used for ascribing knowledge to agents. For us, resources are abstract objects. We make extensive use of ordering and inaccessibility relations on resources, but we do not find it necessary to define a metric. Using principles about resources without using a metric is one of the strengths of our approach.We describe the architecture of a reasoner, built from a finite number of components, who solves a puzzle, involving reasoning about knowing, by explicitly using the notion of resource. Our approach allows the use of axioms about belief ordinarily used in problem solving – such as axiom K of modal logic – without being forced to attribute logical omniscience to any agent. In particular we address the issue of how we can use resource-unbounded (e.g., logically omniscient) reasoning to attribute knowledge to others without introducing contradictions. We do this by showing how omniscient reasoning can be introduced as a conservative extension over resource-bounded reasoning. (shrink)

This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional (...) π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. Utilizing (...) recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

It seems that Whitehead's original problem, as evidenced in his earlier works, was epistemological and metaphysical dualism. His method of extensive abstraction is an attempt to bring the factual world and the abstract world of thought together. So far as his books on nature are concerned, then, Whitehead denied the ultimate reality of a static world and accepted the reality of dynamic relations analogous to the relations found in a biological organism.

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a (...) unique abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$. (shrink)

We enrich intuitionistic logic with a lax modal operator and define a corresponding intensional enrichment of Kripke models M = (W, , V) by a function T giving an effort measure T(w, u) {} for each -related pair (w, u). We show that embodies the abstraction involved in passing from true up to bounded effort to true outright. We then introduce a refined notion of intensional validity M |= p : and present a corresponding intensional calculus iLC-h which gives (...) a natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : | L |= p : } characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of and completely recovered by a suitable semantic interpretation of proofs. (shrink)

Many recent studies of dynamics in formal argumentation within AI focus on the well-known formalism of Dung’s argumentation frameworks (AFs). Despite the usefulness of AFs in many areas of argumentation, their abstract notion of arguments creates a barrier for operators that modify a given AF, e.g., in the case that dependencies between arguments have been abstracted away that are important for subsequent modifications. In this paper we aim to support development of dynamic operators on formal models in abstract argumentation by (...) providing constraints imposed on the modification of the structure that can be used to incorporate information that has been abstracted away. Towards a broad reach, we base our results on the general formalism of abstract dialectical frameworks (ADFs) in abstract argumentation. To show applicability, we present two cases studies that adapt an existing extension enforcement operator that modifies AFs: in the first case study, we show how to utilize constraints in order to obtain an enforcement operator on ADFs that is allowed to only add support relations between arguments, and in the second case study we show how an enforcement operator on AFs can be defined that respects dependencies between arguments. We show feasibility of our approach by studying the complexity of the proposed structural constraints and the operators arising from the case studies, and by an experimental evaluation of an answer set programming (ASP) implementation of the enforcement operator based on supports. (shrink)

Luigi Einaudi was a leading liberal economist, economic historian and political figure. This book provides the English-speaking world with a first critical edition of an unpublished version of Einaudi’s most important epistemological essay. The issues analysed here lie at the core of the problem concerning the nature and scope of economic sciences and the role played by economists in the public sphere, with particular emphasis on the interaction between economists and the ruling class. The earlier version of this essay has (...) also been considered the "historical epilogue" of the Italian tradition in public finance. An extensive reappraisal of this newly discovered document will help to reconsider and cast light on that tradition. This critical edition includes a comprehensive introduction and conclusion, both of which aim to place Einaudi’s essay in the context of his earlier epistemological speculation and the associated debates, and to assess the unsettled questions he left as an enduring heritage for the current generation of social scientists. (shrink)

The specious present (James, 1890/1950) is the phenomenal temporal structure of the representational content of my present experience. This article is a study of the mereological structure of the specious present and what it reveals about the nature of the mind. I argue that the specious present has certain features that cannot be easily explained within the framework of physicalism about the mind — the view that consciousness is nothing over and above what is physical. In particular, the specious present (...) is a phenomenally extended (i.e.non-instantaneous) whole whose parts are conceptual abstractions and are therefore dependent on this whole. However, unlike the specious present, what is physical does not have this special property of the parts being dependent on the whole. Therefore, it is unclear how the specious present can be explained in physicalistic terms. (shrink)