Massimiliano Carrara

Church-Fitch’s Knowability Paradox points to a necessary correlation between ignorance and unknowability. In this paper, we investigate the prospects of quantifying this correlation. We explore various approaches to formalizing the quantitative correlation between unknowns and unknowables and advocate for a specific proposal. Our proposal appeals to the density of true unknowables given the ratio of known to unknown basic truths in a situation. A higher proportion of known basic truths in a situation corresponds to a lower density of true unknowables in that situation. We further examine some implications of this result and discuss potential applications to other philosophical debates.

Logic is often considered neutral, imposing no significant constraints on the world. In this paper, we explore how this purported neutrality can be understood, focusing on a proposal originally sketched by Jc Beall for another purpose. Accordingly, suggesting that logic should be viewed as a tool for theory building, we argue that mere theory closure can be interpreted as a hallmark of logic’s neutrality. The proposal can be connected with a sharp distinction between validity and truth and approached through a metalinguistic framework. We then discuss some complications and possible objections to the account, which gives us the opportunity to further deepen our treatment and connect the proposal with some important issues in the philosophy of logic. In particular, we consider the role of the deduction theorem and the relationship between entailment and implication. We conclude that once correctly interpreted, logic’s neutrality can indeed be understood in terms of theory closure.

This chapter argues against predicative analyses of plurality, which force plurals into the familiar mould of singular logic by turning an apparently plural term standing for several objects into a singular predicate standing for a concept or property. Michael Dummett enlists support from Fregean semantics in favour of a predicative analysis, but his arguments do not stand up, either as exegesis of Frege or on their own merits. As well as facing difficulties in eliminating plural content, predicative analyses are sunk by the equivocity objection: they misrepresent single English predicates as equivocal, by treating the predicate differently according as it combines with singular or plural arguments. Although George Boolos does not offer a predicative analysis, it is argued that his second-order treatment of plurals is also sunk by the equivocity objection. And Ian Rumfitt fails in his attempt to avoid the objection by modifying Boolos's scheme.

Unity and Plurality brings together new work on the logic and ontology of plurality and on the semantics of plurals in natural language. Plural reference is an approach favoured by logicians and philosophers, who take sentences with plurals not to be committed to entities beyond individuals, entities such as classes, sums, or sets. By contrast, linguistic semantics has been dominated by a singularist approach to plurals, taking the semantic value of a definite plural to be a mereological sum or set. Moreover, semantics has been dominated by a particular ontological view of plurality, that of extensional mereology. This volume aims to build a bridge between the two traditions and to show the fruitfulness of nonstandard mereological approaches. A team of leading experts investigates new perspectives that arise from plural logic and non-standard mereology and explore novel applications to natural language phenomena.

This textbook is an accessible, readable, and comprehensive introduction to contemporary analytic metaphysics. It covers all the main topics, including those that are at the forefront of the most recent research. It begins by explaining what metaphysics is and introduces the most important conceptual devices needed to understand and appreciate the ongoing debates. The text follows with core metaphysical questions, such as properties, time, identity, mereology, and modality. It concludes with more specific domains of application, in which the previously introduced concepts are put to work. The contents are organized in self-contained chapters, each of which starts with a stimulating short story, which is further articulated and closes with a summary of the main points, questions for further reflection, and reading suggestions. This textbook is perfect for teaching a basic but comprehensive course in metaphysics for undergraduate students.

A three-valued logic is subclassical when it is defined by a single matrix having the classical two-element matrix as a subreduct. In this case, the language of can be expanded with special unary connectives, called external operators. The resulting logic is called the external version of, a notion originally introduced by D. Bochvar in 1938 with respect to his weak Kleene logic. In this paper we study the semantic properties of the external version of a three-valued subclassical logic. We determine sufficient and necessary conditions to turn a model of into a model of. Moreover, we establish some distinctive semantic properties of.

Dialetheism holds the thesis that certain sentences are dialetheias, i.e. both true and false, and devises several strategies for avoiding trivialism, the (classical) consequence that all sentences are true. Two such strategies are aimed at invalidating one of the most direct arguments for trivialism, viz. Curry's Paradox: a proof that you will win the lottery, a proof that only resorts to naive truth-principles, Conditional Proof (CP), modus ponens (MPP) and the standardly accepted structural rules. The first strategy simply consists in observing that the most well-known dialetheist logic, sometimes referred to as the Logic of Paradox (LP), invalidates MPP. The second strategy consists in rather taking one of the primary senses of 'if' to be captured by an entailment connective which does not validate CP. We argue that both strategies are problematic. © 2021 Elsevier B.V., All rights reserved.

In this chapter we extend the notion of arbitrary reference to individuals to that of arbitrary interpretation. We want to explain how a single arbitrary interpretation of the working mathematician relates to the various possible interpretations in model theory. To this purpose we introduce some arbitrary fictional models. Additionally, we aim to clarify how one can deduce logical consequences of the axioms by reasoning on a single interpretation, even when a theory has non-equivalent elementary models.

The aim of this chapter is to argue that: (a) our semantics of acts of choices (SAC), as developed in Chap. 2, defends second-order logic from claims of ontological commitment; (b) understanding our semantics does not require any prior mathematical concepts; and (c) although SAC is not universally applicable, it still offers significant applicability, especially in mathematics. We conclude the chapter arguing that second-order logic, as interpreted through our semantics, can indeed be considered a genuine logic. One might object that, since only concrete objects can be referred to by arbitrary acts of choice, SAC lacks the logical requirement of universal applicability. However, such limitation is superseded by the structuralist conception of mathematics, according to which arbitrary entities, independently of their very nature, can play the role of mathematical objects.

In this chapter 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. Finally, in the last section of this chapter we show that, adopting a relativistic notion of atom, according to which any individual can play the role of atom, one can reconstruct Cantor’s theorem and prove that, given any infinite, mereology with plural quantification (MPQ) guarantees the existence of large and large infinities. This result seems in conflict with the alleged ontological innocence of MPQ.

This chapter introduces a new approach to plural quantification through the concept of plural arbitrary reference. It highlights the implicit presupposition in mathematical reasoning that any individual in the universe of discourse can be referred to. By introducing a team of ideal agents capable of direct access to any individual, plural arbitrary reference is achieved through simultaneous acts of choice by each agent. This idealized notion of reference provides a basis for understanding plural quantification without second-order entities, maintaining the ontological innocence of second-order logic by distinguishing between acts and entities.

This paper proposes a new dialetheic logic, a Dialetheic Logic with Exclusive Assumptions and Conclusions ( $$\mathsf {DLEAC}$$ DLEAC ), including classical logic as a particular case. In $$\mathsf {DLEAC}$$ DLEAC, 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 $$\mathsf {DLEAC}$$ DLEAC logic. Finally, we show how $$\mathsf {DLEAC}$$ DLEAC supports the dialetheic solution of the liar paradox.

When we count, we often count fractions, too. We contend that fractional counting involves partial entities, which are merely possible parts of entities of the counted kind. The size of these possible parts is measured with respect to the size of a possible member of that kind. Therefore, partialhood is mereomodal, and the logical form of fractional counting claims includes mereological predicates, modal operators, and a measurement functor. Different varieties of modality and forms of measurement are involved, depending on the kinds of entities to be counted and the context. The mereomodal account validates the idea that fractional counting is a way of counting by identity, in continuity with logic-based accounts of non-fractional counting, albeit more complex than them. Such an account also explains why some kinds of entities are not involved in partialhood and cannot be fractionally counted, while others only have marginal involvement in these phenomena. In the last part, we discuss some difficult cases and show that an integrity condition for partial entities is required in the logical form of some fractional counting claims.

This Element delves into the relationship between logic and the sciences, a topic brought to prominence by Quine, who regarded logic as methodologically and epistemologically akin to the sciences. For this reason, Quine is seen as the forefather of anti-exceptionalism about logic (AEL), a stance that has become prevalent in the philosophy of logic today. Despite its popularity and the volume of research it inspires, some core issues still lack clarity. For one thing, most works in the debate remain vague on what should count as logic and what should count as a science. Furthermore, the terms of the comparison are rarely specified and discussed in a systematic way. This Element purports to advance the debate on these crucial issues with the hope of fostering our understanding of the fundamentals of AEL.

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.

A three-valued logic L is subclassical when it is defined by a single matrix having the classical two-element matrix as a subreduct. In this case, the language of L can be expanded with special unary connectives, called external operators. The resulting logic L^e is the external version of L, a notion originally introduced by D. Bochvar in 1938 with respect to his weak Kleene logic. In this paper we study the semantic properties of the external version of a three-valued subclassical logic L. We determine sufficient and necessary conditions to turn a model of L into a model of L^e. Moreover, we establish some distinctive semantic properties of L^e.

One of the goals of the natural sciences– for example biology– is to provide new information about certain phenomena with previously unknown nature. Their contribution to our knowledge is substantial. From this perspective, logic is seemingly not substantial. Sometimes, logic’s insubstantiality is taken for granted while explaining the alleged insubstantiality of other notions. For example, according to truth deflationism, truth is a non-substantial notion in the sense of being a logical property. However, it is not fully clear to what such an insubstantiality amounts. It is also debatable whether logic really is insubstantial. In this paper, we aim to clarify this issue by proposing a formal way of looking at it. In particular, we used the notion of _conservativity_, which has already been used by truth deflationism, for a similar aim. We show that if insubstantiality is read in terms of conservativity, then classical logic is substantial. We then argue that such a verdict of substantiality can be resisted if precise stances on certain _prima facie_ unrelated issues of philosophy of logic are taken, or an anti-exceptionalist view is adopted.