lowe - what is the source of our knowledge of modal truths

What is the Source of Our Knowledge of

Modal Truths?E. J. Lowe

Durham [email protected]

There is currently intense interest in the question of the source of our presumedknowledge of truths concerning what is, or is not, metaphysically possible ornecessary. Some philosophers locate this source in our capacities to conceive orimagine various actual or non-actual states of affairs, but this approach is open tocertain familiar and seemingly powerful objections. A different and ostensibly morepromising approach has been developed by Timothy Williamson, according towhich our capacity for modal knowledge is just an extension, or by-product, ofour general capacity to acquire knowledge of true counterfactual conditionals — acapacity that we deploy ubiquitously in everyday life. Williamson’s account cru-cially involves a thesis to the effect that modal truths can be explained in terms ofcounterfactual truths. In this paper, I query Williamson’s account on a number ofpoints, including this thesis. My positive proposal, which owes a debt to the workof Kit Fine on modality and essence, appeals instead to our capacity to graspessences, understood in a neo-Aristotelian fashion, according to which essencesare expressed by ‘real definitions’.

1. Metaphysical modalities

It seems that, at least sometimes, we can know that something ismetaphysically possible or metaphysically necessary. By ‘metaphysical’

necessity I mean necessity of the strongest possible kind — absolutenecessity — and I take it to be an objective kind of necessity, rather

than being something mind-dependent and reflective of some mentalcondition or ability of a thinker (Lowe 1998, pp. 13–21). It is not to be

confused with epistemic necessity, which is more properly to be calledcertainty. Nor is it to be confused with natural or causal necessity,which may best be understood as a species of relative necessity —

relative, namely, to the natural or causal laws that actually reign.Of course, some philosophers hold that these laws are themselves

metaphysically necessary, in which case the distinction between meta-physical and natural necessity collapses (Shoemaker 1998). But that is,

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to say the least, a contentious thesis, and it is not one that I adhere to(Lowe 2006, pp. 141–55).

A question arises as to whether metaphysical necessity is, or is not,the same as logical necessity. That depends, of course, on exactly what

is understood by the latter. If logically necessary truths are taken to betruths that are, or are logical consequences of, the laws of logic —

whatever precisely we take those ‘laws’ to be — then I think wemust say that the logically necessary truths are only a proper sub-class

of the class of metaphysically necessary truths (Lowe 1998, pp. 14–15).That is to say, all logically necessary truths are ipso facto metaphysic-

ally necessary truths, but not all metaphysically necessary truths arelogically necessary truths. So what would be a plausible example of a

metaphysically necessary truth that is not a logically necessary truth?Some would instantly say: the truth that water is H

2O. And they would

contend that this is also an example of a metaphysically necessarytruth that is knowable only a posteriori. However, I think it is ques-

tionable whether it is true at all, let alone necessarily true, that water isH

2O. (Apart from anything else, any sample of pure water contains

OH– and H3O+ ions as well as H

2O molecules, which explains why

even pure water conducts electricity, albeit only very weakly; so, if

‘Water is H2O’ is understood as asserting water is identical with, or

is wholly composed by, H2O molecules, then it is simply false.) And I

am also somewhat sceptical about the very idea of a posteriori neces-sary truths (Lowe 2007). So I would prefer to venture a different

example of a truth that is metaphysically but not logically necessary:the truth that a uniformly coloured surface is not at once both red

and green. This truth is certainly not a consequence of the laws oflogic. Nor can it even plausibly be said that it follows from those laws

taken together with suitable definitions of the predicates ‘red’ and‘green’: for those predicates, it seems, are in fact primitive and

indefinable.However, perfectly uncontentious examples of interesting (not

purely logical) metaphysically necessary truths are fairly hard tocome by, and even this example is not wholly uncontentious. It is,

by contrast, much easier to come by plausible examples of interestingmetaphysically possible truths.1 Obviously, an example of a metaphys-

ically possible truth is provided by any actual truth, since what is the

1 Note that just as some truths are metaphysically but not logically necessary, so some

truths are logically but not metaphysically possible. Thus, given that it is metaphysically but

not logically necessary that a uniformly coloured surface is not at once both red and green, it

follows that it is logically but not metaphysically possible that a uniformly coloured surface is at

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case is, a fortiori, possibly the case: ab esse ad posse valet consequentia.

The more interesting cases, however, are of metaphysically possible

truths that are not also actual truths. One such truth, very plausibly, is

that I could have had an identical twin brother. Followers of Spinoza

will want to deny this, no doubt, but only because they think that

truth and necessary truth coincide — a thesis that I find quite literally

incredible. Another such truth, even more plausibly, is that the uni-

verse could have contained one more electron than it actually does. Yet

another is that a certain piece of clay, which is now actually spherical in

shape, could now have been cubic in shape.We find it very easy indeed to generate such putative examples of

metaphysically possible truths that are not actual truths. And one

obvious question is: how do we do it? This, it might seem, is essentially

a psychological question and an empirical one too. A more obviously

philosophical question is: how is it possible for us to do it — that is,

how is it possible for beings with our cognitive capacities to come to

know that certain non-actual truths are metaphysically possible? Of

course, we could just maintain, dogmatically, that we possess, as one

of our cognitive capacities, a brute or basic capacity to recognize at

least some metaphysically possible truths as such. But that would be,

or should be, a thesis of the last resort. It would be much more sat-

isfying if we could explain our capacity to know metaphysically pos-

sible truths — that is, more precisely, our capacity to know that

something that is not actually the case could be the case — without

just positing a special capacity to do precisely this. So let us look at

some candidate explanations.

2. Imagination, conceivability, and intuition

Some philosophers appeal to imaginability or conceivability — which

are not, of course, exactly the same thing — as guides to metaphysical

possibility (Gendler and Hawthorne 2002).2 One apparent difficulty

with this sort of approach is that the notions of imaginability and

once both red and green. For, whichever species of modality we are concerned with, ‘h‰p’

entails ‘‰-p’ and ‘‰h‰p’ entails ‘-p’.

2 Some of the critical points that follow are no doubt familiar ones, but it will help to

rehearse them here in order to highlight the differences between such approaches and the one

that I develop later in the paper.

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conceivability are themselves modal notions, these being the notions,respectively, of the ability to imagine and the ability to conceive. So, a

question arises — in, say, the case of imaginability — as to whether it isthe mere psychological possibility of imagining something that is in-

dicative of its metaphysical possibility, or only the metaphysical pos-sibility of doing so. If the latter, we are not really any further forward.

If the former, then either we are in danger of treating metaphysicalpossibility as something mind-dependent, or we must recognize the

very great fallibility of imagination as a supposed guide to metaphys-ical possibility. The latter is indeed surely the case. Imagination is vital

in the composition of works of fiction, but they can be ‘believable’even if they trade in metaphysical impossibilities, such as time-

travellers who change the past. Imagination is also limited in itsscope by its intimate psychological connection with our perceptual

faculties, which are geared to register states of affairs in the actualworld, at the mesoscopic level. Thus, for example, it seems impos-

sible — for us humans, at least — to imagine objects in four-dimensional space, and yet some current theories of physics postu-

late a good deal more than three spatial dimensions at the subatomiclevel.

Some may urge that a way to overcome some of these limitations ordeficiencies of the imagination is to use it in a rationally constrained

way, by recourse to ‘thought-experiments’. And there have been fam-ously compelling thought-experiments in the history of science, such

as Galileo’s thought-experiment to discount the possibility of objectsfalling at different rates according to their weights, with a heavy one

supposedly falling faster than a light one: we just imagine a light and aheavy weight being attached together, to make a still heavier weight,

and observe that the hypothesis under examination now generatescontradictory results, since we should expect the light weight to

retard the fall of the heavy weight and yet also expect the combinedweight to fall even faster than the heavy weight would on its

own (Galileo 1954, pp. 62–4). On the other hand, it is not clear thatimagination really plays any substantive role in this reasoning,

which is apparently just a reductio ad absurdum argument.Thought-experiments which genuinely seem to involve something

like imaginability have, of course, also been widely deployed in phil-osophy, a notorious example being Derek Parfit’s teletransportation

thought-experiment, where we are invited to imagine someone’s bodybeing reduced to atoms, these atoms being beamed to a distant planet,

and the atoms there being reassembled into an exactly similar body

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(Parfit 1984, pp. 199–201). Does the person survive this process or not?

Here we are commonly asked to consult our ‘intuitions’ on the matter.

And, indeed, ‘intuition’, more generally, is often appealed to by phil-

osophers as a source of knowledge of possibility, whether on its own

or in conjunction with imagination.

An ‘intuition’, it seems, is supposed to be a sort of immediate and

unreasoned or non-inferential judgement, carrying a high degree of

conviction.3 But why should intuitions in this sense be supposed to

have much reliability where matters of possibility are concerned? No

doubt, the fact that we all have ‘hunches’ from time to time, in cir-

cumstances in which reliable evidence is not forthcoming, is some-

thing that evolutionary psychology might be expected to explain, and

this might indicate that a faculty for ‘hunches’ is, by and large, an

adaptive cognitive trait. But the point about hunches is that we have

recourse to them when we have nothing more reliable to go on in the

matter of belief-formation, and they are patently often mistaken and

sometimes disastrously so. I suspect that ‘intuitions’ and ‘hunches’ are

pretty much the same thing, and pretty useless as sources of know-

ledge — all the more so to the degree that they concern things distantly

removed from actuality, as is eminently the case in the teletransporta-

tion thought-experiment. Furthermore, intuitions are notoriously

amenable to being ‘massaged’. We do not have a fixed stock of im-

mutable intuitions. Things that seemed intuitively true to our fore-

bears a century or two ago often by no means seem intuitively true to

us now. It may be the case that evolution has equipped us cognitively

with some sort of innate ‘folk physics’, and equivalent innate systems

in other domains, such as ‘folk psychology ’. But if so, they evolved as

successful adaptations to the practical circumstances of human life as

it was lived in prehistoric times, and cannot be regarded as reliable

guides to the truth in the domains in question.Similar points can be made about conceivability as a supposed guide

to metaphysical possibility, where conceivability is, as it should be, dis-

tinguished from imaginability. Long ago, Descartes famously pointed

out the need to distinguish them, utilizing his example of the

3 The doubts that I raise below about the reliability of ‘intuitions’ are not intended to apply

to George Bealer’s highly sophisticated theory of what he calls ‘rational intuitions’ (Bealer

2002, pp. 73–5). However, there is not space enough here for me to discuss Bealer’s very

interesting views and their relation to my own essence-based approach to modal epistemology

presented in later sections of this paper. Suffice it to say that most philosophers who appeal to

‘intuitions’ do not have an account like Bealer’s in mind.

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chiliagon — a thousand-sided figure — for this purpose: we cannotreally imagine a chiliagon, since we cannot distinguish in imagination

between a thousand-sided figure and a thousand-and-one-sided figure,as they would look just the same to us (Descartes 1984, pp. 50–1). But

we can certainly conceive of such a figure, and distinguish in conceptionbetween it and a thousand-and-one-sided figure. Conception, one

might want to say, involves the deployment of concepts in thinking,whereas imagination just deploys perception-based imagery. That,

perhaps, is too simplistic, not least because some forms of thoughtseem to be irreducibly imaginative: consider, for instance, the

thought process involved in thinking how to get from your houseto the nearest railway station — you just imagine the route you

would take. However, propositional thought certainly seems to re-quire the deployment of concepts. Appealing to conceivability as a

guide to possibility is close, if not identical, to appealing to ‘con-ceptual analysis’ for this purpose. So, for example, in the Parfit

teletransportation case we might suppose the real task not to beone of determining what we imagine or intuit the outcome of the

experiment to be, but rather one of assessing whether our conceptof a person is such that the proposition that the person who is

the subject of the experiment survives the process is conceptuallycoherent. A conceptually incoherent proposition, presumably, is

one which is such that, when the concepts involved in it are properly‘analysed’, it turns out to be implicitly self-contradictory or

inconsistent.But there are obvious difficulties with appeals to conceivability

taken in this way. Who is to determine what ‘our’ concept of aperson is? And why should it matter, anyway, what our concept of a

person is: is it not more important what a person is, if indeed therereally are such things as persons? It often turns out that our concepts

of various really existing kinds of things are seriously inadequate to thetrue natures of those things. And sometimes, of course, we have con-

cepts of certain kinds of thing, such as phlogiston, which turn out notto exist at all. What, after all, is a ‘concept’? This is a question that is

not often enough asked by those who happily talk about concepts,conceivability, and conceptual analysis. The nearest I can get to a

quick definition is to say that a concept is a way of thinking of something or kind of things, whether or not a really existent thing or kind of

things (Lowe 2006, pp. 85–6). But, plainly, one may think of some-thing in an inadequate, distorted, or impoverished way. For instance,

the way in which a four-year-old child may think of a triangle is hardly

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likely to be as adequate as that of a mathematician well-versed inmultidimensional algebraic geometry.

Yet another problem with appealing to ‘conceptual analysis’ to re-solve questions of metaphysical possibility is that it is increasingly

being questioned whether many of our concepts really are amenableto ‘analysis’ at all, at least if this means supplying necessary and suf-

ficient conditions for their application (Fodor 1998). The notoriousconcept of a bachelor, supposedly analysable as meaning an unmarried

male, is quite possibly the exception rather than the norm. In any case,setting aside all the foregoing difficulties, the plain fact seems to be

that conceivability cannot always be a good guide to metaphysicalmodality, as certain mathematical examples show.4 Goldbach’s con-

jecture — that every even number greater than 2 is the sum of twoprime numbers — is a mathematical proposition and hence either ne-

cessarily true or necessarily false. But neither it nor its negation has yetbeen proved to be true, and it may indeed be one of those arithmeticaltruths which, according to Godel’s incompleteness proof, is not prov-

able. Be that as it may, the fact surely is that both the conjecture andits negation are conceivably true — why otherwise would we bother to

try to settle the matter? But they cannot both be possibly true, sinceneither is a contingent proposition. Nor would it be wise, I think, to

dismiss this as merely an exceptional example. I think we should takethe lesson to be that we have no good reason to suppose that con-

ceivability is a good guide to metaphysical possibility. At best, the factthat something is conceivable may give us good reason to inquire into

its possibility, as in the case of Goldbach’s conjecture. But that impliesthat the process of establishing its possibility, even if only provisionally

and defeasibly, is a quite different and additional task.

3. Modal knowledge as a special case of counterfactualknowledge

I pass on now to examine a significantly new and prima facie attractiveway of identifying the source of our knowledge of modal truths, de-veloped recently by Timothy Williamson (2007).5 Williamson appeals

4 This applies also to David Chalmers’s notion of ‘ideal conceivability ’, which he defines

thus: ‘S is ideally conceivable when S is conceivable on ideal rational reflection’ (Chalmers

2002, p. 147). Chalmers himself concedes that ‘the mathematical case is the most significant

challenge to scrutability ’ (Chalmers 2002, p. 180).

5 Recent papers discussing Williamson’s approach include Jenkins 2008, Vaidya 2010, and

Roca-Royes 2011.

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not to our capacities of imagination or conception merely as such, but

much more specifically to our capacity to arrive at seemingly reliable

judgements concerning the truth or falsehood of counterfactual con-

ditionals. He allows that this capacity may call on our powers of im-

agination in many cases, but, he suggests, this happens in a suitably

constrained way in counterfactual thinking with the consequence that

such thinking must be acknowledged to be a genuine source of know-

ledge, even if only a defeasible one that may need to be corrected by

further processes of thinking of the same kind. He offers us a typical

example of the product of such thinking in the form of the following

counterfactual, asserted about a rock that is observed to roll down a

hillside only to be blocked in its descent by a bush:

(1) If the bush had not been there, the rock would have ended in

the lake

It seems clear that we engage in counterfactual thinking of this kind

continually in everyday life, and that it is indispensable to us in our

processes of planning future courses of action and predicting possible

outcomes of events. Williamson maintains that our ability to make

judgements of metaphysical possibility and necessity is just a

by-product of this natural way of thinking, which has plausibly de-

veloped in humans as an evolutionary adaptation.

In order to sustain this thesis, Williamson needs to contend, and

does contend, that modal propositions are explicable in terms of

counterfactuals. For this purpose, he does not assume any specific

semantics for counterfactuals, such as the now prevailing Lewis-

Stalnaker semantics according to which, roughly speaking, the coun-

terfactual ‘A.T B’ (read as ‘If A were the case, then B would be the

case’) is true just in case B is true in the closest possible world(s) in

which A is true. However, he does make two crucial assumptions

concerning the logic of counterfactuals, which are these:

NECESSITY: h(A = B) = (A .T B)

and

POSSIBILITY: (A .T B) = (-A=-B)

Williamson then remarks: ‘Although NECESSITY and POSSIBILITY

determine no necessary and sufficient condition for the counterfactual

conditional in terms of necessity and possibility, they yield necessary

and sufficient conditions for necessity and possibility in terms of the

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counterfactual conditional’ (Williamson 2007, p. 156). The necessary

and sufficient conditions in question he shows to be the following,

where ‘o’ denotes any contradiction:6

(h) hA F (‰A .To)

(-) -A F ‰(A .To)

(The labels ‘(h)’ and ‘(-)’ are mine, not Williamson’s.) Appealing to

these alleged equivalences, Williamson then proposes that acquiring

knowledge of modal truths is just a special case of acquiring know-

ledge of counterfactual truths.

4. Some objections to the counterfactual-based account ofmodal knowledge

The foregoing line of thinking is neat enough, but I consider that it

suffers from several fatal flaws. I have no quarrel with Williamson’s

assumption of POSSIBILITY above. But I do not accept NECESSITY. I

consider that the counterfactual conditional ‘A.TB’ is stronger, not

weaker, than the strict conditional ‘h(A = B)’. In fact, I have else-

where defended the following necessary and sufficient condition for

the counterfactual conditional in terms of necessity and possibility

(Lowe 1995):

(.T) (A .T B) F (h(A = B) & (-A _ hB))

Here it may be wondered why the second conjunct in the right-hand

side of (.T) is included. One reason is simply that there are plausible

examples of cases in which the counterfactual (A.T B) fails to hold

because, although the first conjunct in the right-hand side of (.T)

holds, the second does not. Here is one: ‘If there were both rain to-

morrow and no rain tomorrow, then the streets would be dry tomor-

row’. This is a counterfactual with an impossible antecedent and a

contingent consequent. Of course, in some systems of counterfactual

logic, any such counterfactual is deemed to be ‘vacuously ’ true. But I

very much doubt that any native speaker of English would confidently

6 Williamson does not claim that the right-hand and left-hand sides of these alleged

equivalences are synonymous, but remarks that this ‘detracts little from their philosophical

significance’ (Williamson 2007, p. 160). For a challenge to this remark, see Jenkins 2008.

Since I want to deny that the alleged equivalences hold, I can afford to grant that they

would, if true, have the significance that Williamson claims for them.

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assert this particular counterfactual, which strikes me as being plainly

false. However, this is not the only reason for including the second

conjunct in the right-hand side of (.T). It turns out that the system of

counterfactual logic based on (.T) has certain general features which

are attractive, such as the fact that contraposition and strengthening

the antecedent are not unrestrictedly valid for counterfactuals,

whereas they are in a system based on (.T) minus the second conjunct

in its right-hand side.

It is easy to show that if (.T) is correct, then the following equiva-

lences, rather than Williamson’s (h) and (-), are correct, where ‘`’

is any tautology (see Appendix 1):

(h*) hA F (` .T A)

(-*) -A F ‰(` .T ‰A)

Here it may be asked why someone could not simply take the equiva-

lences (h*) and (-*) to explain modal truths in terms of counter-

factuals, just as Williamson’s alleged equivalences (h) and (-) are

supposed to do. That is not feasible, however. For (h*) and (-*) can

hold only in a system of counterfactual logic such as mine which is

reducible to standard modal logic, via a principle like (.T). By con-

trast, Williamson’s (h) and (-) can hold only in a system of coun-

terfactual logic such as David Lewis’s which is not reducible to

standard modal logic. Thus, my equivalences (h*) and (-*) and

Williamson’s (h) and (-) do not have the same logical significance.

His, if correct, entitle him to claim that modal truths are reducible to

counterfactual truths. Mine have no such implication, since they are

just trivial logical consequences, in standard modal logic, of the fact

that counterfactuals, as I specify their truth-conditions, are truth-

functions of certain modal propositions (see again Appendix 1).

In the system of counterfactual logic that I favour, ‘‰(A .T ‰B)’

may be rewritten as ‘(A ST B)’, read as ‘If A were the case, then B

might be the case’. Consequently, if (h*) and (-*) are accepted, ‘It is

necessarily the case that A’ is equivalent to ‘Whatever were the case,

A would be the case’ and ‘It is possibly the case that A’ is equivalent to

‘Whatever were the case, A might be the case’ — understanding the

antecedent ‘Whatever were the case’, as seems quite natural, to be a

way of saying ‘Whether or not X were the case’, for any arbitrarily

chosen proposition X. For ‘Whether or not X were the case, A would

be the case’ just means ‘If X were the case, A would be the case; and if

X were not the case, A would be the case’. And the latter, given (.T),

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is just equivalent to ‘It is necessarily the case that A’ (see Appendix 2).

The same holds, mutatis mutandis, with regard to ‘Whatever were the

case, A might be the case’. It seems to me that my (h*) and (-*) are

considerably more intuitive (for what that’s worth) than Williamson’s

(h) and (-), as principles stating necessary and sufficient conditions

for necessity and possibility in terms of the counterfactual conditional.

I should remark here that Williamson himself also wants to say that

‘something is necessary if and only if whatever were the case, it would

still be the case’ (2007, p. 159), although he expresses this idea as

follows, using ‘propositional quantification’:

(h+) hA F 8p(p .T A)

However, I am not at all convinced that ‘8p(p .T A)’ is the most

natural way of understanding ‘Whatever were the case, A would be the

case’. Clearly, ‘8p(p.T A)’ entails an infinite number of counterfac-

tuals, each of the form ‘X .T A’, one for every proposition X that

there is. Can ordinary speakers seriously be supposed to have this in

mind when they assert ‘Whatever were the case, A would be the case’?

Isn’t the latter, on Williamson’s construal of it, an extraordinarily big

claim to be making?Note that one of the possible values of the variable ‘p’ in

‘8p(p.TA)’ is presumably the infinitely long conjunction of all prop-

ositions, so that ‘8p(p.TA)’ apparently entails ‘If everything were the

case, A would be the case’. Indeed, if propositional quantification

really makes sense and so ‘8p(p .T A)’ expresses a proposition,

then so, by the same token, does ‘8p(p)’, which accordingly expresses

the proposition that everything is the case. (I confess that I have serious

doubts as to whether there really is any such proposition, though,

notwithstanding the fact that we can construct this form of words.)

And then from ‘8p(p.TA)’, by the rule of universal instantiation, we

can infer ‘((8p)p .T A)’, which says ‘If everything were the case, A

would be the case’. And yet it hardly seems credible that an ordinary

speaker who asserts ‘Whatever were the case, A would be the case’

could seriously intend to imply this, even if it makes sense (which, as I

say, I doubt). I do not think it will really do to reply here that ‘((8p)p

.TA)’ is just trivially true, on the grounds that whatever proposition

A is, it is one of those hypothesized to be the case in the antecedent of

this conditional. After all, A’s negation, ‰A, is another one of those

propositions, so on the current proposal we should maintain that ‘If

everything were the case, A would be the case’ and ‘If everything were

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the case, A would not be the case’ are both trivially true. Yet it is hard

to see how either, let alone both, could really be true, and calling them

‘trivially ’ true does not make this any easier, in my view. If things as

strange as these can be said to be ‘trivially true’, then I feel I have lost

my grip on both the notion of truth and the notion of triviality.

For my own part, I am inclined to think that locutions along the

lines of ‘Whatever were the case …’ typically arise in situations such as

the following. Two speakers are discussing in what circumstances a

certain projected cricket match would be called off and one says,

‘Well, if it were to rain, the match would be called off ’. The

other — who is much more sceptical about the match ever being

played — replies as follows: ‘I agree, but equally, if it were not to

rain, the match would also be called off — so, whatever were the

case, it would be called off ’. And this seems a perfectly natural con-

clusion to draw. After all, if the match would be called off in the case

of rain occurring but also in the case of rain not occurring, isn’t that

tantamount to saying that it would be called off whatever the case

might be, since any case that might obtain must either be one of

rain occurring or one of rain not occurring? Anyway, whether or

not I am right about the everyday meaning of ‘Whatever were the

case …’ in these counterfactual contexts, I am content with the ob-

servation that, according to my principles (h*) and (.T), ‘It is ne-

cessarily the case that A’ is equivalent to ‘Whether or not X were the

case, A would be the case’, for any arbitrarily chosen proposition X:

for I think that this equivalence, taken purely on its own terms, is

intuitively plausible.7

It is additionally worth remarking here that, although Williamson’s

NECESSITY and POSSIBILITY do not commit him to it, he is appar-

ently willing to accept the thesis — accepted also by David Lewis —

that ‘any sentence A is logically equivalent to ` .T A, where ` is a

trivial tautology ’ (Williamson 2007, p. 152). This, of course, is in direct

conflict with my principle (h*). But I would urge that logical intuition

and common sense are again on my side in this matter. How, for

instance, could ‘Grass is green’ seriously be supposed to be logically

equivalent to, say, ‘If grass were coloured or not coloured, it would be

green’? (Incidentally, it might be wondered how Williamson’s (h) and

7 Note, by the way, that ‘8p(p .T A)’ entails ‘(‰A .T A)’, which indeed is also equivalent

to ‘hA’ by Williamson’s account. So, according to Williamson, to say that A is necessary is

equivalent to saying that even if A were not the case, A would still be the case. This also strains

my credulity.

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my (h*) can be in conflict, given that the right-hand side of his

principle is just the contrapositive of the right-hand side of mine.

The answer, of course, is that in neither his nor my system of coun-

terfactual logic is a counterfactual logically equivalent to its

contrapositive.)

5. Further difficulties

Williamson’s proposed way of understanding metaphysical necessity

and possibility in terms of counterfactuals has other deficiencies, in

my view, in addition to what I see as being its inherently unintuitive

character. One is that it seems at best very contrived to maintain that

we reason to the necessity of a proposition A by arguing for the truth

of a counterfactual with the negation of A as its antecedent and a

contradiction as its consequent. In fact, according to my own principle

(.T), such a conditional, ‘(‰A .To)’, far from being equivalent to

‘hA’, is necessarily false, whatever proposition A is (see Appendix 3).

How indeed could there be a proposition such that, if it were not true,

a contradiction would be the case? After all, in no circumstances could a

contradiction be the case, and so in none would it be the case. My

principle (.T) concurs with this verdict, because according to it no

true counterfactual can have a contradictory consequent.8 Here it may

be protested that we often argue, in reductio ad absurdum fashion,

from the negation of a proposition to a contradiction, with a view

to establishing the truth of that proposition. That is indeed so, but we

need not suppose that such reasoning also serves to establish, by con-

ditional proof, the truth of a conditional with the proposition’s neg-

ation as antecedent and a contradiction as consequent. We should,

I believe, carefully state the rule of conditional proof so as to avoid this

implication. In any case, a proof by reductio typically only establishes

the truth of a certain proposition, not its necessary truth. Thus, for

example, one may apparently derive a contradiction from the prop-

osition ‘I do not exist’, given that any proposition predicating some-

thing of its subject entails the existence of that subject (so that ‘I do

not exist’ entails ‘I exist’), but this does not entitle us to conclude that

‘I exist’ is a necessary truth — which it plainly is not — only that it is

8 This is because, according to principle (.T), if the consequent of (A .T B) is contra-

dictory and therefore impossible, then in order for (A .T B) to be true its antecedent must be

possible; but (-A & ‰-B) is inconsistent with h(A = B) in standard modal logic.

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true. Furthermore, it seems clear that there are necessary truths whichcould not be known to be such on the basis now being proposed, such

as our earlier example, ‘A uniformly coloured surface is not at onceboth red and green’. For we cannot formally derive a contradiction

from the negation of this proposition, given that the predicates ‘red’and ‘green’ are primitive and indefinable. How, then, is it proposed

that we can come to know the truth of the counterfactual ‘If a uni-formly coloured surface were at once both red and green, a contra-

diction would obtain’? If that is the only way in which one could cometo know that it is necessary that a uniformly coloured surface is not at

once both red and green, then that is not something that we know.And yet we do know this.

Yet another potential difficulty with Williamson’s account is that itseems clear that counterfactuals are endemically context-dependent,

and yet truths of metaphysical necessity and possibility cannot be, ifthey are really wholly mind-independent. (Concerning this worry,

Williamson says that ‘Infection [by vagueness and context-sensitivity]is not automatic’ (2007, p. 175), and goes on to discuss the issue very

briefly.) However, I will not dwell on this objection here, seriousthough I think it is, as I have another and, I consider, even more

fundamental one. On my own account of counterfactuals, as embo-died in my principle (.T), a counterfactual conditional is explicable in

terms of certain modal propositions. However, the account leavesopen what kind of modality may be involved in particular cases.

Some counterfactuals, as is illustrated by Williamson’s own rock-and-bush example (1) above, are causal counterfactuals, because the

modality involved is evidently natural or causal necessity. Other cases,such as examples arising in mathematics, clearly do not involve this

sort of necessity, but instead something like logical necessity. Yetothers, typically arising in philosophical contexts, involve metaphysical

necessity. Williamson, by contrast, proposes to explain the metaphys-ical modalities in terms of a single notion of counterfactual depend-

ence, leaving it quite obscure how the various different species ofmodality, in addition to the metaphysical ones, are to be recovered.

Perhaps he has the resources to do this in his own way. But note thateven if he does, his initial appeal was to our facility with causal coun-

terfactuals such as (1), since these are the ones that we typically needfor purposes of planning and prediction. If our knowledge of meta-

physical necessity and possibility is, however, supposed to be a specialcase of our knowledge of counterfactuals like these, then he is surely

misrepresenting the metaphysical modalities as a species of causal

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modality. On the other hand, if he were to appeal instead to ourfacility with a special class of counterfactuals which are non-causal

in character, this would undermine the alleged evolutionary creden-tials of the theory.

The root trouble is that Williamson’s theory is based on a purelyformal explication of modal propositions in terms of counterfactual

ones, whereas the facility with counterfactuals that he appeals to in hisaccount of how we can acquire modal knowledge draws only on our

ability to handle causal counterfactuals, which are just not relevantwhere the metaphysical modalities are concerned. And yet it is our

knowledge of metaphysical necessity and possibility that is the verysubject at issue. In short, Williamson has put together two ideas to

construct his account of our ability to acquire knowledge of meta-physical necessity and possibility: an appeal to our evolved facility

with causal counterfactuals like (1) and a formal explication ofmodal truths in terms of counterfactuals, in the shape of his principles

(h) and (-). He then proposes that we can know something to bemetaphysically necessary by coming to know a counterfactual truth of

the form prescribed by (h), and come to know this just by deploying thefacility with counterfactuals that evolution has conferred upon us. But

even if that facility delivered up counterfactuals of that prescribedform, how could they be anything other than causal counterfactuals,

with the implication that the ‘necessary truths’ that we would therebycome to know would merely be causally, not metaphysically, necessary

truths? In effect, Williamson has simply avoided the subject that histheory ostensibly aims to address. He has been misled, I think, by the

fact that his formal explication provides an equivalence between amodal truth of the form ‘hA’ and a counterfactual conditional of a

certain form, namely, ‘(‰A.To)’. But many different kinds of modaltruth have the form ‘hA’, differing from one another in respect of the

species of necessity that the modal operator ‘h’ is being taken to rep-resent, and our present concern is only with those that are truths of

metaphysical necessity. Yet the psychological mechanism thatWilliamson proposes as the supposedly reliable source of our know-

ledge of such necessities only seems applicable to counterfactuals thatare distinctively causal, not metaphysical, in character. It is not enough

for evolution to have conferred upon us an ability to formulate, incounterfactual terms, modal propositions which have the form of ex-

pressions of metaphysical necessity: they must also actually expresssuch necessities, and Williamson has done nothing to explain how

they do, much less how we can acquire knowledge of their truth.

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6. Essence and metaphysical modality

Currently dominant accounts of the traditional metaphysical distinc-

tion between essence and accident attempt to explain it in modal terms,

and more specifically in terms of the notions of metaphysical necessity

and possibility. These in turn are commonly explicated in terms of the

language of ‘possible worlds’ (Kripke 1980). Thus, a property F is said

to be an essential property of an object a just in case, in every possible

world in which a exists, a is F. And a’s essence is then said to consist in

the set or sum of a’s essential properties. One difficulty of this ap-

proach, brought to our notice by the work of Kit Fine (1994), is that it

seems grossly to over-generate essential properties. For instance, by

this account, one of Socrates’ essential properties is his property of

being either a man or a mouse and another is his property of being such

that 2 + 2 = 4. It might be objected that these are not genuine proper-

ties anyway and so a fortiori not essential properties of Socrates. But

there are other examples which cannot be objected to on these

grounds, such as Socrates’ property of being the sole member of the

set singleton Socrates, that is, the set {Socrates} whose sole member is

Socrates. Fine urges, plausibly, that it is not part of Socrates’ essence

that he belongs to this set, although it is plausibly the case that it is

part of the essence of singleton Socrates that Socrates is its sole

member. The modal account of essence cannot, it seems, accommo-

date this asymmetry. These points are too well-known for it to be

necessary for me to dwell on them further. Suffice it to say that I

am persuaded by Fine’s objections to the modal account of essence

and accept the lesson that he draws: that it is preferable to try to

explicate the notions of metaphysical necessity and possibility in

terms of the notion of essence, rather than vice versa.9 This may

also enable us to dispense with the language of possible worlds as a

means of explicating modal statements. That would be a good thing,

in my view, since I regard this language as being fraught with onto-

logical difficulties, even if it can sometimes have a heuristic value.However, if we are to take this alternative line of approach, we need,

of course, to provide a perspicuous account of the notion of essence

which does not seek to explicate it in modal terms. Fortunately, we do

9 Recent papers discussing Fine’s views include Zalta 2006 and Correia 2007. To engage

with critiques of Fine’s views here would take me too far away from my main purpose in this

paper. Moreover, although my own conception of essence is similar to his, it also differs in

some important respects.

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have at our disposal some resources wherewith to accomplish this,

drawing on the Aristotelian and Scholastic traditions in metaphysics.10

A key notion here, pointed out and exploited by Fine himself, is that

of a real definition, understood as being a definition of a thing (a res, or

entity), in contradistinction to a verbal definition, which is a definition

of a word or phrase. A real definition of an entity, E, is to be under-

stood as a proposition which tells us, in the most perspicuous fashion,

what E is — or, more broadly, since we do not want to restrict our-

selves solely to the essences of actually existing things, what E is or

would be. This is perfectly in line with the original Aristotelian under-

standing of the notion of essence, for the Latin-based word ‘essence’ is

just the standard translation of a phrase of Aristotle’s which is more

literally translated into English as ‘the what it is to be’ or ‘the what it

would be to be’ (Lowe 2008a, p. 35). We find a similar turn of phrase

in Locke’s Essay, where he tells us that the word ‘essence’, in what he

calls its ‘proper original signification’ just means ‘the very being of any

thing, whereby it is, what it is’ (Locke 1975, III, III, 15).

It will be helpful at this point to proceed by way of examples, the

first of which I borrow from Spinoza. Consider a familiar geometrical

figure, such as a circle. And suppose that someone asks us what a circle

is. This can be understood as a request for a real definition of this kind

of geometrical figure — not a request for the meaning of the English

word ‘circle’, for which we would do well simply to resort to a good

English dictionary. And here, plausibly, is the real definition that is

required (one that will typically be found in textbooks of elementary

geometry):

(C1) A circle is the locus of a point moving continuously in a

plane at a fixed distance from a given point

The given point in question is, of course, the circle’s centre. This

formula or recipe tells us what a circle is, and it does so by revealing

its generating principle — what it takes for there to be or, more exactly,

for there to come into being, a circle.I do not mean to imply here that real definitions must always pro-

ceed by way of revealing generating principles — or what, as we shall

shortly see, Spinoza calls ‘proximate causes’ — since this will be ap-

propriate only in the case of entities that are in a suitable sense capable

10 Had the present paper been more historically oriented, I might also have tried to make

connections at this point with the work of Husserl, but as it is I have neither the space nor, I

fear, the exegetical expertise to pursue this important line of inquiry here.

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of ‘generation’ (not, for instance, in the case of God, as traditionallyconceived, nor, plausibly, in the case of universals, at least if these are

regarded as eternal entities). Moreover, although all of my geometricalexamples will be drawn from Euclidean geometry, nothing of signifi-

cance hinges on this for my present purposes. The question of which

geometry best describes the space of the actual physical world is inlarge measure an empirical one. But we must at least understand

what any given Euclidean or non-Euclidean figure is, by grasping itsessence, if we are to be able to determine, on empirical grounds,

whether or not figures of that type actually characterize regions ofphysical space.

Here is another geometrical example:11

(E1) An ellipse is the locus of a point moving continuously in a

plane in such a fashion that the sum of the distances be-tween it and two other fixed points remains constant

These fixed points are the ellipse’s foci. By contrast, consider thisalternative description of an ellipse (as a type of conic section),

which is equally true of all ellipses:

(E2) An ellipse is the closed curve of intersection between a cone

and a plane cutting it at an oblique angle to its axis greaterthan that of the cone’s side

This, I suggest, tells us a necessary property of all ellipses, but not the

essence of an ellipse — what an ellipse is. For it does not capture an

ellipse’s generating principle. It characterizes an ellipse in terms thatare extrinsic to its nature as the particular kind of geometrical figure

that it is. Certainly, one can make an elliptically shaped surface bycutting a cone in the prescribed fashion, but this procedure does not

really explain why it is that what is so produced is an ellipse. On theother hand, once we understand what an ellipse is, by learning its real

definition, we can go on to understand why it is that the cuttingprocedure in question generates an ellipse, as opposed to any other

11 It may be noted that this definition, which I take from a typical dictionary of mathem-

atics (Baker 1961, p. 120), prevents us from regarding a circle as being a special case of an

ellipse. However, it is debatable whether we should do that anyway. The standard equations of

a circle and an ellipse are certainly different, being x2 + y2 = r2 and x2/a2 + y2/b2 = 1 respectively.

But it is easy to modify definition (E1) so as to avoid this consequence, if that is desired.

Instead of speaking of ‘two other fixed points’, we can speak of ‘a fixed point P and a fixed

point Q’, leaving it open whether these points are identical or different. A corresponding

modification to (E2) is equally easily made, by substituting ‘right or oblique angle’ for ‘oblique

angle’.

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kind of geometrical figure. The reverse does not seem to hold: takingan ellipse to be the shape we get when we cut a cone in the prescribed

fashion does not help us to understand why an ellipse is the locus of apoint moving continuously in a plane in such a fashion that the sum

of the distances between it and two other fixed points remainsconstant.

So, at least, I suggest. The necessary property of all ellipses that I havejust identified — that of being a closed curve of intersection between a

cone and a plane cutting it at an oblique angle to its axis greater thanthat of the cone’s side — holds of all ellipses not purely in virtue of theiressence, but at least partly in virtue of the essence of a quite different

kind of geometrical object, a cone. That is what I mean by saying thatto characterize an ellipse in terms of this property is to characterize it

in terms that are extrinsic to its nature as the particular kind of geo-metrical figure that it is. Here is another way of making this point: an

ellipse can exist even in a purely two-dimensional space, but a conecan exist only in a space of at least three dimensions — hence it cannot

be right to define an ellipse in terms of its relationship to a cone, sinceellipses can exist perfectly well without cones.12 Yet another way of

making the same point is the following: an ellipse evidently does notdepend for its identity on any cone of which it may happen to be asection, but it does depend for its identity on the distances between its

foci and the sum of the distances between them and any point on theellipse.

As I intimated earlier, the view of essence and real definition that Ihave just been articulating is one with a lengthy philosophical pedi-

gree. We find it, for instance, in Spinoza’s On the Improvement of theUnderstanding, where indeed he uses the example of a circle with

which I began:

A definition, if it is to be called perfect, must explain the inmost essence of

a thing, and must take care not to substitute for this any of its

properties. … If a circle be defined as a figure, such that all straight lines

drawn from the centre to the circumference are equal, everyone can see

that such a definition does not in the least explain the essence of a circle,

but solely one of its properties. … If the thing in question be created, [its]

definition must … comprehend [its] proximate cause. For instance, a circle

12 Note that it is still true of such an ellipse that it would be a conic section of the kind

specified in (E2), if cones existed, and in that sense it can still be said to possess the necessary

property in question.

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should, according to this rule, be defined as follows: the figure described by

any line whereof one end is fixed and the other free. … [The] definition of

a thing should be such that all the properties of that thing, in so far as it is

considered by itself, and not in conjunction with other things, can be

deduced from it, as may be seen in the definition given of a circle: for from

that it clearly follows that all straight lines drawn from the centre to the

circumference are equal. (Spinoza 1955, p. 35)

In citing Spinoza here, I do not mean to suggest that I endorse every

aspect of his conceptions of essence and real definition. For my pur-

poses, perhaps the most important feature of the foregoing passage

from On the Improvement of the Understanding is the distinction that

Spinoza draws between the ‘inmost essence’ of a thing, as revealed by

its real definition, and that thing’s ‘properties’, by which he means

necessary features of the thing that are not included in its very essence,

but which may be ‘deduced’ from its real definition. In drawing this

distinction, Spinoza was, of course, merely falling in line with a trad-

ition reaching back, via the mediaeval Scholastics, to Aristotle. It will

be noted, incidentally, that in the cited passage Spinoza uses more

than once the language of explanation — and I have no objection to

that. Real definitions are, doubtless, in some sense explanatory prin-

ciples, serving as they do to ‘improve our understanding’ concerning

the natures of things. But the general notion of explanation is a broad

and multifaceted one, with explanations falling into a number of dif-

ferent species (for instance, logical, mathematical, causal, teleological,

and psychological). The general notion of explanation is not, conse-

quently, a notion that is fit to be appealed to in order to frame a

perspicuous account of essence. Rather, I think we should regard

essence-based explanation just as one more distinctive species of ex-

planation — and this requires us to provide an account of essence

which does not simply appeal to an already assumed notion of ex-

planation. I believe that the Aristotelian approach to essence in terms

of real definition, if carefully and judiciously pursued, achieves pre-

cisely this.

Now, any essential truth is ipso facto a metaphysically necessary truth,

although not vice versa: there can be metaphysically necessary truths

that are not essential truths — understanding an essential truth to be a

truth concerning the essence of some entity. If we can truly affirm that

it is part of the essence of some entity, E, that p is the case, then p is an

essential truth and so a metaphysically necessary truth. Thus, for ex-

ample, it is part of the essence of a certain ellipse, E, that its foci are a

certain distance apart, whence it follows that it is metaphysically

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necessary that E ’s foci are that distance apart. By something’s being a‘part of the essence’ of a certain entity, I just mean that it either is the

whole essence of that entity or else is properly included in its essence.Thus, for instance, since the essence of an ellipse is that it is the locus of

a point moving continuously in a plane in such a fashion that the sum ofthe distances between it and two other fixed points remains constant, and

having foci a certain distance apart is properly included in this essence,it is part of the essence of an ellipse that it has foci a certain distance

apart — but this is obviously not the whole essence of an ellipse, sinceit is also part of the essence of an ellipse that the sum of the distances

between its foci and any point on the ellipse is constant. Consider nowa metaphysically necessary truth such as the fact that an ellipse is the

closed curve of intersection between a cone and a plane cutting it at anoblique angle to its axis greater than that of the cone’s side. It is not part

of the essence of any ellipse that this condition holds, nor is it part ofthe essence of any cone that it does. What is very plausible to contend,

however, is that this metaphysically necessary truth holds in virtue ofthe essences of an ellipse and a cone, which are two quite distinct

essences. It is because of what an ellipse is, and what a cone is, thatthis relationship necessarily holds between ellipses and cones. But it is

not part of anything’s essence that it holds. For ellipses and cones,which are the only things whose essences have a role to play in ex-

plaining why this necessary truth holds, are quite different things. Noris there any such thing as a ‘cone-ellipse’, part of whose essence it

could be that this truth obtains.13 Our proposal concerning metaphys-ically necessary truths is, then, this: a metaphysically necessary truth is

a truth which is either an essential truth or else a truth that obtainsin virtue of the essences of two or more distinct things. On this ac-

count, all metaphysical necessity (and by the same token all meta-physical possibility) is grounded in essence.

A concern that might be raised here is that our example of ellipsesand cones concerns geometrical objects, rather than material ones —

for it might be suspected that our account cannot easily be extended tocover the latter. I think this concern is unfounded. Consider instead,

for instance, material objects of the following two kinds: a bronzestatue and a lump of bronze. I would urge that it is a metaphysically

necessary truth, obtaining in virtue of the essences of such objects —

13 Strictly, of course, I should have said that planes are also involved in the explanation. But

that does nothing to diminish the point being made and there is, in any case, a good reason to

distinguish the role of planes in the explanation from that of cones, since planes must on any

account feature in the essence of ellipses, the latter being essentially planar figures.

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obtaining, that is to say, in virtue of what a bronze statue is and what alump of bronze is — that at any time at which it exists a bronze statue

coincides with a lump of bronze, which is numerically distinct from thatstatue. Likewise, it is a metaphysical possibility, again obtaining in

virtue of the essences of such objects, that the same bronze statueshould coincide with different lumps of bronze at different times.

And I say this despite the protestations of some metaphysicians thatthey do not understand how such things can be the case — how, for

instance, two numerically distinct things, one of them a bronze statueand the other a lump of bronze, can be composed of exactly the same

bronze particles at one and the same time (Olson 2001). If they genu-inely do not understand this, then I say that they have not properly

grasped what a bronze statue is or what a lump of bronze is. For ingrasping those essences they would grasp what the identity and per-

sistence conditions of such objects are, and thereby know that theseare different and that in virtue of these different identity and persist-

ence conditions the truths that they purport not to understand areindeed truths (Lowe 2003, Lowe 2008b).

Of course, I recognize that my view of statue/lump cases is a con-troversial one, which will be disputed by many so-called ‘four-dimen-

sionalist’ theorists of material persistence, even if I am far from beingalone in holding it. But this only goes to show that philosophers can

have honest disagreements about questions of essence. I would onlyseek from my four-dimensionalist opponents an acknowledgement

that if I am right concerning the essences of statues and lumps, thenthere is no mystery as to how and why they can coincide spatiotem-

porally and yet be numerically distinct. If they were to recognize this,they might even be brought to see that their opposition to the ‘three-

dimensionalist’ view — which even they acknowledge to be common-sensical — is unfounded, for one of their chief objections to it is the

alleged mysteriousness of spatiotemporal coincidence between numer-ically distinct material things, to which three-dimensionalism seems to

be committed.While on the topic of the essences of material objects, it might be

appropriate to say something more about the essences of natural kindsof material substance, such as water, briefly touched upon earlier in

connection with the frequently made claim that it is a metaphysicallynecessary, but a posteriori, truth that water is H

2O. As I indicated

then, I am sceptical about such a claim, and correspondingly scepticalthat it is truly part of the essence of water that all and only water is

(chiefly) composed of H2O molecules. This is a thorny issue that is too

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complex to be gone into in detail here, but let me just say that my ownview is that it is only naturally, not metaphysically, necessary that all

and only water is (chiefly) composed of H2O molecules, because the

natural laws governing the chemical behaviour of hydrogen and

oxygen atoms could have been significantly different, with the conse-quence that atoms of those types in chemical combination might not

have composed a substance with any of the typical chemical propertiesof water (a point rather compellingly made in Putnam 1990a; see also

Lowe 2011).It will be recalled that, according to the currently prevailing modal

account of essence, an entity ’s essence consists in the set or sum of itsessential properties, these being the properties that it possesses in

every possible world in which it exists. Hence, according to thisview, an entity ’s essence is a further entity, namely, a set or sum of

certain properties. According to my version of the neo-Aristotelianaccount of essence, however, an entity ’s essence is not some further

entity (Lowe 2008a, pp. 38–40). Rather, an entity ’s essence is just whatthat entity is, as revealed by its real definition. But what E is is not

some entity distinct from E. It is either identical with E (and somescholars think that this was Aristotle’s view) or else it is no entity at all:

and the latter is my own view. On my view, we can quite properly saythat it is part of the essence of a certain entity, E, that it possesses a

certain property, P. But this does not entitle us to say that P is a part ofthe essence of E (Lowe 2008a, p. 39). The latter would imply that E ’s

essence is a further entity, with P as a part, which accords with theorthodox view that E’s essence is a set or sum of certain properties.

But I have rejected that view. We should not, in my opinion, reifyessences. And although I speak of essences as having ‘parts’, I have

already explained what I mean by this, in a way that does not requireus to reify essences. Note that there is a particularly objectionable

feature of the view that an entity ’s essence is some further entity.This is that, since it seems proper to say that every entity has an

essence, the view generates an infinite regress of essences (Lowe2008a, p. 39). Neither the view that an entity is identical with its

own essence, nor my preferred view that an entity ’s essence is notan entity at all, has this defect. And my view, as we shall shortly see,

has an additional advantage when we come to consider the epistem-ology of essence, that is, the proper account of our knowledge ofessence.

Before we come to that, however, we must consider a prior ques-

tion. I have said that an entity ’s real definition reveals its essence —

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tells us what that entity is. But is it the case that every entity has a realdefinition? If real definitions are at all similar to verbal definitions, one

might suppose that at least some entities are ‘really indefinable’. Forinstance, I suggested earlier that the predicates ‘red’ and ‘green’ are

verbally indefinable, meaning thereby that we cannot find complexsynonyms for them employing other terms, in the way that ‘unmarried

male’ is a synonym for ‘bachelor’. Is it similarly true that we cannotsay, non-circularly, what red is? I do not think so, because I think we

can provide red with an ostensive real definition. I can point to acolour-sample and say ‘red is this colour’, or perhaps, more precisely

(since we need to acknowledge the many different shades of red) ‘red isany shade of colour that is similar in hue to this shade’. Notice that the

predicate ‘is similar in hue to this shade’ is certainly not a synonym for‘is red’, so such an ostensive definition will not serve as a verbal def-

inition of the word ‘red’. It might be objected that, by this account, itwill be part of the essence of red that it is similar in hue to this shade,

where ‘this shade’ picks out the ostended shade — and yet, one mightsuppose, red could easily have existed even if this shade had not.

However, by ‘this shade’ I assume that we are referring to a certaincolour-universal, not to a certain colour-trope, even if this kind of

demonstrative reference is a case of ‘deferred’ reference — that is tosay, even though it is secured via ostension of a certain colour-trope.

And it is not objectionable, I think, to hold that there is an essentialconnection between all the determinate colour-universals that are

shades of red. Anyway, I do think we should accept that we humanbeings, at least, can sometimes only provide ostensive real definitions

for entities of certain kinds, but that such definitions are as genuinelyrevelatory of essences as are non-ostensive real definitions.

Note that the foregoing case of colours is not one in which it canplausibly be maintained that a real definition should proceed by way

of revealing the ‘generating principle’ of the entity being defined, aswas appropriate in the Spinozan example of a circle. For one thing, I

am speaking of colour-universals and, as I observed earlier, it is notplausible to talk of ‘generating principles’ in their case. It is true that

we can, in an efficient causal sense, ‘generate’ instances of certainphysical colours by, for example, mixing lights of different wave-

lengths. However, the ‘colours’ of which I am now speaking are phe-nomenal colours, and Locke seems right to have supposed that these

all have simple natures, apparently rendering them amenable only towhat I am calling ‘ostensive’ real definition — at least as far as a

human grasp of their essence is concerned. This, however, then

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raises the question of how I can warrant my earlier claim that it ismetaphysically necessary that a uniformly coloured surface is not at

once both red and green. For how, it may be asked, does this followfrom the supposed ostensive real definitions of red and green? My

initial answer is that in grasping what red is and what green is, bygrasping ostensive real definitions of them, we thereby also grasp

their mutual exclusivity, and thus that what is red is necessarily notgreen. There is no formal logical deduction of this available, as it is

formally deducible from the real definitions of a square and a trianglethat no geometrical figure can be both square and triangular. But, I

suggest, we can still infer, with equal assurance, from appropriateostensive real definitions of red and green, that no uniformly coloured

surface can have both of these colours simultaneously. However, I alsohave a supplementary answer to the question just posed. This is that a

satisfactory real definition of colour will imply that differentcolour-determinables, and different colour-determinates of the same

colour-determinable, are mutually exclusive. Given, then, that red andgreen are different colour-determinables, their mutual exclusivity fol-

lows. These two answers are by no means in tension with each other:indeed, they are mutually complementary. For ostensive real defin-

itions of red and green along the lines proposed above — for example,‘red is this colour’, or ‘red is any shade of colour that is similar in hue

to this shade’ — define them as being colours, whence those definitionsinherit the implications of the real definition of colour.

I acknowledge that some of the claims made in the preceding twoparagraphs are controversial and call for further examination. For

now, however, I would only point out that these claims are not centralto my overall position concerning essence and real definition and that

I am, in principle, open to persuasion that some entities are, as I put itearlier, ‘really indefinable’ — in which case, it seems, the only meta-

physically necessary truths concerning such entities will be perfectlygeneral ones applying to any entity whatever, such as that every entity

is necessarily identical with itself. Clearly, though, red and green are atleast essentially colours, so that even if the difference between them

were deemed to be ‘really indefinable’, we should surely still allowthat they admit of partial real definition. I have highlighted the case

of red and green precisely in order to open up for discussion theimportant question of whether all entities can reasonably be supposed

to have (complete) real definitions. A negative answer to this questionwould, of course, by no means imply that no entities have real defin-

itions (whether partial or complete).

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7. An essence-based account of modal knowledge

Given that all metaphysical modality is grounded in essence, we can

have knowledge of metaphysical modality, provided we can have

knowledge of essence. Can we? Most assuredly we can. We have al-

ready seen this in the case of geometrical figures, such as an ellipse.

Knowing an entity ’s essence is simply knowing what that entity is. And

at least in the case of some entities, we must be able to know what they

are, because otherwise it would be hard to see how we could know

anything at all about them. How, for example, could I know that a

certain ellipse had a certain eccentricity, if I did not know what an

ellipse is? In order to think comprehendingly about something, I surely

need to know what it is that I am thinking about (Lowe 2008a, pp. 35–6).

And sometimes, at least, we surely succeed in thinking comprehend-

ingly about something — for if we do not, then we surely never suc-

ceed in thinking at all, which is absurd. Here it may be objected that

currently prevailing causal or ‘direct’ theories of reference precisely

deny that a thinker must know what it is that s/he is thinking about in

order to be able to think about it. For, on this account, a thinker need

not grasp the ‘sense’ of some referring expression in order to be able

to refer successfully by means of such an expression to its referent:

rather, reference is supposedly secured by a causal connection between

the expression and the thing referred to. This is not the place for me to

enter into a full-scale debate about theories of reference. I would only

urge that, even if the causal approach accommodates some of our

referential practices, it does not make sense to suppose that, for suc-

cessful reference to occur, no one need ever know what it is that they

are thinking about. It suffices for my purposes that at least sometimes

a thinker must be able to know this.Here another objection may be raised, as follows. Sometimes, it may

be conceded, we have a concept of what it is that we are thinking

about: but that is all, and we cannot be entitled ever to suppose

that anything actually falls under that concept. For instance, we

might have the concept of a table, and believe that we are thinking

about such a thing: but for all that, it might well be that in reality there

are no tables, and hence no table for us to be thinking about. But let us

remind ourselves what concepts are. And note that in thus reminding

ourselves we are recalling the essence of a concept: what a concept is. I

earlier proposed that a concept is a way of thinking of some thing or

kind of things. But ways of thinking of things can be more or less

adequate to the nature of the things in question — that is, more or

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less adequate reflections of their essence.14 A child’s way of thinking ofa triangle is less adequate than that of an experienced geometer. A

concept of an entity E is fully adequate only if it captures the wholeessence of E. Now, I concede that a thinker may be able to think of

some entity without fully grasping its whole essence, and this is nodoubt the case with the child’s thought about triangles. But I see no

reason to suppose that we may never fully grasp the whole essence ofany kind of entity. Bear in mind, too, that we want to allow that we

can grasp the essences not only of actually existing things, but also, atleast sometimes, of non-existent things — things such as unicorns and

mermaids, perhaps. So I can happily allow that sometimes, in thinkingabout something, we succeed in grasping its essence and hence in

thinking about it, even though no such thing actually exists. I believethat I know, for instance, what a table is or would be, and hence grasp

the essence of such a thing and can thereby think about tables: but thisdoes not commit me to acknowledging the actual existence of tables. I

might be open to persuasion, by a suitably ingenious philosopher, thattables do not really exist (van Inwagen 1990, Merricks 2001). But if I am

even to be able to understand what such a philosopher is contending, Imust surely know what a table is or would be — what its essence is.

Otherwise, I do not really know what it is whose actual existence thephilosopher is denying, when s/he denies that tables actually exist.

However, only a global sceptic is going to affirm that, quite possibly,nothing that we can think about actually exists, including even our-

selves and our thoughts. And such an all-embracing scepticism isself-undermining and incoherent. So I find nothing in the present

line of attack to suggest that I am wrong in supposing that we can,at least sometimes, grasp the essences of at least some things — includ-

ing some actually existing things — and thereby come to know somemodal truths.

I should stress that nothing of significance turns on my choice hereof tables as an example, which I use merely for purposes of illustration.

I am, in principle, open to persuasion not merely that no tables reallyexist, but even, perhaps, that there is nothing that a table is or would

be — that is, that there is no essence to be grasped in this instance andso no corresponding real definition that is available. Perhaps a phil-

osopher who regards ‘tables’ in much the same way that Wittgenstein

14 I use the term ‘adequate’ here in conscious emulation of its use by seventeenth-century

philosophers such as Descartes and Arnauld, but as the present paper is not primarily a

historical one I am loath to go into the question of how far such philosophers would agree

with the characterization of adequacy that I offer immediately below.

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regarded ‘games’ would seek to persuade me of this. My response here

is not to dig my heels in over the particular case of tables, but only to

insist that there is a limit to how far such a scepticism about essences

can be taken, and that this limit must, once more, fall far short of a

global scepticism. At the same time, I think it should not too readily be

conceded that neither tables nor even games are susceptible to real

definition. Framing a satisfactory real definition is by no means always

an easy matter, and difficulties in doing so should not always be

assumed to be indicative of a misguided quest.15

I mentioned earlier that, according to my account of essence, es-

sences are not entities. This means that grasping an essence — knowing

what something is — is not, by my account, a kind of knowledge by

acquaintance of a special kind of entity, the thing in question’s essence.

All that grasping an essence amounts to, on my view, is understanding

a real definition, that is, understanding a special kind of proposition. To

know what a circle is, for instance, I need to understand that a circle is

the locus of a point moving continuously in a plane at a fixed distance

from a given point. Provided that I understand what a point and a plane

are, and what motion and distance are, I can understand what a circle

is, by grasping this real definition. And bear in mind that I do not

insist that we need fully grasp the whole essence of a thing in order to

be able to think about it to some degree adequately, so that even if I do

not fully grasp what motion, say, is, I can still achieve at least a partial

grasp of what a circle is by means of the foregoing real definition. If, by

contrast, knowledge of essence were knowledge by acquaintance of a

special kind of entity, then indeed we would have cause to be doubtful

about our ability ever to grasp the essences of things. For what mental

faculty of ours could possibly be involved in this special kind of ac-

quaintance? Surely not our faculty of sense perception. Sense percep-

tion may provide us with knowledge by acquaintance of concrete,

physical things, existing in space and time, but hardly with their es-

sences, conceived as further entities somehow grounding modal truths

about those concrete things. If appeal is instead made to some special

intellectual faculty of rational ‘insight’, with essences as its special

objects, then one is open to the charge of anti-naturalistic obscurant-

ism. My own account of what it is to grasp an essence appeals only to

an intellectual ability that, by any account, we must already be

15 In the case of games, Suits 1978 is particularly instructive concerning this. I am indebted

to Colin McGinn for drawing my attention to this remarkable book.

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acknowledged to possess: the ability to understand at least some prop-

ositions, including those that express real definitions.

We now have in place the basic ingredients for a thoroughgoing

epistemology of metaphysical modality — although, of course, much

still remains to be said about the details. Put simply, the theory is this.

Metaphysical modalities are grounded in essence. That is, all truths

about what is metaphysically necessary or possible are either straight-

forwardly essential truths or else obtain in virtue of the essences of

things. An essence is what is expressed by a real definition. And it is

part of our essence as rational, thinking beings that we can at least

sometimes understand a real definition — which is just a special kind

of proposition — and thereby grasp the essences of at least some

things. Hence, we can know at least sometimes that something is

metaphysically necessary or possible: we can have some knowledge of

metaphysical modality. This itself is a modal truth, of course, and one

that obtains in virtue of our essence as rational, thinking beings. And

since we can, it seems clear, grasp our own essence, at least sufficiently

well to know the foregoing modal truth about ourselves, we know that

we can have some knowledge of metaphysical modality. Indeed, to

demonstrate that this is so, and why it is so, has been the ultimate

objective of this paper.16

References

Baker, C. C. T. 1961: Dictionary of Mathematics. London: George

Newnes.

Bealer, George 2002: ‘Modal Epistemology and the Rationalist

Renaissance’. In Gendler and Hawthorne 2002, pp. 71–125.Chalmers, David 2002: ‘Does Conceivability Entail Possibility?’ In

Gendler and Hawthorne 2002, pp. 145–200.

Correia, Fabrice 2006: ‘(Finean) Essence and (Priorean) Modality ’.

Dialectica, 61, pp. 63–84.

Descartes, Rene 1984: Meditations on First Philosophy [1641], in The

Philosophical Writings of Descartes, Volume II, trans. J.

Cottingham, R. Stoothoff, and D. Murdoch. Cambridge:

Cambridge University Press.

16 I am grateful for comments received when earlier versions of this paper were presented

to audiences at the University of Miami, the University of Georgia at Athens, the University of

Nebraska at Omaha, and Durham University. I should also like to thank three anonymous

referees for Mind and the Editor of Mind, who very generously provided me with some

extremely helpful criticisms of preceding drafts of this paper.

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Fine, Kit 1994: ‘Essence and Modality ’. In Tomberlin 1994, pp. 1–16.Fodor, Jerry A. 1998: Concepts: Where Cognitive Science Went Wrong.

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Possibility. Oxford: Clarendon Press.Jenkins, C. S. 2008: ‘Modal Knowledge, Counterfactual Knowledge

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pp. 693–701.Kanzian, Christian (ed.) 2008: Persistence. Frankfurt: Ontos Verlag.

Kripke, Saul A. 1980: Naming and Necessity. Oxford: Blackwell.Le Poidevin, Robin (ed.) 2008: Being: Developments in Contemporary

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Lowe, E. J. 1995: ‘The Truth About Counterfactuals’. The

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Natural Kinds’. Analysis, 67, pp. 286–92.—— 2008a: ‘Two Notions of Being: Entity and Essence’. In Le

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Qualified Defence’. Proceedings of the Aristotelian Society

Supplementary Volume, 85, pp. 1–19.Merricks, Trenton 2001: Objects and Persons. Oxford: Clarendon

Press.Olson, Eric T. 2001: ‘Material Coincidence and the Indiscernibility

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Parfit, Derek 1984: Reasons and Persons. Oxford: Clarendon Press.Putnam, Hilary 1990a: ‘Is Water Necessarily H

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Appendix 1: Proof that hA F (`.T A)

By principle (.T)

(1) (` .T A) F (h(` = A) & (-` _ hA))

But (`= A) is truth-functionally equivalent to A, so that the RHS of

(1) is logically equivalent to

(2) (hA & (-` _ hA))

And (2) is truth-functionally equivalent to hA. Hence, by substitution

of logical equivalents in (1),

(3) (` .T A) F hA

QED

Appendix 2: Proof that ((X.T A) & (‰X.T A)) F hA

By principle (.T)

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(1) ((X .T A) & (‰X .T A)) F (h(X = A) & (-X _ hA) &

h(‰X = A) & (-‰X _ hA))

But (h(X= A) & h(‰X= A)) is logically equivalent to h((X= A)

& (‰X= A)) and hence logically equivalent to hA, since ((X= A) &

(‰X = A)) is truth-functionally equivalent to A. Hence, by substitu-

tion of logical equivalents, the RHS of (1) is logically equivalent to

(2) (hA & (-X _ hA) & (-‰X _ hA))

However, (hA & (-X _ hA)) is truth-functionally equivalent to hA,

whence by substitution of logical equivalents (2) is logically equivalent

to

(3) (hA & (-‰X _ hA))

which is again truth-functionally equivalent to hA. Hence, the RHS of

(1) is logically equivalent to hA and so, by substitution of logical

equivalents in (1),

(4) ((X .T A) & (‰X .T A)) F hA

QED

Appendix 3: Proof that (‰A .To) is necessarily false

By principle (.T)

(1) (‰A .To) F (h(‰A = o) & (-‰A _ ho))

However, (‰A= o) is truth-functionally equivalent to A, so that the

RHS of (1) is logically equivalent to

(2) (hA & (-‰A _ ho))

which is logically equivalent to

(3) (hA & (‰hA _ ho))

But (3) is truth-functionally equivalent to (hA & ho), which is

necessarily false. Hence, (‰A .T o), being logically equivalent to

(3), is likewise necessarily false.

QED

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lowe - what is the source of our knowledge of modal truths

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