‘Logically alien thought’ and the dissolution of ‘sorites’ paradoxes
For
sortal terms, we need a prior -- or at least, a
constitutive -- notion of the conditions under which such a term may be
properly applied. For example, there must be some condition(s) of distribution
for ‘bald’ to apply, and some condition(s) of contiguity for the application of
‘heap’, and so on. How often is this requirement observed, in discussions of
the ‘sorites’?
Discussions
of ‘the ‘sorites’ paradox’ typically involve heaps
with numbers of grains, or heads-of-hair consisting of numbers of hairs, or the
like. But a fairly clear non-case of a heap, such as 100 grains of sand
compressed together under extreme force (such that they becomes a tiny lump
that takes up only a minute amount of space), or an extremely clear non-case of
a heap, such as the same 100 grains (or indeed 1000 or 10000 grains) spread out
widely over the floor of a large hall, might well be (or have been) a heap, if
arranged appropriately: i.e. if heaped together, loosely. This crucial aspect
of what it is to be a heap is unfortunately ignored in most philosophers’
treatments of the matter.[1] Such abstract treatments are
vitiated by the omission.
(An
interesting example partly parallel to the example of a heap needing to have a
certain amount of physical looseness -- of air -- in it to BE a heap; an
example not so often considered; is the vague term, ‘meringue’. A certain
amount of egg whipped up but with very little air present will not constiute a meringue; add (in the right way!) a bunch more air, and you have your meringue...)
The point I have made can be
extended analogically to most standard sorites
paradoxes, probably to all. Think in slightly more detail, for instance, about
baldness: it is of course not just the number of hairs, but their arrangement
(and also their length, and even their thickness, healthiness, etc.) that
crucially matters: Someone whose remaining hair is concentrated in the sideburn
area is far more plausibly adjudged bald than someone with a thin spread over
the crown...
Colour
terms might be thought to be a counter-example: for don’t colours at least
sometimes have a greater simplicity or ‘purity’? We are tempted to say indeed,
that colours ‘in themselves’ are simple, unlike states of hair. But think first
of actual contexts of use of colour-terms: If I ask for a red car, I don’t
expect the tyres or windscreen to be red, but if I ask for a red ball, I expect
it to be (pretty) wholly red. On the other hand, if I ask for a red apple, I might
be happy if it was only partially red, and I wouldn’t expect its interior to be
red; whereas if I wanted my flat painted red, I would expect the paint to be
applied to its interior, not its exterior, etc.
Think
secondly of the following kind of observations, made by Wittgenstein in his Remarks on Colour: ‘The bucket which I
see in front of me is glazed shining white; it would be absurd to call it ‘grey’
or to say ‘I really see a light grey’. But it has a shiny highlight that is far
lighter than the rest of its surface (part of which is turned toward the light
and part away from it), without appearing to be a different colour. (Appearing,
not just being.).’ [2] From which, Wittgenstein concludes
that, ‘The difficulties which we encounter when we reflect about the nature of
colour...are embedded in the indeterminateness of our concept of sameness of
colour.’ [3] Only if one presupposes that our
colour concepts should be determinate -- determinative of their extensions in
all instances -- does one generate a soritical paradox in the first place.
One’s
conclusion must surely be that most discussions of sorites
paradoxes are dangerously de-contextualised.
The paradox in most (perhaps all) cases doesn’t even get off the ground without
the de-contextualization. And if one re-contextualizes, if one considers our
actual practice, looking carefully for instance at the variety of factors which combine to decide whether or not something
is a heap, etc., then philosophical puzzlement largely (even completely)
dissolves. Where the only ‘context’ available by contrast is ‘a philosophical
discussion’, or ‘a desire to solve ‘the paradox itself’’, then one has
condemned oneself to endless debate. When one enables oneself instead to look
at how we actually make judgements vis-à-vis vague predicates, then the puzzle
(thankfully) dissolves in one’s hands.
Thus
for example the bald question, [I] ‘Is
this man bald?’, or even [II] ‘Is this man bald, compared with that man?’,
needs to be seen against a background. If this man has a bald patch, and that
man has a receding hairline, [II] may be ‘undecidable’.
Which counts for more, this or that? Well, it
depends what for. The background against which the question is being asked
may enable us to decide the answer. (For instance, some monkish religious
orders may see a bald patch as perfectly standard, even de rigeur, and thus as little evidence
of baldness.) The desire for context-independent answers to questions like [I]
and [II] is as one with the desire to solve the sorites
paradox ‘itself’. It is a desire we need to grow out of.
Some
readers will not yet be fully satisfied. They will still perhaps be feeling the
pull of the paradox; the general pull, that is, towards thinking that vagueness
poses a philosophical problem that will not
happily dissolve. They will be inclined to the very abstraction that I am
warning against. Let me try to satisfy them by making first a kind of
concession ... by moving away somewhat further from the actual world and actual
(concrete, unconfused, non-metaphysical [4] ) language use: there certainly is a role in the consideration of the sorites for perhaps-peculiar ‘thought experiments’. We can
enrich ‘our diet of examples’ appropriately by considering not just ways in
which we might physically arrange and manipulate grains of sand to produce
cases that tug revealingly on our ‘intuitions’ about heaphood,
but by considering possible radically alternative socio-linguistic practices
involving heaps.
Take
a famous example from Wittgenstein’s corpus: his ‘woodsellers’.[5] These people, who seem to regard
wood piled into heaps as having less quantity (and thus as worth or costing
less than) wood that is spread out to cover a greater area on the ground, have
typically been taken by Wittgenstein’s ‘critics’ as plain stupid/irrational, or
alternatively as manifesting a conceptual impossibility. By Wittgenstein’s
‘followers’, they have been typically taken to demonstrate a true conceptual
relativism, a possible radically-other ‘form of life’.
For
reasons that will become clearer,[6] these responses seem to me
unsatisfactory. Perhaps, though, it is already evident how we might recast the
strange ‘woodsellers’: perhaps they are a
(perhaps-conceivable) would-be real example of the kind of reasoning I
displayed in the opening paragraphs of this paper, above. Perhaps the correct
way to ‘paraphrase’ the woodsellers’ talk, to render
them happily less deeply-strange, is to hear them as talking about heaps. Perhaps they have an evaluative
attitude to heaps of wood of very roughly the kind that (though with a roughly
opposite ‘valence’ to) that we have to meringues. At least in this sense: how
those heaps are ‘made’, or arranged, is of deep import to these people.
Specifically, in this case: if the planks of wood (compare grains of sand) are
de-heaped, though without being completely scattered, they gain value for these
people.
If
this were right, then the woodsellers need not be
sub-human,[7] nor oxymoronic or impossible, nor
even so different that we really cannot speak of them intelligibly (such that
they would not properly even be termed people
at all). They have significantly different preferences or values from us (so
far at least as wood is concerned), but they do not pose a problem of
‘conceptual relativity’, with all the terrible philosophical problems that that
brings in its wake.[8]
I
think that my suggestion is a live option when reading Wittgenstein’s ‘woodsellers’ scenario. It brings into prominence that most
philosophers have tended only to consider a crude set of options, when
confronted with this intriguing and difficult text of Wittgenstein’s. Rarely
has anyone attempted to give, as I have here, a way of filling out
Wittgenstein’s vague (!), lightly-specified
‘object of comparison’ such that these strangers would retain their difference
without being alien to the point of madness or sub-human-ness, etc. . The woodsellers, from my perspective, can be read as different
without having to be read, impossibly and absurdly, as ‘logical aliens’.[9]
And comparing the woodsellers’ case to our own
can yield insights into the (limited[10]) degree to which we can
intelligibly relax the constraints on what we remain willing to call
‘mathematics’, and similarly fundamental aspects of our world-view, of our
life.
But
the particular way of taking forward the comparison essayed here has the added
virtue of helping to dissolve the pull on
one of soritical reasoning. The ‘concession’ I
made above turns out to have helped my case, not hindered it.
Let
me then try to sum up the point that we have now reached. My argument does not,
I believe, contradict the reasoning of ‘New Wittgensteinian’
authors (such as James Conant, Alice Crary and David Cerbone) who have
argued that perhaps the most crucial (and all-too-rarely-understood) aspect of
‘scenarios’ of Wittgenstein’s such as that of the ‘builders’ and the ‘woodsellers’ is that Wittgenstein intends for them to collapse under the philosophical weight we are
tempted, in reading his work, to try to make them bear. The builders and the woodsellers, philosophically-speaking, yield only houses of air, whose collapse leaves no more than empty
rubble behind... When Wittgensteinian considerations
dissolve Frege’s notion of illogical thought -- a
kind of thought that would be ‘radically’ different from our own,
which is yet a kind of thought -- into nothingness, nothing useful remains in
the idea that we may have thought we had of logically-alien thought: of (e.g.)
a ‘wholly’ different mathematics which is yet a mathematics. But Wittgenstein’s
text, I would maintain, also leaves it open
whether there can be anybody whom we might on reflection want to call
‘mathematical strangers’. We might
indeed, that is, call the woodsellers as I have
interpreted them ‘mathematical strangers’, even ‘very-strangers’, but not need to see them as posing a problem of embodying logically
alien thought. There can be a reading of the woodsellers
wherein they could be actual -- and would not pose a philosophical problem for
anyone with any sensible take on these matters.[11]
In
short, I have begun to supply above, I believe, a slightly more-detailed
scenario than Wittgenstein’s own, a scenario that fills out his ‘example’ in a
way that gives it sense. The
appearance of contradiction or paradox in the description of the woodsellers can be dissolved, if we treat them as part of a
dissolution of the would-be paradox of the heap.
It
is very important not to assimilate what I have done so far to any form of objectionable conceptual
relativism or anti-realism. My argument does not amount to the claim that a
heap is no more than what we choose to call a heap. Nor does what I have done
here amount to a form of philosophical quietism.[12] My argument does not leave intact a
pressing philosophical problem of what is to count as baldness or ‘heapness’ that it simply refuses to address. Far from it.
To see why neither of these charges is apposite, it is enough I think to recall
the following remarks of Wittgenstein’s:
‘[S]omeone might object against me: ‘You take the easy way out!
You talk about all sorts of language-games, but have nowhere said what the
essence of a language-game, and hence of language, is: what is common to all
these activities, and what makes them into language or parts of language...’
And this is true.__Instead
of producing something common to all that we call language, I am saying that
these phenomena have no one thing in common which makes us use the same word
for all,__ but that they are related
to one another in many different ways. And it is because of this
relationship, or these relationships, that we call them all ‘language’.’ [13]
I do not have to tell you what
counts as baldness, any more than I have to tell you what counts as
‘game-ness’. If you are a competent practitioner of English, you know. Yet, as the underlined sentence above
-- a sentence so often ignored in accounts of Wittgenstein on
‘family-resemblance’, even, regrettably, by would-be Wittgensteinians
-- makes clear, there is no form of anti-realism here. It is the relationships
of cases of baldness to one another, their complicatedly overlapping
resemblances, that is the ‘basis’ for our understanding of what are and what
are not cases of baldness. The game-ness of games is ‘surveyably’
open to all; if you insist on continuing to ask what it is that makes games games, the best and only needful answer available is: it is
the similarities of games to one another that makes them games.[14] The edges of this concept are
variable (over time and space) and indeed we might well call them vague, but
this vagueness, as I have been arguing throughout, need not pose a problem,
need not be in any sense whatsoever a defect, except for one philosophically
pre-determined that it shall do and be so.[15] Vagueness does not imply
non-existence, and nor need it imply
that the existence of the vague object only comes at the cost of being wholly
and objectionably dependent upon us.
Games
and languages and heaps and baldness are perfectly real. The vague yet
important borderlines where candidates shade off into non-cases does not
require founding, nor does it require policing by philosophers. Competent
language-users can take care of that, as and when they need to, by themselves.
Now
for the last throw of the defender of the sorites.
Isn’t there still reason to think that there is a sorites
paradox, in the cases where other things are
equal, in the cases where the case one is dealing with does become something
like a pure numbers game (a question of numbers of hairs, or of a transition
between colours arrayed in a row, or of picking out the point in an array of
people where ‘tallness’ begins, etc.)? When context and conditions are held
constant, isn’t there still a philosophical problem to be solved?
Let
us again compare Wittgenstein; this is section 61 of the Remarks on Colour: ‘We are inclined to believe [misleadingly] that
the analysis of our colour concepts would lead ultimately to the colours of
places in our visual field, which are independent of any spatial or physical
interpretation; for here there is neither light nor shadow, nor high-light,
etc. etc. .’ I believe that such an inclination, uncomfortably unattractive or
risibly old-fashioned as it may sound, lies at the root of the residual,
hard-to-oust temptation to believe that there is a live sorites
paradox even in cases where there is a pure numbers game or something similar
in play. The unattractiveness of such a belief can I think be thrown into sharp
relief when one considers the following point: ‘tall’ or ‘heap’ or ‘red’ are
tempting candidates for soritical reasoning, but who
would want to argue that ‘a lot’ was a source of paradox? It is plainly not; it
is plainly context- and
purpose-relative. The boundary between ‘a lot’ and ‘not a lot’ is exhausted by the context and purpose of
any utterance featuring these terms. But ‘a lot’ IS a pure numbers game. Unlike
with ‘heap’, for instance, there is no question of the kind of ‘woodsellers’-type points made earlier being relevant. There
is no question of arrangement of the stuff etc. making a difference: if 10
tonnes of sugar is simply called ‘a lot’, then it is a lot whether it is neatly
piled up or accidentally spread all over the warehouse floor.
My
psycho-philosophical diagnosis of what happens when we persist in wanting to
reason soritically is as follows: we do so because we
unwittingly persist in thinking both that there are conditions of application
for the term such as the conditions of distribution and of contiguity relevant
for the ‘bald’ and ‘heap’ cases, respectively, and in thinking that there are not. Only so long as we fail to see
clearly that, when such conditions are
eliminated entirely, leaving (e.g.) only the bare numbers (of hairs, of
grains) and our particular purposes (a lot of cakes might be 3, whereas a lot
of biscuits might be 12 packets’ worth; a lot of sugar in my tea may not be a
lot of sugar at the warehouse, etc.), then the appearance of paradox really
does vanish, ...only that long, do we persist in being bamboozled by the
paradoxes of vagueness. When we simply contemplate number, in the absence of
any and all conditions of distribution and contiguity, there is no question of
that number being ‘a lot’ or ‘a little’. The problem comes -- there starts to
feel like a paradox -- when we start to include such conditions in our
considerations, so as to have a problem at all, but yet want to eliminate them entirely (as ‘impure’?) from the
answers we give to the pressing questions that then start to (at least seem to)
arise: is this a heap, or not? Is this man bald, or not? Is this worm fat, or not? Considerations
of context are through-and-through inevitable, as soon as one has allowed
conditions of distribution and contiguity to enter into one’s considerations at
all; but yet we are tempted to look for a philosophical answer to the paradox
(e.g. the claim that vagueness is merely epistemic; and/or that our language is
in itself flawed; etc.) that can transcend what, following Wittgenstein and Garfinkel,[16] I want to suggest are utterly
unavoidable and normally-utterly-unproblematic indexical features of our
utterances, our practices.
In
short, it only looks like there is the paradox of the heap so long as we -- unawarely -- (and absurdly) want the term ‘heap’ to be both context-bound and
context-independent in its use.
Sainsbury’s
(representative, standard) failure to complicate the alleged sorites paradox of colour by considering actual instances
of use of colour-terms, and by considering actual instances (which will vary)
of encounter with and use of colour-charts and the like, is prototypical. There
is no interesting sorites paradox to be garnered from
imagining an imaginary colour-chart.
One has rather to imagine one actually being looked at, employed. When one does
so, then it becomes much easier for instance to imagine aspect-shifts (which
might come at various points, depending on circumstances) in the colour one was
looking at, such that one would say of two shades that might be abstractly
indistinguishable that the point of transition between (say) red and orange
happened there. That is: I might well
not be making any kind of mistake or producing any kind of myth, if I were to
judge, if forced to pronounce on the matter, that a certain point in a real
colour-chart was where the transition from red to orange was effected. (And the
exercise would be less likely to feel merely forced, if my judgement was
consequential: e.g. if it were part of a set of decisions on a colour-scheme
for a redesign of the Philosophy Department building.) Even if, were I to be presented with the two
shades either side of the transition-point I picked, isolated, in a
psychological experiment, I could not distinguish them from one another.
The same with height: [17] If I were presented with two men
one of whom was 0.1 of an inch smaller than the other, I would most likely
judge either that both were tall or that both were small (Though the
qualification -- ‘most likely’ -- is needed, for it would depend on the
purpose). If I were to judge both as tall, and you were to judge both as small,
there is not yet a paradox any more
than there is if one of us thinks three sugars in a cup of tea a lot and the
other doesn’t. And if we lined up a whole lot of men in a great long line, and
their height gradually dropped, and everything else stayed the same (because of
course in the real world girth might affect judgements of tallness, etc.), then
there would still not need to be any paradox: at some point, if one didn’t
simply reject the whole exercise as artificial and meaningless, one would no
doubt judge that there was no longer tallness in the man before us, even if the
difference between the two men between whom one drew the line would, IF THE TWO
WERE CONSIDERED IN ISOLATION, be insignificant or even imperceptible. Given that the two were NOT being considered
in isolation, there need be no paradox.
In
sum, as Wittgenstein argues extremely powerfully between PI section 65 and 88,
there need be no enduring puzzle around vagueness, if only we do not trap
ourselves into one. There remain of course contexts in which we don’t really
know how to apply an expression, but all this tells us is that our expressions
don’t carry membership conditions with them from the abstract into the
concrete. There may sometimes not be a univocal and unequivocal answer to the
question ‘Is he bald, or not?’, or ‘Is this a heap, or not?’ But why should
this surprise us? Any more than it surprises us or dissatisfies us, outside the
priggish philosopher’s study, to hear people saying things like, ‘Well, he’s a bit bald’, or, simply, ‘Yes and no’.[18]
[1] E.g. It is given no role whatsoever in R.M. Sainsbury’s respected textbook treatment, (in chapter 2 of) his Paradoxes (Cambridge: C.U.P., 1995).
[2] Transl. Anscombe, Berkeley: U Cal. Press, 1977, Part I, section 50. I have emended the translation slightly. See also sections 63, 67 and 59.
[3] Section 56.
[4]
For explication of what this means, see for instance Gordon Baker’s
“Wittgenstein on metaphysical/everyday
use”, Phil Quarterly Vol. 52 (July
‘02), 290-302, reprinted in his Wittgenstein’s Method (
I am not, of course, entertaining or suggesting the risible claim that people never reason soritically to the point of paradox outside Philosophy Departments; I am however suggesting that people do not end up reasoning soritically to the point of paradox once they have overcome metaphysical confusion and abstractions. When people reason soritically in everyday life, the reasoning is most typically brought to an end by their interlocutors responding at some point (roughly) that a point has been stretched beyond breaking-point, that they have taken their argument too far.
[5]
See Remarks on the Foundations of Mathematics (Oxford: Blackwell, 1978
(1956)), Part 1 sections 140-153, especially 149-150.
Alice Crary gives these
people (if that is the right word -- see below) the potentially more fruitful
name, ‘mathematical strangers’. See her powerful discussion in “Wittgenstein
and political philosophy”, The New Wittgenstein (eds. Crary and Read,
[6] And that are already clear in Crary’s (ibid.), and in David Cerbone’s deeply-intriguing “How to do things with wood”, also in Crary and Read (ibid.).
[7] For Sainsbury (ibid.), presumably, the woodsellers would be this, or at best (!) plain stupid. If it is clear and indubitable that only the number of grains or planks matters (and not their arrangement), as Sainsbury very much implies, then the woodsellers are simply wrong. But sometimes, whether something is a heap (or whatever vague term you please) matters; it is not only the quantity of matter that matters. For example, slightly fancifully but not I think absurdly: if there were a democratically-agreed scheme for bald men to be compensated for their allegedly-lower quality of life, it would presumably be their baldness (or otherwise), and not the sheer number of hairs on their head, that was considered to be to the point. A broadly supervaluational account -- suitably deflated by a broadly-Wittgensteinian emphasis on context and practice -- such as that offered by David Houghton (in his “Vagueness, stipulation and context”, UEA Papers in Philosophy New Series no.11 (2000), pp.1-24) may help at this point, in enabling us to recover the ordinary. That is, to see through our philosophical bewitchment, and to see clearly (again) how we can reliably and (crucially) purpose-relatively judge that such-and-such is or is not bald, without having to solve ‘the sorites paradox’ in the abstract.
[8]
As exposed by, for example, Davidson, “On the very idea of a conceptual scheme”
(in Inquiries into truth and interpretation (Cambridge: C.U.P., 1983));
though, contra Davidson, I think that
some of those who he criticizes as conceptual relativists, notably Kuhn, are in
fact quite consonant with the line of thought I am sketching in this paper. For
detail on how to interpret Kuhn as largely invulnerable to correct Davidsonian reasoning, see my Kuhn (
[9] See the discussion below of Conant et al on Wittgenstein (and Frege) on “a hitherto unknown kind of madness” : ‘logically alien thought’.
[10] See again Crary’s and Cerbone’s papers, for a forceful account of the limitation.
[11] The ‘view’ which I seem to be putting forward here may itself seem oxymoronic or radically confused. I shall attempt to show below that it is not, in part by attempting to show that it is not well-read as a view, at all.
[12] Unless perhaps by ‘quietism’ one means to index a line of thought such as that of Cerbone at the close of his (op. cit.).
[13] Philosophical Investigations (Oxford: Blackwell, 1958 (1953); henceforth PI) section 65; underlining mine.
[14] Call this the truth in essentialism, if you like. Hopefully, this ‘truth’ is ‘woolly’ and contentless and ‘question-begging’ enough to seem as trivial and uncontroversial as I mean it to sound.
[15] On this, see especially Wittgenstein’s masterly discussion in for instance sections 68-71, 76-7, 80-84, & 88, of PI.
[16] Harold Garfinkel, the founder of the anti-social-science of ‘ethnomethodology’; see e.g. his Studies in Ethnomethodology (Cambridge: Polity, 1984 (1967)); see also expository work on ethnomethodology and indexicality etc. by Mike Lynch, Wes Sharrock and Rod Watson.
[17] See p. 23 of Sainsbury’s (op.cit.).
[18] My thanks to the members of the Philosophy Evening Reading Group at UEA, especially to John Collins, Cathy Osborne and Nadine Cipa.