Jody Azzouni

An extensive exploration of the special properties of ‘exist’ is here undertaken. Two of several results are: Denying that `exist’ has associated with it a set of necessary and sufficient conditions has seemed to a number of philosophers to imply metaphysical nihilism. This is because it has seemed that without such conditions the target domain of `existence’ is arbitrarily open. I show this is wrong. Second, my analysis sheds light on the puzzling question of what we are asking when we ask of something, `Does it exist?’ and mean that question in an ontically relevant way.

Quine, in his 1980 forward to From a Logical Point of View wrote: ‘The time for revision is past. The book is dated, and its dates are 1953 and 1961’ (p. viii). Quine wrote modestly about himself, as he almost always did. His point—not stated loudly—was that the book was now an important historical document, and so its author had lost the right to tamper with it. The book I’m reviewing may seem different. Although Kripke’s Locke Lectures were given in 1973, the book has appeared in print forty years later. Kripke writes: ‘… the transcript has been available at the Oxford University library for many years … and it has had, since then, a modest life of its own, passed on among members of the profession, sometimes even being discussed and criticized in print’ (pp. xi-xii). Kripke understates the transcript’s impact. It influenced everyone in the profession who wrote on these topics, if only because it was read carefully and absorbed by so many philosophers working in this area. Thomasson, on a back-blurb, writes: ‘For decades getting a copy of these lectures has been a holy grail for philosophers working on fiction’. And in the Oxford advertising, included with my review copy of the book, I read: ‘At last in print—the most famous unpublished philosophy manuscript of recent decades’.

Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented notion of proof, the answer the mathematician will give ilikl..

Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of lack of rigor.

Kant discovered a philosophical problem with mathematical proof. Despite being a priori, its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms.

This chapter begins the analysis of the rule-following problem by the use of disposition-languages, languages with terms that apply exactly the way subjects are disposed to apply them. It’s shown that if empirical circumstances (and the dispositions of the subjects) are felicitous enough, isolated subjects can engage in successful rule-following. That is, they can successfully evaluate the languages they speak as better and worse, and they can use these languages to successfully navigate their worlds. Two examples of rule-following isolated individuals are given, Crusoe 1 and Crusoe 2. Their dispositions are somewhat artificial. Later examples, in later chapters, of isolated rule-following individuals will be much more natural, that is, they will be much more like us.

A feature of agnostic views—views that officially express ignorance about the existence of something —is that they are widely perceived to be epistemically more cautious than views that are committed to the entities in question. This is often seen as giving agnostics a debating advantage: all things being equal, fence-sitters have smaller argumentative burdens. Otávio Bueno argues in this way for what he calls “agnostic nominalism,” the view that we don’t know whether ontologically-independent Platonic objects exist. I show that agnostic nominalism, so called, can be sustained only in ways that don’t give agnostic nominalists debating advantages: the position must either be deduced from antecedently-held broader sceptical assumptions or it requires manufacturing potential referents for one’s terms that artificially generate grounds for scepticism. Neither maneuver leads to an agnostic nominalism with a debating advantage over its nominalist opponents.

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of AU-quantifiers and a model theory for them is given, and an appendix contains a completeness proof.

This monograph presents Azzouni’s new approach to the rule-following paradox. His solution leaves intact an isolated individual’s capacity to follow rules, and it simultaneously avoids replacing the truth conditions for meaning-talk with mere assertability conditions for that talk. Kripke’s influential version of Wittgenstein’s rule-following paradox—and Wittgenstein’s views more generally—on the contrary, make rule-following practices and assertions about those practices subject to community norms without which they lose their cogency. Azzouni summarizes and develops Kripke’s original version of Wittgenstein’s rule-following paradox to make salient the linchpin assumptions of the paradox. By doing so, Azzouni reveals how compelling Kripke’s earlier work on the paradox was. Objections raised over the years by Fodor, Forbes Ginsborg, Goldfarb, Tait, Wright, and many others, are all shown to fail. No straight solution can be made to work. Azzouni illustrates this in detail by showing that a popular family of straight solutions due to Lewis and refined by Williams, “reference magnetism,” fail as well. And yet an overlooked sceptical solution is still available in logical space. Azzouni describes a series of “disposition-meaning” private languages that he shows can be successfully used by a population of speakers to communicate with one another despite their ideolectical character. The same sorts of languages enable solitary “Robinson Crusoes” to survive and flourish in their island habitats. These languages—sufficiently refined—have the same properties normal human languages have; and this is the key to solving the rule-following paradox without sacrificing the individual’s authority over her self-imposed rules or her ability to follow those rules. Azzouni concludes this unusual monograph by uncovering a striking resemblance between the rule-following paradox and Hume’s problem of induction: he shows the rule-following paradox to be a corollary of Hume’s problem that arises when the problem of induction is applied to an individual’s own abilities to follow rules.

When ordinary people - mathematicians among them - take something to follow (deductively) from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof.

Kripke’s version of Wittgenstein’s rule-following paradox has been influential. My concern is with how it—and Wittgenstein’s views more generally—have been perceived as undercutting the individualistic picture of mathematical practice: the view that individuals— Robinson Crusoes —can, entirely independently of a community, engage in cogent mathematics, and indeed have “private languages.” What has been denied is that phrases like “correctly counting” can be applied to such individuals because these normative notions can only be applied cogently in a context involving community standards. I attempt to show that this shocking corollary doesn’t follow even if Kripke’s Wittgensteinian objections to dispositional approaches to rule-following are largely right. My reason for claiming this is that there is another solution to the rule-following paradox, one that doesn’t favor community standards over individual ones. Furthermore, it doesn’t replace truth conditions with assertability conditions; and this latter maneuver is essential to Kripke’s sceptical solution favoring the community over the individual.

One major motivation for nominalism, at least according to Hartry Field, is the desirability of intrinsic explanations: explanations that don’t invoke objects that are causally irrelevant to the phenomena being explained. There is something right about the search for such explanations. But that search must be carefully implemented. Nothing is gained if, to avoid a certain class of objects, one only introduces other objects and relations that are just as nominalistically questionable. We will argue that this is the case for two alleged nominalist views: Field’s fictionalism, and Frank Arntzenius and Cian Dorr’s geometricalism. Central to our competing approach to nominalism is a distinction between terms that refer to objects and ones that instead code empirical phenomena while being referentially empty. We next contrast our approach to nominalism, which uses this term-grained distinction between coding and referring, with approaches that instead attempt to make a sentence-grained distinction between mathematical and non-mathematical content. We show the latter approach fails to be responsive to objections raised by van Fraassen. In the end, only one last approach to nominalism is left standing. 1 Introduction2 Troubles for Mathematical Fictionalism2.1 A distinction2.2 Field’s programme2.3 Parts of nominalistically acceptable entities are nominalistically acceptable2.4 Structural assumptions: How far can these be pushed?2.5 Cases of indispensable theories where the mathematical posits are known not to be real2.6 How does science distinguish between the indispensable structural presuppositions taken to be real and those that aren’t?2.7 Do physicists distinguish between mathematical posits and non-mathematical posits?2.8 Mathematical coding versus genuine metaphysics. Case study: Fields2.9 The upshot3 Troubles for Topologism4 Contrasts: Kitcher, Maddy, Sober, and van Fraassen4.1 Sober’s approach4.2 Maddy’s approach4.3 Kitcher’s approach4.4 Where do we go from here?4.5 Van Fraassen’s objection to Maddy’s approach to ontological commitment5 Conclusion.

A normative constraint on theories about objects which we take to be real is explored: such theories are required to track the properties of the objects which they are theories of. Epistemic views in which observation (and generalizations of it) play a central role, and holist views which see epistemic virtues as applicable only to whole theories, are contrasted in the light of this constraint. It's argued that global-style epistemic virtues can't meet the constraint, although (certain) epistemic views within which observation (and, more generally, thick epistemic access) are centralized do meet it.

It is argued that the blind ascriptive role for the word "true", its use, that is, in conjunction with descriptions of classes of sentences or with proper names of sentences, is one which applies indiscriminately to sentences regardless of whether these are in languages we speak, can understand, or can translate into sentences that we do speak. Formal analogues of the ordinary word "true" as they arise in Tarski's seminal work, and in others, cannot replicate this essential role of the ordinary word "true".

I revisit my earlier arguments for the (trivial) inconsistency of natural languages, and take up the objection that no such argument can be established on the basis of surface usage. I respond with the evidential centrality of surface usage: the ways it can and can't be undercut by linguistic science. Then some important ramifications of having an inconsistent natural language are explored: (1) the temptation to engage in illegitimate reductio reasoning, (2) the breakdown of the knowledge idiom (because its facticity isn't comfortable with every sentence being true and false). Restoring the utility of the knowledge idiom motivates regimentation - the consistentist reinterpretation of natural language inferences (as they occur in real time). It's then sketched how to give the semantics of a natural language despite its inconsistency: Necessary and sufficient truth conditions are dropped in favor of necessary truth conditions and sufficient truth conditions.

Can Quine’s criterion for ontological commitment be comparatively applied across different logics? If so, how? Cross-logical evaluations of discourses are central to contemporary philosophy of mathematics and metaphysics. The focus here is on the influential and important arguments of George Boolos and David Lewis that second-order logic and plural quantification don’t incur additional ontological commitments over and above those incurred by first-order quantifiers. These arguments are challenged by the exhibition of a technical tool—the truncation-model construction of notational equivalents—that compares the ontological role and increased expressive strength of non-first-order ideology to first-order ideology.

Ordinary language and scientific language enable us to speak about, in a singular way (using demonstratives and names), what we recognize not to exist: fictions, the contents of our hallucinations, abstract objects, and various idealized but nonexistent objects that our scientific theories are often couched in terms of. Indeed, references to such nonexistent items-especially in the case of the application of mathematics to the sciences-are indispensable. We cannot avoid talking about such things. Scientific and ordinary languages thus enable us to say things about Pegasus or about hallucinated objects that are true (or false), such as "Pegasus was believed by the ancient Greeks to be a flying horse," or "That elf I'm now hallucinating over there is wearing blue shoes." Standard contemporary metaphysical views and semantic analyses of singular idioms on offer in contemporary philosophy of language have not successfully accommodated these routine practices of saying true and false things about the nonexistent while simultaneously honoring the insight that such things do not exist in any way at all (and have no properties). That is, philosophers often feel driven to claim that such objects do exist, or they claim that all our talk isn't genuine truth-apt talk, but only pretence. This book reconfigures metaphysics (and the role of metaphysics in semantics) in radical ways that allow the accommodation of our ordinary ways of speaking of what does not exist while retaining the absolutely crucial presupposition that such objects exist in no way at all, have no properties, and so are not the truth-makers for the truths and falsities that are about them.

The interaction between intuitions about inference, and the normative constraints that logical principles applied to mechanically-recognizable derivations impose on (informal) inference, is explored. These intuitions are evaluated in a clear testcase: informal mathe­matical proof. It is argued that formal derivations are not the source of our intuitions of validity, and indeed, neither is the semantic recognition of validity, either as construed model-theoretically, or as driven by the subject-matter such inferences are directed towards. Rather, psychologically-engrained inference-packages (often opportunistically used by mathematicians) are the source of our sense of validity. Formal derivations, or the semantic construal of such, are after-the-fact norms imposed on our inference practices.

I continue to explore the language of Crusoe 5, an isolated rule-follower. I show that his language, just like our language, can sustain the use of truth and falsity attributions. That is, it is useful to Crusoe 5, just as it is useful to us, to be able to say that certain groups of statements (or utterances) are true or false. Furthermore, if Crusoe 5 is so inclined, he can develop a truth-conditional semantics for his own language. This is the case even though the sentences of his language are not all underwritten by correspondence facts. These points apply to the natural languages we speak as well.

A distinction between the epistemic practices in mathematics and in the empirical sciences is rehearsed to motivate the epistemic role puzzle. This is distinguished both from Benacerraf's 1973 epistemic puzzle and from sceptical arguments against our knowledge of an external world. The stipulationist position is described, a position which can address this puzzle. Methods of avoiding the stipulationist position by using pure logic to provide knowledge of mathematical abstracta are discussed and criticized.