We introduce an epistemic version of validity and completeness of first order logic, based on the notions of ideal agent and fictional model. We then show how the perspective here considered may help to solve an epistemic puzzle arising from Gödel's second incompleteness theorem.

In section 1 we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of theclassical meaning of the logical constants, and consequently is incompatible with the antirealism of orthodox intuitionism. In section 2 we show how what we call the potential intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability (...) and, on the other, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic. In section 3 we uphold the view that certain consequences of the adoption of atemporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view. (shrink)

Brouwer's theorem of 1927 on the equivalence between virtual and inextensible order is discussed. Several commentators considered the theorem at issue as problematic in various ways. Brouwer himself, at a certain time, believed to have found a very simple counter-example to his theorem. In some later publications, however, he stated the theorem in the original form again. It is argued that the source of all criticisms is Brouwer's overly elliptical formulation of the definition of inextensible order, as well as a (...) certain ambiguity in his terminology. Once these drawbacks are removed, his proof goes through. (shrink)

The present paper deals with natural intuitionistic semantics for intuitionistic logic within an intuitionistic metamathematics. We show how strong completeness of full first order logic fails. We then consider a negationless semantics à la Henkin for second order intuitionistic logic. By using the theory of lawless sequences we prove that, for such semantics, strong completeness is restorable. We argue that lawless negationless semantics is a suitable framework for a constructive structuralist interpretation of any second order formalizable theory (classical or intuitionistic, (...) contradictory or not). (shrink)

This paper proposes a new dialetheic logic, a Dialetheic Logic with Exclusive Assumptions and Conclusions ), including classical logic as a particular case. In \, exclusivity is expressed via the speech acts of assuming and concluding. In the paper we adopt the semantics of the logic of paradox extended with a generalized notion of model and we modify its proof theory by refining the notions of assumption and conclusion. The paper starts with an explanation of the adopted philosophical perspective, then (...) we propose our \ logic. Finally, we show how \ supports the dialetheic solution of the liar paradox. (shrink)

This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers - both new and previously published - it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer's idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical (...) realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer's work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics. (shrink)

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite (...) à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals. (shrink)

In Parts of Classes (1991) and Mathematics Is Megethology (1993) David Lewis defends both the innocence of plural quantification and of mereology. However, he himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out as a whole. In the case of plural quantification . Instead, in the mereological case: (Lewis, 1991, p. 87). The aim of the paper is to (...) argue that one—an innocence thesis similar to that of plural reference is defensible. To give a precise account of plural reference, we use the idea of plural choice. We then propose a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. (shrink)

In Mathematics is megethology. Philosophia Mathematica, 1, 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non expressed (...) in a mereological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological codification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification. (shrink)

In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification (...) are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, ifMPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects. (shrink)

In Parts of Classes [Lewis 1991] David Lewis attempts to draw a sharp contrast between mereology and set theory and to assimilate mereology to logic. He argues that, like logic but unlike set theory, mereology is “ontologically innocent”. In mereology, given certain objects, no further ontological commitment is required for the existence of their sum. On the contrary, by accepting set theory, given certain objects, a further commitment is required for the existence of the set of them. The latter – (...) unlike the sum of the given objects – seems to be an abstract entity whose existence is not directly entailed by the existence of the objects themselves. The argument for the innocence of mereology is grounded on the thesis of “Composition as identity”. Lewis analyses two different versions of the thesis: the first is the Strong composition thesis, according to which certain objects are their sum, where the use of “are” would mean that composition is literally identity. The second version is the Weak composition thesis, according to which composition is analogous, under some aspects, to identity. He criticises the first version of the thesis and argues for the second one. In the paper we argue that (T1) arguments for the ontological innocence of mereology are not conclusive. An obvious objection to the Strong composition thesis is that – given certain objects Xs – they cannot be their sum because none of them is the sum. One could reply to this objection by observing that the “are” in the sentence “The Xs are their sum” is to be understood collectively and not distributively. But the crux is that the collective reading fails to generate a new entity, whereas mereology, in particular in Lewis’ use for the reconstruction of set theory as “megethology”, needs to consider sums as real objects. Besides, we contend that Lewis’ argument for the innocence of mereology based on the Weak composition thesis is a petitio principii. The reason is that the aspects of the analogy between composition and identity, which Lewis emphasises, obtain under the presupposition of the existence of sums. But this is just what a denier of innocence would refuse. (T2) Some arguments against the ontological innocence of mereology show a certain ambiguity in the innocence thesis itself. Some defences of the innocence seem to implicitly presuppose that the sum of certain objects Xs is not a genuine entity. Speaking of the sum of the Xs would be just another way of speaking plurally of the Xs. However, the relevant use of sums in mereology treats them as well determined objects. The relevant innocence thesis takes for granted that, though sums are genuine objects, nevertheless their existence does not require any further commitment. (T3) The innocence thesis, apart from Lewis’ defence, seems to depend on a general conception of the nature of objects and on how the notion of ontological commitment is understood. We think that the thesis is the manifesto of a realistic conception of parts and sums. This conception consists of the following clauses: (i) given any object x, it is well determined which parts it possesses; these are in turn objects whose existence is a necessary consequence of the existence of x. (ii) However any objects Xs are given, they automatically constitute a well determined object x which is their sum; (iii) We can refer singularly and plurally to parts and sums of given objects. Obviously, one might wonder if such a conception is really ontologically innocent. One could object that it is not innocent because clauses (i) – (iii) are not. For example, clause (i) could be considered as an ontological commitment to the existence of sums. But the innocence at issue does not concern the above-sketched conception. The innocence is embedded in the conception itself. In other words, someone who argues for clauses (i) – (iii) takes a point of view from which mereology appears to be innocent. For, such a point of view forces us to consider as well determined the parts of any object and does not allow us to separate the existence of certain objects form the existence of their sum. (T4) is the claim that the alleged innocence of mereology is subject to Quine’s notorious criticisms of the set-theoretical interpretation of second order logic. To the purpose, we construct a mereological model of a substantive fragment of set theory, i.e. the one that grounds the principal model semantics of second order logic. First, we construct a mereological model under the assumption of the existence of infinitely many atoms. Then, we replace this assumption with that of the existence of any infinite object (with or without atoms). Finally, let us make a general point about the innocence thesis of mereology. A conclusive argument for that would be a refutation of the thesis that there are only denumerably many entities. For, since the parts of an infinite object constitute a non-denumerable infinity, such an argument would entail that there could be no infinite without a non-denumerable infinity. However, the thesis that any genuine infinity is a denumerable one has had some important advocates. So, a conclusive argument for the innocence of mereology seems to be highly implausible. (shrink)

In Mathematics is megethology Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis’work is very attractive. However, the alleged (...) innocence of mereology and plural quantification is highly controversial and has been criticized by several authors. In the present paper we propose a new approach to megethology based on the theory of plural reference developed in To be is to be the object of a possible act of choice. Our approach shows how megethology can be grounded on plural reference without the help of mereology. (shrink)

In Parts of Classes David Lewis attempts to draw a sharp contrast between mereology and set theory and he tries to assimilate mereology to logic. For him, like logic but unlike set theory, mereology is “ontologically innocent”. In mereology, given certain objects, no further ontological commitment is required for the existence of their sum. On the contrary, by accepting set theory, given certain objects, a further commitment is required for the existence of the set of them. The latter – unlike (...) the sum of the given objects – seems to be an abstract entity whose existence is not directly entailed by the existence of the objects themselves. The argument for the innocence of mereology is grounded on the thesis of composition as identity. In our paper we argue that: arguments for the ontological innocence of mereology are not conclusive. Some arguments against the ontological innocence of mereology show a certain ambiguity in the innocence thesis itself. The innocence thesis seems to depend on a general conception of the nature of objects and on how the notion of ontological commitment is understood. Specifically, we think that the thesis is the manifesto of a realistic conception of parts and sums. Quine‟s notorious criticism of the set-theoretical interpretation of second order logic seems to be reproducible against Lewis‟defence of mereology. To the purpose we construct a mereological model of a substantive fragment of set theory, adequate to ground the set-theoretical semantics of second order logic. (shrink)

In the 1951 Gibbs lecture, Gödel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naïve proof, to the effect that Gödel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naïve proof, in agreement with Gödel’s conception, superseding the criticisms against the usual notion of naïve proof used by real working mathematicians. We explore the connection between (...) Gödel’s theorem and naïve proof so understood, both from a classical and a dialetheic perspective. (shrink)

This paper investigates the question of how we manage to single out the natural number structure as the intended interpretation of our arithmetical language. Horsten submits that the reference of our arithmetical vocabulary is determined by our knowledge of some principles of arithmetic on the one hand, and by our computational abilities on the other. We argue against such a view and we submit an alternative answer. We single out the structure of natural numbers through our intuition of the absolute (...) notion of finiteness. (shrink)

This book develops a new approach to plural arbitrary reference and examines mereology, including considering four theses on the alleged innocence of mereology. The authors have advanced the notion of plural arbitrary reference in terms of idealized plural acts of choice, performed by a suitable team of agents. In the first part of the book, readers will discover a revision of Boolosʼ interpretation of second order logic in terms of plural quantification and a sketched structuralist reconstruction of second-order arithmetic based (...) on the axiom of infinite, a la Dedekind, as the unique non-logical axiom. The work goes on to analyse the pros and cons of the new interpretation, also with respect to Linneboʼs objections to the thesis that second order logic is genuine logic. A theory of concepts that can be labelled as a theory of logical concepts is expounded. In the second part of the book, the authors consider grounding megethology on plural arbitrary reference and argue that the arguments for the ontological innocence of mereology are not conclusive and that – for a certain use of mereology – a thesis of innocence, similar to that of plural arbitrary reference, is defensible. The work proposes a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. This considered work will appeal to scholars from branches of analytic philosophy, logic and the philosophy of mathematics in particular. (shrink)

The entailment connective is introduced by Priest (2006b). It aims to capture, in a dialetheically acceptable way, the informal notion of logical consequence. This connective does not “fall foul” of Curry’s Paradox by invalidating an inference rule called “Absorption” (or “Contraction”) and the classical logical theorem called “Assertion”. In this paper we show that the semantics of entailment, given by Priest in terms of possible worlds, is inadequate. In particular, we will argue that Priest’s counterexamples to Absorption and Assertion use (...) in the metalanguage a dialetheically unacceptable principle. Furthermore, we show that the rejection of Assertion undermines Priest’s claim that the entailment connective expresses the notion of logical consequence. (shrink)

In To be is to be the object of a possible act of choice the authors defended Boolos’ thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that—in a sense to be explained—can be labeled as a theory of logical concepts. (...) Within this theory, we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege’s traditional logicism. (shrink)

MartinoEnrico.* * Intuitionistic Proof Versus Classical Truth, The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics. Logic, Methodology and the Unity of Science; 42. Springer, 2018. ISBN: 978-3-319-74356-1 ; 978-3-030-08971-9, 978-3-319-74357-8. Pp. xiii + 170.

PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which (...) is radically alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts. (shrink)

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the (...) existence of individuals rather than on the existence of concepts. (shrink)

L'arte è l'espressione intima--ma concepita per essere condivisa--della sensibilità di alcuni individui che avvertono l'urgenza di manifestare il proprio sentire rispetto al reale, o al non-reale. Gli artisti e, di conseguenza, le loro opere non possono non accogliere le sollecitazioni provenienti dall'attualità che li anima elaborando le complessità del mondo contemporaneo. È quindi necessario intrecciare come in una mappa neuronale lo studio delle opere d'arte con altre discipline, come antropologia, filosofia, biologia, perché una più ampia libertà di indagine consente di (...) strutturare con maggiore armonia la storia e la critica d'arte. Realizzare dialoghi plurimi permetterà così una visione più piena e approfondita dell'arte stessa."--Back cover. (shrink)

Recently, José María Ariso and Samuel Laves have critically debated whether killing innocent and non‐threatening people [=WK] is a universal moral certainty. One of the main topics of their discussion concerns the case of the pisa‐suaves, children born in the context of the Colombian civil war who grew up with the FARC guerrillas. While Laves argues that such children hold WK, Ariso rejects his claim and stresses that pisa‐suaves have no moral code of conduct. In my work, I side with (...) Laves and contend that WK is a universal moral certainty. Accordingly, I maintain that (i) either pisa‐suaves possess WK and thereby count as human moral agents, or (ii) if they do not possess any moral code of conduct, then they cannot count as human moral agents and, in turn, their existence does not undermine WK's universality. (shrink)

Over the past two decades, virtue ethicists have begun to devote increasing attention to applied ethics. In particular, the application of virtue ethical frameworks to the environmental ethics debate has flourished. This chapter reviews recent contributions to the literature in this field and highlights some strengths and weaknesses of thinking about climate change through a virtue ethical lens. Section “Two Benefits of Virtue Ethical Approaches to Climate Change” explores two benefits of applying virtue ethics to climate change: (a) we can (...) better capture the phenomenology of our moral experience, and (b) we avoid the problem of inconsequentialism. Section “A Catalogue of Environmental Virtues” analyzes various practical proposals that have been put forward in the form of specific environmental virtues. Section “An Objection to Virtue-Oriented Approaches to Climate Change” reconstructs a fundamental objection to the idea of using a virtue ethical normative approach to tackling the urgent and imminent dangers of climate change. (shrink)

The article presents the various phases in which one of the most eminent journals of the history of philosophy, the Archiv für Geschichte der Philosophie (1888–), dealt with Leibniz’s philosophy and his intellectual legacy. In particular, this study compares the main moments of historiographical interest and disinterest for this subject to the specific attitudes of the journal during the long 20th century.

De nombreuses recherches ont mis en évidence le fait que les épicuriens n’étaient pas complètement hostiles à la poésie en général mais qu’ils refusaient probablement les compositions ou procédures poétiques qui ne conduisent pas à la fin naturelle du plaisir « catastématique », c’est-à-dire le bonheur. Dans cet article, nous nous demanderons donc si les épicuriens ont inventé une poésie cathartique de type positif ou s’ils ont simplement rejeté toutes les formes de catharsis poétique en les décrivant comme des expériences (...) négatives. Nous constaterons dans ce travail de recherche, divisé en trois parties, que la première hypothèse est plus probable que la seconde. Dans un premier temps, nous démontrerons que les épicuriens ont effectivement reconnu l’existence d’une catharsis positive, transmise par la raison et l’enseignement de la philosophie, ayant la capacité de faire disparaître les émotions négatives qui entravent l’accès au bonheur (l’amour, la peur du divin, le désir excessif, etc.). L’idée développée, dans un second temps, établit qu’Épicure et ses disciples pourraient s’être opposés à la catharsis tragique décrite dans la Poétique d’Aristote, parce qu’elle implique de passer par des passions négatives telles que la pitié et la terreur. Cette expérience esthétique entrerait alors en contradiction avec l’idéal épicurien du plaisir comme absence de trouble et d’inquiétude psychique. Enfin, dans un troisième temps, nous exposerons l’idée selon laquelle les épicuriens pourraient avoir accepté une seule forme de catharsis poétique : celle facilitant le processus de purification effectué par la raison et l’enseignement de la philosophie. Un exemple possible de cette pratique est le poème De rerum natura de Lucrèce dans lequel la poésie n’est pas considérée comme cathartique en soi, mais si et seulement si elle assiste la philosophie épicurienne dans sa conquête du bonheur. (shrink)

William King’s De Origine Mali has recently started to attract some attention in early modern scholarship. In a recent paper devoted to King’s theory of free will, Kenneth Pearce identifies a “lacuna” in his text, namely the fact that King “never explicitly describes the process whereby election leads to action” (Pearce, “William King on Free Will”, 4). In this paper, I analyse King’s theory of ‘election’ (roughly, free choice) and Pearce’s interpretation of it. I discuss his claim that there is (...) a lacuna in King’s account and argue that the text provides us with important suggestions on how election generates action. Therefore, no speculative proposal to fill the lacuna is needed. This, in turn, allows me to develop a reading of King’s text that avoids a textual puzzle unsolved in Pearce’s interpretation: while he maintained that every election is “with reason”, my account can take King’s text at face value and explain how some elections are reasonable while others are not. (shrink)

Nel presente contributo è nostra intenzione sondare un binomio che appare fondamentale in funzione dialogica interculturale e interreligiosa, ossia la relazione tra logos e mythos, principalmente in vista dell’incontro tra cristianesimo e tradizione induista. È nostra convinzione che le forme complementari di «teo-logia» e «teo-mythia» possa essere esplorato con frutto, tanto in funzione specifica dell’incontro con la tradizione induista, quanto per una più ampia e profonda inculturazione del vangelo.

Il presente studio indaga lo sviluppo della dottrina teologica degli Atomisti antichi, secondo cui gli dèi esistono, ma non si occupano in tutto o in parte di noi umani. Ritenuta per lungo tempo espressione di ateismo, essa è presentata qui come una teoria filosofica solida ed eticamente rilevante, perché offre una concezione positiva della natura divina. Il volume compie un’indagine sistematica dei testi teologici degli Atomisti, partendo dai frammenti di Democrito e di Epicuro, fino agli scritti degli Epicurei di età (...) romana, come Demetrio Lacone e Filodemo, preservati soprattutto dai papiri di Ercolano. Oltre alle immancabili somiglianze che intercorrono tra questi autori, si sottolineano anche alcune profonde differenze tra le loro idee. In particolare, si propone l’ipotesi che esistano tre “teologie”, o meglio tre varianti teologiche, nell’Atomismo antico: l’idea di Democrito che gli dèi sono mortali e aiutano gli uomini saggi; la concezione di Epicuro, che stima dio un vivente del tutto indifferente alle vicende umane e dedito alla sola contemplazione di sé; la variante degli Epicurei successivi, che considera la divinità come dedita ad azioni e occupazioni che non ledono la sua perfezione. Attraverso le loro ipotesi sulla divinità, gli Atomisti individuano, inoltre, alcuni modelli di beatitudine, al quale l’essere umano deve cercare di assimilarsi per vivere felicemente. Sulla base di tale ulteriore proposta, si ricavano alcune potenziali ricadute morali delle teologie atomiste e si suggerisce che queste possano proporre alternative “stranianti” di pensiero / di prassi, capaci di influire costruttivamente sulle nostre vite. (shrink)