The Monist’s call for papers for this issue ended: “if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible”. We use the Grelling-Nelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping (...) theories of semantics, the usual account for the possibility of non intentional semantics, doesn’t seem able to account for the indefinitely extensible productivity of natural language. (shrink)
The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefiniteextensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefiniteextensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefiniteextensibility of the concept set, plausible principles permit the derivation of modal collapse.
This essay endeavors to define the concept of indefiniteextensibility in the setting of category theory. I argue that the generative property of indefiniteextensibility for set-theoretic truths is identifiable with the elementary embeddings of large cardinal axioms. A modal coalgebraic automata's mappings are further argued to account for both reinterpretations of quantifier domains as well as the ontological expansion effected by the elementary embeddings in the category of sets. The interaction between the interpretational and metaphysical (...) modalities of indefiniteextensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebraic automata are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefiniteextensibility. (shrink)
It is argued that Gabriel Uzquiano's approach to set-theoretic indefiniteextensibility is a version of in rebus structuralism, and therefore suffers from a vacuity problem.
Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefiniteextensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)
Logical Indefinites.Jack Woods - 2014 - Logique Et Analyse -- Special Issue Edited by Julien Murzi and Massimiliano Carrara 227: 277-307.details
I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.
Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this (...) article, I show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)
It is widely alleged that metaphysical possibility is “absolute” possibility Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215; Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity “necessity in the highest degree”. Van Inwagen claims that if P is metaphysically possible, then it is possible “tout court. Possible simpliciter. Possible period…. possib without qualification.” And Stalnaker writes, “we can agree (...) with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, and most others who allow themselves to talk about possible worlds at all, that metaphysical necessity is necessity in the widest sense.” What exactly does the thesis that metaphysical possibility is absolute amount to? Is it true? In this article, I argue that, assuming that the thesis is not merely terminological, and lacking in any metaphysical interest, it is an article of faith. I conclude with the suggestion that metaphysical possibility may lack the metaphysical significance that is widely attributed to it. (shrink)
In two rarely discussed passages – from unpublished notes on the Principles of Philosophy and a 1647 letter to Chanut – Descartes argues that the question of the infinite extension of space is importantly different from the infinity of time. In both passages, he is anxious to block the application of his well-known argument for the indefinite extension of space to time, in order to avoid the theologically problematic implication that the world has no beginning. Descartes concedes that we (...) always imagine an earlier time in which God might have created the world if he had wanted, but insists that this imaginary earlier existence of the world is not connected to its actual duration in the way that the indefinite extension of space is connected to the actual extension of the world. This paper considers whether Descartes’s metaphysics can sustain this asymmetrical attitude towards infinite space vs. time. I first consider Descartes’s relation to the ‘imaginary’ space/time tradition that extended from the late scholastics through Gassendi and More. I next examine carefully Descartes’s main argument for the indefinite extension of space and explain why it does not apply to time. Most crucially, since duration is merely conceptually distinct from enduring substance, the end or beginning of the world entails the end or beginning of real time. In contrast, extension does not depend on any enduring substance besides itself. (shrink)
Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...) scope of quantifiers reveals a natural way out. (shrink)
This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types (...) of intention, when the latter is interpreted as a modal mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. Chapter 3 provides an abstraction principle for epistemic intensions. Chapter 4 advances a two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the modal $\mu$-calculus to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter 6 advances a solution to the Julius Caesar problem based on Fine's "criterial" identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The epistemic two-dimensional truthmaker semantics developed in chapter 4 is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapter 8 examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter 9 examines the modal profile of $\Omega$-logic in set theory. Chapter 10 examines the interaction between epistemic two-dimensional semantics and absolute decidability. Chapter 11 avails of modal coalgebraic automata to interpret the defining properties of indefiniteextensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and metaphysical modalities thereof. The hyperintensional, epistemic two-dimensional truthmaker semantics developed in chapter 4 is applied in chapters 8, 10, and 11. Chapter 12 provides a modal logic for rational intuition. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)
In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...) the philosophy of mathematics that rules out the maximalist view in many different ontological debates. (shrink)
I distinguish between a metaphysical and a logical reading of Generality Relativism. While the former denies the existence of an absolutely general domain, the latter denies the availability of such a domain. In this paper I argue for the logical thesis but remain neutral in what concerns metaphysics. To motivate Generality Relativism I defend a principle according to which a collection can always be understood as a set-like collection. I then consider a modal version of Generality Relativism and sketch how (...) this version avoids certain revenge problems. (shrink)
The effective altruism movement aims to save lives in the most cost-effective ways. In the future, technology will allow radical life extension, and anyone who survives until that time will gain potentially indefinite life extension. Fighting aging now increases the number of people who will survive until radical life extension becomes possible. We suggest a simple model, where radical life extension is achieved in 2100, the human population is 10 billion, and life expectancy is increased by simple geroprotectors like (...) metformin or nicotinamide mononucleotide by three more years on average, so an additional 750 million people survive until “immortality”. The cost of clinical trials to prove that metformin is a real geroprotector is $65 million. In this simplified case, the price of a life saved is around eight cents, 10 000 times cheaper than saving a life from malaria by providing bed nets. However, fighting aging should not be done in place of fighting existential risks, as they are complementary causes. (shrink)
Abstract: The field of life extension is full of ideas but they are unstructured. Here we suggest a comprehensive strategy for reaching personal immortality based on the idea of multilevel defense, where the next life-preserving plan is implemented if the previous one fails, but all plans need to be prepared simultaneously in advance. The first plan, plan A, is the surviving until advanced AI creation via fighting aging and other causes of death and extending one’s life. Plan B is cryonics, (...) which starts if plan A fails, and assumes cryopreservation of the brain until technical capabilities to return it to life appear. Plan C is digital immortality in the sense of collecting data about the person now so future AI will be able to recreate a model of a person. Plan D is the hope based on some unlikely scenarios of infinite survival, like so-called “quantum immortality”. All these plans have personal and social perspective. The personal aspect means efforts of the increasing chances of personal survival via taking care about one’s own health, signing cryocontract or collecting digital immortality data. The social aspect means the participation in collective work towards creation and increase of the availability of life extension technologies, which includes funding scientific research, promotion of life extension value and direct performing of research and implementation, as well as preventing global catastrophic risk. All plans converge at the end, as their result is the indefinite survival as an uploaded mind inside an ecosystem, created by a superintelligent AI. (shrink)
According to certain kinds of semantic dispositionalism, what an agent means by her words is grounded by her dispositions to complete simple tasks. This sort of position is often thought to avoid the finitude problem raised by Kripke against simpler forms of dispositionalism. The traditional objection is that, since words possess indefinite (or infinite) extensions, and our dispositions to use words are only finite, those dispositions prove inadequate to serve as ground for what we mean by our words. I (...) argue that, even if such positions (emphasizing simple tasks) avoid the traditional form of Kripke's charge, they still succumb to special cases of the finitude problem. Furthermore, I show how such positions can be augmented so as to avoid even these special cases. Doing so requires qualifying the dispositions of interest as those possessed by the abstracted version of an actual agent (in contrast to, say, an idealized version of the agent). In addition to avoiding the finitude problem in its various forms, the position that results provides new materials for appreciating the role that abstracting models can play for a dispositionalist theory of meaning. (shrink)
Dâwûd al-Qarisî (Dâvûd al-Karsî) was a versatile and prolific 18th century Ottoman scholar who studied in İstanbul and Egypt and then taught for long years in various centers of learning like Egypt, Cyprus, Karaman, and İstanbul. He held high esteem for Mehmed Efendi of Birgi (Imâm Birgivî/Birgili, d.1573), out of respect for whom, towards the end of his life, Karsî, like Birgivî, occupied himself with teaching in the town of Birgi, where he died in 1756 and was buried next to (...) Birgivî. Better known for his following works on Arabic language and rhetoric and on the prophetic traditions (hadith): Sharḥu uṣûli’l-ḥadîth li’l-Birgivî; Sharḥu’l-Ḳaṣîdati’n-nûniyya (two commentaries, in Arabic and Turkish); Şarḥu’l-Emsileti’l-mukhtalifa fi’ṣ-ṣarf (two commentaries, in Arabic and Turkish); Sharḥu’l-Binâʾ; Sharḥu’l-ʿAvâmil; and Sharḥu İzhâri’l-asrâr, Karsî has actually composed textbooks in quite different fields. Hence the hundreds of manuscript copies of his works in world libraries. Many of his works were also recurrently printed in the Ottoman period. One of the neglected aspects of Karsî is his identity as a logician. Although he authored ambitious and potent works in the field of logic, this aspect of him has not been subject to modern studies. Even his bibliography has not been established so far (with scattered manuscript copies of his works and incomplete catalogue entries). This article primarily and in a long research based on manuscript copies and bibliographic sources, identifies twelve works on logic that Karsî has authored. We have clarified the works that are frequently mistaken for each other, and, especially, have definitively established his authorship of a voluminous commentary on al-Kâtibî’s al-Shamsiyya, of which commentary a second manuscript copy has been identified and described together with the other copy. Next is handled his most famous work of logic, the Sharhu Îsâghûcî, which constitutes an important and assertive ring in the tradition of commentaries on Îsâghûcî. We describe in detail the nine manuscript copies of this work that have been identified in various libraries. The critical text of Karsî’s Sharhu Îsâghûcî, whose composition was finished on 5 March 1745, has been prepared based on the following four manuscripts: (1) MS Kayseri Raşid Efendi Kütüphanesi, No. 857, ff.1v-3v, dated 1746, that is, only one year after the composition of the work; (2) MS Bursa İnebey Yazma Eser Kütüphanesi, Genel, No.794B, ff.96v-114v, dated 1755; (3) MS Millet Kütüphanesi, Ali Emiri Efendi Arapça, No. 1752, ff.48v-58r, dated 1760; (4) MS Beyazıt Yazma Eser Kütüphanesi, Beyazıt, No. 3129, ff.41v-55v, dated 8 March 1772. While preparing the critical text, we have applied the Center for Islamic Studies (İslam Araştırmaları Merkezi, İSAM)’s method of optional text choice. The critical text is preceded by a content analysis. Karsî is well aware of the preceding tradition of commentary on Îsâghûcî, and has composed his own commentary as a ‘simile’ or alternative to the commentary by Mollâ Fanârî which was famous and current in his own day. Karsî’s statement “the commentary in one day and one night” is a reference to Mollâ Fanârî who had stated that he started writing his commentary in the morning and finished it by the evening. Karsî, who spent long years in the Egyptian scholarly and cultural basin, adopted the religious-sciences-centered ‘instrumentalist’ understanding of logic that was dominant in the Egypt-Maghrib region. Therefore, no matter how famous they were, he criticized those theoretical, long, and detailed works of logic which mingled with philosophy; and defended and favored authoring functional and cogent logic texts that were beneficial, in terms of religious sciences, to the seekers of knowledge and the scholars. Therefore, in a manner not frequently encountered in other texts of its kind, he refers to the writings and views of Muhammad b. Yûsuf al-Sanûsî (d.1490), the great representative of this logical school in the Egyptian-Maghrib region. Where there is divergence between the views of the ‘earlier scholars’ (mutaqaddimûn) like Ibn Sînâ and his followers and the ‘later scholars’ (muta’akhkhirûn), i.e., post-Fakhr al-dîn al-Râzî logicians, Karsî is careful to distance himself from partisanship, preferring sometimes the views of the earliers, other times those of the laters. For instance, on the eight conditions proposed for the realization of contradiction, he finds truth to be with al-Fârâbî, who proposed “unity in the predicative attribution” as the single condition for the realization of contradiction. Similarly, on the subject matter of Logic, he tried to reconcile the mutaqaddimûn’s notion of ‘second intelligibles’ with the muta’akhkhirûn’s notion of ‘apprehensional and declarational knowledge,’ suggesting that not much difference exists between the two, on the grounds that both notions are limited to the aspect of ‘known things that lead to the knowledge of unknown things.’ Karsî asserts that established and commonly used metaphors have, according to the verifying scholars, signification by correspondence (dalâlat al-mutâbaqah), adding also that it should not be ignored that such metaphors may change from society to society and from time to time. Karsî also endorses the earlier scholars’ position concerning the impossibility of quiddity (mâhiyya) being composed of two co-extensive parts, and emphasizes that credit should not be given to later scholars’ position who see it possible. According to the verifying scholars (muhaqqîqûn), it is possible to make definition (hadd) by mentioning only difference (fasl), in which case it becomes an imperfect definition (hadd nâqis). He is of the opinion that the definition of the proposition (qadiyya) in al-Taftâzânî’s Tahdhîb is clearer and more complete: “a proposition is an expression that bears the possibility of being true or false”. He states that in the division of proposition according to quantity what is taken into consideration is the subject (mawdû‘) in categorical propositions, and the temporal aspect of the antecedent (muqaddam) in hypothetical propositions. As for the unquantified, indefinite proposition (qadiyya muhmalah), Karsî assumes that if it is not about the problems of the sciences, then it is virtually/potentially a particular proposition (qadiyya juz’iyyah); but if it is about the problems of the sciences, then it is virtually/potentially a universal proposition (qadiyya kulliyyah). This being the general rule about the ambiguous (muhmal) propositions, he nevertheless contends that, because its subject (mawdû‘) is negated, it is preferable to consider a negative ambiguous (sâliba muhmalah) proposition like “human (insân) is not standing” to be a virtually/potentially universal negative (sâliba kulliyyâh) proposition. He states that a disjunctive hypothetical proposition (shartiyya al-munfasila) that is composed of more than two parts/units is only seemingly so, and that in reality it cannot be composed of more than two units. Syllogism (qiyâs), according to Karsî, is the ultimate purpose (al-maqsad al-aqsâ) and the most valuable subject-matter of the science of Logic. For him, the entire range of topics that are handled before this one are only prolegomena to it. This approach of Karsî clearly reveals how much the ‘demonstration (burhân)-centered’ approach of the founding figures of the Muslim tradition of logic like al-Fârâbî and Ibn Sînâ has changed. al-Abharî, in his Îsâghûjî makes no mention of ‘conversion by contradiction’ (‘aks al-naqîd). Therefore, Karsî, too, in his commentary, does not touch upon the issue. However, in his Îsâghûjî al-jadîd Karsî does handle the conversion by contradiction and its rules. Following the method of Îsâghûjî, in his commentary Karsî shortly touches on the four figures (shakl) of conjuctive syllogism (qiyâs iqtirânî) and their conditions, after which he passes to the first figure (shakl), which is considered ‘the balance of the sciences’ (mi‘yâr al-‘ulûm), explaining the four moods (darb) of it. In his Îsâghûjî al-jadîd, however, Karsî handles all the four figures (shakl) with all their related moods (darb), where he speaks of fife moods (darb) of the fourth figure (shakl). The topic of ‘modal propositions’ (al-muwajjahât) and of ‘modal syllogism’ (al-mukhtalitât), both of which do not take place in the Îsâghûjî, are not mentioned by Karsî as well, either in his commentary on Îsâghûjî or in his Îsâghûjî al-jadîd. Karsî proposes that the certainties (yaqîniyyât), of which demonstration (burhân) is made, have seven, not six, divisions. After mentioning (1) axioms/first principles (awwaliyyât), (2) observata/sensuals (mushâhadât), (3) experta/empiricals (mujarrabât), (4) acumenalia (hadthiyyât), (5) testata (mutawâtirât), and (6) instictives (fitriyyât), that is, all the ‘propositions accompanied by their demonstrations,’ Karsî states that these six divisions, which do not need research and reflection (nazar), are called badîhiyyât (self-evidents), and constitute the foundations (usûl) of certainties (yaqîniyyât). As the seventh division he mentions (7) the nazariyyât (theoreticals), which are known via the badîhiyyât, end up in them, and therefore convey certainty (yaqîn). For Karsî, the nazariyyât/theoreticals, which constitute the seventh division of yaqîniyyât/certainties, are too numerous, and constitute the branches (far‘) of yaqîniyyât. Every time the concept of ‘Mughâlata’ (sophistry) comes forth in the traditional sections on the five arts usually appended to logic works, Karsî often gives examples from what he sees as extreme sûfî sayings, lamenting that these expressions are so widespread and held in esteem. He sometimes criticizes these expressions. However, it is observed that he does not reject tasawwuf in toto, but excludes from his criticism the mystical views and approaches of the truth-abiding (ahl al-haqq), shârî‘â-observant (mutasharri‘) leading sufis who have reached to the highest level of karâmah. (shrink)
The problem of surviving the end of the observable universe may seem very remote, but there are several reasons it may be important now: a) we may need to define soon the final goals of runaway space colonization and of superintelligent AI, b) the possibility of the solution will prove the plausibility of indefinite life extension, and с) the understanding of risks of the universe’s end will help us to escape dangers like artificial false vacuum decay. A possible solution (...) depends on the type of the universe’s ending that may be expected: very slow heat death or some abrupt end, like a Big Rip or Big Crunch. We have reviewed the literature and identified several possible ways of survival the end of the universe, and also suggest several new ones. There are seven main approaches to escape the end of the universe: use the energy of the catastrophic process for computations, move to a parallel world, prevent the end, survive the end, manipulate time, avoid the problem entirely or find some meta-level solution. (shrink)
This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three (...) books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions. (shrink)
In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density (...) matrices induced by the experiments or `measurements' is the Lüders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM. (shrink)
Radical enactive and embodied approaches to cognitive science oppose the received view in the sciences of the mind in denying that cognition fundamentally involves contentful mental representation. This paper argues that the fate of representationalism in cognitive science matters significantly to how best to understand the extent of cognition. It seeks to establish that any move away from representationalism toward pure, empirical functionalism fails to provide a substantive “mark of the cognitive” and is bereft of other adequate means for individuating (...) cognitive activity. It also argues that giving proper attention to the way the folk use their psychological concepts requires questioning the legitimacy of commonsense functionalism. In place of extended functionalism—empirical or commonsensical—we promote the fortunes of extensive enactivism, clarifying in which ways it is distinct from notions of extended mind and distributed cognition. (shrink)
This paper brings together several strands of thought from both the analytic and phenomenological traditions in order to critically examine accounts of cognitive enhancement that rely on the idea of cognitive extension. First, I explain the idea of cognitive extension, the metaphysics of mind on which it depends, and how it has figured in recent discussions of cognitive enhancement. Then, I develop ideas from Husserl that emphasize the agential character of thought and the distinctive way that conscious thoughts are related (...) to one another. I argue that these considerations are necessary for understanding why forms of cognitive extension may diminish our cognitive lives in different ways. This does not lead to a categorical rejection of cognitive enhancement as unethical or bad for human flourishing, but does warrant a conservative approach to the design and implementation of cognitive artifacts. (shrink)
The chapter considers two semantic issues concerning will-sentences: Stalnaker’s Asymmetry and modal subordination in Karttunen-type discourses. The former points to a distinction between will and modal verbs, seeming to show that will does not license non-specific indefinites. The latter, conversely, suggests that will-sentences involve some kind of modality. To account for the data, the chapter proposes that will is semantically a tense, hence it doesn’t contribute a quantifier over modal alternatives; a modal feature, however, is introduced in the interpretation of (...) a will-sentence through a supervaluational strategy universally quantifying over possible futures. That this is not part of will’s lexical semantics is shown to have consequences that ultimately contribute to explain Stalnaker’s Asymmetry. Furthermore, that a modal quantification is present in the interpretation of a will-sentence is shown to imply the availability of modal subordination in Karttunen-type discourses. (shrink)
Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.
In a recent publication in this journal, Asle Kiran and Peter-Paul Verbeek (hereafter K&V) argue that extension theory and the notion of trust it implies are flawed. In this commentary, I defend extension theory against their critique. I first briefly introduce extension theory, then reconstruct K&V’s five arguments against extension theory and demonstrate that four of their five arguments are misplaced.
In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint (...) the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of Formal Inconsistency is examined. (shrink)
This article argues, first, that there is plenty of agreement among philosophers on philosophically substantive claims, which fall into three categories: reasons for or against certain views, elementary truths regarding fundamental notions, and highly conditionalized claims. This agreement suggests that there is important philosophical progress. It then argues that although it's easy to list several potential kinds of philosophical progress, it is much harder to determine whether the potential is actual. Then the article attempts to articulate the truth that the (...) deniers of philosophical progress are latching on to. Finally, it comments on the significance of the agreement and progress. (shrink)
In Kant’s logical texts the reference of the form S is P to an “unknown = x” is well known, but its understanding still remains controversial. Due to the universality of all concepts, the subject as much as the predicate is regarded as predicate of the x, which, in turn, is regarded as the subject of the judgment. In the CPR, this Kantian interpretation of the S-P relationship leads to the question about the relations between intuition and concept in judgment. (...) In contrast to intuition, if no concept, due to its universality, refers immediately to an object, how should one understand the relations of S and P to one another, as well as their relations to intuition, which corresponds to the possible individuality of the object in general = x? To answer this question, it is necessary to understand Kant’s notion of extension, and to prove its irreducibility to the Port-Royal notion of extension as well as to the Fregean one. (shrink)
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)
Contemporary discussions do not always clearly distinguish two different forms of vagueness. Sometimes focus is on the imprecision of predicates, and sometimes the indefiniteness of statements. The two are intimately related, of course. A predicate is imprecise if there are instances to which it neither definitely applies nor definitely does not apply, instances of which it is neither definitely true nor definitely false. However, indefinite statements will occur in everyday discourse only if speakers in fact apply imprecise predicates to (...) such indefinite instances. (What makes an instance indefinite is, it should be clear, predicate-relative.) The basic issue in the present inquiry is whether this indefiniteness ever really occurs; the basic question is, Why should it ever occur? (shrink)
Extended cognition holds that cognitive processes sometimes leak into the world (Dawson, 2013). A recent trend among proponents of extended cognition has been to put pressure on phenomena thought to be safe havens for internalists (Sneddon, 2011; Wilson, 2010; Wilson & Lenart, 2014). This paper attempts to continue this trend by arguing that music perception is an extended phenomenon. It is claimed that because music perception involves the detection of musical invariants within an “acoustic array”, the interaction between the auditory (...) system and the musical invariants can be characterized as an extended computational cognitive system. In articulating this view, the work of J. J. Gibson (1966, 1986) and Robert Wilson (1994, 1995, 2004) is drawn on. The view is defended from several objections and its implications outlined. The paper concludes with a comparison to Krueger’s (2014) view of the “musically extended emotional mind”. (shrink)
Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are (...) category-theoretically dual to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)
In their article 'Causes and Explanations: A Structural-Model Approach. Part I: Causes', 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.
This paper will defend the claim that, under certain circumstances, the material vehicles responsible for an agent’s conscious experience can be partly constituted by processes outside the agent’s body. In other words, the consciousness of the agent can extend. This claim will be supported by the Extended Mind Thesis (EMT) example of the artist and their sketchpad (Clark 2001, 2003). It will be argued that if this example is one of EMT, then this example also supports an argument for consciousness (...) extension. Clark (2009) rejects claims of consciousness extension. This paper will challenge Clark and argue that he fails to show that the material vehicles responsible for consciousness must be internal to the agent. (shrink)
This paper evaluates the Natural-Kinds Argument for cognitive extension, which purports to show that the kinds presupposed by our best cognitive science have instances external to human organism. Various interpretations of the argument are articulated and evaluated, using the overarching categories of memory and cognition as test cases. Particular emphasis is placed on criteria for the scientific legitimacy of generic kinds, that is, kinds characterized in very broad terms rather than in terms of their fine-grained causal roles. Given the current (...) state of cognitive science, I conclude that we have no reason to think memory or cognition are generic natural kinds that can ground an argument for cognitive extension. (shrink)
Plants from a handful of species provide the primary source of food for all people, yet this source is vulnerable to multiple stressors, such as disease, drought, and nutrient deficiency. With rapid population growth and climate uncertainty, the need to produce crops that can tolerate or resist plant stressors is more crucial than ever. Traditional plant breeding methods may not be sufficient to overcome this challenge, and methods such as highOthroughput sequencing and automated scoring of phenotypes can provide significant new (...) insights. Ontologies are essential tools for accessing and analysing the large quantities of data that come with these newer methods. As part of a larger project to develop ontologies that describe plant phenotypes and stresses, we are developing a plant disease extension of the Infectious Disease Ontology (IDOPlant). The IDOPlant is envisioned as a reference ontology designed to cover any plant infectious disease. In addition to novel terms for infectious diseases, IDOPlant includes terms imported from other ontologies that describe plants, pathogens, and vectors, the geographic location and ecology of diseases and hosts, and molecular functions and interactions of hosts and pathogens. To encompass this range of data, we are suggesting inOhouse ontology development complemented with reuse of terms from orthogonal ontologies developed as part of the Open Biomedical Ontologies (OBO) Foundry. The study of plant diseases provides an example of how an ontological framework can be used to model complex biological phenomena such as plant disease, and how plant infectious diseases differ from, and are similar to, infectious diseases in other organism. (shrink)
Norm theory (Kahneman & Miller, 1986) described a tendency for people to associate stronger regret with a negative outcome when it is a result of an exception (abnormal behavior) compared to when it is a result of routine (normal behavior). In two pre-registered studies, we conducted a replication and extension of three classic experiments on past behavior exception/routine contrasts (N = 684). We successfully replicated Kahneman and Miller’s (1986) experiments with the classic hitchhiker-scenario (Part 1) and car accident-scenario (Part 2). (...) In both cases, participants examined negative outcomes and tended to indicate a protagonist who deviated from own past behavior as more regretful than another who followed routine. Pre-registered extensions also showed effects for ratings of social norms, negative affect, and perceived luck. We did not find support for the Miller and McFarland (1986) experiment robbery scenario (Part 3) using a compensation measure, in that compensation to a victim of a robbery was not significantly different comparing exceptional and routine circumstances. However, a pre-registered extension showed that robbery under exceptional circumstances was regretted more than robbery under routine circumstances. We discuss implications for current and future research. (shrink)
In Humanity’s End: Why We Should Reject Radical Enhancement, Nicholas Agar presents a novel argument against the prospect of radical life-extension. Agar’s argument hinges on the claim that extended lifespans will result in people’s lives being dominated by the fear of death. Here we examine this claim and the surrounding issues in Agar’s discussion. We argue, firstly, that Agar’s view rests on empirically dubious assumptions about human rationality and attitudes to risk, and secondly, that even if those assumptions are granted, (...) the fears that Agar adverts to are unlikely to dominate people’s lives if and when radical life-extension is made possible. Further, we claim that the structure of the decision-making process around life-extension is unlikely to be the way that it would have to be in order for Agar’s claims about fear of death to make sense. Finally, we argue that Agar is implicitly committed to a narrow conception of human value. In response, we suggest that the pursuit of life-extension can itself be seen as an expression of certain important aspects of our distinctively human nature. (shrink)
It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion of logical (...) extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)
The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the -fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as logics of formal inconsistency and logics of formal undeterminedness.
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)
The main task of this paper is to understand if and how static images like photographs can represent and/or depict temporal extension (duration). In order to do this, a detour will be necessary to understand some features of the nature of photographic representation and depiction in general. This important detour will enable us to see that photographs (can) have a narrative content, and that the skilled photographer can 'tell a story' in a very clear sense, as well as control and (...) guide the attention of the spectator of the photograph. The understanding and defence of this claim is a secondary aim of this paper, and it will then allow us to provide a good treatment of the particular case of photographic representation and depiction of temporal extension. (shrink)
If two self-connected individuals are connected, it follows in classical extensional mereotopology that the sum of those individuals is self-connected too. Since mainland Europe and mainland Asia, for example, are both self-connected and connected to each other, mainland Eurasia is also self-connected. In contrast, in non-extensional mereotopologies, two individuals may have more than one sum, in which case it does not follow from their being self-connected and connected that the sum of those individuals is self-connected too. Nevertheless, one would still (...) expect it to follow that a sum of connected self-connected individuals is self-connected too. In this paper, we present some surprising countermodels which show that this conjecture is incorrect. (shrink)
In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics , we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.
This chapter explores to what extent some core ideas of predictive processing can be applied to the phenomenology of time consciousness. The focus is on the experienced continuity of consciously perceived, temporally extended phenomena (such as enduring processes and successions of events). The main claim is that the hierarchy of representations posited by hierarchical predictive processing models can contribute to a deepened understanding of the continuity of consciousness. Computationally, such models show that sequences of events can be represented as states (...) of a hierarchy of dynamical systems. Phenomenologically, they suggest a more fine-grained analysis of the perceptual contents of the specious present, in terms of a hierarchy of temporal wholes. Visual perception of static scenes not only contains perceived objects and regions but also spatial gist; similarly, auditory perception of temporal sequences, such as melodies, involves not only perceiving individual notes but also slightly more abstract features (temporal gist), which have longer temporal durations (e.g., emotional character or rhythm). Further investigations into these elusive contents of conscious perception may be facilitated by findings regarding its neural underpinnings. Predictive processing models suggest that sensorimotor areas may influence these contents. (shrink)
For well over two decades, Andy Clark has been gleaning theoretical lessons from the leading edge of cognitive science, applying a combination of empirical savvy and philosophical instinct that few can match. Clark’s most recent book, Supersizing the Mind: Embodiment, Action, and Cognitive Extension, brilliantly expands his oeuvre. It offers a well-informed and focused survey of research in the burgeoning field of situated cognition, a field that emphasizes the contribution of environmental and non-neural bodily structures to the production of intelligent (...) behavior. The situated research program, fledgling though it may be in some respects, has reached an age at which its philosophical stock can reasonably be taken; and Clark is just the person to take it. Supersizing the Mind consists of three main divisions. The first develops the case for the distinctively extended view of cognition, according to which the human mind or cognitive system literally comprises elements beyond the boundary of the human organism. The second responds to critics of the extended outlook: Frederick Adams, Kenneth Aizawa, Keith Butler, Brie Gertler, Rick Grush, and me, among others. The third major division evaluates nonextended strands in the situated program, in particular, those that emphasize the role of the non-neural body in cognition. (shrink)
The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties.
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