Massimiliano Carrara

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.

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.

P.T. Geach has maintained (see, e.g., Geach (1967/1968)) that identity (as well as dissimilarity) is always relative to a general term. According to him, the notion of absolute identity has to be abandoned and replaced by a multiplicity of relative identity relations for which Leibniz's Law - which says that if two objects are identical they have the same properties - does not hold. For Geach relative identity is at least as good as Frege's cardinality thesis which he takes to be strictly connected with relative identity - according to which an ascription of cardinality is always relative to a concept which specifies what, in any particular case, counts as a unit. The idea that there is a close connection between relative identity and Frege's cardinality thesis has been issued again quite recently by Alston and Bennett in (1984). In their opinion, Frege's cardinality thesis is not only similar to relative identity - as Geach maintains - but it implies it. Moreover, they agree with Geach in claiming that a commitment to Frege's cardinality thesis forces a parallel commitment to relative identity. Against Geach, Alston and Bennett we will claim that (Tl): "Frege's cardinality thesis is similar to relative identity" is false and that therefore (T2) "Frege's cardinality thesis implies relative identity" is false as well.

According to Frege a proposition—or, in his terms, a thought—is an abstract structured entity constituted by senses which satisfies, at least, the three following properties: it can be semantically assessed as true or as false, it is the object of so called propositional attitudes and it can be grasped. What Frege meant by 'grasping' is the peculiar way in which we can have epistemic access to propositions. The possibility for propositions to be grasped is put by Frege as a warrant for their existence: to challenge their graspability would amount to jeopardise their ontological reality. But is it true, as Frege uncritically maintained, that the "graspability requirement" is satisfied as far as propositions are concerned? This is the topic of the present work. A negative answer to the above mentioned question has been given in recent time by the representatives of what has come to be labelled the "cognitive turn" in analytical philosophy. People such as Fodor and Johnson-Laird patently denied the possibility for propositions, conceived Frege, to be accessed by the grasping relation. What grounds their position is, to put it roughly, the following train of thought: in order for something to be the target of the grasping relation it must enter the mind. Nothing which is different from a mental entity can enter the mind. Therefore, what can be grasped must be mental. The upshot of this move implies, among other things, the rejection of that radical anti-psychologism which was characteristic of the forefathers of the analytical tradition. In our work we shall try to resist their conclusion by showing that it is not necessary to zero the distinction between propositions and mental entities in order to provide an adequate account of the grasping relation. What one has to give up, instead, is only Frege's late Platonism of the "third realm" which, in our view, is a wholly unnecessary and dispensable accretion of his picture. For, as we shall show, if Platonism is in place it is difficult to provide an account of the grasping relation which makes no use of the "representationalist hypothesis" — i.e. of the hypothesis that mediate our access to whatever can be given to us. But representationalism, once in place, makes the theoretical role of the notion of sense dispensable or purely additional.

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is.

The general question (G) How do we categorize artifacts? can be subject to three different readings: an ontological, an epistemic and a semantic one. According to the ontological reading, asking (G) is equivalent to asking in virtue of what properties, if any, a certain artifact is an instance of some artifact kind: (O) What is it for an artifact a to belong to kind K? According to the epistemic reading, when we ask (G) we are investigating what properties of the object we exploit in order to decide whether a certain artifact belongs to a certain kind. (G) thus becomes: (E) How can we know that artifact a belongs to kind K? Finally, (G) can also be read as a question concerning the semantics of artifact kind terms. The semantic reading of (G) is: (S) What kind of reference do artifact kind terms have, if any? In this editorial we expand on the different answers to (O), (E) and (S) that are given in the selected literature on the topic. The result should give us an overall picture of the possible answers to (G)

In this paper we consider the emerging position in metaphysics that artifact functions characterize real kinds of artifacts. We analyze how it can circumvent an objection by David Wiggins (Sameness and substance renewed, 2001, 87) and then argue that this position, in comparison to expert judgments, amounts to an interesting fine-grained metaphysics: taking artifact functions as (part of the) essences of artifacts leads to distinctions between principles of activity of artifacts that experts in technology have not yet made. We show, moreover, that our argument holds not only in the artifactual realm but also in biology: taking biological functions as (part of the) essences of organs leads to distinctions between principles of activity of organs that biological experts have not yet made. We run our argument on the basis of analyses of artifact and biological functions as developed in philosophy of technology and of biology, thus importing results obtained outside of metaphysics into the debate on ontological realism. In return, our argument shows that a position in metaphysics provides experts reason for trying to detect differences between principles of activities of artifacts and organs that have not been detected so far.

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.

The goal of the paper is to analyse some specific features of a very central concept for top-level ontologies for information systems: i.e. the concept of artefact. Specifically, we analyse the relation to be a copy of that is strongly linked to the notion of artefact and—as we will demonstrate—could be useful to distinguish artefacts from objects of other kinds. Firstly, we outline some intuitive and commonsensical reasons for the need of a clarification of the notion of artefact in ontologies for information systems, and we analyse some characterisations of the notion given by two top-level ontologies (Cyc and Wordnet). Secondly, we introduce and critically analyse Tzouvaras’ notion of copy. Thirdly, we try to complete an analysis of copy by distinguishing three kinds of copies: replicas (Tzouvaras’ notion of copy), rigid copies, and functional copies. With the help of these three notions we outline a first and preliminary distinction between artefacts, objects of art and natural objects

According to strong composition as identity, the logical principles of one–one and plural identity can and should be extended to the relation between a whole and its parts. Otherwise, composition would not be legitimately regarded as an identity relation. In particular, several defenders of strong CAI have attempted to extend Leibniz’s Law to composition. However, much less attention has been paid to another, not less important feature of standard identity: a standard identity statement is true iff its terms are coreferential. We contend that, if coreferentiality is dropped, indiscernibility is no help in making composition a genuine identity relation. To this aim, we analyse as a case study Cotnoir’s theory of general identity, in which indiscernibility is obtained thanks to a revisionary semantics and true identity statements are allowed to connect non-coreferential terms. We extend Cotnoir’s strategy for indiscernibility to the relation of comaternity, and we show that, neither in the case of composition nor in that of comaternity, indiscernibility contibutes to show that they are genuine identity relations. Finally, we compare Cotnoir’s approach with other versions of strong CAI endorsed by Wallace, Bøhn, and Hovda, and canvass the extent to which they violate coreferentiality. The comparative analysis shows that, in order to preserve coreferentiality, strong CAI is forced to adopt a non-standard semantic treatment of the singular/plural distinction.

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.

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.

In "The Limits of Science" N. Rescher introduces a logical argument known as the Knowability Paradox, according to which, if every true proposition is knowable, then every true proposition is known, i.e. if there are unknown truths, there are unknowable truths. Rescher argues that the Knowability Paradox, giving evidence to a limit of our knowledge (the existence of unknowable truths) could be used for arguing against perfected science. In this article we present two criticisms against Rescher's argument.