Review of 'Between Logic and Intuition'

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suggestive but sketchy response to Mates-style objections. There are odd omissions: e.g., such highly relevant and well-known works as Christopher Peacocke’s A Study of Concepts (Cambridge, MA: MIT Press, ) and Ned Block’s ‘Advertisement for a Semantics for Psychology’ (Midwest Studies in Philosophy, vol. X, Minneapolis: University of Minnesota Press, , pp. – ) are not even mentioned! But taken for what it purports to be — viz., a polemical tract aimed at restoring the dialectical balance between internalist and externalist views of cognitive content —Segal’s book is reasonably successful. Its more original parts, most notably the specific arguments against Putnam and Burge, should stimulate much fruitful discussion and serve as valuable therapy against the sclerosis of intuitions that comes from a one-sided diet of examples. Department of Philosophy The Ohio State University 350 University Hall 230 N. Oval Mall Columbus, Ohio 43210 USA

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Between Logic and Intuition: Essays in Honor of Charles Parsons, edited by Gila Sher and Richard Tieszen. Cambridge: Cambridge University Press, . Pp. viii + . H/b £.. Charles Parsons is one of our most distinguished colleagues. Any of us interested in logic, philosophy of mathematics, Frege or Kant would want to take Parsons’s views seriously, and we are all indebted to him for the part he played in making the Journal of Philosophy a premier organ of our discipline. It is thus just that he be honoured by a Festschrift. Between Logic and Intuition is well above average in the quality of the essays it contains, and thus does Parsons the honour he is due. One wishes Parson had been invited to comment on these papers. Let us begin with a nod of thanks toward each contributor. The book starts with Hilary Putnam’s two Tarski lectures at Berkeley in , the first on the liar, and the second on how to be less realistic than Quine or Gödel about sets. Arnold Koslow sets out an algebraic description of truth and its ilk. Vann McGee presents a provocative discussion of the prospects for forcing ‘all’ to mean absolutely everything. James Higgenbotham lays out linguistic evidence on the naturalness of second order logic in an informative way. Gila Sher explores the roots in Skolem for Quine’s indeterminacy. Isaac Levi expounds the standards of rational health. Carl Posy reads Kant on intuition against a background of Leibniz’s understanding of intuition. Michael Friedman extends his debate with Parsons on whether intuition is logical or perceptual by taking us on a very informative tour through post-Kantian, mostly nineteenth-century, developments in geometry; this is an essay to be used, partly,

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as a reference work. Michael Resnik thinks through some issues about intuition in an enlightening way. Richard Tieszen defends Gödel’s rather rationalist epistemology against Quine. George Boolos asks us how sure we should be of the larger sets of which Zermelo-Fraenkel set theory assures us. As always, Boolos is entertaining as well as enlightening; his death is much to be mourned. W. W. Tait gives us a strikingly unplatonist reading of Cantor on sets and the paradoxes. Mark Steiner contributes a short discussion of whether Frege took the natural numbers as a natural kind. Penelope Maddy describes a way of building up proper classes analogous to the way Kripke built up extensions and anti-extensions for truth predicates. Solomon Feferman and Geoffrey Hellman defend their predicative foundations for arithmetic; they say there is an absolute notion of finitude from which an enlightening analysis of the structures of the natural numbers unfolds. Some of these essays drew me in more than others; this remark is not about the value of the essays, but about my taste and experience. In his first Tarski lecture, the one on the liar, Putnam gives prominence to Hans Herzberger’s work as well as Parsons’s. This is a mark of Putnam’s, and Parsons’s, excellent judgement. But it is striking that they do not mention Herzberger’s ‘New Paradoxes for Old’ (Proceedings of the Aristotelian Society, , –, pp. –), one of the most insightful pieces on the liar in the last century. Herzberger’s thesis there was that machinery introduced to solve the paradox of the liar can always be rejigged into a new version of the paradox. Thinking it through, this means that there is no final solution to the liar. Putnam, following Parsons, says that all insight into the liar is schematic. Perhaps they mean that such insights are intrinsically open, that is, written with free variables whose universe of discourse cannot be completed. To illustrate, begin with a liar paradox, a sentence that says of itself that it is not true, and impeccable reasoning to the effect that it both is, and is not, true. As Fred Fitch argued in this journal in , self-reference is too important to our critical armoury to be utterly surrendered; we must still be able to say, for example, that any statement of Russell’s theory of types violates itself. So we should set ourselves against any complete proscription of self-reference. At this point it is common to propose limited ranges of reference tout court, not just self-reference. So we get particular levels, types or contexts; call them , ,  and so on, not to suggest any particular order or arrangement of them, but more to suggest an indefinite extent of levels or contexts. We are told we cannot speak of truth period, but only of truth at level , or truth in context of . Then ‘This sentence is not true at level ’ is a banal truth at level . But to give the wiring diagram of this machinery to solve the liar, we use a predicate like ‘is a level’ or ‘is a context’ conceived with an indefinite extension, like ‘is a quark’ or ‘tastes better than glue’. If we are entitled to name members of the extension of ‘is a context’, then we are also entitled to variables of quantification ranging over that extension and beyond. We then rejig this machinery as a sentence which says of itself that it is true in no context. Call this

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sentence Fred. We can imbed Fred in reasoning no less impeccable than our original paradox. Let c be any old context. Suppose Fred is true in c. Then from Fred by universal instantiation, Fred is not true in c. But c was any context. So by universal generalization, Fred is true in no context. That’s what Fred says, so at least Fred is true here and now in this, our present, context. Whence from Fred by universal instantiation Fred is not true here and now. So Fred is both true and not true here and now. We have rejigged machinery for solving the liar into a new version of the liar. Herzberger’s thesis is that this rejigging can always be done, that, as it were, any hierarchy of types, levels or classes is topped by a new liar. We might try to put Putnam’s thesis as that each such new liar initiates yet a new hierarchy of types, contexts or whatever restrictions on reference we tried. There is no last liar, solved by some hierarchy beyond it, and there is no last hierarchy, topped by no liar. (Consider ‘This sentence is true in no hierarchy’.) The notion of an intrinsically schematic insight is an effort to articulate this instability, this absence both of a final liar and a full hierarchy. As it were, while the dialectic of the liar has a thesis, from Herzberger, and an antithesis, from Putnam, it has no synthesis; there is no absolute. We have yet to articulate what this void means for the unfolding of reason. Van McGee’s ‘Everything’ also tickled my fancy. His essay really comes to life with his exposition of Putnam’s use of Skolem. Let A be the set of everything named or ostendible, or causally connected however remotely to something name or ostendible. It is arguable that A is countable. U is the collection of absolutely everything; it is uncountable, and by contemporary lights too big to be a set. Let L be a countable first order regimentation of (part of) English in quantificational style. Let S be the Skolem hull of U got by closing A under the Skolem functions fixed by the (presumably intended) interpretation of L in U. If A is indeed countable, so is S. And yet S and U agree on all sentences of L, and even on which open sentences of L are true of which individuable objects. So how, Putnam asked, can we tell whether we’re quantifying over S or U? Arthur Prior once responded to use-mongering by proposing a connective with the introduction rules of disjunction and the elimination rules of conjunction. Nuel Belnap replied with something like existence and uniqueness conditions for operators. The existence (or, perhaps, coherence) condition for an operator is that ‘the’ system with it be a conservative extension of the system without it. The uniqueness condition is that if to the system with the operator we add a duplicate of the operator subject to the same axioms as the operator, the two are demonstrably co-extensive. (Quine notes on page  of From a Logical Point of View that truth has the uniqueness property, and that identity does too on page  of The Ways of Paradox.) J. H. Harris proved in  that all the operators of first order quantification theory (axiomatized natural deduction style) meet Belnap’s uniqueness condition, and McGee notes that this result extends to second order quantification. A direct syntactical proof that an operator meets Belnap’s uniqueness con-

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dition typically appeals to instances of axioms for one copy of the operator in which the other also occurs. Quine’s proof cited above that identity meets the condition is an instructive example. This phenomenon leads McGee into a subtle and instructive examination of what he calls the open-endedness of a rule of inference. For example, since substitutivity of identity is in the first instance second order, in first order theories it is usually put as an axiom schema with an instance for each formula of the language of the theory. That might raise a question about instances of substitutivity in extensions of the original language. In the balance of his essay McGee seems mostly exercised by what look like rather infinitary open ends in languages taken not as vehicles for communication amongst folks like us, but rather as abstract combinatorial mathematical objects. Fair enough. Still, there remains the question by Putnam out of Skolem that got us going. How do we know whether we’re quantifying over U or S? Two is the unique even prime because any even prime is identical with two. One might be forgiven for similarly hoping that if the universal quantifier meets Belnap’s uniqueness condition, then universal quantification over U rather than over S is the only one to obey the logic of universal quantification. That would have been a tidy answer to Putnam’s question. But it is not forthcoming. The linguistic commingling noted above has a worldly echo in model theoretic equivalents of Belnap’s uniqueness condition. For example, a theory with a binary predicate R (like the identity predicate) meets Belnap’s condition if and only if in any two models for the theory with the same domain, R has the same extension in that domain. None of this fixes the domain, let alone uniquely. Similarly, the logic of quantification does not settle whether U or S is our universe of discourse. McGee is well aware of this. His response is, ‘The rules of inference do not determine the range of quantification. What they do ensure is that the domain of quantification in a given context includes everything that can be named within that context. This includes even contexts in which there are no restrictions on what can be named. In such contexts, the quantifiers range over everything’. The end of universal instantiation is so open as to admit a name for each thing. To rule out nonstandard models, McGee needs the converse (to infer the universal quantification from the totality of its intended instances), which will in general require infinitely large proofs, and in the case of U, proofs too big to be sets, and so much too big to fit into all spacetime. What this plethora of names really seems to come to is a fairly bald denial of the suggestion in Putnam above that A exhausts the things we can in any serious sense name; the slippage between Putnam and McGee may be the difference between languages usable for communication versus languages as combinatorial mathematical abstractions. Of course, if interpretation of quantification requires a model in the sense we inherit from Tarski, then since the domain of a model has to be a set (so we have settled laws for handling models) and U is not a set (that’s the conven-

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tional wisdom for handling the paradoxes of set theory), U was a nonstarter as our universe of discourse from the outset. If we’re worried about whether we’re quantifying over S or U, let’s take the quantifiers not as varying as we shift with Tarski from domain to domain, but rather with Frege as genuine logical constants in the sense that the only sort of interpretation we countenance has U and U alone as the domain. That might mean leaning less on conventional set theory, but if Putnam scares us enough, perhaps the price is worth paying. Department of Philosophy (mc 267) 1420 University Hall University of Illinois at Chicago 601 South Morgan Street Chicago, IL 60607 USA

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Ontology of Mind, by Helen Steward. Oxford: Oxford University Press. Pp. . H/b £.. P/b £.. This ambitious and thoughtful book targets widely held views in contemporary philosophy of mind about the ontological underpinnings of our ordinary, folk-psychological attributions of mental states. Philosophers of mind would profit from responding to the challenge it presents. But the book’s audience extends well beyond that field. Indeed, the misunderstandings Steward sees as endemic to much current philosophy of mind are taken up directly only in the last thirty or so pages; the preceding discussion—while it certainly makes frequent contact with issues in philosophy of mind—can be read as an autonomous essay on the metaphysics of events, states, and processes, and on their different roles in the metaphysics of causation. Throughout, the book is rich in stimulating ideas and detailed, often quite intricate arguments. Even though the central ones, I think, fall short of their mark, the effort of working through them is very rewarding. Here is the central argument, in a nutshell. Steward contends that statements like () are quite literally incomprehensible: () Ned’s belief that it is cold outside=the brain-state described by D. (We bequeath to the neuroscience of the future the task of finding an appropriate substitute for ‘D’.) Why? Partly because we do not possess a rich enough conception of what sort of thing a belief (or desire, intention, etc.) is to invest such identities with content, without the help of some philosophical theory. Put another way, our ordinary understanding of folk psychology and its idioms does not extend so far as to include an understanding of what it could mean to identify a belief with a brain-state. Steward argues that the only way that philosophical theorizing could make up for this lack is by insisting that beliefs such as Ned’s belief that it is cold out-