- Submit
-
Browse
- All Categories
- Metaphysics and Epistemology
- Value Theory
- Science, Logic, and Mathematics
- Science, Logic, and Mathematics
- Logic and Philosophy of Logic
- Philosophy of Biology
- Philosophy of Cognitive Science
- Philosophy of Computing and Information
- Philosophy of Mathematics
- Philosophy of Physical Science
- Philosophy of Social Science
- Philosophy of Probability
- General Philosophy of Science
- Philosophy of Science, Misc
- History of Western Philosophy
- Philosophical Traditions
- Philosophy, Misc
- Other Academic Areas
- More
Switch to: References
Citations of:
Add citations
You must login to add citations.
|
|
|
We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) |
|
|
From the advent of general purpose, Turing-complete machines, the relation between operators, programmers and users with computers can be observed as interconnected informational organisms (inforgs), henceforth analysed with the method of levels of abstraction (LoAs), risen within the philosophy of information (PI). In this paper, the epistemological levellism proposed by L. Floridi in the PI to deal with LoAs will be formalised in constructive terms using category theory, so that information itself is treated as structure-preserving functions instead of Cartesian products. (...) |
|
|
We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and. |
|
|
|
|
|
|
|
|
The role played by continuous morphisms in propositional modal logic is investigated: it turns out that they are strictly related to filtrations and to suitable variants of the notion of a free algebra. We also employ continuous morphisms in incremental constructions of (standard) finitely generated free ????4-algebras. |
|
|
|
|
|
|
|
|
Following the guiding thread of Peirce’s use of diagrammatic syntax in his system of existential graphs , which depends crucially on the role of the Sheet of Assertion, we introduce the notion of Sheet of Indication as the basis for a general diagrammatic semantics applicable to a wide range of diagrams. We then show how Peirce’s EG-alpha graphs may be understood as instances of SIs and how logically coherent models of the graphs are represented in the SI semantics. |
|
|
Brandom's method of analyzing pragmatic relations among different practices and vocabularies through meaning-use diagrams is used to specify how Laruelle's nonphilosophical suspension of the Principle of Sufficient Philosophy may be distinguished from the philosophical auto-critiques of such thinkers as Badiou and Derrida. A superposition of diagrams modeling philosophical sufficiency on the one hand and supplementation through the Other on the other provides a schematic representation of the core duality of what Laruelle calls The-Philosophy. In contrast to this self-implicating and self-reproducing (...) |
|
|
We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) |
|
|
We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) |
|
|
Recent work on the hole argument in general relativity by Weatherall has drawn attention to the neglected concept of models’ representational capacities. I argue for several theses about the structure of these capacities, including that they should be understood not as many-to-one relations from models to the world, but in general as many-to-many relations constrained by the models’ isomorphisms. I then compare these ideas with a recent argument by Belot for the claim that some isometries “generate new possibilities” in general (...) |
|
|
|
|
|
Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us (...) |
|
|
|
|
|
Ernest Nagel Lecture, Columbia University, Sept. 27, 2007. |
|
|
The biennial DEON conferences are designed to promote interdisciplinary cooperation amongst scholars interested in linking the formal-logical study of normative concepts and normative systems with computer science, artificial intelligence, linguistics, philosophy, organization theory and law. In addition to these general themes, DEON 2016 encouraged a special focus on the topic "Reasons, Argumentation and Justification.". |
|
|
Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...) |
|
|
This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...) |
|
|
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...) |
|
|
Following the development of the selectionist theory of the immune system, there was an attempt to characterise many biological mechanisms as being ‘selectionist’ as juxtaposed with ‘instructionist’. However, this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky’s principles-and-parameters language-acquisition mechanism together under the ‘selectionist’ umbrella, even though Chomsky’s mechanism and embryonic development are significantly different from the selectionist mechanisms of biological evolution and the immune system. Surprisingly, there is an abstract way using two dual mathematical (...) |
|
|
|
|
|
Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) |
|
|
Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about (...) |
|
|
We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the “extensional projections” of some sequential algorithms. We define a model of PCF where morphisms are “extensional” sequential algorithms and prove that any equation between PCF terms which holds in (...) |
|
|
We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a \-monotone \-complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral (...) |
|
|
Higher types can readily be added to set theory, Bernays-Morse set theory being an example. A type for each ordinal is added in [2]. Adding higher types to set theory provides a neat solution to the problem of how to handle higher type categories. We give the basic definitions, and prove cocompleteness of some higher type categories. MSC: 14A15. |
|
|
In the philosophical literature on Yang-Mills theories, field formulations are taken to have more structure and to be local, while curve-based formulations are taken to have less structure and to be nonlocal. I formalize the notion of locality at issue and show that theories with less structure are nonlocal. However, the amount of structure had by some formulation is independent of whether it uses fields or curves. The relevant difference in structure is not a difference in set-theoretic structure. Rather, it (...) |
|
|
In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras. |
|
|
Dramatic changes or revolutions in a field of science are often made by outsiders or 'trespassers,' who are not limited by the established, 'expert' approaches. Each essay in this diverse collection shows the fruits of intellectual trespassing and poaching among fields such as economics, Kantian ethics, Platonic philosophy, category theory, double-entry accounting, arbitrage, algebraic logic, series-parallel duality, and financial arithmetic. |
|
|
In this paper we present a new categorical approach which attempts to provide an original understanding of QM. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about —rather than as problems that need to be bypassed in order to allow a restoration of a classical “common sense” understanding of what there is. In particular, we discuss a solution to (...) |
|
|
Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →. |
|
|
In this paper we attempt to consider quantum superpositions from the perspective of the logos categorical approach presented in [26]. We will argue that our approach allows us not only to better visualize the structural features of quantum superpositions providing an anschaulich content to all terms, but also to restore —through the intensive valuation of graphs and the notion of immanent power— an objective representation of what QM is really talking about. In particular, we will discuss how superpositions relate to (...) |
|
|
|
|
|
In this paper we address a deeply interesting debate that took place at the end of the last millennia between David Mermin, Adan Cabello and Michiel Seevinck, regarding the meaning of relationalism within quantum theory. In a series of papers, Mermin proposed an interpretation in which quantum correlations were considered as elements of physical reality. Unfortunately, the very young relational proposal by Mermin was too soon tackled by specially suited no-go theorems designed by Cabello and Seevinck. In this work we (...) |
|
|
1. Logic, determinism and free will. The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological and logical character; my concern here is to limit attention to two arguments from logic. To begin with, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to (...) |
|
|
Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present (...) |
|
|
Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the (...) |
|
|
A reconstruction of Kant’s Copernican Revolution is given using the linguistic version of conceptual schemes together with mathematical tools from Model Theory. In response to Davidson’s criticisms against conceptual relativism, untranslatable conceptual schemes are shown to exist. Arguments from the Tractatus are used to formulate the conditions required for an isomorphism between a representation and the structure of reality. It is argued that the scientific accuracy of a representation does not require a structural isomorphism with reality. Finally, a conception of (...) |
|
|
Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...) |
|
|
Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...) |
|
|
This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is not to (...) |
|
|
Following the development of the selectionist theory of the immune system, there was an attempt to characterize many biological mechanisms as being "selectionist" as juxtaposed to "instructionist." But this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky's language-acquisition mechanism as all being "selectionist." Yet Chomsky's mechanism (and embryonic development) are significantly different from the selectionist mechanisms of biological evolution or the immune system. Surprisingly, there is a very abstract way using two dual mathematical logics to (...) |
|
|
Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (...) |
|
|
Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...) |
|
|
This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...) |
|
|
The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested (...) |
|
|
Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) |
PhilPapers logo by Andrea Andrews and Meghan Driscoll.
This site uses cookies and Google Analytics (see our terms & conditions for details regarding the privacy implications). Use of this site is subject to terms & conditions.
All rights reserved by The PhilPapers Foundation
Server: philpapers-web-5ff6c685cb-rskd8 uwo