The Oxford Translation of Aristotle was originally published in 12 volumes between 1912 and 1954. It is universally recognized as the standard English version of Aristotle. This revised edition contains the substance of the original Translation, slightly emended in light of recent scholarship three of the original versions have been replaced by new translations and a new and enlarged selection of Fragments has been added. The aim of the translation remains the same: to make the surviving works of Aristotle readily (...) accessible to English speaking readers. (shrink)

Amodal completion is the representation of occluded parts of perceived objects. We argue for the following three claims: First, at least some amodal completion-involved experiences can ground knowledge about the occluded portions of perceived objects. Second, at least some instances of amodal completion-grounded knowledge are not sensitive, that is, it is not the case that in the nearest worlds in which the relevant claim is false, that claim is not believed true. Third, at least some instances of (...) amodal completion-grounded knowledge are not safe, that is, it is not the case that in all or nearly all near worlds where the relevant claim is believed true, that claim is in fact true. Thus, certain instances of amodal completion-grounded knowledge refute both the view that knowledge is necessarily sensitive and the view that knowledge is necessarily safe. (shrink)

Amodal completion is usually characterized as the representation of those parts of the perceived object that we get no sensory stimulation from. In the case of the visual sense modality, for example, amodal completion is the representation of occluded parts of objects we see. I argue that relationalism about perception, the view that perceptual experience is constituted by the relation to the perceived object, cannot give a coherent account of amodal completion. The relationalist has two options: construe (...) the perceptual relation as the relation to the entire perceived object or as the relation to the unoccluded parts of the perceived object. I argue that neither of these options are viable. (shrink)

Outstanding translations by leading contemporary scholars--many commissioned especially for this volume--are presented here in the first single edition to include the entire surviving corpus of works attributed to Plato in antiquity. In his introductory essay, John Cooper explains the presentation of these works, discusses questions concerning the chronology of their composition, comments on the dialogue form in which Plato wrote, and offers guidance on approaching the reading and study of Plato's works. Also included are concise introductions by Cooper and Hutchinson (...) to each translation, meticulous annotation designed to serve both scholar and general reader, and a comprehensive index. This handsome volume offers fine paper and a high-quality Smyth-sewn cloth binding in a sturdy, elegant edition. (shrink)

We consider the following generalization of the notion of a structure recursive relative to a set X. A relational structure A is said to be a Γ-structure if for each relation symbol R, the interpretation of R in A is ∑math image relative to X, where β = Γ. We show that a certain, fairly obvious, description of classes ∑math image of recursive infinitary formulas has the property that if A is a Γ-structure and S is a further relation on (...) A, then the following are equivalent: For every isomorphism F from A to a Γ-structure, F is ∑math image relative to X, The relation is defined in A by a ∑math image formula with parameters. (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system is deducible (...) by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.

Each had met his match, and happily, in the other. This book is the story of an elective affinity.

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept (...) of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.

Current discussion of scientific realism and antirealism often cites Pierre Duhem’s argument for the underdetermination of theory choice by evidence. Participants draw on an account of his underdetermination thesis that is familiar, but incomplete. The purpose of this article is to complete the familiar account. I argue that a closer look at Duhem’s The aim and structure of physical theory suggests that the rationale for his underdetermination thesis comes from his philosophy of scientific language. I explore how an understanding of (...) physical laws as symbolic is meant to support the thesis. In the course of my argument, I point out that Duhemian underdetermination is not meta-practical but grounded in the practice of science, specifically in the scientist’s use of instruments and measurement techniques. Measurement has a significant limitation, according to Duhem: it always involves approximation and a degree of experimental error. Consequently, it cannot overcome the gap between the ordinary, concrete language of observation and the mathematical language of science. Moreover, Duhem argues that the use of instruments in experiment invokes whole groups of theories. I contend that, ultimately, this reliance on auxiliary assumptions—which makes possible the use of instruments—is the foundation of his thesis and that recognizing this completes the familiar account of his underdetermination argument.Author Keywords: Duhem; Suppes; Underdetermination; Measurement; Physical theory. (shrink)

Guy Debord (1931-1994) was the most influential member of the Situationist International, the avant-garde group that helped trigger the May 1968 revolt in France. His book The Society of the Spectacle is considered by some to be the most important theoretical book of the 20th century. But while Debord's written work is some of the most notorious in the world of political and cultural radicalism, his films have remained tantalisingly inaccessible. Knabb's translation accompanies that long awaited English versions of these (...) films. Illustrated with 62 stills. (shrink)

This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

: In the first edition of his book on the completeness of Kant’s table of judgments, Klaus Reich shortly indicates that the B-version of the metaphysical exposition of space in the Critique of pure reason is structured following the inverse order of the table of categories. In this paper, I develop Reich’s claim and provide further evidence for it. My argumentation is as follows: Through analysis of our actually given representation of space as some kind of object, the metaphysical exposition (...) will show that this representation is secondary to space considered as an original, undetermined and as such unrepresentable intuitive manifold. Now, following Kant, the representation of any kind of object involves diversity, synthesis and unity. In the case of our representation of space as formal intuition, this involves, firstly, a manifold a priori, i.e. space as pure form, delivered by the transcendental Aesthetic, secondly, a figurative, productive synthesis of that manifold, and, thirdly, the unity provided by the categories. Analysing our given representation of space – the task of the metaphysical exposition – amounts to dismantling its unity and determine its characteristics with respect to the categories. (shrink)

The Quantified Argument Calculus (Quarc) is a formal logic system, first developed by Hanoch Ben-Yami in (Ben-Yami 2014), and since then extended and applied by several authors. The aim of this paper is to further these contributions by, first, providing a philosophical motivation for the truth-valuational, substitutional approach of (Ben-Yami 2014) and defending it against a common objection, a topic also of interest beyond its specific application to Quarc. Second, we fill the formal lacunae left in the original presentation, which (...) did not incorporate identity systematically into Quarc, and although it proved the soundness of the system did not prove its completeness. (shrink)

This paper contains a completeness proof for the system ILW, a rather bewildering axiom system belonging to the family of interpretability logics. We have treasured this little proof for a considerable time, keeping it just for ourselves. Johan’s ftieth birthday appears to be the right occasion to get it out of our wine cellar.

We prove completeness of the propositional modal logic S 4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and . Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, , and (...) we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in with an open representative. We prove completeness of the modal logic S 4 for the algebra . A corollary to the main result is that non-theorems of S 4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in. (shrink)

The paper discusses methodological guidelines for evaluating mechanistic explanations. According to current accounts, a satisfactory mechanistic explanation should include all of the relevant features of the mechanism, its component entities and activities, and their properties and organization, as well as exhibit productive continuity. It is not specified, however, how this kind of mechanistic completeness can be demonstrated. I argue that parameter sufficiency inferences based on mathematical model simulations provide a way of determining whether a mechanism capable of producing the phenomenon (...) of interest can be constructed from mechanistic components organized, acting, and having the properties described in the mechanistic explanation. (shrink)

In this paper we present BTC, which is a complete logic for branchingtime whose modal operator quantifies over histories and whose temporal operators involve a restricted quantification over histories in a given possible choice. This is a technical novelty, since the operators of the usual logics for branching-time such as CTL express an unrestricted quantification over histories and moments. The value of the apparatus we introduce is connected to those logics of agency that are interpreted on branching-time, as for instance (...) Stit Logics. (shrink)

This paper uses the notion of Galois-connection to examine the relation between valuation-spaces and logics. Every valuation-space gives rise to a logic, and every logic gives rise to a valuation space, where the resulting pair of functions form a Galois-connection, and the composite functions are closure-operators. A valuation-space (resp., logic) is said to be complete precisely if it is Galois-closed. Two theorems are proven. A logic is complete if and only if it is reflexive and transitive. A valuation-space is complete (...) if and only if it is closed under formation of super-valuations. (shrink)

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...) of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

In the Eudemian Ethics II 1, 1219a34–b8, Aristotle defines happiness as ‘the activity of a complete life in accordance with complete virtue’. Most scholars interpret a complete life as a whole lifetime, which means that happiness involves virtuous activity over an entire life. This article argues against this common reading by using Aristotle’s notion of ‘activity’ (energeia) as a touchstone. It argues that happiness, according to the Eudemian Ethics, must be a complete activity that reaches its end at any and (...) every moment. The upshot of this reading is that life reaches completeness within a lifetime and that death cannot be the requirement for making life complete. (shrink)

The type of completeness Whitehead and Russell aimed for in their Principia Mathematica was what I call descriptive completeness. This is completeness with respect to the propositions that have been proved in traditional mathematics. The notion of completeness addressed by Gödel in his famous work of 1930 and 1931 was completeness with respect to the truths expressible in a given language. What are the relative significances of these different conceptions of completeness for traditional mathematics? What, if any, effects does incompleteness (...) of the type Gödel had in mind have for the evaluation of the Principia Mathematica project and other projects like it? (shrink)

Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. -/- Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. -/- Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. -/- The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise is (...) logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. -/- As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. -/- But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.

Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.

Response to Gover (2015) on Trogdon and Livingston (2015) on artwork completion.

Halldén completeness closely resembles the relevance property. To prove Halldén completeness in terms of Kripke-style semantics, the van Benthem–Humberstone theorem is often used. In relevant modal logics, the Halldén completeness of Meyer–Fuhrmann logics has been obtained using the van Benthem–Humberstone theorem. However, there remain a number of Halldén-incomplete relevant modal logics. This paper discusses the Halldén completeness of a wider class of relevant modal logics, namely, those with some Sahlqvist axioms.

We develop an abstract proof calculus for logics whose sentences are ‘Horn sentences’ of the form: $(\forall X)H \Rightarrow c$ and prove an institutional generalization of Birkhoff completeness theorem. This result is then applied to the particular cases of Horn clauses logic, the ‘Horn fragment’ of preorder algebras, order-sorted algebras and partial algebras and their infinitary variants.

Investigation of a puzzle concerning complete yet authorless artworks.

We show that early vision can use monocular cues to rapidly complete partially-occluded objects. Visual search for easily detected fragments becomes difficult when the completed shape is similar to others in the display; conversely, search for fragments that are difficult to detect becomes easy when the completed shape is distinctive. Results indicate that completion occurs via the occlusion-triggered removal of occlusion edges and linking of associated regions. We fail to find evidence for a visible filling-in of contours or surfaces, (...) but do find evidence for a "functional" filling-in that prevents the constituent fragments from being rapidly accessed. As such, it is only the completed structures—and not the fragments themselves—that serve as the basis for rapid recognition. (shrink)

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set (...) of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism. (shrink)

A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 11, posed by D. Scott, H. Barendregt, and G. Mitschke) if every PCA can be completed. A negative answer to this question was given by Klop in [12, 11]; moreover he provided a sufficient condition for completability of a PCA (M, ·, K, S) in the form of ten axioms (inequalities) on terms of M. We prove that just one (...) of these axiom (the so called Barendregt's axiom) is sufficient to guarantee (a slightly weaker notion of) completability. (shrink)

This paper investigates different modes of organizing for corporate social responsibility (CSR). Based on insights from organization theory, we theorize two ways to organize for CSR. “Complete” organization for CSR happens within businesses and depends on the availability of certain organizational elements (e.g., membership, hierarchy, rules, monitoring, and sanctioning). By contrast, “partial” organization for CSR happens when organizers do not have direct access to all these organizational elements. We discuss partial organization for CSR by analyzing how standards and cross-sector partnerships (...) make selective use of organizational elements. We maintain that an important feature of the increasing institutionalization of CSR—not only within businesses but also among non-governmental, governmental, and professional actors—is the rise of partial forms of organization. We discuss the contributions to this Special Issue in the context of our theorization of complete/partial organization for CSR and outline avenues for further research. (shrink)

We show that basic hybridization makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i$\end{document} in propositional and first-order hybrid logic. This means: interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i\alpha _a$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _a$\end{document} is an expression of any type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document}, as an expression of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document} that rigidly returns the value that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_a$\end{document} receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic. (shrink)

The claim defended in the paper is that the mechanistic account of explanation can easily embrace idealization in big-scale brain simulations, and that only causally relevant detail should be present in explanatory models. The claim is illustrated with two methodologically different models: Blue Brain, used for particular simulations of the cortical column in hybrid models, and Eliasmith’s SPAUN model that is both biologically realistic and able to explain eight different tasks. By drawing on the mechanistic theory of computational explanation, I (...) argue that large-scale simulations require that the explanandum phenomenon is identified; otherwise, the explanatory value of such explanations is difficult to establish, and testing the model empirically by comparing its behavior with the explanandum remains practically impossible. The completeness of the explanation, and hence of the explanatory value of the explanatory model, is to be assessed vis-à-vis the explanandum phenomenon, which is not to be conflated with raw observational data and may be idealized. I argue that idealizations, which include building models of a single phenomenon displayed by multi-functional mechanisms, lumping together multiple factors in a single causal variable, simplifying the causal structure of the mechanisms, and multi-model integration, are indispensable for complex systems such as brains; otherwise, the model may be as complex as the explanandum phenomenon, which would make it prone to so-called Bonini paradox. I conclude by enumerating dimensions of empirical validation of explanatory models according to new mechanism, which are given in a form of a “checklist” for a modeler. (shrink)