Massimiliano Carrara

Sometimes mereologists have problems with counting. We often don't want to count the parts of maximally connected objects as full-fledged objects themselves, and we don't want to count discontinuous objects as parts of further, full-fledged objects. But whatever one takes "full-fledged object" to mean, the axioms and theorems of classical, extensional mereology commit us to the existence both of parts and of wholes – all on a par, included in the domain of quantification – and this makes mereology look counterintuitive to various philosophers. In recent years, a proposal has been advanced to solve the tension between mereology and familiar ways of counting objects, under the label of Minimalist View. The Minimalist View may be summarized in the slogan: "Count x as an object iff it does not overlap with any y you have already counted as an object". The motto seems prima facie very promising but, we shall argue, when one looks at it more closely, it is not. On the contrary, the Minimalist View involves an ambiguity that can be solved in quite different directions. We argue that one resolution of the ambiguity makes it incompatible with mereology. This way, the Minimalist View can lend no support to mereology at all. We suggest that the Minimalist View can become compatible with mereology once its ambiguity is solved by interpreting it in what we call an epistemic or conceptual fashion: whereas mereology has full metaphysical import, the Minimalist View may account for our ways of selecting "conceptually salient" entities. But even once it is so disambiguated, it is doubtful that the Minimalist View can help to make mereology more palatable, for it cannot make it any more compatible with commonsensical ways of counting objects.

We consider a “polarized” version of bi-intuitionistic logic [5, 2, 6, 4] as a logic of assertions and hypotheses and show that it supports a “rich proof theory” and an interesting categorical interpretation, unlike the standard approach of C. Rauszer’s Heyting-Brouwer logic [28, 29], whose categorical models are all partial orders by Crolard’s theorem [8]. We show that P.A. Melliès notion of chirality [21, 22] appears as the right mathematical representation of the mirror symmetry between the intuitionistic and co-intuitionistc sides of polarized bi-intuitionism. Philosophically, we extend Dalla Pozza and Garola’s pragmatic interpretation of intuitionism as a logic of assertions [10] to bi-intuitionism as a logic of assertions and hypotheses. We focus on the logical role of illocutionary forces and justification conditions in order to provide “intended interpretations” of logical systems that classify inferential uses in natural language and remain acceptable from an intuitionistic point of view. Although Dalla Pozza and Garola originally provide a constructive interpretation of intuitionism in a classical setting, we claim that some conceptual refinements suffice to make their “pragmatic interpretation” a bona fide representation of intuitionism. We sketch a meaning-asuse interpretation of co-intuitionism that seems to fulfil the requirements of Dummett and Prawitz’s justificationist approach. We extend the Brouwer-Heyting-Kolmogorov interpretation to bi-intuitionism by regarding co-intuitionistic formulas as types of the evidence for them: if conclusive evidence is needed to justify assertions, only a scintilla of evidence suffices to justify hypotheses.

Following the speech act theory, we take hypotheses and assertions as linguistic acts with different illocutionary forces. We assume that a hypothesis is justified if there is at least a scintilla of evidence for the truth of its propositional content, while an assertion is justified when there is conclusive evidence that its propositional content is true. Here we extend the logical treatment for assertions given by Dalla Pozza and Garola by outlining a pragmatic logic for assertions and hypotheses. On the basis of this extension we analyse the standard logical opposition relations for assertions and hypotheses. We formulate a pragmatic square of oppositions for assertions and a hexagon of oppositions for hypotheses. Finally, we give a mixed hexagon of oppositions to point out the opposition relations for assertions and hypotheses.

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.

The Knowability Paradox starts from the assumption that every truth is knowable and leads to the paradoxical conclusion that every truth is also actually known. Knowability has been traditionally associated with both contemporary verificationism and intuitionistic logic. We assume that classical modal logic in which the standard paradoxical argument is presented is not sufficient to provide a proper treatment of the verificationist aspects of knowability. The aim of this paper is both to sketch a language $$\mathcal {L}_{\Box,K}^{P}$$, where alethic and epistemic classical modalities are combined with the pragmatic language for assertions $$\mathcal {L}^{P}$$, and to analyse the result of the application of our framework to the paradox.

According to the so-called strong variant of Composition as Identity (CAI), the Principle of Indiscernibility of Identicals can be extended to composition, by resorting to broadly Fregean relativizations of cardinality ascriptions. In this paper we analyze various ways in which this relativization could be achieved. According to one broad variety of relativization, cardinality ascriptions are about objects, while concepts occupy an additional argument place. It should be possible to paraphrase the cardinality ascriptions in plural logic and, as a consequence, relative counting requires the relativization either of quantifiers, or of identity, or of the is one of relation. However, some of these relativizations do not deliver the expected results, and others rely on problematic assumptions. In another broad variety of relativization, cardinality ascriptions are about concepts or sets. The most promising development of this approach is prima facie connected with a violation of the so-called Coreferentiality Constraint, according to which an identity statement is true only if its terms have the same referent. Moreover - even provided that the problem with coreferentiality can be fixed - the resulting analysis of cardinality ascriptions meets several difficulties.

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.

Logical orthodoxy has it that classical first-order logic, or some extension thereof, provides the right extension of the logical consequence relation. However, together with naïve but intuitive principles about semantic notions such as truth, denotation, satisfaction, and possibly validity and other naïve logical properties, classical logic quickly leads to inconsistency, and indeed triviality. At least since the publication of Kripke’s Outline of a theory of truth , an increasingly popular diagnosis has been to restore consistency, or at least non-triviality, by restricting some classical rules. Our modest aim in this note is to briefly introduce the main strands of the current debate on paradox and logical revision, and point to some of the potential challenges revisionary approaches might face, with reference to the nine contributions to the present volume.For a recent introduction to non-classical theories of truth and other semantic notions, see the excellent Beall a ..

Some forms of analytic reconstructivism take natural language (and common sense at large) to be ontologically opaque: ordinary sentences must be suitably rewritten or paraphrased before questions of ontological commitment may be raised. Other forms of reconstructivism take the commitment of ordinary language at face value, but regard it as metaphysically misleading: common-sense objects exist, but they are not what we normally think they are. This paper is an attempt to clarify and critically assess some common limits of these two reconstructivist strategies.

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.

Identity criteria are used to confer ontological respectability: Only entities with clearly determined identity criteria are ontologically acceptable. From a logical point of view, identity criteria should mirror the identity relation in being reflexive, symmetrical, and transitive. However, this logical constraint is only rarely met. More precisely, in some cases, the relation representing the identity condition fails to be transitive. We consider the proposals given so far to give logical adequacy to inadequate identity conditions. We focus on the most refined proposal and expand its formal framework by taking into account two further aspects that we consider essential in the application of identity criteria to obtain logical adequacy: contexts and granular levels.

A logical argument known as Fitch’s Paradox of Knowability, starting from the assumption that every truth is knowable, leads to the consequence that every truth is also actually known. Then, given the ordinary fact that some true propositions are not actually known, it concludes, by modus tollens, that there are unknowable truths. The main literature on the topic has been focusing on the threat the argument poses to the so called semantic anti-realist theories, which aim to epistemically characterize the notion of truth; according to those theories, every true proposition must be knowable. But the paradox seems to be a problem also for epistemology and philosophy of science: the conclusion of the paradox – the claim that there are unknowable truths – seems to seriously narrow our epistemic possibilities and to constitute a limit for knowledge. This fact contrasts with certain views in philosophy of science according to which every scientific truth is in principle knowable and, at least at an ideal level, a perfected, “all-embracing”, omniscient science is possible. The main strategies proposed in order to avoid the paradoxical conclusion, given their effectiveness, are able to address only semantic problems, not epistemological ones. However, recently Bernard Linsky (2008) proposed a solution to the paradox that seems to be effective also for the epistemological problems. In particular, he suggested a possible way to block the argument employing a type-distinction of knowledge. In the present paper, firstly, we introduce the paradox and the threat it represents for a certain views in epistemology and philosophy of science; secondly, we show Linsky’s solution; thirdly, we argue that this solution, in order to be effective, needs a certain kind of justification, and we suggest a way of justifying it in the scientific field; fourthly, we show that the effectiveness of our proposal depends on the degree of reductionism adopted in science: it is available only if we do not adopt a complete reductionism.

The goal of [3] is to sketch the construction of a syntactic categorical model of the bi-intuitionistic logic of assertions and hypotheses AH, axiomatized in a sequent calculus AH-G1, and to show that such a model has a chirality-like structure inspired by the notion of dialogue chirality by P-A. Melliès [8]. A chirality consists of a pair of adjoint functors L ⊣ R, with L: A → B, R: B → A, and of a functor (.)* : A → Bop(0,1) satisfying certain conditions. The definition of the logic AH in [3] needs to be modified so that our categories A and B are actually dual. With this modification, a more complex structure emerges.

Nicholas Rescher, in The Limits of Science (1984), argued that: «perfected science is a mirage; complete knowledge a chimera». He reached the above conclusion from a logical argument known as Fitch’s Paradox of Knowability. The argument, starting from the assumption that every truth is knowable, proves that every truth is also actually known and, given that some true propositions are not actually known, it concludes, by modus tollens, that there are unknowable truths. Prima facie, this argument seems to seriously narrow our epistemic possibilities and to constitute a limit for knowledge (included scientific knowledge). Rescher’s above quoted conclusion follows the same sort of reasoning. Recently, Bernard Linsky exploited a possible way to block the argument employing a type-distinction of knowledge. If the Knowability paradox is blocked, then Rescher’s conclusion cannot be drawn. After an introduction to the paradox, we suggest, in our paper, a possible way of justifying a type-solution for it in the scientific field. A noteworthy point is that the effectiveness of this solution depends on the degree of reductionism adopted in science: the given solution is available only if we do not adopt a complete reductionism in science so that there is just one kind of scientific knowledge and, consequently, of scientific justification. Otherwise Rescher's argument still works.

From a logical point of view, identity criteria should mirror the identity relation in being reflexive, symmetrical, and transitive. However, the relation representing the identity condition fails to be transitive in many cases. We consider the proposals given so far to give logical adequacy to inadequate identity conditions. We focus on the most refined proposal and expand its formal framework by taking into account two further aspects that we consider essential in the formal treatment of identity criteria: contexts and granular levels.

The Knowability Paradox is a logical argument that, starting from the plainly innocent assumption that every true proposition is knowable, reaches the strong conclusion that every true proposition is known; i.e. if there are unknown truths, there are unknowable truths. The paradox has been considered a problem for every theory assuming the Knowability Principle, according to which all truths are knowable and, in particular, for semantic anti-realist theories. A well known criticism to the Knowability Paradox is the so called restriction strategy. It bounds the scope of the universal quantification in (KP) to a set of formulas whose logical form avoids the paradoxical conclusion. Specifically, Tennant suggests to restrict the quantifier in (KP) to propositions whose knowledge is provably inconsistent. He calls them Anti-Cartesian propositions and distinguished them in three kinds. In this paper we will not be concerned with the soundness of the restriction proposal and the criticisms it has received. Rather, we are interested in analyzing the proposed distinction. We argue that Tennant’s distinction is problematic because it is not completely clear, it is not grounded on an adequate logical analysis, and it is incomplete. We suggest an alternative distinction, and we give some reasons for accepting it: it results logically grounded and more complete than Tennant’s one, inclusive of it and independent from non-epistemic notions.

This is an excerpt from the contentIn the introduction to part I of the symposium we stated that a rational agent could be thought of as an agent who has good reasons for its actions. In formal analyses of economic, medical, political, military and forensic decisions rationality, that is the “goodness” of those reasons, is inextricably intertwined with probability. Typically, those analyses concern decisions in a particular class of uncertain situations, namely “risky” situations, where all the relevant available alternative actions are known, each or most actions are non-deterministic , and the probability of each outcome is known to some degree of approximation. As yet, formal analyses have only begun to scratch the surface of the subtleties implicated in uncertain, but not risky, situations, where the available courses of actions, and/or their possible outcomes, and/or the probabilities of those outcomes, are not all

The Knowability Paradox is a logical argument to the effect that, if there are truths not actually known, then there are unknowable truths. Recently, Alexander Paseau and Bernard Linsky have independently suggested a possible way to counter this argument by typing knowledge. In this article, we argue against their proposal that if one abstracts from other possible independent considerations supporting reasons for typing knowledge and considers the motivation for a type-theoretic approach with respect to the Knowability Paradox alone, there is no substantive philosophical motivation to type knowledge, except that of solving the paradox. Every attempt to independently justify the typing of knowledge is doomed to failure.

The book's aim is to give a working representation of what metaphysics is today. The historical contributions reveal the roots of metaphysical themes and how today's methods are linked to their Aristotelian and Leibnizian past. The volume also touches on the relationships between ontological and linguistic analysis, the questions of realism and ontological commitment, the nature of abstract objects, the existential meaning of particular quantification, the primitiveness of identity, the question of epistemic versus ontological vagueness, the necessity of origin, the nature of natural necessity, the possibility of intermittent existence, the notion of a temporal part and its place in an account of persistence, the question of identity and change across time and possible worlds, and many more. Readership: A toolbox for any researcher in metaphysics and an essential source for any PhD student with ontological interests.