This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

Why do we remember the past and not the future? What does it mean for time to "flow"? Do we exist in time or does time exist in us? In lyric, accessible prose, Carlo Rovelli invites us to consider questions about the nature of time that continue to puzzle physicists and philosophers alike. For most readers this is unfamiliar terrain. We all experience time, but the more scientists learn about it, the more mysterious it remains. We think of it (...) as uniform and universal, moving steadily from past to future, measured by clocks. Rovelli tears down these assumptions one by one, revealing a strange universe where at the most fundamental level time disappears. He explains how the theory of quantum gravity attempts to understand and give meaning to the resulting extreme landscape of this timeless world. Weaving together ideas from philosophy, science and literature, he suggests that our perception of the flow of time depends on our perspective, better understood starting from the structure of our brain and emotions than from the physical universe. (shrink)

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, (These are examples of concepts that are taken to satisfy naive rules such as the naive truth schema and naive comprehension, and that therefore are (...) compatible with a solution to paradox cast in the logics considered below. Other notions of similar kind can be added to the list.) if one is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. This article shows that this claim is highly misleading, if not simply false. Under natural assumptions, some well-established approaches to recapture are incorrect. (shrink)

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal {M}$, or an axiomatization S thereof, we find a modal logic M such that a modal sentence $\varphi $ is a theorem of M if and only if the sentence $\varphi ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal {M}$ or a theorem of S under all such translations. (...) To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of noncongruent modal logics whose internal logic is nonclassical with respect to this semantics. (shrink)

Quantum gravity poses the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. The loop and spinfoam approach, presented in this book, is one of the leading research programs in the field. The first part of the book discusses the reformulation of the basis of classical and quantum Hamiltonian physics required by general relativity. The second part covers the basic technical research directions. Appendices include a detailed history of the (...) subject of quantum gravity, hard-to-find mathematical material, and a discussion of some philosophical issues raised by the subject. This fascinating text is ideal for graduate students entering the field, as well as researchers already working in quantum gravity. It will also appeal to philosophers and other scholars interested in the nature of space and time. (shrink)

In this paper, I discuss some of Maximilian de Gaynesford’s arguments regarding indexicals. Although I agree with his treatment of the first singular personal pronoun as a prototype of demonstrative expressions, I challenge his refusal to treat indexicals as complex demonstratives. To offer an alternative to this refusal I try to develop a common ground from different theories that consider indexicals as linguistic constructions that embed a nonlinguistic element, following an original idea in Frege’s latest writings. These views form the (...) backdrop on which we can put forward the claim of treating all indexicals as complex demonstratives. In the central part of the paper, I criticize each of de Gaynesford’s arguments against the reduction of indexicals to complex demonstratives. Besides, I propose a new definition of the concept of “demonstration” as a nonlinguistic feature of all indexicals in their referential uses, to contrast de Gaynesford’s rejection of the idea that demonstrations are an essential feature of indexicals. Eventually, I strengthen my claim by distinguishing indexicals from proper names and definite descriptions on the ground that only perceptual indexicals necessarily require an accompanying demonstration. However, the main point of the paper is a negative one, that is the rejection of de Gaynesford’s arguments against the reduction of indexicals to complex demonstratives. More work is needed to reach a positive conclusion on this topic. (shrink)

In this paper, we present one of the main starting points of naturalism in ethics: Geach’s challenge against non-cognitivism. We try to find an answer to Geach’s challenge in the notion of family resemblance applied to ethics. In doing so we recover a not much-discussed influence of Moore on Wittgenstein’s conception of family resemblance, which leads us to define Wittgenstein as non-non-cognitivist in ethics. -/- Pre print (some changes in the published edition).

In the paper we investigate typed axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. The (...) framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting. (shrink)

Relational quantum mechanics (RQM) is an interpretation of quantum mechanics based on the idea that quantum states do not describe an absolute property of a system but rather a relationship between systems. There have recently been some criticisms of RQM pertaining to issues around intersubjectivity. In this article, we show how RQM can address these criticisms by adding a new postulate which requires that all of the information possessed by a certain observer is stored in physical variables of that observer (...) and thus is accessible by measurement to other observers. This makes it possible for observers to reach intersubjective agreement about quantum events that have occurred in the past. We suggest a possible ontology for a version of RQM employing this postulate; this ontology upholds the principle that quantum states are always relational, but it also postulates a set of quantum events that are not strictly relational. We show that the new postulate helps address the preferred basis problem in RQM. (shrink)

The existence of a close connection between results on axiomatic truth and the analysis of truth-theoretic deflationism is nowadays widely recognized. The first attempt to make such link precise can be traced back to the so-called conservativeness argument due to Leon Horsten, Stewart Shapiro and Jeffrey Ketland: by employing standard Gödelian phenomena, they concluded that deflationism is untenable as any adequate theory of truth leads to consequences that were not achievable by the base theory alone. In the paper I highlight, (...) as Shapiro and Ketland, the irreducible nature of truth axioms with respect to their base theories. But, I argue, this does not immediately delineate a notion of truth playing a substantial role in philosophical or scientific explanations. I first offer a refinement of Hartry Field’s reaction to the conservativeness argument by distinguishing between metatheoretic and object-theoretic consequences of the theory of truth and address some possible rejoinders. In the resulting picture, truth is an irreducible tool for metatheoretic ascent. How robust is this characterizaton? I test it by considering: a recent example, due to Leon Horsten, of the alleged explanatory role played by the truth predicate in the derivation of Fitch’s paradox; an essential weakening of theories of truth analyzed in the first part of the paper. (shrink)

Keith Donnellan wrote his paper on definite descriptions in 1966 at Cornell University, an environment where nearly everybody was discussing Wittgenstein’s ideas of meaning as use. However, his idea of different uses of definite descriptions became one of the fundamental tenets against descriptivism, which was considered one of the main legacies of the Frege–Russell– Wittgenstein view; and I wonder whether a more Wittgensteinian interpretation of Donnellan’s work is possible.

Is reality three-dimensional and becoming real (Presentism), or is reality four-dimensional and becoming illusory (Eternalism)? Both options raise difficulties. I argue that we do not need to be trapped by this dilemma. There is a third possibility: reality has a more complex temporal structure than either of these two naive options. Fundamental becoming is real, but local and unoriented. A notion of present is well defined, but only locally and in the context of approximations.

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion (...) of logical consequence: we study the corresponding theory of validity by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth. Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence. (shrink)

This volume contains essays that explore explicit and implicit communication through linguistic research. Taking as a framework Paul Grice's theories on "what is said," the contributors explore a number of areas, including: the boundary between semantics and pragmatics; the concept of implicit communication; the idea of the logical form of our assertions; the notion of conventional meaning; the phenomenon of deixis, which refers to when an utterance require context in order to be understood fully; the treatment of definite descriptions; and (...) the different kinds of pragmatic processes. (shrink)

Contrary to claims about the irrelevance of philosophy for science, I argue that philosophy has had, and still has, far more influence on physics than is commonly assumed. I maintain that the current anti-philosophical ideology has had damaging effects on the fertility of science. I also suggest that recent important empirical results, such as the detection of the Higgs particle and gravitational waves, and the failure to detect supersymmetry where many expected to find it, question the validity of certain philosophical (...) assumptions common among theoretical physicists, inviting us to engage in a clearer philosophical reflection on scientific method. (shrink)

The article intends to tackle two key questions: what solidarity means and why it is considered a valuable trait for a society to exhibit. In so doing, it aims to bridge the literature gap between solidarity’s nature and its normative value. In this article, an original hypothesis is explored, i.e. that solidarity ought to be understood in functionalist terms, and that we should want it accordingly because solidarity discharges the crucial function of societal cohesion.

ABSTRACT ABSTRACT: I show that Aristotelian physics is a correct and nonintuitive approximation of Newtonian physics in the suitable domain in the same technical sense in which Newton’s theory is an approximation of Einstein’s theory. Aristotelian physics lasted long not because it became dogma, but because it is a very good, empirically grounded theory. This observation suggests some general considerations on intertheoretical relationships.

This article argues that the meaning of σύνεσις in the classical period has been inadequately understood, and consequently its historical significance has likely been misplaced. The traditional view is that the word possessed two basic meanings. First and foremost, σύνεσις meant a general ability to understand. Second and less frequently, it meant moral conscience or some such ability to judge the morality of human choice and action. However, by considering anew the attestations of σύνεσις and its grammatically related forms, it (...) will be shown that σύνεσις never meant moral conscience, but instead often denoted a hermeneutic virtue by which we interpret the deeper significance of things said and done. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants of Kripke–Feferman truth. The first, \, features a consistent but partial truth predicate. The second, \, an inconsistent but complete truth predicate. It is known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability, under natural assumptions, coincides with (...) definitional equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of \ and \ raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)

This paper provides a commentary on Truth and Method III.1.

We present here the first translation into a modern language of the anonymous commentary on the second book of the Nicomachean Ethics. It is an evidence of the style of exegetical work that was being done in the Peripatetic schools during the 2nd–3rd century AD, and a testimony to a particular version of 2nd century Aristotelianism. Even if the comment is not continuous, one gets the impression of listening to a good lecturer illustrating Aristotle’s text. He paraphrases it at times, (...) makes comparisons with other passages of the corpus, illustrates the structure and the logic of the discourse, and shows the difference between Aristotle’s positions and those of other schools. In the most theoretically interesting parts of the commentary the anonymous discusses aporiai, and opposes Aristotle’s ethical theories to other schools, especially to the Stoics. (shrink)

L’article présente une brève histoire de l’interprétation évolutionniste des traités d’Aristote à partir de la tendance évolutionniste de Jaeger. Il se concentre surtout, en premier lieu, sur la naissance et le développement des interprétations dialectiques des traités aristotéliciens à partir des années soixante du XX e siècle et, en second lieu, sur les critiques adressées à cette tendance par les interprètes contemporains du début du XXI e siècle. La section finale propose une approche alternative : selon cette interprétation, Aristote ne (...) renonce pas, y compris dans ses Éthiques, à la recherche de la définition. Dans l’ Éthique à Nicomaque, il parvient à définir les principaux concepts et phénomènes moraux en suivant la méthode permettant de fonder les définitions qui est décrite dans les Seconds Analytiques, livre II. Cela n’exclut pas que la méthode « dialectique » ou « endoxale » ait un certain rôle dans l’ Éthique à Nicomaque. (shrink)

La investigación presente analizará las semejanzas y relaciones encontradas entre los planteamientos de Edmund Husserl y Simone Weil respecto al fenómeno religioso de la idolatría. Para ello, señalaremos en primer lugar el tratamiento que la religión podría tener dentro del marco fenomenológico husserliano; posteriormente, nos enfocaremos en el análisis de una de las publicaciones póstumas de autoría weiliana. Respecto al filósofo austríaco, se recurrirá a textos de distintos períodos de su obra, por lo que implícitamente se contemplará una continuación entre (...) sus denominados análisis estáticos y los genéticos. (shrink)

We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules are all valid in fixed-point models; the second by ‘\-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that (...) to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth. (shrink)

Should college teachers still teach works with immoral content? What if the works are by deeply immoral thinkers? This guide is intended to help us answer these sorts of pedagogical questions by articulating the pertinent moral issues and then suggesting strategies for navigating them.