Deflating Logical Consequence [61]

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Issue Table of Contents
The Philosophical Quarterly, Vol. 61, No. 243 (April 2011) pp. 225-446
Front Matter http://www.jstor.org/stable/10.2307/23012894?origin=JSTOR-pdf USING MEREOLOGICAL PRINCIPLES TO SUPPORT METAPHYSICS [pp. 225-246] http://www.jstor.org/stable/10.2307/23012895?origin=JSTOR-pdf IS THERE COLLECTIVE SCIENTIFIC KNOWLEDGE? ARGUMENTS FROM EXPLANATION [pp. 247-269] http://www.jstor.org/stable/10.2307/23012896?origin=JSTOR-pdf `CHOCOLATE' AND OTHER KIND TERMS: IMPLICATIONS FOR SEMANTIC EXTERNALISM [pp. 270-292] http://www.jstor.org/stable/10.2307/23012897?origin=JSTOR-pdf HOW TO USE A CONCEPT YOU REJECT [pp. 293-319] http://www.jstor.org/stable/10.2307/23012898?origin=JSTOR-pdf DEFLATING LOGICAL CONSEQUENCE [pp. 320-342] http://www.jstor.org/stable/10.2307/23012899?origin=JSTOR-pdf KORSGAARD'S CONSTITUTIVE ARGUMENTS AND THE PRINCIPLES OF PRACTICAL REASON [pp. 343-362] http://www.jstor.org/stable/10.2307/23012900?origin=JSTOR-pdf MACROSCOPIC ONTOLOGY IN EVERETTIAN QUANTUM MECHANICS [pp. 363-382] http://www.jstor.org/stable/10.2307/23012901?origin=JSTOR-pdf DISCUSSIONS
PERCEPTUAL CONTENT AND SENSORIMOTOR EXPECTATIONS [pp. 383-391] http://www.jstor.org/stable/10.2307/23012902?origin=JSTOR-pdf VIOLATING REQUIREMENTS, EXITING FROM REQUIREMENTS, AND THE SCOPE OF RATIONALITY [pp. 392-399] http://www.jstor.org/stable/10.2307/23012903?origin=JSTOR-pdf CRITICAL STUDY
AFTER TRUTH GIVES WAY [pp. 400-409] http://www.jstor.org/stable/10.2307/23012904?origin=JSTOR-pdf BOOK REVIEWS

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DEFLATING LOGICAL CONSEQUENCE Author(s): Lionel Shapiro Source: The Philosophical Quarterly (1950-) , April 2011, Vol. 61, No. 243 (April 2011), pp. 320-342 Published by: Oxford University Press on behalf of the Scots Philosophical Association and the University of St. Andrews Stable URL: https://www.jstor.org/stable/23012899 REFERENCES Linked references are available on JSTOR for this article: https://www.jstor.org/stable/23012899?seq=1&cid=pdfreference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms

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The Phibsophical Quarterly Vol. 61 No. 243 April 2011 ISSN 0031-8094 doi: 10.nn/j.1467-9213.2010.678.x

DEFLATING LOGICAL CONSEQUENCE By Lionel Shapiro

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth

predicate is merely a device that allows us to express a certain kind of generality. I argue that a

parallel approach is available in the case of logical consequence. Just as deflationism about truth offers an alternative to accounts of truth's nature in terms of correspondence or justification, dflationism about consequence promises an alternative to model-theoretic or proof-theoretic accounts

of consequence's nature. I then argue, against considerations put forward by Field and Beall, that Curry's paradox no more rules out deflationism about consequence than the liar paradox rules out deflationism about truth.

Deflationists about truth argue that an appreciation of the expressive role of

the predicate 'is true' undercuts the demand for a metaphysically substan tive account of the nature of truth. In contrast, deflationism about a second

notion central to logic, logical consequence, appears not to have been pursued. Its possibility goes unmentioned even by those theorists who have explored what deflationists about truth should say about consequence.1 This is how Stewart Shapiro ('The Guru p. 126) frames the options: 'The issue concerns the notion or notions of consequence available to the various deflationists. In broad terms, there are two different approaches to con sequence: deductive and semantic.' In this paper, I shall argue that a third approach should be on the table - a deflationary approach. In §1, I present a version of deflationism about truth, and use it as a model in formulating a deflationism about logical consequence. §11 responds

to a series of objections to deflationism about logical consequence, ob jections based on what may appear to be a significant disanalogy with deflationism about truth. §111 identifies some other factors that have made

the possibility of deflating logical consequence easy to overlook. Finally, in 1 See J. Azzouni, Tracking Reason: Proof, Consequence, and Truth (Oxford UP, 2004); J. Beall, Spandrels of Truth (Oxford UP, 2009); H. Field, Saving Truth from Paradox (Oxford UP, 2008); C. Gauker, 'Semantics for Deflationists', inj. Beall and B. Armour-Garb (eds), Deflationism and

Paradox (Oxford UP, 2005), pp. 148-76; S. Shapiro, 'The Guru, the Logician, and the De flationist: Truth and Logical Consequence', Nous, 37 (2003), pp. 113-32. © The Author © 2010 201 o The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

Published by Blackwell Publishing, 9600 Garsington Road, Oxford 0x4 2DQ, UK, and 350 Main Street, Maiden, ma 02148, USA

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DEFLATING LOGICAL CONSEQUENCE

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§IV, I consider the ramifications for logic of deflating logical consequence. It is widely held that revisions to classical logic are mandated if the expressive role which deflationists about truth attribute to the predicate 'is true' is to be maintained in the face of the liar paradox. I shall criticize recent arguments by Hartry Field and Jc Beall (themselves deflationists about truth) according

to which Curry's paradox shows that nothing can play the expressive role a deflationist about logical consequence attributes to the predicate 'is a consequence of'. I. 'MERE EXPRESSIVE DEVICE' DEFLATIONISM I.i. The truth predicate

The variety of deflationism I shall consider applying to logical con

is one that has been applied to truth in a tradition of thought tracing Quine. According to this tradition, the essential role of our truth pred

to enable a 'semantic ascent' that allows us to simulate quantificat sentence position by generalizing instead over objects, either sent (pace Quine) propositions.2 For example, to serve as a universal ge tion over the infinite number of worldly claims Snow is white or snow is not white

Oxygen is an element or oxygen is not an element etc.

we can use the generalization Every sentence of the form 'p v —1// is true.

(Or: every proposition with the corresponding structure is true.) Similar explanations can be given for the truth predicate's usefulness in our

generalizations about people's utterances or beliefs, as in 'Some of what he

said is not true'.

How does the truth predicate manage to serve the expressive function just

explained? According to Quinean tradition, it does so in virtue of an equi

valence between and 'The sentence is true'. (Henceforth, I shall focus 2 W.V. Quine, Philosophy of Logic (Harvard UP, 1970), pp. 10-12. For some elaborations of this theme, see S. Leeds, 'Theories of Reference and Truth', Erkenntnis, 13 (1978), pp. 111-29; D. Grover, 'On Two Deflationary Truth Theories', in J.M. Dunn and A. Gupta (eds), Truth or Consequences (Dordrecht: Kluwer, 1990), pp. 1—17; P. Horwich, Truth, 2nd edn (Oxford UP, 1998); Field, Postscript to 'Tarski's Theory of Truth', in his Truth and the Absence of Fact (Oxford

UP, 2001), p. 28, and Saving Truth from Paradox, p. 18; Azzouni, Tracking Reason, ch. 1; Beall, Spandrels of Reason, pp. 1-2. See M. David, 'Quine's Ladder: Two and a Half Pages from the Philosophy of Logic', Midwest Studies in Philosophy, 32 (2008), pp. 274-312, for a careful critical

examination.

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LIONEL SHAPIRO

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on sentences, though my claims could also be stated in terms of tional truth.) At the very least, the equivalence encompasses the fo two inference rules, which I shall call 'the T-rules':

T-Intro. ^ T-Elim. p is true p

The T-rules should suffice to underwr

uted to the truth predicate in the gener

also need to account for the predicat texts, e.g., 'If a sentence is true, the everything he said is true, then oxyg logic of the conditional, this may re

instead require what Beall calls the full '

the intersubstitutability of'/;' for 'The s sentential contexts, including the antece except where noted, I shall assume that ive role in generalizations to the T-rules

The above story about the truth pred is often taken to have 'deflationary' c it is only the need for an otherwise u explains why we cannot do without a p of truth. Once we understand how th new kind of generalization, we have a

performs the function that is its raison rests entirely on the logical features of

T-rules). The deflationist contends th

these logical features require the proper

amenable to any substantive character To understand the full import of thi need to know what counts as a substan nature. Fortunately, a sufficient condi

poses. Any non-trivial answer to th

having the property consist in?' count property's nature.4 If the deflationar mains no reason to aim for a theory of

correspondence to reality, ideal verifi

semantic norms.

3 Beall, Spandrels of Truth, p. vii; see also Field, Saving Truth from Paradox, pp. 209-10.

4 See CJ.G. Wright, 'Truth: a Traditional Debate Reviewed', in S. Blackburn and

K. Simmons (eds), Truth (Oxford UP, 1999), pp. 203-38, at p. 209, citing Horwich, Truth, p. 2. Wright (pp. 205-6, 218-19) notes that reductive analysis is not the only kind of substantive characterization of truth rejected by deflationists. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

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Laid out explicitly, the deflationary argument sketched above consists of five main steps.

Ti. If it were not for the need to express a certain kind of generality, we would have no need for the predicate 'is true'. Instead of predicating 'is true' of a sentence, we could employ the sentence itself

T2. To explain how 'is true' allows us to express the kind of generality in question, we need only make use of the predicate's logical features, namely, the T-rules T3. There is no reason to think that the T-rules require 'is true' to express a property whose nature is amenable to substantive characterization T4. From (Ti), (T2) and (T3), it follows that there is no reason to think that

in order to understand how 'is true' serves the function that is its raison

d'etre, we must take this predicate to express a property whose nature is amenable to substantive characterization

T5. Hence we have no reason to hold that 'is true' expresses a property whose nature is amenable to substantive characterization.

The inference from (T4) to (T5) is based on a methodological principle of parsimony. Here is how Marian David summarizes the above argument (which he does not defend): a 'richer conception of truth' than that supplied

by the T-rules 'is not warranted by the need to explain [why we have the truth predicate]; hence, not warranted'.5 I shall call the view arrived at by this argument mere expressive device de flationism (MED deflationism for short). The label is meant to distinguish this

view from varieties of deflationism that involve stronger commitments. Unlike the 'minimalism' defended by Paul Horwich {Truth, p. 135), MED

deflationism does not amount to a 'theory of truth', a collection of

'principles about the property of truth on the basis of which all the facts about

truth are to be explained'. For example, MED deflationists need not see it as

their task to derive generalizations such as 'Every sentence of the form "p v —ip" is true' from more basic principles.6 MED deflationism is also more

modest than Horwich's view in a second important respect: it need not be taken as explaining what the truth predicate's meaning consists in, or what possession of the concept of truth consists in (Horwich, Truth, pp. 36-7, but cf. p. 126 n. 5). Finally, unlike some versions of deflationism, MED de flationism does not assert that the truth predicate is conservative over the 5 David, 'Quine's Ladder', p. 284. For similar remarks, see M. Williams, 'Meaning and Deflationary Truth', Journal of Philosophy, 96 (1999), pp. 545-64, at pp. 547-8, and Armour Garb and Beall, 'Deflationism: the Basics', in Armour-Garb and Beall (eds), Deflationary Truth (Chicago: Open Court, 2005), pp. 1-29, at p. 12. 6 A. Gupta, 'A Critique of Deflationism', Philosophical Topics, 21 (1993), pp. 57-81, argues

that Horwich fails in this task.

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LIONEL SHAPIRO

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truth-free fragment of the language, or that the truth predicate cann ineliminably in any kind of explanation.7

It is not my aim in this paper to defend MED deflationism abou Even if premises (Ti) and (T2) are granted, the above line of rea open to at least two challenges. An opponent of deflationism migh premise (T3), arguing that a predicate's satisfaction of the T-rules

need of explanation in terms of a substantive characterization of the of the property the predicate expresses. One might also question the

parsimony in inferring (T5) from (T4).8 Nevertheless, MED def

about truth is a prominent philosophical position, one whose attractio felt by many who oppose it. My aim is to argue that to the extent th deflationism counts as a plausible and attractive approach to truth, it

count as an equally plausible and attractive approach to logical con This is because a story essentially parallel to the one I have just t the truth predicate can be told about the consequence predicate. 1.2. The consequence predicate

For initial expository purposes, I mean by 'consequence predicat place predicate expressing a relation between a single premise and

sion. This is the relation that obtains between two sentences whenever the

second 'is a logical consequence of' the first, or, equivalently, whenever there exists a 'valid argument' with the first sentence as sole premise and the

second as conclusion. Shortly, I shall explain how the proposal encompasses multiple-premise consequence. My thesis is that an argument parallel to (Ti)—(T5) supports MED defla tionism concerning the consequence predicate. I expect, however, that this argument will strike the reader as differing in significant ways from the original one concerning the truth predicate. My aim in this section is simply

to present the argument concerning the consequence predicate, calling attention to the structure it shares with its model. The significance of the differences between the two arguments will be the topic of §11.

As before, we start with a claim about the predicate's essential expressive

function.

Ci. If it were not for the need to express a certain kind of generality, we would have no need for the predicate 'is a consequence of'. In place of '^2 is a logical consequence of ,S[', we could employ sentences i'i and s2 themselves, joined by a suitable sentential connective. 7 See S. Shapiro, 'Proof and Truth: Through Thick and Thin', Journal of Philosophy, 95 (1998), pp. 493-521, for the problems besetting deflationary views which embrace these addi

tional theses.

8 Azzouni, Tracking Reason, ch. 1, rejects the inference from (T4) to (T5), while endorsing

both claims.

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DEFLATING LOGICAL CONSEQUENCE 325

This is because our language contains, or could be expanded to c binary sentential connective 'that ... entails that (In embracin connective, as §111 will explain, my proposal departs from Quine.)

Graham Priest, I suggest that 'Those who do not like using

"entails" as a connective can read instead "if... then logically . then it follows logically that ..." or some similar locution, with

tional suitably qualified adverbially'.9 As does Priest, I shall write ' place of the explicit 'logically entails', just as I shall often abbreviat

consequence' as 'consequence'. To illustrate claim (Ci), instead of

'Snow is black and oxygen is an element' has 'Snow is black' a consequence

we can simply say That snow is black and oxygen is an element entails that snow is black.

It is only in generalizations that semantic ascent from talk of snow and oxygen to talk of sentences becomes necessary. Thus we can achieve the effect of universally generalizing over the infinite number of claims

That snow is black and oxygen is an element entails that snow is black That water is wet and books are animals entails that water is wet etc.

by using the generalization

Every sentence of the form 'p a q' has the respective '// as a logical consequence.

Similarly, using the generalization

He denied a logical consequence of something she said we can achieve the effect of existentially generalizing over That water is wet entails that snow is white, and she said 'Water is wet' but he denied 'Snow is white'

That snow is white and water is wet entails that snow is white, and she said 'Snow is white and water is wet' but he denied 'Snow is white' etc.

In order for the consequence predicate to play the expressive role just described, it should suffice that we have an equivalence encapsulated by what I shall call 'the C-rules'. 9 G. Priest, In Contradiction, 2nd edn (Oxford UP, 2006), p. 82. Priest follows A.R. Anderson and N.D. Belnap, Entailment: the Logic of Relevance and Necessity, Vol. 1 (Princeton UP, 1975), p. 491. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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LIONEL SHAPIRO

^ That p entails that q 'p' has V as a consequence

C-Intro. t ,, , - C-Elim. - — 3 p has q as a consequence 1 hat p entails that q

Again I shall not worry about whether a stronger equivalence is needed to account for embedded contexts. However, I should explain how the pro posal can be extended to handle multi-premise consequence relations. For a finite number of premises, we can use conjunction:10 Cn-Intro.

CL-Elim.

That (p\ and ... pn) entails that q

ip\' and ... jbn' have 'q' as a consequence

jb\ and ... have 'q' as a consequence That (pi and ... pn) entails that q

The MED deflationist can exploit the generalizing function of the truth predicate to obtain a version of the C-rules which accommodates a con sequence relation between infinitely many premises and a conclusion. Here the equivalence is between 'Premise set T has "q" as a consequence' and 'That all members of T are true entails that q\

The proposal, then, is that the consequence predicate is an expressive device whose essential role is to allow us to frame useful generalizations in virtue of the semantic ascent underwritten by the G-rules.

C2. To explain how 'is a consequence of' allows us to express the kind of generality in question, we need only make use of the predicate's logical features, namely, the C-rules. If correct, this proposal can be construed in the same deflationary fashion as

the familiar view regarding the truth predicate. For we should be able to argue as follows.

C3. There is no reason to think that the C-rules require 'is a consequence

of' to express a relation whose nature is amenable to substantive characterization

C4. From (Ci), (C2) and (C3), it follows that there is no reason to think that in order to understand how 'is a consequence of' serves the function that is its raison d'etre, we must take this predicate to express a relation whose nature is amenable to substantive characterization

C5. Hence we have no reason to hold that 'is a consequence of' expresses a relation whose nature is amenable to substantive characterization.

10 In §IV, I shall point out a context in which the two-premise entailment locution 'That p\ and that p2 together entail that q' may need to be interpreted otherwise than as 'That (p\ and p'is false f is false not-p

F-Intro. , F-Elim.

The deflationist then argues that the falsity predic

accounted for without assuming that falsity is whose nature is amenable to substantive charact

flationary argument concerning falsity runs par

except for the fact that the F-rules involve a sente negation. MED deflationists about falsity should allow that substantive investigation into the logic of negation.

investigation yields any substantive characteriza falsity property. Similarly, as I shall show in §IV

consequence face the substantive task of investigati the entailment connective. Again there is no reason

tion will yield any substantive characterizati

consequence relation. Logical enquiry concerns w

example, we may debate whether that p entails tha

long as instances of this schema are construed connective, they are not even relational claims, underlying nature of a relation. Here is another Anderson and Belnap observe (p. 485), the falsity negation connective in the same relation as that in 11 See esp. Beall, Spandrels of Truth, pp. 1-2.

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I.IONEL SHAPIRO

33°

stands to the 'null connective'. Suppose, then, that I am mistaken, studying the logical behaviour of negation sentences is indeed tan

to studying the nature of falsity. In that case, studying the logical be of sentences in general should be tantamount to studying the nature o

Yet deflationists are not alone in denying that logic is the study of th of truth. In short, the need for substantive enquiry into the logic of

and entailment does not undercut MED deflationism about falsit consequence.

It may be objected that the 'deflationary' proposal leaves room for more kinds of substantive enquiry than the logical kind just discussed, and that some of these appear akin to traditional enquiry into the nature of con sequence. After all,

(a) Will 'deflating consequence' not leave a metaphysical question about the nature of entailment facts?

(b) Will recent debates between pluralists and monists about logical con sequence not survive, transposed into a new key?12

(c) Will there not remain substantive questions about the connection between entailment and inference?

In each case, I shall argue that if the question threatens MED deflationism

about logical consequence, a parallel question threatens MED deflation ism about falsity. Hence if I am right that MED deflationists about truth

should advocate a similar deflationism about falsity, the above three

questions can give a deflationist about truth no reason to resist deflationism

about consequence. As regards falsity, the question that corresponds to (a) is whether defla tionism about falsity leaves room for substantive debate about the nature of negative facts. I shall now argue that regardless of how a deflationist about

truth and falsity answers this question, a deflationist about truth and con sequence may answer question (a) in the same way. Many deflationists about

truth and falsity view debate about the nature of negative facts as just the sort that deflationism undermines. Such a deflationist may allow that there are facts, regarding these as true propositions.13 On this view, we may indeed

ask what being a negative proposition consists in. But deflationism about truth is supposed to entitle us to reject the question 'What does being a true

negative proposition consist in?'. Hence deflationists about truth who re gard negative facts as true negative propositions should repudiate substan tive enquiry into the nature of negative facts. Since such deflationists should regard entailment facts as true entailment propositions, they should likewise 12 Questions (a) and (b) were posed by a referee. 13 See J. Dodd, An Identify Theoiy of Truth (Basingstoke: Palgrave Macmillan, 2000). © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

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answer 'No' to question (a). A second kind of deflationist about truth and falsity might countenance substantive enquiry into the nature of facts, yet

deny that there are any negative facts. It is not clear why such a theorist should take the deflationist about consequence to be committed to the exist ence of entailment facts. Finally, a third kind of deflationist about truth and falsity might concede that there is a substantive question about the nature of negative facts. In that case, however, it is not clear why a 'Yes' answer to (a) should count as an objection to deflationism about consequence. When it comes to questions (b) and (c), the deflationist about consequence should be unconcerned about answering 'Yes'. Deflationists about truth do not purport to 'deflate' all philosophical debates that have appeared to concern the nature of truth. Rather, they argue that such debates, where genuine, have not actually concerned the nature of truth.14 In this regard, deflationism about consequence is no different. Hence we should not expect deflating logical consequence to undermine debates between logical plural ists and monists.15 From the deflationary perspective, a pluralist about logical consequence can be understood as holding that we should recognize the possibility of multiple 'entailment connectives', just as some argue that there can be multiple 'negation connectives'.16 Of course, just as a pluralist about negation owes an explanation of what makes something count as a 'negation connective', a pluralist about entailment owes an explanation of what makes something count as an 'entailment connective'. Likewise, deflationists about consequence should agree that there is room for substantive theorizing about the relation our use of the entailment connective bears to our practice of inference. Here, too, the comparison with negation is instructive. A theorist who holds that something substantive can be said about the relation our use of the negation connective bears to our practices of rejection or denial is not thereby debarred from advocating a defla tionary approach to the property of falsity.17 II. 3. Can the consequence predicate's essential function lie in generalization?

A third objection from disanalogy challenges MED deflationism's starting point, its identification of the consequence predicate's essential expressive function. Provided our language has an entailment connective, the objector 14 Horwich, Truth, pp. 7-8, 118-ig. 15 See J. Beall and G. Restall, Logical Pluralism (Oxford UP, 2006); Priest, Doubt Truth to be a Liar (Oxford UP, 2006), ch. 12; Field, 'Pluralism in Logic', Review of Symbolic Logic, 2 (2009),

PP- 342-59-. 16 Beall, in his 'Deflationism and Gaps: Untying "Not"s in the Debate', Analysis, 62 (2002),

pp. 299-305, argues that deflationists about truth should make use of multiple negation connectives.

17 For such a view of negation, see H. Price, 'Why "Not"?', Mind, 99 (1990), pp. 221-38. © 201 o The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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LIONEL SHAPIRO

points out, the kind of generalization facilitated by the conse

predicate can instead be performed using a truth predicate. Rath generalize about the consequences of sentences, we can generalize ab truth of sentences of the form 'That p entails that q\

My response is that a predicate's 'essential function' should not b

fused with a function that 'makes it essential' to have the predicate. Ra

as is reflected in how I formulated (Ti) and (Gi), it is the predicate

d'etre. Even if semantic ascent for purposes of generalizing over entail

can be performed (albeit more cumbersomely) without a consequen

dicate, it may still be the case that were it not for that function, we wou

no need to employ the predicate. Similarly, we might be able to ge into sentence position using either substitutional quantifiers or ob quantifiers that bind propositional variables. This would be consiste

the MED deflationist's claim that we would have no need for a truth

predicate were it not for how it allows us to simulate such generalization using first-order objectual quantifiers.18

Here again the parallel between consequence and falsity is a close one. It is no objection to MED deflationism about falsity that instead of general izing about the falsity of sentences, we can generalize about the truth of their negations (or, perhaps equivalently, about the sentences' own un truth19). Of course, it would be good to have an explanation for the fact that

we employ falsity and consequence predicates, but not, for example, a predicate that applies to a pair of sentences iff their disjunction is true. But

there seems no reason to hold that an explanation of this fact about our language must be based on the natures of the falsity property and the con sequence relation. III. WHY THE PROPOSAL IS EASILY OVERLOOKED

I have argued that extending deflationism from truth to logical consequen no more problematic than extending deflationism from truth to falsity.

two circumstances have made this option easy to overlook, even for theorists who have recognized an entailment connective.

First, there is the history of discussion of the relation betwe

consequence predicate and an entailment connective. I start with Carn

remarks on C.I. Lewis' 'relation of strict implication'.20 Accordin

18 This way of making the point was suggested by a referee. 19 The T-rules themselves do not underwrite the inference from the truth of a senten

negation to the sentence's untruth (Priest, In Contradiction, pp. 69-71). Still, deflationists endorsed this inference, which is underwritten by the condition I called 'transparency' in 20 C.I. Lewis and C.H. Langford, Symbolic Logic (New York: Century, 1932), p. 124. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

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Carnap, Lewis' '/> -3 q should be interpreted as a 'quasi-syntactical' sentence of the object-language, whose 'correlated syntactical sentence' is the overtly

meta-linguistic '"q" is a consequence of 21 Whereas the MED deflation ist claims that were it not for a need to generalize, we could suffice with an

entailment connective, Carnap is concerned to show that this connective, though 'not inadmissible', is unnecessary in view of the availability of the consequence predicate. Carnap also charges, against Lewis, that use-mention confusion may be 'partly to blame for the fact that the fundamental difference between the sentential connections (e.g., implication) and the syntactical relations be tween sentences (e.g., the consequence relation) is frequendy overlooked' (§42 [translation modified]; cf. §69). Quine endorses this complaint. The

verb 'implies', he insists, is a two-place 'semantical predicate' whose argument-places take sentence names rather than sentences.22 Lacking Carnap's

tolerance, Quine indeed questions the admissibility of a corresponding connective. Hence when he subsequently comes to develop his deflationary account of the truth predicate, he is in no position to treat the consequence predicate as a device of semantic ascent underwritten by the C-rules.

Interestingly, when Quine compares his predicate 'implies' with the connective he rejects, he finds no difference with regard to the expression of

generality. The generality of logical laws, Quine writes, would be expressed using schematic letters, whether we use an entailment connective

1. p^>(pvq) or use Quine's purportedly 'rectified' form 2. implies r(]> v v|/n.

What Quine does not point out is that in (2), but not in (1), we can replace the schematic letters with first-order variables bound by a quantifier. Nor is

this difference noted in Anderson and Belnap's 'Grammatical Propaedeu tic', the seminal text responsible for rehabilitating the entailment connective in the tradition of relevance logic and paraconsistent logic more generally.23

Responding to Quine, Anderson and Belnap argue that we should regard the difference between (1) and (2) as merely grammatical and of no deeper 21 R. Carnap, Logische Syntax der Sprache, tr. A. Smeaton as The Logical Syntax of Language

(London: Routledge, 1937), §63 and §69. 22 Quine, 'Three Grades of Modal Involvement', Proceedings of the Xlth International Congress of

Philosophy, Vol. xrv (Amsterdam: North-Holland, 1953), pp. 163-6, and Word and Object (MIT Press, i960), §41. 23 Anderson and Belnap, Entailment, pp. 473-92; see J.M. Dunn and G. Restall, 'Relevance Logic', in D.M. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol. vi (Dord recht: Kluwer, 2002), pp. 1-128, at pp. 2-5; Priest, In Contradiction, p. 82. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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LIONEL SHAPIRO

334

logical significance. It is thus no surprise that they too, and those who

them, fail to consider the thought I have been suggesting. This i

thought that semantic ascent from (i), with its entailment connective,

with its consequence predicate, has an expressive role comparable w

of Quinean semantic ascent from '/>' to ' "p" is true'. A second reason why the deflationary option has been easy to overlo that there is already a familiar relation between a sentential connectiv

the consequence predicate. This is the (single-premise) 'deduction th that obtains for various connectives ' -> '.

3- 'P' b V iff b lP ?'•

Here the turnstile ' expresses some proof- or model-theoretically relation that is intended to capture the extension of the logical c

uence relation. If »' is our own language's entailment connect should be able to formulate the turnstile's extensional adequacy thus:

4. y b y -> q

Yet the prominence in proof and model theory of biconditional (3), with it

turnstiles on both sides, tends to obscure what might be suggested b biconditional (4), where the turnstile appears only on the left. This is

possibility of using (4), or the C-rules themselves, to formulate a deflation

account of the relation whose extension the turnstile is being invoked capture.

IV. CURRY'S PARADOX AND THE C-RULES

It is widely acknowledged that the liar paradox places restrictions

logic of any sufficiently resourceful language that contains a truth pr

governed by the T-rules.24 Curry's paradox turns out to place ad restrictions on the logic of a language that also contains a conseque dicate governed by the C-rules. I shall now argue, against recent cl 24 See J. Beall and M. Glanzberg, 'Where the Paths Meet: Remarks on Truth

dox', Midwest Studies in Philosophy, 32 (2008), pp. 169-98; Field, Saving Truth from P H. Leitgeb, 'What Theories of Truth Should Be Like (But Cannot Be)', Philosophy C

(2007), pp. 276-90. Gupta, 'Do the Paradoxes Pose a Special Problem for Deflationis

Beall and Armour-Garb (eds), Deflationism and Paradox, pp. 133-47, at pp. 142-5, mig to deny that the T-rules impose any logical restrictions in the presence of a liar sent

the theory he and Belnap defend, however, introducing the T-rules (which they reflecting partial definitions, sometimes circular, of the truth predicate) into an classical natural deduction system requires introducing index restrictions: Gupta an

The Revision Theory of Truth (MIT Press, 1993), pp. 126, 157-60. I shall not here explor this avenue could be exploited by a deflationist about consequence. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

335

two prominent truth deflationists, Field and Beall, that these restrictions can

be satisfied. This brings to light what is from the present perspective a tension in Field's and Beall's positions. On the one hand, they insist on re vising classical logic in the face of the liar paradox, so as not to compromise

the role of 'is true' in enabling generalizations. Faced with Curry's paradox, on the other hand, they are willing to compromise what I have argued is the similar role of 'is a consequence of' in generalizations. My aim is to explain why doing so is unnecessary. The significance of this conclusion does not depend on my further proposal that this expressive role of the consequence

predicate can be exploited to defend MED deflationism about the con sequence relation.25

The arguments by Field and Beall I have in mind deploy Gurry's paradox against the traditional principle that logical consequence preserves truth.26 In

particular, both authors defend a conclusion I shall call 'the Field- Beall thesis'. For the sake of readability, I shall join them from here on in appear

ing to talk about an object-language distinct from the meta-language I am using (and use italics for quotation). But the expressions Tlx), Cons(x,y) and a

should really be regarded as abbreviations of 'x is true', ly is a consequence ofx', and 'and'. Field-Beall thesis: let P be any arbitrary sentence, and suppose the language

contains a transparent truth predicate T, a predicate Cons, and a

connective >-> such that

(CURRY) there is a sentence A" equivalent to Tr K? >-¥ P

(T-PRES) (i) Cons(rA1, riT) implies TA1 >-> l'rPP (ii) the fact that condition (i) holds vindicates the thesis that logical consequence preserves truth.27 Then P is true.

Condition (T-PRES) is satisfied if >-» is a connective that is related to our language's consequence predicate according to the C-rules. In what follows, 25 Cf. Gupta, 'Do the Paradoxes Pose a Special Problem for Deflationism?', who argues

that in so far as the paradoxes pose a challenge for deflationists about truth, this is on account

of their insight about the expressive role of 'true', not on account of the deflationary con sequences they draw from it. 26 Field includes MED deflationists about truth (such as himself) among those theorists who cannot 'accept even that valid arguments all actually preserve truth, let alone that they do so of necessity' (Saving Truth from Paradox, p. 286; cf. pp. 377-8 and 'Pluralism in Logic', pp. 349-53).

According to Beall, MED deflationism about truth implies that 'the claim that validity is [truth-preserving] needs to be rejected, and I reject it' (Spandrels of Truth, p. 35).

27 In effect, Beall observes that the material conditional ('hook') of his logic satisfies (i) of

(T-PRES), but not (ii). This explains why, when reiterating his rejection of the claim that consequence preserves truth, he adds the qualification 'in anything beyond the hook sense' (Spandrels of Truth, p. 37). © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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I shall call any such connective an 'entailment connective'.28 MED tionism about consequence requires that our language contains, or c expanded to contain, an entailment connective. If the Field-Beall t

true, therefore, this (assuming also a transparent truth predicate) rule

MED deflationism about consequence. For it is unacceptable that th of an arbitrary sentence P should follow from the mere existenc sentence K equivalent to Tvt? >-» P.

In explaining how this obstacle can be avoided, I shall proceed by examining a more direct argument for the conclusion that Curry's rules out an entailment connective. Unlike the argument via the Fie

thesis, the direct argument does not explicitly invoke a conse predicate, nor does it presuppose that the language's truth predicate is

transparent.29 After responding to the direct argument in §IV.i, explain in §IV.2 how my response undermines a key assumption Field's and Beall's argument for the Field-Beall thesis. For this pap

poses, it suffices that I vindicate the possibility of an entailment conn

against the direct and indirect challenges from Gurry's paradox. I n argue that the expression 'that ... entails that...' or any other actual expression has the logical features my vindication requires. IV. i. Avoiding the direct argument against an entailment connective

Let K be a sentence equivalent to TrK? —> P, where —> is an entailmen nective. (To simplify my presentation, I shall pretend that these are n

equivalent sentences, but the same sentence.) Curry's paradox can formulated as the following natural deduction proof, presented in form using the Lemmon-style sequent format.30

I [I] I [2] I [3]

TT rfr^p p

[4]

rip-^p

[5]

VIP

[6]

p

Ass

1 T-Elim (with pretence noted abov 1, 2 MP 1, 3 CP 4 T-Intro (with pretence noted above)

4, 5 MP

How might we block this derivation? When the arrow is his own proposed 28 Here I depart from Priest, whose 'entailment connective' violates C-Elim. See fn. 37

below.

29 A third argument, this time invoking a predicate Cons governed by the C-rules, but not a truth predicate, would start with a sentence K* equivalent to Cons^IC", rP1). Cf. H. Deutsch, 'Diagonalization and Truth Functional Operators', Analysis, 70 (2010), pp. 215-17. 30 E.J. Lemmon, Beginning Logic (London: Nelson, 1965). Sequents are numbered in brackets to the left of each sequent's conclusion; numbers farther to the left index the sequent's

assumptions. In my example, sequents [i]-[3] have as sole assumption.

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DEFLATING LOGICAL CONSEQUENCE

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conditional, Field rejects Conditional Proof (Saving Truth from Paradox, p. 282). But this is not an option for an entailment connective. The rationale behind

our deduction rules is that when we derive B from assumption A, we have established that B is a consequence of A, and hence the entailment A —> B.

As Anderson and Belnap put it, 'we clearly want to be able to assert A —> B whenever there exists a deduction of B from A'.31 Our only remain

ing option, it would appear, is to reject Modus Ponens for our entailment connective. Yet theorists of an entailment connective have regarded Modus Ponens as equally essential to such a connective.32 So it might appear that there are two jointly unsatisfiable demands on an entailment connective. However, there are well known natural deduction systems, substructural

logics without contraction rules, that block the above derivation despite featuring both Conditional Proof and a version of Modus Ponens. Field, who

notes this, characterizes such logics as 'radical' and 'very hard to get [his] head around' (Saving Truth from Paradox, pp. 10-11, 282-3). I shall now argue

that a certain perspective on natural deduction rules (one that comes naturally to the MED deflationist about consequence) helps to dispel this appearance, at least where —> is an entailment connective.33

First, the substructural version of MP. Here is a simple use of the rule, where, as in line [3] of the above Curry derivation, both premises depend on the same assumption.

j [j] c Ass j [k] A —> B (some rule) j [k+i] A (some rule) A standard Lemmon-style system licenses us to add the sequent

j [k+2] B k, k+iMP

In a substructural deduction system, in contrast, we are in add the sequent

j;j [k+2] B k, k+i MP

where 'j; j' records a twofold dependence of B on assumption C, in

31 Anderson and Belnap, Entailment, p. 7. See also A. Weir, 'Naive Truth and Soph Logic', in Beall and Armour-Garb (eds), Deflationism and Paradox, pp. 218—49, at p. 2 32 Again see Anderson and Belnap, p. 7. Priest claims that obeying MP is 'analytic

of what implication is', in a context where he is speaking of 'the entailment con

species of 'implication connective' (In Contradiction, pp. 83, 86). But see fn. 28 on his te

33 Unlike Field, Beall does not dismiss contraction-free substructural logics per s he does not think they show how consequence could be truth-preserving. (This i does not question what I shall call in §IV.2 'the Field-Beall assumption'.) © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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338

LIONEL SHAPIRO

premises [k] and [k+i] respectively.34 Use of a 'structural contraction then required in order to derive

j [k+3] B k+2 Weak Contraction Thus our Curry derivation is blocked in any substructural deduction system that lacks structural contraction rules.

Concerning either version of MP, we may ask what the significance is of

the rule's licensing us to write down a certain sequent. Field answers, in effect (Saving Truth from Paradox, pp. 163-4), that our Lemmon-style sequents

state consequence relations understood as norms of 'conditional assertion'. In the above instance, he understands the standard MP rule as telling us this:

Given that we may assert A —» B on the assumption of C, and that we may assert A on the assumption of C, we may assert B on the assumption

of C.

When he considers the MP rule's substructural version, however, Field rightly finds it hard to get his head around the idea that it might instead only be legitimate to assert B on the double assumption of C. For what might that mean?

Perhaps, though, there is a better way than Field's for the MED defla tionist to understand our two versions of the MP rule. As a first step, I suggest that when we use this rule in a standard natural deduction system, what we are doing (in the above simple example) is purporting to register an entailment:

5. That A —» B is a consequence of C, and that A is a consequence of C, together entail that £ is a consequence of C.

As used in a substructural deductive system, in contrast, the same MP step should be understood as purporting to register a different entailment:

6. That A —» B is a consequence of C, and that A is a consequence of C, to gether entail that it is a consequence of C that B is a consequence of C.

Here I have used a two-premise entailment locution together with a consequence predicate. Moreover, in (6) I have also used the form 'It is a consequence of s that p\ where the first argument place takes a name and the second a sentence.35 We can try to use semantic descent to obtain entailments in which only our binary connective —» appears. 34 S.L. Read, Relevant Logic: a Philosophical Examination of Inference (Oxford: Blackwell, 1988); J. Slaney, 'A General Logic', Australasian Journal of Philosophy, 68 (1990), pp. 74-88; G. Restall, On Logics Without Contraction, Univ. of Queensland PhD thesis, 1994, http://consequendy.org/ papers/onlogics.pdf.

35 It is a case of what Prior calls a 'predicate at one end and connective at the other': A.N. Prior, Objects of Thought (Oxford: Clarendon Press, 1971), p. 135. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

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In the case of a standard natural deduction system, it does not matter which of the following options we choose as corresponding to (5):

5a. ((C->(/l->i3))A(C-»4))-»(C-»£) In a substructural natural deduction system that lacks contraction rules, in contrast, only the second of the following candidates for a 'consequence' free entailment corresponding to (6) is derivable:

6a. ((C-»(4-*JS9)a(C-»^)->(C->(C->5)) 6b. (C(4->£))-» ((C-» 4)-»(C-»(C-*£))) In short, then, I propose that in our simple example, the substructural MP step should be construed as purporting to register entailment (6b).36 In defence of contraction-free logics of entailment, I now suggest that (6b)

is indeed a more compelling entailment than (5a), (5b) or (6a). Each sub stitution instance of (6b) is an instance of the more general (C -» (A —» Bj) —> ((Z) —» A) —» (C —> (D —> B)j). We can convince ourselves of the latter entail

ment using a derivation that employs only instances of CP and of a rule I shall call 'Trans' which reflects the transitivity of —» : Ass

Ass Ass

2, 3 Trans

3, 4 CP 1, c; Trans

2, 6 CP

[8] (C->lA->£))->((D->A)->(C->(D->B))) 1,7 CP In contrast, there is no obvious way to convince ourselves of (5a), (5b) or (6a)

using a derivation each of whose steps is as immediately compelling as instances of CP and Trans. In particular, any derivation using the standard MP rule involves an MP step that purports to register an entailment akin to

(5a) or (5b)itself

36 Weir, in 'Naive Truth and Sophisticated Logic', agrees that step [3] of our Curry derivation registers a spurious entailment of form (5), but proposes a different modification to

the MP rule (pp. 227, 236). In our simple example of an MP step, his version requires an additional premise to the effect that C is not 'indeterminate'. Moreover, (6b) fails in Weir's logics, where the failure of structural (generalized) transitivity (pp. 227-8) requires restrictions on the rule Trans for the conditional (pp. 242, 247). Cf. Field, Saving Truth from

Paradox, p. 283m © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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LIONEL SHAPIRO

34°

IV. 2. Avoiding the Field-Beall thesis

So far, I have argued that there is no good reason to hold that a containing a Curry sentence cannot also contain an entailment con This is because the contraction-free substructural logics that ma connective possible should not be difficult to get our heads around construe natural deduction rules, such as MP, as serving to genera

entailments, such as (6b). But I still need to make explicit how contract

logics yield a response to the threat to the C-rules posed by the F

thesis.

A feature of contraction-free logics is the failure of the entailmen Restall (On Logics Without Contraction, pp. ix, 42) has dubbed 'Pseudo Ponens':

PMP. ((A ^ B) a A) ^ B. It should be noticed that (PMP) follows by C-Elim from

PMP*. B is a consequence of (A —> B) a A.

This means that the deflationist about consequence who uses a contraction free logic preserves a version of the natural deduction rule MP at the price of rejecting (PMP*). By the same token, though, giving up (PMP*) is a price our deflationist should be willing to pay. This is because whatever dia gnosis the contraction-free logician offers for the seeming obviousness of (PMP) is automatically available to the deflationist as a diagnosis for the seem ing obviousness of (PMP*). For example, Priest has urged that the plaus ibility of (PMP) derives from an illicit slide between the following two claims, each stated using devices of semantic ascent:37 7. That A —> B is true entails that B is a consequence of A

8. That ^4 —> is true and A is true entails that B is true.

Anderson and Belnap (p. 7) defend the MP rule by invoking an alleged pragmatic upshot of (7): 'we expect also that a rule of modus ponens ... will obtain for —» , in the sense that whenever A —> B is asserted, we shall be entitled to infer B from A'. We can agree with them without joining Field (Saving Truth from Paradox, p. 284) in insisting on a corresponding upshot of

(8), namely, that 'one can legitimately assert B on the assumption of A and

A -» B\

37 Priest, 'Sense, Entailment and Modus Ponens', Journal of Philosophical Logic, 9 (1980), pp. 415-35, at pp. 432-3. Though Priest rejects (PMP), he is committed to affirming (PMP*) (cf. In Contradiction, pp. 82-3), and thus to rejecting the rule C-Elim. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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DEFLATING LOGICAL CONSEQUENCE

If we reject (PMP*), we can avoid the Field-Beall thesis and the threat which it poses to deflationism about consequence. This is because Field's and Beall's argument for the thesis depends crucially on the following assumption:

Field-Beall assumption: for any connective >-» that satisfies condition (T-PRES), B must be a consequence oi(A >^> B) a A. As above, if T is transparent, an entailment connective —> must satisfy (T-PRES). In that case the Field-Beall assumption would yield that P is a consequence of (TrIP —» P) a 7r/T\ Using (T-PRES) and the transparency of truth, we could then infer ((TrIO —> P) a T^IP) —» P, which can be shown

to yield our arbitrary P in a few simple steps.38 Field and Beall would therefore be right that in a language with a transparent truth predicate and

a sentence satisfying (CURRY), there can be no connective satisfying (T-PRES), and hence no consequence predicate governed by the C-rules. But this problem disappears once we allow that our entailment connective —» violates the Field-Beall assumption.

Rejecting the Field-Beall assumption should not preclude recognizing some conditional => which satisfies the condition that B is a consequence of (A => B) a A. Perhaps Field is right that 'The absence of a conditional with modus ponens [in this robust sense] is enough to prevent any semblance of ordinary reasoning' (Saving Truth from Paradox, p. 369). The respective con

ditionals introduced by Field and Beall themselves arguably do satisfy the above condition. Unlike an entailment connective, these conditionals do not satisfy (T-PRES).

I conclude that Curry's paradox need not rule out deflating logical consequence. What it does show is that extending MED deflationism from truth to consequence requires facing up to an additional logical challenge. Just as Field rejects excluded middle and Beall accepts contradictions in order to maintain the T-rules, a deflationist about consequence may be led to reject (PMP*) in order to maintain the C-rules. Such a logical revision offers two rewards. First, it allows us to recognize the expressive role which

the consequence predicate, like the truth predicate, plays in facilitating 38 Field, Saving Truth from Paradox, pp. 377-8; Beall, Spandrels of Truth, p. 35. Field sees a

second problem with a connective —> satisfying (T-PRES). An 'acceptable' theory of arith metic in a language with such a connective and a transparent truth predicate, he argues (pp. 286-90), would prove its own soundness, contrary to Godel's second incompleteness theorem. But Field recognizes that to serve as inductive step in the soundness proof, the claim that MP for —> preserves truth must take the following form (pp. 377-80, slightly simplified):

V*V)>V£[(}> is a conditional sentence with antecedent x and consequent z) a (x and y are true) —> z is true].

But if we reject PMP for our connective satisfying (T-PRES), we reject this claim as well. © 2010 The Author The Philosophical Quarterly © 2010 The Editors of The Philosophical Quarterly

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341

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generalizations. Secondly, it promises a more thoroughgoing imple

tion of the deflationary attitude, the attitude which seeks to reveal no seeming metaphysical weight as mere expressive devices.39 University of Sydney & University of Connecticut

39 Versions of the paper were presented at Logica 2009 (Czechia), the 2009 meetin Australasian Association of Philosophy, and the University of Sydney. For comm

suggestions, I wish to thank Arif Ahmed, Jc Beall, Nuel Belnap, Mark Colyvan, Ole H Jenann Ismael, Mark Jago, James Justus, Michael Lynch, Julien Murzi, Daniel Nolan,

Peregrin, Huw Price, Stephen Read, Kevin Scharp, Stewart Shapiro, Nicholas J.J. Shawn Standefer, Shunsuke Yatabe and two referees.

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