anti-realist truth and concepts of superassertibility

JIM EDWARDS

ANTI-REALIST TRUTH AND CONCEPTS OF SUPERASSERTIBILITY

ABSTRACT. Crispin Wright offers superassertibility as an anti-realist explication of truth. A statement is superassertible, roughly, if there is a state of information available which warrants it and it is warranted by all achievable enlargements of that state of information. However, it is argued, Wright fails to take account of the fact that many of our test procedures are not 'sure fire', even when applied under ideal conditions. An alternative conception of superassertibility is constructed to take this feature into account. However, it is then argued that when this revised concept of superassertibility is taken as the truth predicate of probability statements, statements whose test procedures are paradigmatically not sure fire, then any anti-realist theory of the sense of such probability statements cannot be compositional, in Dummett's sense of 'compositional'.

1. INTRODUCTION

Michael Dummett has made famous the idea that an anti-realist about some disputed subject matter holds that truth is evidentially constrained, in contrast to a realist who holds that propositions about the subject matter in question may be true even though no evidence is available of their truth, or false even though all available evidence points to their truth. Thus a Dummettian anti-realist endorses, for any assertoric discourse, all instances of the schema:

It is true that P --+ P is knowable.

Such a characterisation of truth is, however, negative - notwithstanding that the schema is seen to arise from constraints on the theory of meaning, constraints themselves grounded by the premise that meaning must be manifest in use. The schema tells us, in effect, what it is that the realist asserts about truth - viz. that truth can exceed what is knowable - which the anti-realist denies. But the schema and the familiar argument for it do not tell us what truth for an anti-realist positively is. This, has become apparent, I think, now that Crispin Wright has taken us further by giving us a positive account of anti-realist truth. 1 He has argued that given any discourse for which the above schema holds a priori, we can identify, subject to a certain condition to be discussed below, the property of being true,

Synthese 109: 103-120, 1996. @ 1996 Kluwer Academic Publishers. Printed in the Netherlands.

104 JIM EDWARDS

for any statement of that discourse, with the property of that statement of being superassertible. Wright defines his notion of superassertibility on the back of the more familiar notion of a warrant to assert a statement, and we shall come to his definition shortly. Given Wright's identification of truth with superassertibility, we can see why all true statements are knowable - - because truth is superassertibility and their being superassertible implies their being knowable. 2

As Wright notes, 3 his definition of superassertibility introduces a range of conceptions of superassertibility, not a single concept. However, the particular conception of superassertibility which he favours is not, I think, attractive to an anti-realist. It does not properly allow for the defeasibility of statements because it ignores the fact that certain of our best test procedures even when applied in ideal conditions are not sure fire', they may give misleading results. I shall revise his definition of superassertibility to allow for this feature.

The interest of Wright's argument for identifying truth with some ver- sion of superassertibility is not confined to the issue of anti-realism. The argument for the identification goes through for any region of discourse of which the schema

P --+ P is knowable

holds a priori. The anti-realist, or course, thinks this is every region of assertoric discourse. But those who think that truth may be verification transcendent in some area, say mathematical discourse, may also think that in other areas truth is evidentially constrained. For example, they may think that aesthetic or moral facts cannot be unknowable. If so Wright's argument will go through locally, so to speak, and the truth of aesthetic or of moral judgements will be, subject to a certain condition to be introduced below, their superassertibility.

2. WRIGHT'S CONCEPTION OF SUPERASSERTIBILITY

Wright's fullest definition of superassertibility runs as follows (Wright 1986 pp. 298-9):

'P' is superassertible just in case the world will, in sufficiently favourable circumstances, permit the generation in an investigating subject, S, of a set of beliefs {B1,..., B~ } with the following characteristics:

(a) S has adequate grounds for regarding each of {B1,..., B~} as an item of S's knowledge.

ANTI-REALIST TRUTH AND CONCEPTS OF SUPERASSERTIBILITY 105

(b) The status of each of{B1,. . . , B~ } as an item of S"s knowledge will survive arbitrary close and extensive inyestigation.

(c) The state of information constituted by {B1,.. . , B~ } warrants the assertion of 'P ' . (d) The case provided by {BI , . . . ,B~} for 'P' is not, in fact, defeasible; i.e. no

{B1,..., B,~,..., B~ } containing {B1,..., Bn } and satisfying (a) and (b) for some S, yet failing to warrant 'P', can be achieved in this world, no matter how favourable the circumstances for the attempt.

Clearly there are key phrases in this definition which call for fleshing out in a fuller account. And different fleshings-out will give rise to different specific conceptions of superassertibility. To get Wright's own conception of superassertibility we shall have to say more about what constitutes 'favourable circumstances' both for the search for an enlarged state of information {(B1, . . . , Bn,.. . , Bz} which fails to warrant P referred to in (d), and for the investigation of {B1, . . . , Bn} referred to in (b). And we shall need to ask what is the range of enlarged states of information {B1,..., Br~,..., Bz } which can be achieved in such favourable circumstances?

The following is a platitude connecting warrants to truth:

(w) To have a warrant for P is to have a warrant to claim that P is true.

So if superassertibility is to be identified as truth, then the following must be a priori:

(W s) To have a warrant for P is to have a warrant to claim that P is superassertible.

(W s) provides the constraint which Wright uses to generate his particular interpretation of 'favourable circumstances' for further investigation concerning ' P ' and also the range of enlarged states of information which can be achieved in those favourable circumstances referred to in clauses (b) and (d) of the above definition. We want the range to be such that having a warrant to assert P is having a warrant to assert that clauses (b) and (d) are satisfied.

If we allow the range to be all possible outcomes which are logically consistentwith having had a warrant for P , then clauses (b) and (d) will fail. The warrant for P will be, in interesting cases, defeasible. So having had a warrant for P is logically consistent with outcomes which overturn the warrant, either by undermining it (clause (b) fails), or by building a counter- case (clause (d) fails). Hence, if the range were all logically possible investigations whose outcomes are consistent with having had a warrant for P , (W s) would fail. On the other hand, at the other extreme, confining

106 JIM EDWARDS

the range of investigations and their outcomes to those which actually occur would not help. For having a warrant for P does not imply that any further relevant investigations will be undertaken, and so superassertibility might be too easily achieved. We need some middle course between (i) all logically possible investigations whose outcomes are consistent with having had a warrant for P and (ii) whatever further investigations actually take place.

Wright solution is that to have a warrant for P is to have a warrant to assert that further investigation will satisfy (b) and (d) provided theproce- dures employed to investigate whether P are optimal and the conditions in which those procedures are applied are favourable. What does Wright understand by 'optimal procedures applied in favourable circumstances'? Wright, in effect, conjoins two ideas, which I shall argue in the next section, should not be conjoined.

Firstly, he so understands 'optimal procedures in favourable circumstances' that, supposing P to be actually true, if an investigation yields results which fail to warrant P, or results which overturn a previous warrant for P , then it follows that that the procedures were not 'optimal' or the circumstances not 'favourable' for the investigation of P , or both - since the investigation delivered the wrong result. The enlarged state of information resulting from the investigation is, ex hypothesi, misleading as to P , 50 it is not a state of knowledge concerning P. Wright comments on any state of information which is misleading as to whether P that

it ceases to be germane to issues concerning superassertibility. (Superassertibility, to stress, requires stable assertibility under accretions to knowledge.) (Wright 1986, p. 301, footnote ~3).

Knowledge, of course, connects directly with truth. Thus Wright so understands 'optimal procedures applied in favourable conditions' that if the outcome of an investigation is misleading as to whether P , i.e. not an accretion to knowledge of P, then that is sufficient to show that the procedures were not optimal, or the conditions not favourable, or both.

A second ingredient in Wright's conception of 'optimal procedures applied in favourable circumstances' is needed if superassertibility is to provide a notion of truth acceptable to an anti-realist. So far the notion of superassertibility presupposes a prior notion of truth, since we have made it a necessary condition of procedures being optimal and circumstances being favourable that they deliver the truth about P. An anti-realist treats the notion of warranted assertion as semantically basic, or at least coeval with truth. The assertion of P is warranted, paradigmatically, as a result of the application of optimal test procedures in favourable circumstances - even if in a particular case this amounts to no more than having a looksee.


So, if truth as superassertibility is to emerge as subordinate to, or at most coeval with, a semantically basic notion of warranted assertibility, then Wright needs the optimal procedures and favourable conditions associated with a given P to be operational, i.e. for their realisation to be recognisable independently of knowing whether P is in fact the case or not. Of course, a judgement that a procedure is optimal and the conditions favourable is still defeasible. But the overturn conditions are themselves operationally recognisable. The judgement may be overturned by such further discoveries as that the wires of the test apparatus were not connected as per the manufacturer's instructions, or the lighting conditions were not standard, and the like. 4

How can 'optimal procedures applied in favourable conditions' satisfy both of these requirements? How can their fulfilment be operationally recognisable when there is also the non-effective overturn condition, that the investigations yield the true result vis avis P, which seemingly makes 'optimal procedures and favourable conditions' non-operational? To com- bine both ideas Wright needs to assume:

If operationally recognisable optimal procedures are conducted in operationally recognisable favourable circumstances, then their outcomes will reflect what is the case concerning whether or not P.

With this assumption, Wright's conception of superassertibility can conjoin the two ideas: the idea that associated with P there is a set of operationally recognisable best test procedures and circumstances for their application, and the idea that these procedures applied in these conditions do not yield misleading results visa vis P. Thus Wright's conception of superassertibility, in effect, presupposes that our operationally defined optimal test procedures applied in operationally defined favourable conditions are 'sure fire', they never yield misleading results. In the next section I will develop a alternative notion of superassertibility, one which does not make this presupposition.

A further feature of Wright's conception of superassertibility will carry over to the alternative conception of superassertibility to be developed in the next section. Wright recognises that his notion ofsuperassertibility involves an idealisation. Wright understands "all enlarged states of information which may be generated in favourable circumstances"

as embracing everything which we should possess if, stationed in time as we are, we were nevertheless per impossible to investigate everything which, at any given time, there is opportunity to investigate and if the information thereby accumulated were never to be lost. (Wright 1992, p. 68)

In practice the execution of one investigative procedure may remove the possibility of applying some other procedure - e.g. dissolving the sample

108 JIM EDWARDS

in nitric acid as required by one procedure makes it impossible to carry out another procedure which required it be dissolved in sulphuric acid. In practice the conditions for carrying out each of the relevant investigative procedures may be so far from being optimal as to be absent. And in practice information is lost, we forget.

As Wright also recognises, this idealisation in his conception of superassertibility will carry over to the notion of knowledge employed in the anti-realist's claim

P --+ P is knowable

If the truth of P is its superassertibility, as understood by Wright, then we have

P -~ there is a warrant for P

but, supposing P , nothing follows about the availability of this warrant to investigation in practice, nor about whether the warrant would be sustained when the issue is subject to further investigation in practice. It only follows that if the investigations wereper impossible to leave no stone unturned and no information were ever lost, then a warrant for P would be discovered and would be sustained. Thus, given that the availability of a warrant has been idealised, the resulting sense of"knowable" in the anti-realist's claim

P --+ P is knowable

is correspondingly idealised. Wright is, of course, not claiming that if S has a warrant to assert P ,

then, it follows that, under the per impossible conditions described, the warrant would not be overturned (i.e. that clauses (b) and (d) would be satisfied). For S may have a warrant for P , and yet P be false. Wright is claiming that if S has a warrant to assert P , then S has a warrant to assert that, under theper impossible conditions described, the warrant would not be overturned (that clauses (b) and (d) would be satisfied).

3. A RIVAL CONCEPTION OF SUPERASSERTIBILITY

I will introduce a second conception of superassertibility, which I shall dub superassertibility* to distinguish it from Wright's conception of superassertibility, a conception which will require a modification of Wright's general defini t ion- in particular of clauses (b) and (d). Superassertibility* is intended to recognise that operationally recognisable optimal procedures applied


in operationally recognisable favourable conditions are not in general 'sure fire'. Having abandoned Wright's presupposition, the best we can hope for is

If operationally recognisable optimal procedures are conducted in operationally recognisable favourable circumstances, then outcomes which reflect what is the case concerning whether or not P are more likely than outcomes which do not reflect what is the case concerning whether or not p.5

To illustrate, suppose we are investigating whether a certain coin is unbiased, then an optimal investigative procedure is repeatedly tossing the coin, and the circumstances which are favourable to such a test are easily recognised to be met. Suppose the coin is indeed unbiased. Then possible outcomes of rtms of tosses in which the number of heads is closer to the number of tails are more likely than possible outcomes in which the number of heads is further from the number of tails. (One cannot say, more simply, that outcomes with equal numbers of heads and tails are most likely, because a run of tosses may consist of an odd number of trials wherein such an outcome is not possible.) Clauses (b) and (d) of Wright's conception of superassertibility, with optimal procedures andfavourable circumstances as now understood, would fail, but for the wrong reasons. The obvious point is, supposing the coin is indeed unbiased and is inves- tigated by operationally recognisable optimal procedures in favourable circumstances, then enlarged states of information are naturally possible in which the distribution of heads and tails is misleading, i.e. the distribution does not correspond to the facts concerning the bias of the coin. Hence clause (b) or clause (d), whichever is in question, is not satisfied, and the original statement is not superassertible. Of course, we do not want the original statement to fail to be superassertible* on these grounds, not if superassertibility* is to be identified with truth.

If the coin is indeed unbiased, then there will be further enlargements, the results of more extensive series of tosses in favourable circumstances, which will correct the misleading impression of the former investigations. But this point is irrelevant to the identification of truth and superassertibility, as the damage has already been done. Clause (b) required that the pedigree stand up to arbitrary investigation, not that it stand up to maximal investigation, whatever that might be, but could lapse in intermediate states of investigation. And clause (d) required that no extension produced by operationally defined optimal procedures in favourable circumstances fail to warrant P , not that maximal extensions so failed.

To define superassertibility* I propose to adapt clauses (b) and (d) to the new conception of operationally recognisable investigative procedures and favourable circumstances, a conception freed from the assumption that such

110 JIM EDWARDS

procedures are 'sure fire'. Let us abbreviate 'operationally recognisable optimal procedures for investigating P in operationally recognisable ideal circumstances' to ' O ( P ) ' - h e r e ' P ' is a schematic letter taking the place of the proposition under test. Thus clause

Co) The status of each of {B~, . . . , Bn } as an item of S's knowledge will survive arbitrary close and extensive investigation.

is replaced by

(b,) Outcomes of any O(B1 , . . . , Bn) which warrant {B1, . . . , Bn} are more likely than outcomes which do not.

and clause

(d) The case provided by Bl , . . . , Bn for ' P ' is not, in fact, defeasible; i.e. no {B1, . . . , Bn, . . . , Bz} containing {B1, . . . , B~} and satisfying (a) and (b) for some S, yet failing to warrant 'P' , can be achieved in this world, no matter how favourable the circumstances for the attempt.

'is replaced by 6

(d*) Possible enlargements of { B 1 , . . . , B n } generated by O(P) which continue to warrant P are more likely than possible enlargements which do not warrant P.

A range of outcomes or possible enlargements is quantified over in (b*) and in (d*). This range is confined to investigations resulting in states of information which are actually possible for creatures given our cognitive abilities. But the range of associated favourable circumstances may be counterfactual. If I have a warrant to assert that this coin is unb iased- say I've tossed it fifty time - and I then dissolve it in nitric acid, no favourable opportunities for further relevant investigations are possible, and clause (d*) is in practice inapplicable. Our anti-realist will want there to be a relevant range of favourable circumstances in such a case - those which would have obtained had the coin still been available to be tossed. Further, the relevant range of favourable circumstances will be idealised in the same way as was Wright's above. Long and ever longer runs of tosses will in practice wear away or distort the coin, so that the original bias would no longer be under test. Optimal tests in favourable circumstances postulate no such changes, and are therefore idealised.

Before an anti-realist can accept superassertibility* as a conception of truth, he will need to be satisfied that he can legitimately draw the key

ANTI-REALIST TRUTH AND CONCEPTS OF SUPERASSERTIBILITY 1 1 1

distinction by which superassertibility* is distinguished from superassertibility. Superassertibility* employs the idea that the most likely outcome of O(P) is the one which is veridical with respect to P. Superassertibility* therefore recognises the possibility of outcomes of O(P) which are misleading as to/9. Thus superassertibility*, unlike superassertibility, presupposes that, having a present warrant for/9, we can distinguish between:

(i) the possible results of further projected O(P) which fail to warrant P and are misleading as to P ,

and

(ii) the possible results of further projected O(P) which fail to warrant P and which show our present warrant for P to have been misleading.

A realist can make this distinction because he holds that truth is only contingently connected to evidence. But can an anti-realist draw this distinction between possible future states of information? The difficulty is to make logical room for (i). For if the outcome of further investigations do not warrant P , then, it is clear, our warrant for P will then have been overturned. So at the future stage when the sum total of investigations to date do not warrant P , (ii) takes precedence over (i) and the warrant for P is overturned. But the anti-realist knew, at the earlier time when he had the original warrant for P that this would be the case if the then future investigation went against P. So what content can the anti-realist now give to the possibility of the outcome of future O(P) being misleading as to P? The anti-realist seems to have no logical room for possibility referred to in case (i).

However, logical room is made for the possibility referred to in case (i) when we recall that the enlarged state of information is itself defeasible. Yet further investigation may restore the warrant for P , and thereby show the intermediate state to have been misleading as to whether P. At no stage can we have warrant to assert that our total present state of information regarding P is misleading. But at each stage we recognise that our present best case may be overturned in the future, and therefore that it may be misleading. And we knew all this about future states of information when we had our original warrant for P. So an anti-realist could recognise then a distinction between (i) a possible future state which misleadingly fails to warrant P (but would rightly but falsely be taken by those in it to overturn P), and (ii) a future state which will show our present warrant for P to have been mistaken. He can think of future total states of information regarding P as dividing into cases (i) and (ii), and he can apply

112 JIM EDWARDS

the distinction retrospectively to past total states of information regarding P , but the language game requires him to think of all present such states of information as being veridical.

4. CONCEPTIONS OF SUPERASSERTIBILITY AND TRUTH

I said above that Wright proposes to identify anti-realist truth with superassertibility subject to a certain condition. The condition in question is that there is no other property of the statements of the discourse, a property differing in extension from their superassertibility, which has a better claim to be truth. If no other candidate can demonstrate to the anti-realist a better claim to the title, then truth is superassertibility by default, as it were. I wish to make the same claim, but for superassertibility*.

My strategy in this section will be to follow through the case which Wright makes (Wright 1992, Chapter Two) for superassertibility being the default truth predicate for a discourse, and to show that a parallel case holds equally for superassertibility*. Hence if the argument above recommends superassertibility* to the anti-realist, then superassertibility* takes the place of superassertibility as the default truth predicate for a discourse. To this end I shall replace, so far as possible, 'superassertibility' in my exposition of Wright's argument by the schematic letter 'S' , since, I claim, 'S' may be replaced in these sections of the argument indifferently by ' superassertibility' or by superassertibility*, without change of cogency.

Wright gives a sufficient and necessary condition for S being the truth predicate of the discourse in ques t ion-viz , that the following schema, the Equivalence Schema, be a priori:

(E s) P ++it is that P .

But to test (E s) we look for cases in which the left hand side is true and the right hand side is not true, or vice versa. Hence, if the candidate up for the honorific title of being the truth predicate for the discourse is S, then we are asking, in effect, whether the following is a priori:

(E ss) It is S that P ++ it is S that it is S that P.

Wright's necessary condition for S being a truth predicate is that (E ss) be a pr ior i .

(E ss) being a priori is not a sufficient condition for S being a truth predicate for the discourse m question. The point is that there may be, as Wright puts it, a rival candidate in the field with a stronger claim. Suppose there is such a rival truth predicate, call it "T", which differs in extension


from S. Then supposing (E ss) holds a priori, (E s) may still fail. (E s) may fail because its left and right hand sides differ in T-value - since ex hypothesi we have supposed T and S to differ in extension. Hence S is not a truth predicate because it fails to satisfy (Es). Thus passing the test of (E ss) is only a necessary condition of (E s) holding a priori, and hence only a necessary condition of S being a truth predicate.

We can see an example of a concept whose claim to be a truth predicate is over-ridden in this way. Instances of the following schema hold apriori:

(E ww) It is warranted here and now that P ++ it is warranted here and now that it is warranted here and now that P .

where we suppose the context C to fix the values of "here and now". But in this case we do not have

(E w) P ++ it is warranted here and now that P

because S itself provides a rival over-ridding candidate for being the truth predicate of the discourse. Thus we may we warranted in certain circumstances C in asserting P . And it may be S that in the circumstances C we were warranted in asserting P - all enlarged states of information concur in the judgement that P was warranted in C. But P is not S - in some enlarged states of information the warrant for P itself is overturned. Hence (E w) fails, its left and right hand sides differ in truth value, when truth is taken to be S. Hence being warranted is not a truth predicate, despite (E ww) being a priori. Thus S is a rival candidate which over-rules the claim of mere waranted assertibility to be a truth predicate.

The argument of the last paragraph is without prejudice to the claim of S itself for the role of truth predicate. It may be that its own claim is over- ridden by a stronger candidate. Hence (E sS) is only a necessary condition of S being a truth predicate.

Thus Wright takes (ES), if true apriori, to be a necessary and sufficient condition for S being a truth predicate. And a necessary condition of (E s) being apriori is (E ss) being a priori. However, Wright has an argument to show that a sufficient condition for (E ss) is:

(W s) To have a warrant for P is to have a warrant for the claim that P is S, and conversely.

Wright mediates the step from (W s) to (E ss) by the following argument (Wright 1992, p. 54 footnote 17). In what follows let A = ' P ' , and B = ' P is superassertible':

114 JIM EDWARDS

suppose that to have a warrant for A is to have a warrant for B and vice versa, but, for reductio, that A is superassertible, while B is not. Let I be a total state of information in virtue of which A is superassertible; i.e., I warrants A and so does any improvement I*, of I. By hypothesis, I also warrants B. Since B is not superassertible, there must therefore be some improvement, I*, of I , which fails to warrant B. Since any such I* warrants A, the supposition is contradicted. This shows that (i) coincidence in assertibility conditions suffices for (ii) a pair of statements both being superassertible if either is.

T h e a r g u m e n t f r o m ( W s) to (E ss) goes th rough for superasser t ib i l i ty*,

mod i f i ed to a l low for the d i f fe rence b e t w e e n superasser t ibi l i ty* and super-

assert ibil i ty. Aga in , let A = ' P ' , and B = ' P is superasser t ib le* ' :

Suppose (i) that to have a warrant for A is to have a warrant for B and vice versa- i.e., any outcome of any procedure which warrants A also warrants B, and vice versa. It follows that any recognisable optimal test procedure applied in recognisably favourable circumstances aimed at discovering whether A is also a reeognisable optimal test procedure applied in recognisably favourable circumstances aimed at discovering whether/3, and vice versa. In short, O(A) = O(B). Suppose (ii), for reductio, that A is superassertible*, while B is not. Let I be a total state of information in virtue of which A is superassertible*; i.e., I warrants A and all enlarged states of information 1" which are generated by O(A) and which are most likely, also warrant A. By the corollary of supposition (i), I also warrants/3, and I* warrants/3 and is the most likely outcome of an 0(/3). Hence/3 is superassertible*, contrary to supposition (ii). Hence, if a warrant for A is a warrant for/3, and vice versa, then A is superassertible * iff B is superassertible*.

T h e pos i t ion w e have r eached is that ( W s) impl ies (ESS), and (E sS) is a

n e c e s s a r y cond i t ion o f (ES), w h i c h is a n e c e s s a r y and sufficient condi t ion

for S to be a truth pred ica te . So the final pos i t ion is that it is a n e c e s s a r y cond i t ion o f S be ing a truth p red ica te that ( W s) ho ld apriori. A g a i n I shall

a rgue that the case for (W S) is as g o o d wi th S = superasser t ibi l i ty* as it is

wi th S = superasser t ibi l i ty . T h e case for ( W S) wi th S = superasser t ib i l i ty can be m a d e quite briefly. 7

(W s) To have a war ran t for P is to have a war ran t for the c l a im that P is superasser t ib le , and converse ly .

R igh t to left is trivial. A n y war ran t to c l a im that P is superasser t ib le w o u l d

o b v i o u s l y be a war ran t to asser t P . Le f t to r ight is es tab l i shed b y the fo l lowing:

It is a platitude that any warrant for P is a warrant to assert that P is true. It is therefore a warrant to assert that the outcomes of any investigations which reflect the truth about P will warrant P. Therefore a warrant for P is a warrant to assert that any optimal procedures applied in favourable circumstances whose outcomes reflect the truth about P will warrant P. Hence it is a warrant to assert that P is superassertible.

It ma t t e r s not to the c o g e n c y o f this a rgumen t that w e have used the no t ion o f truth b y appea l ing to a p la t i tude l inking war ran t and truth. For the issue is w h e t h e r superasser t ib i l i ty (or superasser t ib i l i ty*) is a defaul t t ruth


predicate for an anti-realist, and we have presupposed only that there is a notion o f truth in play in the discourse in question, not that it is identical to superassertibility (or to superassertibility*).

The case for (W s) with S = superassertibility* is a little more complex. Right to left is again trivial. And left to right is established by the following

argument:

Again, a warrant to assert P is, platitudinously, a warrant to assert that P is true. It is therefore a warrant to assert that the outcome of any investigations which would overturn the warrant for P, even outcomes generated by O(P), would be misleading as to whether P. (In more detail: from the standpoint of present evidence, such hypothetical outcomes are rightly judged to be misleading outcomes as to whether P, notwithstanding that in such an enlarged state of information it would then be correct to deny P, the former warrant having been overturned.) However, we agreed in the previous section, that the only way an anti-realist can make sense of the idea that future outcomes of O(P) might be misleading as to P was in terms of the idea that further O(P) would restore the warrant for P by showing the intermediate stage to have been misleading. Thus, judging from the standpoint of the original warrant for P, and provided that the holder of the warrant is an anti-realist, he has a warrant to assert that the if the hypothetical outcomes of further O(P) go against P, then, idealising as in the last section by supposing that yet further such investigation is always possible, further diligent investigation will always restore the warrant for P. On the standard account of likelihood, this is equivalent to saying that any further O(P) will most likely warrant P. Thus having a warrant to assert P, our anti-realist has a warrant to assert that p is superassertible*.

The above argument has made essential use o f the idea that the holder o f

the warrant for P is an anti-realist. A realist holding a warrant for P may suppose, consistently with that warrant to assert P , that the outcomes o f all further investigations, even operationally recognisable optimal procedures applied in operationally recognisable favourable conditions, run forever against P . O f course, the realist will agree that someone in receipt o f such

outcomes would have no warrant to assert P . Thus the case which Wright makes for an anti-realist to take superassert-

ibility as a default conception o f truth is paralleled by an equally cogent case for superassertibility*. When we add the results o f the previous sections - viz. superassertibility implausibly assumes that optimal procedures applied in favourable circumstances are sure fire - then I claim for superassertibility* a superior title to being the default truth predicate. Superassertibility* trumps superassertibility for an anti-realist.

5. SUPERASSERTIBILITY* AND THE PROSPECTS FOR A COMPOSITIONAL THEORY OF THE SENSES OF PROBABILITY STATEMENTS

The concept o f superassertibility* was developed to offer an anti-realist a truth predicate which recognised that optimal procedures applied in

116 JIM EDWARDS

favourable conditions need not be sure fire. Probability statements are a paradigm of such statements. However, in this section I shall show that probability statements, when their truth is taken as superassertibility*, fail to meet a condition which Dummett has famously laid upon any theory of sense: viz. compositionality. I f the truth o f probability statements is their superassertibility*, then the theory of sense of such statements is to an extent holistic.

Dummett writes, o f any theory of m e a n i n g - verificationist, pragmatist, or realist:

The principle of compositionality is most easily illustrated by the logical constants. On a compositional meaning-theory, to know the meaning of 'or', for example, is to be able to derive, from the meanings of any sentences A and B, the meaning of 'A or B', where the meaning of a sentence consists in what counts as verifying it, or in the consequences of accepting it as true, or in the condition for it to be true. (Dummett 1991, p. 222.)

In addition, compositionality requires that the meanings (= senses) of the sentences A and B are given by the theory prior independently of the sense o f 'A or B'. Given that we are taking truth to be superassertibility*, the semantically basic notion of the theory of sense will the conditions under which sentences are verified, i.e. what we have been calling their conditions of warranted assertion. Transposing to the case we shall be concerned with, the existential quantifier, a necessary condition for a area o f discourse being amenable to a verificationist and compositional theory o f sense is the following:

A complex sentence of the form '(3x)Ax' is warranted just in case a simpler sentence of the form 'Ax' with some object assigned as the value of 'x' is warranted. 8

I f this and certain other conditions are satisfied, a compositional account can be given of the sense of complex sentences whose main constant is the existential quantifier. The sense of the more complex sentence ' (3x)Ax ' , i.e., a condition for its warranted assertion, is derived from the sense o f the simpler sentence 'Ax ' , i.e. from a condition for its warranted assertion, because the former is derived from the latter by existential introduction. We shall find that statements of probability do not meet the above condition.

Consider the following probability statement:

(S) This coin has a bias to heads of 0.5.

It is standard to take (S) as equivalent to:

For any small number e, there is a number n such that for all sequences of tosses of the coin of length m > n, the difference between F(H/m) and 0.5 is less than

ANTI-REALIST TRUTH AND CONCEPTS OF SUPERASSERTIBILITY l 17

where F(H/m) is the ratio o f heads in the sequence o f m tosses, and where we suppose the sequence o f tosses is idealised in that there is no alteration in the state o f the coin, no wear or distortion. Consider also the denumerably infinite sequence o f probabili ty statements:

(S I) This coin has a bias to heads o f 0.501

(S 2) This coin has a bias to heads o f 0.5001



etc.

It is obvious that any evidence, i.e. anyfinite sequence o f toss, we might otherwise take to warrant (S) will equally warrant (sn), (S ~+1), (S~+2),

etc. for some value o f n.

Taking truth to be superassertibility*, it is easy to see that (S1), ($2), ($3), etc. are contraries o f (S):

Assume (S) is superassertible*. Then the most likely outcome of all expansions m of our sequence of tosses carried out in favourable circumstances is one in which F(H/m) is closest to 0.5. But, given the standard statistical understanding of what it means to say that this outcome is most likely, it follows that for each small number e there is a large number k such that for all sequences of tosses of length greater than k, F(H/k) differs from 0.5 by less than e. For each (Si), we can choose a value of k such that sequences of length greater than an appropriate value of k will overturn the claim that the most likely results of sequences of tosses of length greater than k are those predicted by (Si). Thus (S i) is not superassertible* if(S) is superassertible*. And this holds for each (Si).

The following principles are basic to epistemology:

If a state of information warrants the assertion of P , and Q is a contrary of P , then that state of information warrants the denial of Q.

And

No state ofinfornaation can warrant both the assertion and the denial of any statement.

We are forced to the conclusion that no state o f information can warrant the assertion o f (S). For we have agreed that any state o f information we might otherwise take to warrant (S) would also warrant (Sn), (sn+l), (sn+2), etc. for some value o f n, and we have agreed that these are contraries o f (S), whence that state o f information would, per impossible, warrant both the assertion and the denial o f (S).

118 JIM EDWARDS

What may be warranted is not (S) itself but

(S') This coin has a bias to heads of 0.5 4- 5.

where 5 defines some proportionately small vague interval around 0.5. And a further extension to the idealised sequence of tosses may continue to warrant (S') by warranting

(S") This coin has a bias to heads of (b 4, A.

where ¢~ 4, A is another vague interval significantly smaller than and lying within 0.5 4, 5. The extended series of tosses continues to warrant (S') since (S") implies (S'). But the only outcome which would warrant (S) itself would be the outcome of an actually completed infinite sequence of tosses, an ideal±sat±on too far for any anti-realist. 9

An anti-realist will acknowledge that he understands the statement (S), and others of the same form which assign precise values to the bias of the coin. This is one way in which an anti-realist distinguishes himself from a logical positivist. Such sentences have sense for the anti-realist, even though they are not decidable, because they have decidable logical relations to structurally and semantically related sentences. Just so the anti- realist understands, for example, Ooldbach's Conjecture without knowing whether it is provable or disprovable. 10

One such sentence inferentially related to (S) is the following existen- tially quantified statement

(s @hi((0.5 i 5.) _< n _< (0.5 - 5)) & this coin has a bias to heads of n]

Since (S) and '((0.5 4. 5.) _ 0.5 < (0.5 - 5))' imply (S e) by &- Introduction and Existential Introduction. A problem arises for the anti- realist because he has, we may suppose, a warrant to assert (S') and (S') is inferentially precisely equivalent to (se), so he has a warrant to assert (se). But compositionality requires of an anti-realist theory of sense that if (S e) is warranted, then for some value assigned to 'n ' , the following is warranted

((0.5 4- 5.) <_ n _< (0.5 - 5)) & this coin has a bias to heads of n

And the requirement of compositionality of sense, applied now to '&' will in turn require that, with that value assigned to 'n ' , the following is warranted:

This coin has a bias to heads of n


But w e have seen that no s t a t emen t o f this f o rm can be war ran ted . H e n c e ,

I c o n c l u d e , an an t i - r ea l i s t t h e o r y o f sense for p r o b a b i l i t y s t a t emen t s w h i c h

iden t i f i e s t ruth as superasse r t ib i l ty* canno t be c o m p o s i t i o n a l . 11

NOTES

* I would like to thank John Divers, Alex Miller, Nick Zangwill, and the two anonymous referees of this journal for comments upon earlier drafts of this paper. l Mainly in "Can a Davidsonian Meaning-Theory he Construed in Terms of Assertibility?", reprinted in Wright 1986, pp. 287-316, and in chapter two of Wright 1992. 2 Hiliary Putnam, particularly as a result of Putnam (1981) and Putnam (1983), has been credited with a notion of truth such that P is true just in case P would be justified under ideal epistemic circumstances. Wright effectively criticises this notion (1992, p, 46). Such a notion of ideal epistemic circumstances is dubiously legitimate on the following ground. l f a person were, per impossible, in such ideal epistemic circumstances visa vis P, then it would be true that they were in such ideal circumstances. So, given the notion of truth under consideration, there would have to be ideal circumstances under which this was apparent to them- i.e. apparent to them that their epistemic circumstances visa vis P are ideal. But this last, double idealisation seems dubiously coherent. Even supposing someone could in fact have all the evidence concerning P before them, it doesn't follow that it even makes sense to suppose that they could know that this was their position, know that there was nothing further to be turned up concerning P by diligent investigation. We could invent examples in which all the evidence is available to our investigator, but no evidence available that this is all the evidence. Wright's conception of superassertibility does not assume that an empirical enquiry can, even as an idealisation, be completed. 3 Wright (1992, p. 68). 4 Are the optimal operational procedures for determining whether P fixed by the sense of P and such that any competent language user who understands P would recognise them? Or might the optimal procedures and even the sense of P change holistically with the discovery of esoteric theory which bears on the question whether P? Such questions rightly exercise anti-realists. 5 I would have preferred to say "outcomes which correspond to the facts concerning P are more likely than outcomes which do not correspond to the facts concerning P". However, superassertibility* is intended to be congenial to anti-realists, and some anti-realists define their position by eschewing talk of correspondence to the facts. I am persuaded by Wright (1992, passim) that they are ill-advised to do so, and that they should distinguish themselves from realists in quite other ways. However, for the purposes of this paper it is sufficient to remark that nothing hangs on the choice of phrase, because Dummettian anti-realists have no quarrel with talk of statements corresponding to the facts. 6 In clause (d) the phrase "satisfying (a) and (b)" requires only that S have adequate grounds for {B l , . . . , B,~,. . . , B~ } and extensive investigation does not overturn S 's warrant for {BI , . . . , B ~ , . . . , B~ }. 7 In his 1992 chapter 2 the case Wright makes for (W s) is protracted. The brief argument which follows is not taken from Wright. 8 I only claim that this is a necessary condition for a compositional, verificationist theory of the senses of sentences whose main constant is an existential quantifier. Much else is needed for compositionality. The interested reader should consult Dummett 1991, especial- ly chapters 12 and 13.

120 JIM EDWARDS

9 No state of evidence could directly warrant (S), short of a completed infinity of tosses. This allows that there could be an indirect warrant for (S) arising from the laws of motion and the relevant symmetry of the coin. But I conjecture that any indirect evidence for (S) would raise essentially the same problems. To warrant (S) as against all (S i) the indirect evidence would have to be to the effect that the coin is perfectly symmetrical. The problem of warranting the statement that the coin is perfectly symmetrical on the basis of measuring procedures which have a finite margin of error is exactly analogous to the problem of verifying (S) directly on the basis of tossing the coin. 10 The point is clearly made by Neil Tennant in his 1987 chapter 11. 11 Should we conclude more strongly that any theory of the sense of probability statements which takes warranted assertion as its basic semantic concept is not compositional, whether of not truth is identified as superassertibility* ? Superassertibility* was used in the argument to show that (S ~), ($2), ($3), etc.are contraries of (S). Without an account linking truth to warranted assertibility, which superassertibility* was intended to provide, the judgement of contrariety is ungrounded. The bald intuition that (S i), ($2), ($3), etc. are contraries of (S) might turn out to rest upon a realist notion of truth.

REFERENCES

Dummett, M. A. E.: 1991, The Logical Basis of Metaphysics, Duckworth, London. Putnam, H.: 1981, Reason, Truth and History, Cambridge University Press, Cambridge. Putnam, H.: 1983, Realism and Reason, Cambridge University Press, Cambridge. Tennant, N.: 1987, Anti-Realism andLogic, Oxford University Press, Oxford. Wright, C.: 1986, Realism Meaning and Truth, Blackwell, Oxford. Wright, C.: 1992, Truth and Objectivity, Harvard University Press, London.

Department of Philosophy University of Glasgow Glasgow G 12 8 QQ United Kingdom

anti-realist truth and concepts of superassertibility

Documents