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Relatively about: Loose Composites and Loose Ends Author(s): Joseph S. Ullian Source: Linguistics and Philosophy, Vol. 7, No. 1, Coherence (Feb., 1984), pp. 83-100 Published by: Springer Stable URL: http://www.jstor.org/stable/25001155 . Accessed: 14/06/2014 00:51 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . 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]. . Springer is collaborating with JSTOR to digitize, preserve and extend access to Linguistics and Philosophy. http://www.jstor.org This content downloaded from 195.78.109.162 on Sat, 14 Jun 2014 00:51:40 AM All use subject to JSTOR Terms and Conditions

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Relatively about: Loose Composites and Loose EndsAuthor(s): Joseph S. UllianSource: Linguistics and Philosophy, Vol. 7, No. 1, Coherence (Feb., 1984), pp. 83-100Published by: SpringerStable URL: http://www.jstor.org/stable/25001155 .

Accessed: 14/06/2014 00:51

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Linguistics and Philosophy.

http://www.jstor.org

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Page 2: Coherence || Relatively about: Loose Composites and Loose Ends

JOSEPH S. ULLIAN

RELATIVELY ABOUT: LOOSE COMPOSITES AND LOOSE ENDS

Nelson Goodman's paper 'About'1 made a giant contribution to the

analytical study of relevance. It set forth a crisp criterion determining what statements are about what, and it made a compelling case for the criterion's

acceptance. Goodman's basic notion was absolutely about. Roughly, a statement S is absolutely about an object k if S has a consequence T that uses a name of k but S does not have as further consequence the

generalization of T with respect to that name. Thus a statement that implies either 'Maine is crowded' or 'The Pine Tree State is crowded' is thereby absolutely about Maine, provided it does not also imply 'Everything is crowded'. Put another way, for S to be absolutely about k, S must "attribute

something" to k that it does not attribute to everything. Say that T, which S

implies, is the explicit vehicle of the attribution. Then in Goodman's phrase, S implies T differentially with respect to k. This concept of differential

consequence was the key to Goodman's explication of 'absolutely about'. But consider a statement like

(i) Aroostook County grows potatoes.

This does not pass the test for being absolutely about Maine. Yet if (i) has the

company of

(ii) Aroostook County is in Maine,

we can derive

(iii) Some county in Maine grows potatoes.

So Goodman, whose example this is, urges that though (i) is not absolutely about Maine, it can well be regarded as about Maine relative to (ii). On the other hand, (ii) is absolutely about Maine. Yet it took company of (i) for (ii) to give us (iii). So, symmetrically, (ii) has call to be regarded about Maine relative to (i), on top of its being absolutely about Maine on its own.

Thus we find ourselves with a relative notion. (i) was about Maine relative to (ii), but not relative to 'Potatoes are hazardous'. (ii) was about

Maine relative to (i), but not, one would imagine, relative to 'Caribou is frozen'. How to develop this notion of relatively about? "Tentatively,... S is about k relative to Q if and only if there is some statement T that follows

differentially with respect to k from S and Q together but not from either

Linguistics and Philosophy 7 (1984) 83-100. 0165-0157/84/0071-0083$01.80 ? 1984 by D. Reidel Publishing Company

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84 JOSEPH S. ULLIAN

alone" (p. 260). Thus the conjunction of S and Q must "attribute

something" to k that is no work of just one of them; and it must do so

differentially with respect to k. This clearly gives what we want for (i), (ii), and Maine.

Goodman was quick to point out a serious flaw in the tentative account.

As it stands, almost any statement about anything would be about Maine relative to almost any other statement that is absolutely about Maine. For example, 'Ghana is tropical' would have to be counted as about Maine relative to 'Maine prospers', since the conjunction (of the two) .. .follows differentially with respect to Maine from the two together but not from either alone

(pp. 260-261).

The tentative account was not sufficiently discriminating; it tolerated

consequences that were simply "loose composites" of what the given statements implied separately. Evidently something tighter is needed: a criterion that is fulfilled only when the two statements have a genuinely joint consequence that is differentially implied with respect to the specific object.

Put loosely, we want to know when a pair of statements together provides more information about the object k than is provided by the statements taken singly. When the statements collaborate in such joint work they are about k relative to each other, or just "relatively about k". The problem is

obviously important for relevance, and it is crucial to the study of coherence as well. For when discourse coheres it tells us more than we can glean from its individual conponents; the statements of the discourse interact with one

another, so that the content of the whole is greater than the sum of the contents of its parts. This is what distinguishes coherent discourse from

mere lists, essays from almanacs. Goodman's analysis of 'relatively about' was partial. He considered only

cases where there is no nested quantification, and even for that class of cases he left some stones unturned. But he did provide a foundation upon which the construction of a full analysis might be attempted. Such an attempt is the

subject of the present paper. It is hoped that our brief sketch has enabled the uninitiated reader to gain

some grasp of the main notions in Goodman's inquiry. Of course study of 'About' would bring a much firmer grasp, and would bestow all the further

pleasures to be expected form a classic piece of philosophical analysis. More particularly, we hope that the reader is open to the thought that

there is some intuitive basis for distinguishing pairs of statements that should be counted as relatively about k from pairs that should not be so

counted. This is the distinction that we will seek to explicate. There is no doubt that the distinction, antecedent to analysis, is far from wholly clear.

We will be striving to sharpen it as we examine various cases.

Our focus will be on quantificational formulas rather than on statements

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in the natural language. One advantage is that this lets us deflect the issue of how to regiment particular English sentences. That is a separate problem, and the less we are diverted by it the better. In any case, our analysis must

proceed in terms of logical form, so we simplify our task by beginning there.

Moreover, concentrating on the logical forms themselves frees us from distractions. We do not want our fledgling intuitions befogged by vaguely sensed connections, or by other cloudy matter that should remain irrelevant to our inquiry. As partial palliative, we will offer some examples in footnotes that are intended to bring our formulas to life; thus we will not be totally removed from the sphere of concrete application. But for us, the

quantificational formulas will be father to the English embodiments, and not the other way around.

Our study starts with the problem about loose composites. Raised for the full realm of quantificational formulas, this turns out to be a more difficult

problem than one might have expected. The analysis is in terms of unitary formulas; both it and they may have independent interest. We distinguish "simple" formulas, those that might be taken to convey single "pieces" of information, from composite formulas. We find, however, that this dis tinction alone is not enough to allow us to explicate 'relatively about' in a

way that accords with even our rudimentary intuitions. We face various

troubling cases, not yet ruled out, where we are hesitant to regard a pair of formulas as relatively about k. These give rise, successively, to a restriction

involving identity, a demand for "essential consequences", and a condition about partial generalization. With these adopted, the explication appears to fit with intuition in most cases. But it is seen that by some lights disturbing cases still remain.

1. UNITARY AND CONJUNCTIVE FORMULAS

A main part of our task is to characterize those consequences of a pair of formulas that are not "loose composites" of their separate consequences. This is harder than it might appear. Suppose we were to say that a

consequence of r'- s12 is a loose composite if it is equivalent to some

conjunction of a consequence of 4 with a consequence of f. Then every consequence y of ro* -1 would count as a loose composite. For any such y is

equivalent to the conjunction of r- ^ v y', which 4 implies because r'b-$

implies y, with rq v y', implied by $f. Clearly, a more delicate analysis is needed. What seems to be desired is a

criterion for distinguishing formulas that are themselves "simple" from formulas that are "composite". ('Simple' is not the most precise term that could be used at this point, but we see that as an advantage. Sharper terms,

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86 JOSEPH S. ULLIAN

like 'indissoluble', prejudice the issue too much.) We take it that there is a presystematic distinction here that is ripe for explication. If we draw the distinction in a reasonable way, we can confine our attention to the "simple"

consequences of a conjunction, retrieving all that follows from the

conjunction without singling out any composites, loose or otherwise. There are three obvious requirements to be met. We will want the

distinction between simple and composite to provide a partitioning of the class of all formulas; every formula should count as one or the other, but

none as both. The divisions should be closed under logical equivalence. It would be disturbing to find equivalent formulas in opposite camps, since we are as much concerned with content as with form. We might even think of a

simple formula as a formula that conveys a "piece of information", relative to the construals accorded its parameters. Thus we will be looking beyond bare syntax, and should not expect our explication to yield a decidable

property of formulas. Finally, we will require that every composite formula be equivalent to a conjunction of simple ones, thereby assuring the

retrievability lately mentioned. All information, to pursue that line of

thought, is divisible into constituent pieces. We will be considering formulas of lower predicate calculus that may

contain identity and constant symbols but no n-ary function symbols with n i. We find it convenient to regard conjunction as applicable to n formulas at a time, for all n > 2; similarly for alternation (i.e., disjunction).

We will confine attention to formulas in which the only truth-functional connectives are conjunction, alternation and negation, and in which

negation applies to atomic formulas only. Since every formula has an equivalent of this kind and we will be honoring our equivalence require

ment, this is no limitation.

The components of a quantification r(3a)p1 or r(a)4 l are those formulas, not themselves conjunctions or alternations, from which r4 is built by 0 or

more applications of conjunction and alternation. So they are just the

deepest formulas in the truth-functional makeup of 4, except that negation signs are not sundered. Accordingly,

(1) (3x)(- Fx v (3y)Gyx (z)(- Gzz v (3z)(w)(Gzw v Fx)))

has exactly three components: '-Fx', '(3y)Gyx', and the universal quantification with respect to 'z'; that universal quantification has the

components '-Gzz' and '(3z)(w)(Gzw v Fx)'; that last existential quantification has only '(w)(Gzw v Fx)' as component; and that universal

quantification has 'Gzw' and 'Fx' as components. We will call a quantification pure if the variable of the quantifier occurs

free in every component of the quantification; if not pure, impure. Thus in

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(1) both of the universal quantifications are impure, while the three existential quantifications are pure.

Let us call an existential quantification frank if it has no equation as a

component, a universal quantification frank if it has no negated equation as a component. Say that a formula is frank if all quantifications occurring in it are. Every formula is readily transformable into a frank equivalent: e.g., both '(3x)(x = y.Fx)' and '(x)(x # y v Fx)' are equivalent to the frank

'Fy'. Much that follows is based on the analysis given by Goodman in 'About',

in particular on his characterization of "explicitly unitary consequences". That analysis stopped short of nested quantification. Our class of unitary formulas may be seen as providing a natural generalization of a class that is

crucial, though not quite explicit, in Goodman's treatment. We define recursively the class of unitary formulas. (i) Atomic formulas

and their negations are unitary; (ii) any alternation of (> 2) unitary for mulas is unitary; (iii) if n > 1 and 4i is unitary for each i

- n, then r(3a)

(l'-.. .'4n) is unitary if it is pure and frank (if n = 1 this is just r(3a)411);

(iv) a pure and frank universal quantification of a unitary formula is again unitary; (v) no formula is unitary unless it is so by (i)-(iv).

So for an existential quantification r(3a)4l to be unitary 4 must, truth-functionally, be a conjunctional normal formula built from the

quantification's components; and for r(a)4l to be unitary 4 must be an alternation of the components, if there is more than one. Conjunction signs, in Goodman's word, are all "captive" within existential quantifications.

Unitary formulas contain no impure quantifications; this assures that there are no vacuous quantifications and that no scopes of quantifiers are condensable by "rules of passage" (other than those for negation) alone.

Notice, though, that while the unitary '(3x)(3y)(Fxy Gy)' shows no

impurity, the result of permuting its quantifiers does. What is again unitary is its permute's "purification", '(3 y)(Gy-(3x)Fxy)'.

The need for demanding that quantifications be pure is obvious. There is no less need for demanding frankness, if the class of unitary formulas is to be saved from embarrassing bloatedness. Lacking either of these de

mands, we would have a class whose closure under logical equivalence is the class of all formulas! What meets the eye in the case of frankness is

equivalence of 'Fc Gc' to '(3x)(x = c Fx Gx)'. But then since 'x = c' is

equivalent to '(y)(y f x v -y = c)', 'Fc Gc' is equivalent again to '(3x) ((y)(y 7 x- v -y = c) Fx- Gx)'. The reader is invited to verify the exclama

tory claim made a moment ago - less immediate for frankness than for

purity. Preparedness to add redundant clauses comes in handy in the proof. Whether or not a formula is unitary is clearly determinable by inspection;

whether a formula has a unitary equivalent is not in general decidable, but it

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88 JOSEPH S. ULLIAN

is at least verifiable if true. It is to the formulas with unitary equivalents that we will be looking, since it seems plausible that any simple formula should have a unitary equivalent. Every formula is readily transformable into an

equivalent that is either unitary or a conjunction of unitary formulas, thanks to the distributivity of universal quantification over conjunction, the rules of passage, and various truth-functional routines.

There is a further threat. Thanks to the possibility of redundant padding, any formula that is a truth-function of quantifications is sure to have a

unitary equivalent. For example,

(x)Fx (3x) Gx,

surely no candidate for simplicity, is equivalent to the unitary

(2) (3z)((x)(Fx v (3y)(Jxyz. -

Jxyz)).(3x)(Gx-Hxz v - Hxz))

and also to the less prolix

(3z)((x)(Fx v (3y)(Jxyz. - Jxyz)). Gz).

From (2) one can discern a general method for manufacturing such undesired unitary equivalents. Often less work suffices;

(3x)Fx-(3x)Gx

is equivalent to both

(3x)(Fx (3 y)(Gy Hxy v - Hxy))

and

(3x)(Fx- Gx v (3 y)(y f x. Gy)).

A formula is redundant if it is equivalent to a result of deleting some occurrences of its alternants, conjuncts, and quantificational components; nonredundant if such deletions always destroy equivalence. It may be

necessary to drop more than a single component occurrence from a redundant formula to reach an equivalent. '(3x)(Fx v Gx-Fx v Gx)' is a

case in point, given how we defined 'component'. Clearly we need to discriminate against redundancy. What suggests itself

is this:

(*) The formulas to be regarded simple are those with nonredundant

unitary equivalents.

It should be clear that the italicized word has force. Removal of redun

dancies from a unitary formula may cause impurity, and subsequent "purification" may leave a formula that is no longer unitary. So a unitary

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formula may have no nonredundant unitary equivalent, which we surely hope is the case with (2) above.

Since nonredundancy, unlike its opposite, is not in general confirmable, we can maintain (*) only at a price. We apparently have to give up the

verifiability of simplicity, i.e., the effective enumerability of the class of

simple formulas. Under (*) there may be simple formulas whose simplicity cannot be established. Verifiability of nonsimplicity, or compositeness, was all but surrendered as soon as we looked to unitary equivalents.3 ((*) makes

simplicity a "Z2" property of formulas.) We saw that every formula is equivalent either to a unitary formula or to a

conjunction of unitary formulas. It is equally true that every formula is

equivalent either to a nonredundant unitary formula or to a conjunction of such formulas, though no longer by purely mechanical transformations.

Now conjunctions of nonredundant formulas may themselves be redun

dant; so what is more worth recording is the slightly stronger fact that every formula is equivalent either to a nonredundant unitary formula or to a nonredundant conjunction of ( - 2) unitary formulas. We call formulas that are equivalent to such nonredundant conjunctions conjunctive. Thus formulas not passing as simple under (*) are always conjunctive. Presys tematically speaking, conjunctiveness of a formula might seem to attest to its being composite. This suggests another principle:

Composite formulas are those that are conjunctive.

By the partition requirement, this has the same force as

(**) The simple formulas are those that are not conjunctive.

(On its face, (**) makes simplicity a "12" property of formulas.) (*) and

(**) both carry a certain plausibility. Can we maintain them both?

Certainly not in full generality. The inconsistent 'Fx - Fx' has non redundant unitary equivalents, such as '(3x)(Fx. - Fx)' and 'x f x'. But we have no real interest in deriving inconsistent consequences. We could keep both (*) and (**) for consistent formulas if we found that no consistent

conjunctive formula has a nonredundant unitary equivalent. Truth-func tional logic offers hope that this condition might hold, since the desired

disjointness is readily established there. For the nonce, call a formula basic if it is built from statement letters and their negations by alternation alone.

The ready theorem is that no basic formula is equivalent to a nonredundant

conjunction of basic formulas. So far so good. Further hope, of a lesser kind, springs from exploration of such quantificational equivalences as that of

(x)(y)Fxy

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90 JOSEPH S. ULLIAN

to

(x)(3y)Fxy.(x)(y)(-Fxy v (z)Fxz).

For the latter, though a conjunction of nonredundant unitary formulas, is seen to harbor redundancy. But we can retain our optimism for only so long.

When we examine

(3) (3x)(Fx.(y)(-Fy v x = y))

and its equivalent

(4) (3x)Fx-(x)(-Fx v (y)(-Fy v x = y))

we find a case of the feared overlap - noting that 'x = y' is not a component of (3), so not a violation of frankness.4

It might still be hoped that this unwelcome phenomenon is somehow the fault of identity. Identity causes trouble, as we find in the next section.

Perhaps we can still embrace both (*) and (**) for consistent formulas

lacking identity. But even this hope collapses upon consideration of

(5) (3x)((y)(-Fy v Gyx).(y)(Fy v -Gyx)-(3y)-Gyx)

(6) (3x)((y)(-Fy v Gyx).(y)(Fy v -Gyx)).(3x) - Fx.

(5) is nonredundant and unitary, (6) a nonredundant conjunction of unitary formulas, and the two are equivalent.5

So at least one of (*) and (**) must be discarded. Which of them has the

stronger claim is arguable; much could depend on further findings. It seems reasonable enough to regard (3) as simple, despite (4). At least what we get from (3) by supplanting 'x = y' by 'Gxy' is surely to be classed as simple, since it is not bothered by equivalence to the formula similarly issuing from

(4). The case for (5) is less clear; are we to take (6) as a defeat for its

simplicity or (5) itself as assurance of it? Can we regard (5) as a simple consequence of the conjunction (6) even though it is equivalent to that

conjunction? Or should it be seen as a loose composite, despite its form? No decisive intuition presents itself. If we find conjunctive formulas that we are bound somehow to see as simple then (**) must be abandoned; if we find nonredundant unitary formulas that we are bound to see as composite then

(*) is untenable. It is possible that neither can be sustained. Our own

preference at this point is to accept (*) with something of an experimental spirit.

2. THE IDENTITY CRISIS

We recall the details of some definitions from 'About'. The formula 3 is said

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to follow from a differentially with respect to k if a implies 3, f3 contains some

expression designating k, and a does not imply the generalization of 3 with

respect to any pSrt of that expression. When the designating expression is a

constant, what the last clause comes to is that a must not imply the result of

supplanting all occurrences of that constant in ,f by occurrences of a new constant. For short, we will say that a k-implies f3, or that f3 is a

k-consequence of a or is k-implied by a, if f3 follows from a differentially with respect to k. a is then absolutely about k iff a has a k-consequence.

Goodman framed these definitions in terms of statements rather than

formulas. Since the parameters of our formal language may be interpreted as we wish, we are free to regard our formulas as making statements and their closed terms and predicates as designating or selecting objects. With Goodman, we will regard a pair of formulas 4 and t as about k

relative to each other (or as relatively about k) iff there is a formula X that (i) is

k-implied by rb- i1, (ii) follows from neither 4) nor i alone, and (iii) fulfils certain special conditions. For us, one of these special conditions will be that

X is simple. If we accept (*), this means that X is equivalent to a

nonredundant unitary formula. That is very close to the special condition

required on Goodman's analysis, an analysis which considered no X with nested quantifiers and so avoided some complexities. It still leaves out a small portion of Goodman's special condition; we will turn to that in Section 4.

In our analysis we will limit explicit consideration to cases in which the

object k in the definitions lately cited is designated by a constant; further, we will be taking that constant to be 'k'. Our analysis is fully applicable to whatever other cases one might wish to consider. In particular, it applies when the focal object is the extension of a predicate letter; most of the

examples given below are suitable for that case. We pause to emphasize the rationale for demanding that the consequence

"X" be simple. If it is not then it is equivalent to a certain conjunction of

simple formulas. Any conjunct of this conjunction that is not k-implied by rI ' 41 may be dropped as irrelevant for our purpose. Now if each remaining

conjunct of the conjunction is a consequence of one or the other of ) and 4, then the relevant part of X is itself either implied by just one of them or else a

conjunction of separate consequences of ) and 4. If the latter, it may well be regarded as a "loose composite". The only other alternative is that there remains at least one simple conjunct of the conjunction-equivalent-to-x that is a consequence of neither 4 nor 4 alone but only of their conjunction,

which k-implies it. So it is plausible to demand such a simple k-consequence in the first place. Wanted, then, are simple k-consequences of r- bi1 that do not follow

from either 4 or 4 alone; pending further fleshing out of the "special

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92 JOSEPH S. ULLIAN

conditions", their existence is what testifies that 4 and iq are about k relative to each other. On to the fleshing out.

Let 4> be 'Fk' and q be either '(3x)Gx' or 'Gd'. On its face, 4 and ql should not be relatively about k; for how can iq combine with 'Fk' to deliver

anything "new" about k? Yet r'* q1 has the simple consequence

(7) Gk v (3x)(Fx-(3y)(x + y Gy)).6

(7) is no consequence of either ) or i alone, and it is a k-consequence of their conjunction inasmuch as that conjunction does not imply

Gc v (3x)(Fx.(3y)(x # y Gy)).

This means that our specifications must be tightened. One path worth exploring is suggested in a footnote in 'About' (p. 262).

From (7) we can obtain a further consequence of r'. ql by instantiating a constant - 'k' - in one of the existential quantifications - the quantification with respect to 'x'. The instantiation yields

(8) Gk v Fk.(3y)(k f y. Gy),

which is no longer unitary. In fact (8) expands into the conjunction of

Gk v Fk,

a consequence of 4 alone, with

Gk v (3y)(k f y Gy),

a consequence of if. So (8) has all the earmarks of a "loose composite", and

(7) is hiding behind its skirts. Perhaps we want to discriminate against consequences, like (7), in which instantiation of a constant in an existential

quantification yields a further consequence of the pair of formulas under

scrutiny. Goodman noted that this restriction gives unwanted results. Take the pair

of formulas 'Hkd' and 'Gd'. We do want these to count as about

k relative to each other, in view of their joint k-consequence '(3x) (Hkx Gx)'.7 But the discriminatory restriction frowns on this k-con

sequence, since its instantiation 'Hkd- Gd' is also forthcoming. (The astute reader may realize that even with the restriction the pair of formulas passes as relatively about k on the criteria invoked to this point - but they do so

only on account of consequences that we are about to disqualify.) The restriction appears to be too strong.

Still more crucially, the proposed restriction does not bring enough relief. The pair 'Fk' and '(3x)Gx' (or equally the pair 'Fk' and 'Gd') has also the

simple k-consequence

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Gk v (x)(3y)(x + y-Fy v Gy).8

The restriction is powerless to block this from certifying that these formulas

are, after all, relatively about k. Almost as upsetting is the discovery that the

conjunction of 'Fk' and 'Fd' has the simple k-consequence

k = d v (x)(3y)(y -

x-Fy)9

that follows from neither 'Fk' nor 'Fd' alone. We do not want to count 'Fk' and 'Fd' as about k relative to each other, but the restriction is no help here

either. The drastic remedy that we propose is the requirement that our

k-consequence "X" must lack identity. One can find precedent, of a sort, for such a demand in Hilary Putnam's 'Formalization of the Concept "About"'.10 Putnam pursues a line of explication far from Goodman's. He bases his analysis on Carnapian state-descriptions, using a language in which each of the supposed finitely many individuals is denoted by a unique constant. And for such a language over a finite universe, the identity predi cate adds nothing. Admittedly, Putnam intends his analysis to be extendible to cases with infinitely many individuals; but short of such extension, his

analysis renders identity otiose. Precedent or not, we see no good way of avoiding our drastic step. We

lose much less in the way of desired consequences than one might expect. We

briefly survey some typical losses. The formulas 'k = c' and 'c f d' might be taken to be about k relative to each other, in view of their k-consequence 'k . d'; but no longer for us.11 The formulas 'Fkc Hc' and'Fkd.-Hd' have

the k-consequence '(x)(3y)(y : x Fky)'; this might or might not seem to

testify that they should count as about k relative to each other, but they will not be so counting by our lights.12 Finally, consider 'Fk' and '-Fd'.

Together they yield 'k : d', whose testimony we now rule out.13 Do we want to regard 'Fk' and '-Fd' as relatively about k? If we do we can take

solace in the observation that they have not yet been denied that status. For

they have the identity-free k-consequence

(9) -Hkd v (3x)(Fx-(3y)(-Fy Hxy)).14

3. ESSENTIAL CONSEQUENCES

Our solace, if it was that, has short duration. For it dawns on us that if (9) shows the pair 'Fx' and '-Fd' to be about k relative to each other, then no

less does

(10) -Hkd v (3x)(Fx.(3y)(Gy Hxy))

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94 JOSEPH S. ULLIAN

show our old antagonists 'Fk' and 'Gd' to be so also.15 What has gone wrong? A first suggestion is that (10) contains the new predicate letter 'H', which seems somehow like cheating. Perhaps we should ban k

consequences with new predicate letters. This would not be enough. For 'Hab v Fk' and 'Hab v Gd' have as simple k-consequence the alter nation

(11) Hab v -Hkd v (3x)(Fx (3y)(Gy Hxy)),16

yet they have no more claim than 'Fk' and 'Gd' to be called relatively about k. Again we have need of further remedy.

To this end we recall the notion of essential consequence that was introduced in another connection.17 We say that a implies B essentially, or that 3 is an essential consequence of a, if the following condition is fulfilled: for any p-ary predicate letter ir (or constant IL) occurring in 3, no result of

supplanting a positive number of occurrences of rr (of ,) in 3 by a new p-ary predicate letter (a new constant) is implied by a. The identity sign does not count here as a predicate letter, but that is beside the point now for us. New

predicate letters or constants are ones occurring in neither a nor 3. Since

change of the two occurrences of 'H' in (10) to a new binary predicate letter

yields but another consequence of 'Fk Gd', (10) is no essential con

sequence thereof. Since like change of the last two occurrences of 'H' in

(11) yields again a consequence of 'Hab v Fk. Hab v Gd', (11) is not among its essential consequences. Note that no essential consequence of a can contain a predicate letter or constant that does not occur in a.18

We must require our k-consequence "X" to be an essential consequence of the focal conjunction. This will free us from our worries about (10) and

(11), as well as ending any solace still felt from (9). This new requirement as

sures, by the way, that if X contains an expression designating k then X is

k-implied by the conjunction. The argument is trival. As usual, we suppose the designating expression to be 'k'. If X were not k-implied, the

conjunction would imply the result of supplanting all occurrences of 'k' in X by a new constant, so that X would be no essential consequence of the

conjunction. Our stiffened criterion for ruling k and iI to be about k relative to each

other can be stated fairly crisply, if we take 'essential k-consequences' to denote essential consequences that are also k-consequences - or

equivalently, essential consequences that contain an expression designating k. It is that r'_ -l have a simple essential k-consequence lacking identity that follows from neither 4 nor $ alone. For all its stiffness, we know of no

important case where this criterion fails to sustain an intuitive judgment to the effect that two formulas are about k relative to each other. Indeed, the

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only known cases of any kind where support for such judgment is not

forthcoming are those lost to the identity proviso; and most of them were

pretty dubious anyway. But our criterion is still not quite stiff enough.

4. FURTHER PROBLEMS

We now turn to the small portion of Goodman's "special condition" that we have neglected so far. The pair '(x)Hkx' and '(x)Jkx' seem to offer quite separate items of information about k; they seem ill qualified to count as about k relative to each other. Their consequence '(x)(Hkx Jkx)' is plainly a loose composite on any plausible standard. Yet they have also the lesser

consequence '(3 x)(Hkx-Jkx)', and it is a simple essential k-consequence of them.19 Must we then concede that the formulas in the pair are, after all, relatively about k?

Goodman's solution, nearly enough, was to bar consideration of con

sequences in which "merely exchanging the existential quantifier for a universal one results in a statement that still follows" (p. 262). This was part of his analysis of what it was for conjunction signs to be "captive"; existential quantifications of conjunctions were not to be mere dilutions of available universal quantifications. As the example above shows, the restriction is desirable.

As stated, however, it seems not to go far enough. For the pair '(3x)Hkx' and '(x)Gx' yields the simple essential k-consequence '(3x)(Hkx. Gx)', but not the stronger '(x)(Hkx Gx)'. Yet it is a strain to suppose that this pair should count as relatively about k; its second formula might as well be any universal quantification.20 Or again, consider '(x)Hkx' and '(3x)Gx'. They too yield '(3x)(Hkx Gx)' without yielding its universally quantified coun

terpart, yet they too have doubtful claim to be counted as about k relative to each other.21

It must be recorded that the cases just cited have given rise to differing reactions. There have been some questioned who find it acceptable to

regard the two pairs as relatively about k, surprising as this may seem. These

judgments, however, appear to spring more from allegiance to some

proviously embraced analysis than from any direct intuitions about the cases themselves.

Goodman banned existential quantifications that may be generalized in toto. We give a natural extension of his restriction by banning existential

quantifications that may be generalized even in part. Our analysis is for formulas of unlimited quantificational complexity. We are still confining our attention to formulas whose only truth-functions are conjunction, alter nation, and negation, and in which negation applies to none but atomic

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96 JOSEPH S. ULLIAN

formulas. We call a formula y a bolster of formula P if y results from 3 by replacing some existential quantification

'(3 V)(1 ? ? ? n)' in ( by

(3 V)(1.. ?<k-1 .k+1 ? ? n)-(V)k I,

with 1 _ k ' n and 'V' a metalinguistic variable ranging over variables of the formal language. For n = 1 this means replacing r(3 V),1) by ( V)i11.

The Xi may themselves be conjunctions. So to get a bolster of 3 one drops some conjunct from inside an existential quantification in , and conjoins its universal quantification to what remains of the existential quantification. It is easily shown that any bolster of / implies /3. Bolsters of 3 that are

equivalent to ( are called improper bolsters; those that are not (the more

typical case) are proper bolsters. We want to disqualify the "testimony" of any consequence of r-4./1

that has a proper bolster also derivable from r- 41. This discredits '(3x) (Hkx Gx)' in both of the cases three paragraphs back; in the former because of its bolster '(3x)Hkx.(x)Gx', in the latter because of

'(3x) Gx (x) Hkx. A consequence a of a set of formulas r will be said to fulfil the bolster

condition for F if r implies no proper bolster of any nonredundant unitary equivalent of a. The new restriction is that our simple consequence of F)' 4.1

must fulfil the bolster condition for {I, lp}.22 So on the analysis to this point, 4 and 4, are to count as about k relative to each other if they have a simple essential k-consequence lacking identity that fulfils the bolster condition for {4, 4/} and follows from neither ) nor 4, alone.23

We will not argue at any length for the desirability of the bolster

condition, but we will offer a few observations. If a consequence a of rF'- 41

fails the condition then r'4. 41 implies some proper bolster y of an equivalent of a. If y is simple we can investigate it as a source of the sought testimony; otherwise we can investigate the simple formulas whose conjunction is

equivalent to y. We thus replace the testimony of dubiously manufactured

consequences of r'. 4/1 - those that fail the bolster condition for {(, 4/} - by

that of more reliable informants. Nonredundant unitary equivalents of a given formula may differ in

bolsterability. Say that 4 is '(3x)Fkx' and 4 is '(x)(Gx v Hx)'. No bolster of

their simple consequence

(3x)(Fkx Gx) v (3x)(Fkxc Hx)

is implied by r4- 4i1, but a proper bolster of its simple equivalent

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(3x)(Fkx Gx v Hx)

is implied. Or more dramatically, take 4b as '(3x)(-Fkx - Gkx) v

(3x)(FkxI Hkx)' and 4 as '(x)(-Hkx v Gkx)'. Then the consequence

(3x)(Fkx v -Hkx- Gkx v -Fkx Hkx v - Gkx)

has no bolster that is implied by r bl', yet its equivalent

(3x)(-Fkx v Hkx -Gkx v Fkx -Hkx v Gkx)

does. There are nontrivial simple formulas that have improper bolsters. One is

(3x)((z)(y)(-Gyzx v Fxy)-(z)(y)(Gyzx v -Fxy) (3 y)(3z) Gyzx),

which is equivalent to the result of changing its rightmost quantifier to '(z)'. Our investigations have indicated no reason for discriminating against such

consequences - hence the word 'proper' in the bolster condition. We have imposed a heavy battery of conditions in our battle to ward off

unwelcome consequences. We have been threatened again and again with

having to count some pair of formulas as relatively about k even though our intuitions would bid us rule otherwise. Some of these intuitions are less than

firm, since the concept being explicated has been so little developed. And as noted above, there are differences of opinion. In particular, the work done by the bolster condition is seen as unnecessary by some.

But for the rest of us, there is trouble coming; the fort can be held for only so long. Consider the pair 'Fk' and '(x)(Gx v Hx)'. On its face, this should no more be counted as relatively about k than should the pair '(3x)Hkx' and

'(x)Gx', which led to the bolster condition. Yet this newly considered pair yields

(12) Gk v (3x)(Fx Hx),

a consequence that fulfils every condition that has been set down thus far. So as our criterion now stands, 'Fk' and '(x)(Gx v Hx)' count as relatively about k.24 Are there remedies available?

We could broaden the bolster condition by removing the word 'non redundant'. (12) is equivalent to the unitary

Gk v (3x)(Fx- Gx v Hx-Gx v Hx),

which has a derivable proper bolster, allowing the broadened bolster condition to deflect (12). But this broadened bolster condition would have us rule out far too much. For take 4q and 4 as '(3x)- Fx v (3x)Gkx' and

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98 JOSEPH S. ULLIAN

'(x)Fx'. We do want these to count as relatively about k, since they produce '(3x)Gkx'.25 Yet this last formula has the redundant unitary equivalent

(3x)( Gkx Gkx v Fx),

one of whose proper bolsters is implied by r - .i1. Thus the broadened

bolster condition rules against this 4 and qf. It is, in short, too broad.

Perhaps there is some effectual variant of the bolster condition that is not too broad; but if so, it is still evading us.

A different expedient would be resurrection of the condition involving instantiation of constants, briefly considered in Section 2. We could have derived 'Gk v -Fk Hk' in place of (12); so the "instantiation condition" would have declared against (12). That condition's shortcomings have

already been noted. But in addition, there are cases all too much like the one that gave us (12) against which the instantiation condition is powerless.

Consider the pair '(x)(3 y)Jkxy' and '(x)(Gx v Hx)'. Together they yield the

simple k-consequence

(13) (3x)(3y)(Jkxy. Gy) v (x)(3 y)(Jkxy. Hy), which appears to pass all tests.26 Yet this consequence is clearly beyond the reach of the instantiation condition. So that is not the answer either.

A third way to bar the door on (12) - and (13) - would be to demand that

the formulas 4) and qf share some parameter - either a predicate letter or a

constant. The lack of such sharing was part of what made the case of (12) so

stark. As far as it goes, this new restriction seems to have no harmful effects.

Still, it is both unpleasantly ad hoc and glaringly inadequate. For if 'Fk' and

'(x)(Gx v Hx)' should not count as about k relative to each other, neither

should 'Fk-(3x)Gx' and '(x)(Gx v Hx)'.27 The "sharing condition" leaves too many loopholes to recommend itself.

Conceivably the lesson is that the qualitative distinction between those

pairs that are relatively about k and those pairs that are not is one that is

simply not sharp. Perhaps the distinction should give way to a quantitative one: what - or how much - about k can be derived from the conjunction that cannot be derived from the formulas of the pair individually? If (12) be seen as telling little about k, then perhaps we can answer the question by 'Not much' when we consider 'Fk' and '(x)(Gx v Hx)'.28

Alternatively, perhaps we should just bite the bullet when it comes to (12) and (13). We can console ourselves with the thought that there was probably no uniformity of intuition about them anyway. Where intuition is uniform and clear, the analysis forged above does seem to agree with it. "Relatively about" has proved to be a remarkably elusive concept, and the survival of a

mild anomaly or two in its analysis may just be unavoidable. Still, it is hard to

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doubt that there are improvements in that analysis that are waiting to be found. They will be welcomed.

NOTES

1 N. Goodman, 'About', Mind 70 (1961), 1-24. The paper is reprinted together with some useful discussion in Goodman's Problems and Projects (Bobbs-Merrill, Indianapolis and New York, 1972). Page references are to the paper's reprinted version. 2 This is the standard use of quasi-quotation that is familiar from Quine. For most logical notation we will be following Quine's Methods of Logic, 3rd ed. (Holt, New York, 1972). 3 Under (*) the nonsimple formulas cannot be effectively enumerable. Let 4) be any formula

lacking 'F and 'x'. Then it can be seen that the conjunction of 'Fx' with 4 is simple by (*) iff 4 is either valid or inconsistent. Since the class of consistent nonvalid formulas is not effectively enumerable, the claim follows. We have no proof that (*) forbids effective enumerability of the simple formulas, but strongly conjecture that it does. 4 We have not forgotten those who are looking forward to English readings of formulas. (3)

might say 'There's exactly one rabbit'; correspondingly, (4) comes to 'There's at least one rabbit and there's at most one rabbit'. 5 We can read (5) as 'There's something done by all and only rabbits that isn't done by everything', (6) as 'There's something done by all and only rabbits, but not everything is a rabbit'. 6 d) might say 'Mycroft is a rabbit' and ) 'There are friendly beings' or 'Baskerville is friendly'.

Are these relatively about Mycroft? (7) now says 'If Mycroft isn't friendly then there's a rabbit from which some friendly being is distinct'. 7 Example: From 'Mycroft likes Baskerville' and 'Baskerville is a cat' we derive 'Mycroft likes a cat'. 8 In terms appropriate to the earlier example, this says 'If Mycroft isn't friendly then there is

more than one thing that is either a rabbit or friendly'. 9 From 'Mycroft is a rabbit' and 'Perseus is a rabbit' we can derive 'If Mycroft isn't Perseus then there are at least two rabbits'. 10

Philosophy of Science 25 (1958), 125-130. 11

From 'Mycroft is Peter' and 'Peter isn't Perseus' we infer 'Mycroft isn't Perseus'. 12

Together, 'Mycroft likes Cottontail, who is flop-eared' and 'Mycroft likes Mopsy, who isn't

flop-eared' yield 'Mycroft's likes number at least two'. 13

'Mycroft is a rabbit' and 'Baskerville is no rabbit' secure 'Mycroft isn't Baskerville'. 14 We might read (9) as 'If Mycroft likes Baskerville then there's a rabbit who likes a

nonrabbit'. 15 For (10): 'If Mycroft likes Baskerville then a rabbit likes a cat'. 16 'Hab' could say 'Alice likes Boris', and (11) that either that's so or else our reading of (10) is. 17 See this author's 'Wanton Embedding Revised and Secured'. Journal of Philosophy 77

(1980), 487-495. 18 A formula is said to be self-essential if it is an essential consequence of itself. It was shown in the paper cited above that, if we allow use of identity, every formula has a self-essential

equivalent. The properties of being self-essential and of being nonredundant share a certain flavor, but they are independent of each other. 19 So from 'Mycroft sniffs everything' and 'Mycroft fears everything' we get the lesser

consequence 'Mycroft sniffs something he fears'. 20 An English example of the pair might be 'Mycroft sniffs something' and Everything is

interesting'; the consequence is then 'Mycroft sniffs something interesting'. 21 The pair is now 'Mycroft sniffs everything' and 'Something is interesting', with the

consequence still as just above.

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22 Clearly, this implies Goodman's condition in all cases where changing an existential

quantifier to a universal one strengthens the formula. 23 Fulfilment of the bolster condition for given {4, i,} is a "12" property of formulas. So our analysis now makes the relation of being relatively about k "E3" (ignoring any problem about determining what denotes k). 24 From 'Mycroft is a rabbit' and 'All carnivores are cunning' we derive 'If Mycroft is a carnivore then there's a cunning rabbit'. Or, changing the second "premise" to 'Whatever is

heavy sinks', we get the consequence 'If Mycroft is heavy then there's a rabbit that sinks'. 25 'If everything is interesting then Mycroft has an idea' together with 'Everything is

interesting' yield the obvious conclusion. 26 For the pair: 'Everything reminds Mycroft of something' and 'Whatever isn't heavy floats'; for (13): 'If nothing reminds Mycroft of anything heavy, then everything reminds him of

something that floats'. The reader with good will will suspend worries about referential opacity when considering our reading of (13). 27 Manufacture of other such unwelcome pairs is easy. 'Fk v Lc' and '(x)(Gx v Hx) v Jc' is another example. 28 The Putnam paper cited above raised the question of how much information about an

object is contained in various statements.

Dept. of Philosophy, Washington University, St. Louis, MO 63130, U.S.A.

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