Franz Berto

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

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.

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the imagined non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics.

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.

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.

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.

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....

World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic side, with a representation operator, (R), and on the semantic side, with particularly anarchic non-normal worlds. This combination easily invalidates unwelcome “logical omniscience” in- ferences of standard epistemic logic, such as belief-consistency and closure under entailment. Some open questions are then raised on the best strategies to regiment (R) in order to express more vertebrate kinds of conceivability.

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

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.

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.

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.

Noneism a is form of Meinongianism, proposed by Richard Routley and developed and improved by Graham Priest in his widely discussed book Towards Non-Being. Priest's noneism is based upon the double move of building a worlds semantics including impossible worlds, besides possible ones, and admitting a new comprehension principle for objects, differerent from the ones proposed in other kinds of neo-Meinongian theories, such as Parsons' and Zalta's. The new principle has no restrictions on the sets of properties that can deliver objects, but parameterizes the having of properties by objects to worlds. Modality is therefore explicitly built in - so the approach can be conveniently labeled as "modal noneism". In this paper, I put modal noneism to work by testing it against classical issues in modal logic and semantics. It turns out that - perhaps surprisingly - the theory performs well in problems of transworld identity, which are frequently considered to be the difficult ones in the literature; faces a limitation, albeit not a severe one, when one comes to transworld individuation, which is often taken as an easy issue, if not a pseudo-problem; and may stumble upon a real trouble when dealing with what I shall call 'extensionally indiscernible entities' - particular nonexistent objects modal noneism is committed to.