Stewart Shapiro

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”.

Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professionalbasketball players. The main feature of Shapiro's account is that the extensions of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis is that in some cases, a competent speaker ofthe language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. Shapiro calls this open texture, borrowing the term from Friedrich Waismann.The formal model theory has a similar structure to the supervaluationist approach, employing the notion of a sharpening of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The ultimate goal of the technical aspects of the work is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox.Later chapters deal with what passes for higher-order vagueness - vagueness in the notions of 'determinacy' and 'borderline' - and with vague singular terms, or objects. In each case, the philosophical picture is developed by extending and modifying the original account. This is followed with modifications to the model theory and the central meta-theorems.As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak. But vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. Vagueness is also due to the kinds of beings we are. There is no need to blame the phenomenon on any one of those aspects.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

In this paper, we outline and critically evaluate Thomas Hofweber’s solution to a semantic puzzle he calls Frege’s Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber’s proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber’s analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different solution to Frege’s Other Puzzle is required.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting.

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms.

One is often said to be reasoning well when they are reasoning logically. Many attempts to say what logical reasoning is have been proposed, but one commonly proposed system is first-order classical logic. This Element will examine the basics of first-order classical logic and discuss some surrounding philosophical issues. The first half of the Element develops a language for the system, as well as a proof theory and model theory. The authors provide theorems about the system they developed, such as unique readability and the Lindenbaum lemma. They also discuss the meta-theory for the system, and provide several results there, including proving soundness and completeness theorems. The second half of the Element compares first-order classical logic to other systems: classical higher order logic, intuitionistic logic, and several paraconsistent logics which reject the law of ex falso quodlibet.

The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting three necessary and independently motivated restrictions on the metalinguistic predicates ‘atom’ and ‘non-atom’: both are sort, property, and context-relative. With these restrictions in place, Schein’s problematic assumption becomes evident: his presumed singularist analysis of ‘non-atom’ conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage.

According to Ole Hjortland, Timothy Williamson, Graham Priest, and others, anti-exceptionalism about logic is the view that logic “isn’t special”, but is continuous with the sciences. Logic is revisable, and its truths are neither analytic nor a priori. And logical theories are revised on the same grounds as scientific theories are. What isn’t special, we argue, is anti-exceptionalism about logic. Anti-exceptionalists disagree with one another regarding what logic and, indeed, anti-exceptionalism are, and they are at odds with naturalist philosophers of logic, who may have seemed like natural allies. Moreover, those internal battles concern well-trodden philosophical issues, and there is no hint as to how they are to be resolved on broadly scientific grounds. We close by looking at three of the founders of logic who may have seemed like obvious enemies of anti-exceptionalism—Aristotle, Frege, and Carnap—and conclude that none of their positions is clearly at odds with at least some of the main themes of anti-exceptionalism. We submit that, at least at present, anti-exceptionalism is too vague or underspecified to characterize a coherent conception of logic, one that stands opposed to more traditional approaches.

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.

Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist.

We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that various logics are constitutive for thought within particular practices, but none are constitutive for thought as such.

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.