We need to understand the impossible. Francesco Berto and Mark Jago start by considering what the concepts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition have in common. They are all concepts which divide the world up more finely than logic does. Logically equivalent sentences may carry different meanings and information and may differ in how they're believed. Fictions can be inconsistent yet meaningful. We can suppose impossible things without collapsing into total incoherence. Yet for the leading (...) philosophical theories of meaning, these phenomena are an unfathomable mystery. To understand these concepts, we need a metaphysical, logical, and conceptual grasp of situations that could not possibly exist: Impossible Worlds. This book discusses the metaphysics of impossible worlds and applies the concept to a range of central topics and open issues in logic, semantics, and philosophy. It considers problems in the logic of knowledge, the meaning of alternative logics, models of imagination and mental simulation, the theory of information, truth in fiction, the meaning of conditional statements, and reasoning about the impossible. In all these cases, impossible worlds have an essential role to play. (shrink)

It is a venerable slogan due to David Hume, and inherited by the empiricist tradition, that the impossible cannot be believed, or even conceived. In Positivismus und Realismus, Moritz Schlick claimed that, while the merely practically impossible is still conceivable, the logically impossible, such as an explicit inconsistency, is simply unthinkable. -/- An opposite philosophical tradition, however, maintains that inconsistencies and logical impossibilities are thinkable, and sometimes believable, too. In the Science of Logic, Hegel already complained against “one of the (...) fundamental prejudices of logic as hitherto understood”, namely that “the contradictory cannot be imagined or thought” (Hegel 1931: 430). Our representational capabilities are not limited to the possible, for we appear to be able to imagine and describe also impossibilities — perhaps without being aware that they are impossible. -/- Such impossibilities and inconsistencies are what this entry is about... (shrink)

Drawing on different suggestions from the literature, we outline a unified metaphysical framework, labeled as Modal Meinongian Metaphysics (MMM), combining Meinongian themes with a non-standard modal ontology. The MMM approach is based on (1) a comprehension principle (CP) for objects in unrestricted, but qualified form, and (2) the employment of an ontology of impossible worlds, besides possible ones. In §§1–2, we introduce the classical Meinongian metaphysics and consider two famous Russellian criticisms, namely (a) the charge of inconsistency and (b) the (...) claim that naïve Meinongianism allows one to prove that anything exists. In §3, we have impossible worlds enter the stage and provide independent justification for their use. In §4, we introduce our revised comprehension principle: our CP has no restriction on the (sets of) properties that can characterize objects, but parameterizes them to worlds, therefore having modality explicitly built into it. In §5, we propose an application of the MMM apparatus to fictional objects and defend the naturalness of our treatment against alternative approaches. Finally, in §6, we consider David Lewis’ notorious objection to impossibilia, and provide a reply to it by resorting to an ersatz account of worlds. (shrink)

This article seeks to examine Ricœur's reflection on metaphor through an intertextual reading. This reading relates The Rule of Metaphor (1975) with lines of thought developed in a series of lectures held at the Centre de recherches phénoménologiques in Paris, from 1973–1974, on the theme From Language to Image, and an essay which appeared in 1978, “The Metaphorical Process as Cognition, Imagination and Feeling”. This work starts with an analysis of Ricœur's interpretation of the Aristotelian theory of metaphor, a theory (...) which the French philosopher principally develops in The Rule of Metaphor. In the other two papers, Ricœur develops his metaphorology through a dialogue with Kant and he reformulates in linguistic terms the doctrine of productive imagination and transcendental schematism, which become the ontological sphere on which metaphorical and narrative activity are founded. Through the long journey of language, Ricœur's hermeneutics builds a bridge between Aristotle and Kant. This article examines this link to understand and explain further, also at the epistemological level, the relationship between metaphor and narrative. (shrink)

Modal Meinongianism—the version of Meinongianism invented by Graham Priest—presupposes that we can think about absolute impossibilities. I defend the view that we can, by tidying up a couple of loose ends in Priest’s arguments to this effect from his book Towards Non-Being.

Recent controversies about the level of replicability of behavioral research analyzed using statistical inference have cast interest in developing more efficient techniques for analyzing the results of psychological experiments. Here we claim that complementing the analytical workflow of psychological experiments with Machine Learning-based analysis will both maximize accuracy and minimize replicability issues. As compared to statistical inference, ML analysis of experimental data is model agnostic and primarily focused on prediction rather than inference. We also highlight some potential pitfalls resulting from (...) adoption of Machine Learning based experiment analysis. If not properly used it can lead to over-optimistic accuracy estimates similarly observed using statistical inference. Remedies to such pitfalls are also presented such and building model based on cross validation and the use of ensemble models. ML models are typically regarded as black boxes and we will discuss strategies aimed at rendering more transparent the predictions. (shrink)

Philosophical dialetheism, whose main exponent is Graham Priest, claims that some contradictions hold, are true, and it is rational to accept and assert them. Such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC). But all the classic formulations of the LNC are, in a sense, not questioned by a typical dialetheist, since she is (cheerfully) required to accept them by her own theory. The goal of this paper is to develop a formulation of the (...) Law which appears to be unquestionable, in the sense that the Priestian dialetheist is committed to accept it without also accepting something inconsistent with it, on pain of trivialism—that is to say, on pain of lapsing into the position according to which everything is the case. This will be achieved via (a) a discussion of Priest's dialetheic treatment of the notions of rejection and denial; and (b) the characterization of a negation via the primitive intuition of content exclusion. Such a result will not constitute a cheap victory for the friends of consistency. We may just learn that different things have been historically conflated under the label of 'Law of Non-Contradiction'; that dialetheists rightly attack some formulations of the Law, and orthodox logicians and philosophers have been mistaken in assimilating them to the indisputable one. (shrink)

A conflict of interest is a situation in which a person has competing loyalties or interests that make it difficult to fulfil his or her duties impartially. Conflict of interest is not categorically improper in itself but requires proper management. A SCOPUS literature search was performed for publications on the efficacy/safety of Phospho-Di-Esterase Inhibitors for treating erectile dysfunction. A categorization tool was used to review and classify the publications as supportive/not-supportive for the discussed active ingredient and reporting or not reporting (...) a COI for that specific drug or for the remaining PDEIs. Multivariable binary logistic regression was performed. In the 419 selected records the prevalence of supportive opinions was higher when a COI for the index label was declared. The CoOpCaT showed good internal consistency, discriminative validity and intra/inter-rater agreement. The strongest predictor for a supportive opinion was the total number of financial COIs for the index label. A mild protective effect of the total number of financial COIs for any competitor label was noted. Financial COIs have frequently been associated with bias, and the measures currently adopted to restrain it lack effectiveness. Some evidence for monitoring and/or compensating this bias is reported here, but the ultimate solution remains distant. (shrink)

Reflection on the natural/artificial, real/imaginary, subjective/objective dicotomies is increasingly the subject of debate on systems of communication, which make more and more widespread use of expert and/or intelligent advanced technologies.This paper analyses different forms of communication operating in a socio-technical system. The analysis is concerned with changes in creativity and participation of the human being in the decision and production processes within various contexts of social life. It is thus important to verify the possibility of an interpolation between vital world, (...) social system, and information technologies in order to understand a possible interrelation between therational and therelational components in the presentsocio-technical system. (shrink)

Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, (...) non-technical style. (shrink)

This book is both an introduction to and a research work on Meinongianism. “Meinongianism” is taken here, in accordance with the common philosophical jargon, as a general label for a set of theories of existence – probably the most basic notion of ontology. As an introduction, the book provides the first comprehensive survey and guide to Meinongianism and non-standard theories of existence in all their main forms. As a research work, the book exposes and develops the most up-to-date Meinongian theory (...) (called modal Meinongianism), applies it to specific fields, and discusses its open problems. Part I of the book provides a historical introduction to, and critical discussion of, the dominant philosophical view of existence: the “Kantian-Fregean-Quinean” perspective. Part II is the full-fledged introduction to the Meinongian theories of existence as a real property of individuals: after starting with the so-called naïve Meinongian conception and its problems, it provides a self-contained presentation of the main neo-Meinongian proposals, and a detailed discussion of their strengths and weaknesses. Part III develops a specific neo-Meinongian theory of existence employing a model-theoretic semantic framework. It discusses its application to the ontology and semantics of fictional objects, and its open problems. The methodology of the book follows the most recent trends in analytic ontology. In particular, the meta-ontological point of view is largely privileged. (shrink)

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.

Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...) operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it. (shrink)

There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does not (...) hold unrestrictedly – that in peculiar circumstances the same thing may at the same time be and not be, and contradictions may obtain in the world. This book opens with an examination of the famous logical paradoxes that appear to speak on behalf of contradictions (e.g., the Liar paradox, the set-theoretic paradoxes such as Cantor’s and Russell’s), and of the reasons for the failure of the standard attempts to solve them. It provides, then, an introduction to paraconsistent logics – non-classical logics in which the admission of contradictions does not lead to logical chaos –, and their astonishing applications, going from inconsistent data base management to contradictory arithmetics capable of circumventing Gödel’s celebrated Incompleteness Theorem. The final part of the book discusses the philosophical motivations and difficulties of dialetheism, and shows how to extract from Aristotle’s ancient words a possible reply to the dialetheic challenge. How to Sell a Contradiction will appeal to anyone interested in non-classical logics, analytic metaphysics, and philosophy of mathematics, and especially to those who consider challenging our most entrenched beliefs the main duty of philosophical inquiry. (shrink)

We outline a neo-Meinongian framework labeled as Modal Meinongian Metaphysics (MMM) to account for the ontology and semantics of fictional discourse. Several competing accounts of fictional objects are originated by the fact that our talking of them mirrors incoherent intuitions: mainstream theories of fiction privilege some such intuitions, but are forced to account for others via complicated paraphrases of the relevant sentences. An ideal theory should resort to as few paraphrases as possible. In Sect. 1, we make this explicit via (...) two methodological principles, called the Minimal Revision and the Acceptability Constraint. In Sect. 2, we introduce the standard distinction between internal and external fictional discourse. In Sects. 3–5, we discuss the approaches of (traditional) Meinongianism, Fictionalism, and Realism—and their main troubles. In Sect. 6 we propose our MMM approach. This is based upon (1) a modal semantics including impossible worlds (Subsect. 6.1); (2) a qualified Comprehension Principle for objects (Subsect. 6.2); (3) a notion of existence-entailment for properties (Subsect. 6.3). In Sect. 7 we present a formal semantics for MMM based upon a representation operator. And in Sect. 8 we have a look at how MMM solves the problems of the three aforementioned theories. (shrink)

Accounts of propositions as sets of possible worlds have been criticized for conflating distinct impossible propositions. In response to this problem, some have proposed to introduce impossible worlds to represent distinct impossibilities, endorsing the thesis that impossible worlds must be of the same kind; this has been called the parity thesis. I show that this thesis faces problems, and propose a hybrid account which rejects it: possible worlds are taken as concrete Lewisian worlds, and impossibilities are represented as set-theoretic constructions (...) out of them. This hybrid account (1) distinguishes many intuitively distinct impossible propositions; (2) identifies impossible propositions with extensional constructions; (3) avoids resorting to primitive modality, at least so far as Lewisian modal realism does. (shrink)

In his famous work on vagueness, Russell named “fallacy of verbalism” the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories – the non-adjunctive and relevant approaches –, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of (...) their underlying semantics. The meaning of “intuitive reading” is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense – therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency – and this could be a welcome result for a dialetheist. (shrink)

This research is published within the project ‘The Logic of Conceivability’, funded by the European Research Council, Grant Number 681404.

We address an argument by Floridi (Synthese 168(1):151–178, 2009; 2011a), to the effect that digital and analogue are not features of reality, only of modes of presentation of reality. One can therefore have an informational ontology, like Floridi’s Informational Structural Realism, without commitment to a supposedly digital or analogue world. After introducing the topic in Sect. 1, in Sect. 2 we explain what the proposition expressed by the title of our paper means. In Sect. 3, we describe Floridi’s argument. In (...) the following three sections, we raise three difficulties for it, (i) an objection from intuitions: Floridi’s view is not supported by the intuitions embedded in the scientific views he exploits (Sect. 4); (ii) an objection from mereology: the view is incompatible with the world’s having parts (Sect. 5); (iii) an objection from counting: the view entails that the question of how many things there are doesn’t make sense (Sect. 6). In Sect. 7, we outline two possible ways out for Floridi’s position. Such ways out involve tampering with the logical properties of identity, and this may be bothersome enough. Thus, Floridi’s modus ponens will be our (and most ontologists’) modus tollens. (shrink)

In two papers earlier in his career, Siméon Rajaona—one of Madagascar's most famous intellectuals—argues that Westerners have tended to distort the Malagasy worldview by interpolating Western notions into their understanding of it. As a result, the authentic characteristics of the Malagasy mind have been missed by many in the West. He claims that when compared to Westerners, Malagasy have a distinct notion of truth, a different style of reasoning, a different conceptual connection with the world, and a distinct ethical system. (...) His work on this topic is pioneering and insightful. We think that Rajaona is correct on some points but that others are overestimated. In the essay, we explain his work and raise challenges for most of his claims and express agreement with him in parts. While we express skepticism about his claims involving truth, reasoning, and conceptual connection, we agree with him that there is a distinctive Malagasy ethics, though it has analogues in the West. At the end of the paper, we sketch what we take to be distinct elements in the Malagasy worldview relative to Rajaona’s claims. (shrink)

Es mag überraschen, in einer der Religion und Offenbarung gewidmeten Schrift ein einleitendes Kapitel moralischen Inhalts zu finden: Die »Theorie des Willens«, die bedeutendste Erweiterung in der zweiten Auflage der Offenbarungskritik. Die Einschiebung dieses Kapitels macht die Beziehung zwischen Moral und Religion noch deutlicher, wodurch sich Fichte auf eine charakteristische Tendenz der deutschen Aufklärung bezieht. Ein erster Grund dieser fichteschen Stellungnahme liegt in der geistigen Bildung des jungen Gelehrten, der zur moralischen Thematik und zur Begeisterung für Kants moralisches System aus (...) einem ursprünglichen theologischen und religiösen Interesse heraus kommt: In den unmittelbar der Offenbarungskritik vorangehenden Jahren ist dieses Interesse besonders dem Problem der Gnade und der menschlichen Freiheit gewidmet. Ich beschränke mich auf die drei wichtigsten Schriften der jugendlichen fichteschen Periode, oder besser der sogenannten »vorkantischen« Periode, die Predigten An Mariä Verkündigung und Ueber die Absichten des Todes Jesu und das Fragment Einige Aphorismen über Religion und Deismus ; in all diesen Schriften ist die Annahme der sich auf den Menschen auswirkenden Gnade Gottes, die ihm Versöhnungs- und fortschreitende Annäherungsgelegenheiten zur moralischen Vollkommenheit gibt, der Bedingung unterworfen, daß der Mensch von seiner moralischen Verantwortlichkeit nicht abgehalten wird, daß er sich sozusagen nicht einer untätigen Erwartung des göttlichen Gefallens überläßt. Er soll dagegen, je nach der Wohltat, die er von Gott bekommen hat, das Bessere anstreben, und das ist auch das Kriterium, wonach Gott ihn beurteilen wird. Dieses aktive Bemühen um die eigene Rettung kann aber nur aus der Beachtung und aus dem Gehorsam gegenüber der Stimme des eigenen Gewissens, und nicht aus der Ausübung einer nur formalen Gerechtigkeit oder aus der Befolgung besonderer Kultusformen hervorgehen. Wie Fichte mit ausdrücklichem Bezug auf Paulus behauptet, wird man vor Gott nicht durch Werke gerechtfertigt, sondern durch die Ausübung eines Glaubens, der in seiner vollkommensten Form von der christlichen Religion vertreten wird und vor allem ein und dasselbe mit einer »Verfassung der Seele« ist; das führt zu jener »genauen Beobachtung unsrer eignen Seele«, die uns erlaubt, die von Gott in unsere Seele gebrachten Veranlassungen zur moralischen Besserung mit Klarheit zu erfassen. Die wahre Religion hat die Rolle, die Sinnlichkeit auszurotten und die »sanften Neigungen des Menschen, Güte, Sanftmuth, Gefälligkeit« zu erwecken: Fichte spricht von einer »Religion des Herzens«, aus dessen Güte und Wohlwollen eine »warme, lebendige Überzeugung« kommt, die auf ein inneres »Gefühl des Guten« gegründet ist und die religiöse Wahrheiten zur moralischen Triebfeder macht. In den kurz nach der Lektüre der ersten Kritik Kants geschriebenen Aphorismen ist das Gleichgewicht zwischen intellektuellen und sinnlichen Bestandteilen durch den von Fichte hervorgehobenen Widerspruch zwischen dem Gesichtspunkt des Herzens und den Forderungen der Spekulation in Frage gestellt: In der Spekulation würde man verlangen, eine rein vernünftige Erklärung des Glaubens zu geben, dessen innigste Natur aber, als der Bestrebung zu einem vertrauten und persönlichen zuerst auf die Empfindung gegründeten Verhältnis zu Gott einer solchen Beanspruchung eben nicht gerecht werden kann. Nur der von Kant herausgestellte Unterschied zwischen praktischer und spekulativer Vernunft wird Fichte die Mittel für die Überwindung dieses Widerspruchs geben, und damit auch die Möglichkeit, die Harmonie zwischen Philosophie und Religion wieder zu bestätigen: In einer bemerkenswerten Stelle der Aphorismen sieht Fichte nämlich eine völlige Übereinstimmung zwischen der kantischen Kritik und den Grundsätzen der Apostel über die Grenze der menschlichen Vernunft in den Untersuchungen der Glaubensdinge. (shrink)

The question of the mimesis metaphorike in Aristotle is developed, in the interpretative perspective of this work, through an itinerary that defines it in relation to poetical thought, ontology and in its gnoseological and epistemological aspects. The pages that the philosopher has explicitly dedicated to the metaphor might be considered as examples in a sort of reflection-guide to understand the link existing, according to Aristotle, between language, thought and being. At the same time they offer a fundamental theoretical element to (...) proceed in the search for that unity by analogy that has occupied a great part of the Aristotelian interpretative tradition. The hermeneutical perspective of this work represents a critical position in relation to the most common interpretations of the theory of the metaphor in Aristotle (that explain it in linguistic-semiotic terms) and tries to retrieve and bring to light the metaphysical horizon and the “referentialist” vocation that mark the reflection of the Greek philosopher. Yet, notwithstanding this background, Aristotle’s reflection manages to generate a realism that is not quashed in the opposition object-subject, inner world-outside world, but that is able, even today, to offer fundamental parameters to think of the relationship between language, world and thought in terms that are not any longer those of a world defined through a totalising linguistic dimension. In this sense, the Aristotelian doctrine of the metaphor constitutes today a fundamental theoretical paradigm to think deep and critically the kind of relation that might exist between arts and truth and, also, between language and world. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

I take into account Ferraris’ attempt at reversing the traditional order of explanation going from thought to language and writing, as exposed in Documentalità. The reversal is supposed to provide a new ontology of social objects that dispenses with Searle’s notion of (collective) intentionality. The book’s motto is «[social] object = written act». What does that identity sign mean? Given that social objects are not identical with documents taken as mere material objects, they must be identical with documents taken as (...) (systems of) signs, sÚmbola. Can one explain how a sign refers to what it is a sign of, without resorting to intentionality? I’m not sure – but I am pretty sure that the Derridean notion of arch-writing is not going to help. (shrink)

Impossible Worlds focuses on an exciting new theory in philosophy, with applications in metaphysics, logic, and the theory of meaning. Its central topic is: how do we meaningfully talk and reason about situations which, unbeknownst to us, are impossible? This issue emerges as a central problem in contemporary philosophical accounts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition. The book is written bytwo of the leading philosophers in the area and contains original research of relevance to professional philosophers (...) and logicians working in metaphysics, philosophy of language, formal logic, and adjacentareas. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

The present paper is meant to summarise and enlighten the theoretical implications of the twin theories of text comprehension and of text compression. Compatibility and non-exclusiveness of particle-like analysis of language and wave-like analysis of intentionality are also demonstrated within the newly established quantum linguistics framework. The informative state of language is viewed as being relatively stable; once activated and subject to motion, therefore reaching a communicative state, different phenomena occur, which may be observed, analysed and visualised through CPP-TRS observational (...) devices. Relativity theory may therefore be organised in terms of quanta with continuity and no contradiction. (shrink)

Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogeneous, simple units, the atoms or cells. At each time unit, the cells instantiate one of a finite set of states. They evolve in parallel at discrete time steps, following (...) state update functions or dynamical transition rules: the update of a cell state obtains by taking into account the states of cells in its local neighborhood (there are, therefore, no actions at a distance). Secondly, CA are abstract, as they can be specified in purely mathematical terms and implemented in physical structures. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems. Despite functioning in a different way from traditional, Turing machine-like devices, CA with suitable rules can emulate a universal Turing machine, and therefore compute, given Turing's Thesis, anything computable.... (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

A counterpossible conditional is a counterfactual with an impossible antecedent. Common sense delivers the view that some such conditionals are true, and some are false. In recent publications, Timothy Williamson has defended the view that all are true. In this paper we defend the common sense view against Williamson’s objections.