Calculus of Qualia 3: Godel Incompleteness, CQ Transcendence

A Calculus of Qualia (CQ) was proposed (on PhilPapers). The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader's mind and is non-referential, *it does not even refer to itself*. The meaning and behavior of these terms is radically different. In this paper in this series of papers, we discuss Godel Incompleteness and the incompleteness or not of CQ

Indeterministic finite-precision physics and intuitionistic mathematics

In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In this thesis, we discuss some motivations for dropping this assumption, which we believe partly arises from the usual point-based approach to the mathematical continuum. We focus on the case of classical mechanics specifically, but the ideas could be extended to other theories as well. We analyse the alternative theory of classical mechanics presented by Gisin and Del Santo, which suggests that physical quantities can equivalently be thought of as being only determined up to finite precision at each point in time, and that doing so naturally leads to indeterminism. Next, we investigate whether we can use intuitionistic mathematics to mathematically express the idea of finite precision of quantities, arriving at the cautious conclusion that, as far as we can see, such attempts are thwarted by conceptual contradictions. Finally, we outline another approach to formalising finite-precision quantities in classical mechanics, which is inspired by the intuitionistic approach to the continuum but uses classical mathematics

The Asymmetric Nature of Time

This open access monograph offers a detailed study and a systematic defense of a key intuition we typically have, as human beings, with respect to the nature of time: the intuition that the future is open, whereas the past is fixed. For example, whereas it seems unsettled whether there will be a fourth world war, it is settled that there was a first world war. The book contributes, in particular, three major and original insights. First, it provides a coherent, non-metaphorical, and metaphysically illuminating elucidation of the intuition. Second, it determines which model of the temporal structure of the world is most appropriate to accommodate the intuition, and settles on a specific version of the Growing Block Theory of time (GBT). Third, it puts forward a naturalistic foundation for GBT, by exploiting recent results of our best physics (viz. General Relativity, Quantum Mechanics, and Quantum Gravity). Three main challenges are addressed: the dismissal of temporal asymmetries as non-fundamental phenomena only (e.g., thermodynamic or causal phenomena), the epistemic objection against GBT, and the apparent tension between GBT and relativistic physics. It is argued that the asymmetry between the open future and the fixed past must be grounded in the temporal structure of the world, and that this is neither precluded by our epistemic device, nor by the latest approaches to Quantum Gravity (​e.g., the Causal Set Theory). Aiming at reconciling time as we find it in ordinary experience and time as physics describes it, this ​innovative book ​will raise the interest of both academic researchers and ​graduate students working on the philosophy of time. More generally, it ​presents contents of interest for all metaphysicians and non-dogmatic philosophers of physics. This is an open access book

A luz evanescente da contingência : operadores temporais, modalidades, e sensibilidade a contextos de avaliação

Orientador: Walter Alexandre CarnielliTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: O que queremos dizer quando falamos que algo vai acontecer amanhã, ou quando falamos que algo pode acontecer amanhã? Quais são as condições que precisam ser cumpridas para avaliarmos certas predições sobre nosso futuro, como dizendo (ou tendo dito) algo verdadeiro ou falso? Ou ainda, o que consideramos ter sido expressado por falantes, quando estes fazem afirmações sobre como o futuro irá se desdobrar, ou poderá se desdobrar? São a estas questões que esta tese se dedica. Em seu objetivo mais geral, pretendemos familiarizar o leitor com as ferramentas necessárias para se avaliar as diversas teorias envolvendo semânticas para uma linguagem temporal-modal, e como elas se saem com respeito ao problema dos futuros contingentes. Assim sendo, o Capítulo 1 inicia a tese apresentando o problema dos futuros contingentes, sendo então seguido por um levantamento das principais teorias semânticas para uma linguagem temporal-modal. No Capítulo 2, fornecemos as definições de teorias T×W, e preparamos o terreno para uma discussão aprofundada (a ser exposta no Capítulo 3) sobre a abordagem de MacFarlane a respeito do problema (2003, 2008, 2014), que toma 'futuros contingentes' como expressões cujo estatuto-de-verdade (se são verdadeiras, falsas, ou nem verdadeiras, nem falsas) é determinado não apenas pelo contexto de uso da sentença, mas também pelo contexto de avaliação de onde se avalia este uso. No capítulo final (4), apresentamos finalmente nossa meta particular: o objetivo desta seção é o de investigar o que seria necessário para um modal (de possibilidade) se comportar como sendo sensível a contextos de avaliação. Sugerimos que um modal com este comportamento torna-se saliente no discurso ordinário, especialmente quando ele se encontra em modo indicativo: opondo-se assim usos contendo "pode", de locuções semelhantes utilizando "poderia", quando o modal se encontra em modo subjuntivo. Nós acreditamos que esta sugestão se torna sensivelmente persuasiva através de impasses envolvendo asserções contendo o primeiro modal, porém inexistentes com afirmações semelhantes envolvendo o modal em modo subjuntivo. Essencialmente, argumentamos que ao se tomar a abordagem de MacFarlane sobre asserções - entendendo que quando alguém assere um conteúdo p, ele implica se comprometer em defender sua afirmação de qualquer contexto de avaliação (caso seja contestado) -, e se igualmente supomos que um falante assere um conteúdo cuja forma lógica é '????', se segue então que este falante acaba se representando como comprometido em defender uma alegação que não pode em princípio ser avaliada como verdadeira. Por outro lado, como se mostra, o mesmo não ocorre com afirmações envolvendo o modal subjuntivo (poderia ser,), já que ele não é sensível a contextos de avaliação, e portanto a conjunção pode em princípio ser avaliada como verdadeiraAbstract: What do we mean when we say that something will happen tomorrow, or when we say that something can happen tomorrow? What are the conditions that need to be fulfilled in order to make us evaluate a prediction about our future as saying (or having said) something true or false? Or else, what do we take speakers to have imparted when they make claims involving how the future will unfold, or how the future may unfold? It is to these questions that this dissertation is devoted. Within its most general goal, it intends to acquaint the reader with the tools required to assess the many distinct theories involving the semantics of temporal-modal languages, and how they square with the problem of future contingents. Thus accordingly, Chapter 1 sets out by framing the problem of future contingents, which is then followed by a detailed survey of distinct theories regarding the semantics of a temporal-modal language: we provide the relevant definitions, and assess their merits and downfalls. In Chapter 2, we provide the definitions of T×W theories of time, and set the stage for a thorough discussion (to be carried out in Chapter 3 ) about John MacFarlane's distinctive take on the problem (2003, 2008, 2014), which views 'future contingents' as expressions whose truth-status (whether they are true, false, or neither) depends not only on a context of use of the sentence, but also on a context of assessment from which one evaluates this use. The final chapter (4) will then embody our particular goal: the gist of this section is to investigate what it would take for a possibility-like modal to be sensitive to contexts of assessment. We suggest that a modal behaving as such becomes more salient in ordinary discourse when the modal is phrased in the indicative mood: thus opposing uses of 'can' from similar locutions using 'could', with the modal phrased in the subjunctive mood. The suggestion, we believe, becomes sensibly compelling in light of a puzzle involving defectiveness of assertions embedding such modals, as opposed to non-defectiveness of similar assertions involving 'could'. Essentially, it is argued that if we take MacFarlane's view of assertions - by understanding that whenever one flat out asserts a content p, one imparts being committed in defending his claim from any context of assessment (if challenged) -, and further suppose that a speaker asserts a content taking the logical form of '???? ' - where '?' is a future contingent and '?' is the assessment-sensitive modal -, then the defectiveness springs from the fact that the speaker is representing himself as committed in defending a claim that cannot be expected to ever have grounds to be assessed as true. On the other hand, as we show, the same does not occur with similar assertions involving 'could', since the modal is not assessment-sensitive, and thus the whole conjunction can be expected to be assessed as trueDoutoradoFilosofiaDoutor em Filosofia142478/2010-5CNP

The Asymmetric Nature of Time

This open access monograph offers a detailed study and a systematic defense of a key intuition we typically have, as human beings, with respect to the nature of time: the intuition that the future is open, whereas the past is fixed. For example, whereas it seems unsettled whether there will be a fourth world war, it is settled that there was a first world war. The book contributes, in particular, three major and original insights. First, it provides a coherent, non-metaphorical, and metaphysically illuminating elucidation of the intuition. Second, it determines which model of the temporal structure of the world is most appropriate to accommodate the intuition, and settles on a specific version of the Growing Block Theory of time (GBT). Third, it puts forward a naturalistic foundation for GBT, by exploiting recent results of our best physics (viz. General Relativity, Quantum Mechanics, and Quantum Gravity). Three main challenges are addressed: the dismissal of temporal asymmetries as non-fundamental phenomena only (e.g., thermodynamic or causal phenomena), the epistemic objection against GBT, and the apparent tension between GBT and relativistic physics. It is argued that the asymmetry between the open future and the fixed past must be grounded in the temporal structure of the world, and that this is neither precluded by our epistemic device, nor by the latest approaches to Quantum Gravity (​e.g., the Causal Set Theory). Aiming at reconciling time as we find it in ordinary experience and time as physics describes it, this ​innovative book ​will raise the interest of both academic researchers and ​graduate students working on the philosophy of time. More generally, it ​presents contents of interest for all metaphysicians and non-dogmatic philosophers of physics. This is an open access book

Logics of Responsibility

The study of responsibility is a complicated matter. The term is used in different ways in different fields, and it is easy to engage in everyday discussions as to why someone should be considered responsible for something. Typically, the backdrop of these discussions involves social, legal, moral, or philosophical problems. A clear pattern in all these spheres is the intent of issuing standards for when---and to what extent---an agent should be held responsible for a state of affairs. This is where Logic lends a hand. The development of expressive logics---to reason about agents' decisions in situations with moral consequences---involves devising unequivocal representations of components of behavior that are highly relevant to systematic responsibility attribution and to systematic blame-or-praise assignment. To put it plainly, expressive syntactic-and-semantic frameworks help us analyze responsibility-related problems in a methodical way. This thesis builds a formal theory of responsibility. The main tool used toward this aim is modal logic and, more specifically, a class of modal logics of action known as stit theory. The underlying motivation is to provide theoretical foundations for using symbolic techniques in the construction of ethical AI. Thus, this work means a contribution to formal philosophy and symbolic AI. The thesis's methodology consists in the development of stit-theoretic models and languages to explore the interplay between the following components of responsibility: agency, knowledge, beliefs, intentions, and obligations. Said models are integrated into a framework that is rich enough to provide logic-based characterizations for three categories of responsibility: causal, informational, and motivational responsibility. The thesis is structured as follows. Chapter 2 discusses at length stit theory, a logic that formalizes the notion of agency in the world over an indeterministic conception of time known as branching time. The idea is that agents act by constraining possible futures to definite subsets. On the road to formalizing informational responsibility, Chapter 3 extends stit theory with traditional epistemic notions (knowledge and belief). Thus, the chapter formalizes important aspects of agents' reasoning in the choice and performance of actions. In a context of responsibility attribution and excusability, Chapter 4 extends epistemic stit theory with measures of optimality of actions that underlie obligations. In essence, this chapter formalizes the interplay between agents' knowledge and what they ought to do. On the road to formalizing motivational responsibility, Chapter 5 adds intentions and intentional actions to epistemic stit theory and reasons about the interplay between knowledge and intentionality. Finally, Chapter 6 merges the previous chapters' formalisms into a rich logic that is able to express and model different modes of the aforementioned categories of responsibility. Technically, the most important contributions of this thesis lie in the axiomatizations of all the introduced logics. In particular, the proofs of soundness & completeness results involve long, step-by-step procedures that make use of novel techniques

The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines