Author: Bill Vallicella

More on Translating ‘Something Exists’ and a Response to Brightly

I issued the following challenge: translate 'Something exists' into standard first-order predicate logic with identity. This is the logic whose sources are Frege and Russell. So I call it Frege-Russell logic, or, to be cute, 'Fressellian' logic.  My esteemed commenters don''t see much of a problem here.  So let me first try to explain why I see a problem.  I then  consider David Brightly's proposal.

1. First of all, 'Something exists' cannot be rendered as 'For some x, x exists.'  This is because 'exist(s)' is not an admissible first-level predicate in Frege-Russell logic.  The whole point of the Fressellian approach is to make 'exist(s)' disappear into the machinery of quantification. There is no such propositional function as 'x exists.'  'For some x, x exists' is gibberish, syntactic nonsense in Frege-Russell logic. 

2. But the following is not gibberish: 'For some x, x = x.'  So one will be tempted to say that 'Something exists' can be rendered as 'For some x, x = x,' ('Something is self-identical') and 'Everything exists' as 'For all x, x = x' ('Everything is self-identical'). 

But this won't work either.  It is true that everything that exists is self-identical, and vice versa.  But it doesn't follow, nor is it true, that existence is self-identity. Here is one consideration.  When I say of Tom that he exists, I am not saying that he is self-identical. Suppose I hear a false rumour to the effect that Tom is no more.  But then I encounter him in the flesh.  I exclaim, "You still exist!"  Clearly, "You are still self-identical" does not mean the same.  If I said that, Tom might retort, "What the hell, man, were you worried that I had become legion?"  In some circumstances, that a man should continue in existence is surprising.  But we are never surprised by a man's continuing in self-identity.

Furthermore, when Tom ceases to exist, he does not become self-diverse.  Loss of existence is not loss of self-identity.  To put the point in formal mode, after his demise 'Tom' continues to refer to one and the same individual, Tom.  The bearer of the name is gone, but not the reference. Otherwise it could not be true that Tom is gone.  There is also a modal consideration.  Tom is a contingent being: he exists but he might not have existed.  If existence is self-identity, then Tom's possible nonexistence is Tom's possible self-diversity — which is absurd.  It makes prima facie sense to say of an individual that it might not have existed or that it no longer exists; but it make no sense at all to say of an individual that it might not have been self-identical or that it is no longer self-identical.  If Tom might not have existed, then it is Tom who might not have existed.  But if Tom might not have been self-identical, then it is not Tom who might not have been self-identical.

So, even if everything that exists is self-identical and conversely, existence is not self-identity.  When we say that something exists we are not saying that something is self-identical, and when we say that everything exists we are not saying that everything is self-identical.  I conclude that 'Something exists' is not expressible in the terms of the Frege-Russell system.  As for 'Everything exists,' it is surely a presupposition of the whole Frege-Russell approach: the approach presupposes that Meinong was wrong to speak of nonexistent objects.  But this presupposition cannot be expressed, cannot be 'said,' in Fressellian terms.

We are in the following curious predicament.  Something that must be true if if the Fresselian system is to be tenable — that everything exists, that there are no nonexistent objects — is not expressible within the system.

3. David Brightly accepts my challenge to give a Frege-Russell translation of 'Something exists.'  He writes:

And as a Fressellian I accept the challenge. That property is Individual aka Object, the concept at the root of the Porphyrean tree. We can say 'Something exists' with ∃x.Object(x), ie, there is at least one object. Likewise ∀x.Object(x) (which is always true, even when the box is empty) says 'Everything exists' and its negation (which is always false) says 'Some thing is not an object'. But both these last are unenlightening—because always true and always false, respectively, they convey no information, make no distinction, are powerless to change us.

I asked: which property  is it whose instantiation is the existence of something?  David's answer is that it is the property or concept Individual or Object.  And so I take David to be saying something like the following. "Just as the existence of cats is the being-instantiated of the concept cat, the existence of something is the being-instantiated of the concept Object."

David mentions the tree of Porphyry:

Tree-of-Porphyry

David speaks of the 'root' of the tree where I speak of its apex. No matter.  However we visualize it, upside down or right side up, David's suggestion is that Object or Substance (as above) is a summum genus, a supreme genus. It is a concept superordinate to every concept, a concept under which everything falls.

Operating with a scheme like this, we can, in the spirit of Frege's dialogue with the illustrious Puenjer, reduce every existential proposition (or at least every general existential proposition) to a predication by climbing Porphyry's tree.  Thus:

Cats exist –> Some mammal is a cat
Mammals exist –> Some animal is a mammal
Animals exist –> Some  living thing is an animal
Living things exist –> Some  body is a living thing
Bodies exist –> Some  substance is a body
Substances exist –> Some Objects are substances.

The point of these translations is to dispense with 'existst(s)' by showing how propositions of the form Fs exist can be replaced salva veritate with propositions of the form Some G is a F, where G is superordinate to F.  This amounts to the elimination of existence in favor of the logical quantity, someness.

We have now climbed to the tippy-top of the tree of Porphyry. We have ascended to a concept superordinate to every concept (except itself) a genus generalissimum, a most general genus.  And what concept might that be? Such a concept must have maximal extension and so will have minimal intension. It will be devoid of all content, abstracting as it does from all differences. Frege in his dialog with Puenjer suggests something identical with itself as the maximally superordinate concept. 'There are men' and 'Men exist' thus get rendered as 'Something identical with itself is a man.' (63)  Something identical with itself is equivalent to Brightly's Object.

4. Now why can't I accept the Frege-Brightly view? Well, I've already shown that 'Everything exists' cannot be translated as 'Everything is self-identical.'  But this is tantamount to having established that the concept whose instantiation is the existence of everything cannot be the concept self-identical something or the concept Object.

Another way to see this is by considering two individuals at the very bottom of the Porphyrean tree.  So consider my cats, Max and Manny.  In respect of being cats, mammals, beasts, animals, living things, material substances, and self-identical somethings, they do not differ.  They do not differ quidditatively.  But they do differ: they differ in their very existence.  Each has his own existence.  Max is not Manny, and Manny is not Max.  That is not a mere numerical difference; it is a numerical-existential difference.  Since each cat has its own existence, the existence of either cannot be the being-instantiated of any quidditative concept. All such concepts abstract from existence.  The same goes for all individuals.  Individuals exist.  But the existence of individuals is not the being-instantiated of any concept. If you want, you can think of existent (self-identical something) as a highest genus, but Existence — that in virtue of which things  exist and are not nothing — is not a highest genus.  And it is Existence that is the topic.  There are no instances of Existence.  Existing things are not a kind of thing.

The Frege-Russell theory fails utterly as a theory of Existence.

As sure as I am sitting here, I am sure that I will not convince the Londonistas.  That fact is more grist for the (meta)philosophical mill.

First John Derbyshire, then Naomi Riley

John Fund in Censoring Naomi Riley comments on the latter's dismissal by the The Chronicle of Higher Education:

Earlier this week, the Chronicle of Higher Education, the trade paper for faculty members and administrators in universities, fired Naomi Schaefer Riley, a paid blogger for its website. Her crime? She had the courage to respond to a Chronicle story called “Black Studies: ‘Swaggering Into the Future,’” which stated that “young black-studies scholars . . . are less consumed than their predecessors with the need to validate the field or explain why they are pursuing doctorates in their discipline.” The article used five Ph.D. candidates as examples of those “rewriting the history of race.” Riley looked at the subject areas of the five proposed dissertations and concluded that they were “obscure at best . . . a collection of left-wing victimization claptrap at worst.”

John Fund goes on to make a number of obvious points in protest of the illiberalism of contemporary liberals.

But Fund neglects to comment on the irony of publishing his piece in National Review Online, which recently defenestrated John Derbyshire.  (My posts on Derbyshire are in the Race category.)  What makes it worse is that NRO is supposedly a conservative publication.  We have a supposedly conservative publication publishing a piece that criticizes The Chronicle for dumping a blogger who bravely  spoke her mind and expressed some unpleasant truths that many acknowledge but few have the courage to express.  But this same publication did exactly the same thing to John Derbyshire.  We expect craven acquiescence to race-baiters from politically correct liberals, but not from so-called conservatives such as Rich Lowry and Andrew McCarthy.

Why doesn't Fund stick up for Derbyshire? (Perhaps he has in some other venue.)  I could be wrong, but Derbyshire is a more substantial commentator on the passing scene than the blogger Riley.

Saturday Night at the Oldies: Travel, Travail, Transition

Johnny Cash, I've Been Everywhere, man, crossed the deserts bare, man/I've breathed the mountain air, man/Of travel I've had my share, man/I've been everywhere.
Pete Seeger, Passing Through.  "Yankee, Russian, white or tan, Lord a man is just a man/We're all brothers and we're only passing through."
Soggy Mountain Boys, I am a Man of Constant Sorrow
Peter, Paul, and Mary, 500 Miles
EmmyLou Harris, Wayfaring Stranger
Ralph Stanley, Will the Circle be Unbroken
Karla Bonoff, The Water is Wide.  Is there a better treatment?
Tom Waits, Shiver Me Timbers.  If you've read Jack London's Martin Eden, you will figure out what this song is about.

More on Asserting and Arguing

James Anderson comments astutely via e-mail:

I have a worry about your post Asserting and Arguing.

You seem to affirm all of the following:

(1) An assertion is a mere assertion unless argued.
(2) Mere assertions are gratuitous.
(3) The premises of arguments are assertions.
(4) One cannot argue for every premise of every argument.

This is an accurate summary except for (3).  I did not say that the premises of arguments are assertions since I allow that the premises of an argument may be unasserted propositions.  The constituent propositions of arguments considered in abstracto, as they are considered in formal logic, as opposed to arguments used in concrete dialectical situations to convince oneself or someone else of something, are typically unasserted.

Since the conclusion of an argument cannot be any stronger (or less gratuitous) than its premises, doesn't it follow from these claims that the conclusion of every argument is gratuitous?

Well, if the conclusion follows from the premises, then it has the support of those premises, and is insofar forth less gratuitous than they are.  Your point is better put by saying that, if the premises are gratuitious, then the conclusion canot be ultimately non-gratuitous, but only proximately non-gratuitous.

You distinguish between 'making' arguments and 'entertaining' arguments, but that doesn't offer a way out here because the kind of argument required in (1) and (3) is a 'made' argument rather than an 'entertained' argument.

Isn't the answer here to reject (1) and to grant that some assertions (e.g., the assertion that your cats are on the desk) can be neither mere assertions nor argued assertions?  We need a category like 'justified' assertions:  no justified assertion is a mere assertion and not every justified assertion is an argued assertion.

Professor Anderson has put his finger on a real problem with the post, and I accept his criticism.  I began the post with the sentence, "Mere assertions remain gratuitous until supported by arguments."  But that is not quite right.  I should have written:  "Mere assertions remain gratuitous until supported, either by argument, or in some other way."  Thus my assertion that two black cats are lounging on my writing table  is not a mere assertion although it is and must be unargued; it is an assertion justified by sense perception.

Expressed more clearly, the main point of the post was that ultimate justification via argument alone cannot be had.  Sooner or late one must have recourse to propositions unsupportable by argument.  Argument does not free us of the need to make assertions.  (I am assuming that there is no such thing as infinitely regressive support or circular support.  Not perfectly obvious, I grant: but very plausible.)

 

Keezer on Kats

Bill Keezer writes,

I had a cat once like which there will never be another. He supposedly was my wife’s cat. He decided I was his human. He considered it his divine right to walk up in my lap when I was studying statistical mechanics and lie down on the book. I would walk up the street to cross the highway to the convenience store and he would come up to me, let me carry him across and then once down, go hunt in the field until it was time to come back. At such time he would come up to me, let me pick him up and carry him back across the highway. Once down on the other side, he would disappear until we reached home. He was a phenomenal hunter. He grew up on a farm and until was sent to us would produce several mouse heads a day. A mouse got in the house once and hid behind the refrigerator. I pulled the fridge out, dropped the cat in, and came back a bit later. Problem solved. When we lived in San Francisco, we had a stair with right angles to a glass front outside door. He would scare all the callers because he was pure panther-black and moved like a much bigger cat. The pumas in the zoo remind me of his walk. He died of infectious feline anemia and rather broke my heart. 

So I envy you your cats. We now have my son’s dog and very expensive living room furniture. I don’t think cats are in our future. But that is OK. We had cats afterwards and it wasn’t the same.

 Enjoy them.

Asserting and Arguing

Mere assertions remain gratuitous until supported by arguments. Quod gratis asseritur, gratis negatur.  That which is gratuitously assertible is gratuitously deniable.  Thus one is right to demand arguments from those who make assertions.  It is worth pointing out, however, that  the difference between making an assertion and giving an argument is not absolute. Since no argument  can prove its own premises, they must remain mere assertions from within the context of the argument. No doubt they too can be supported by further arguments, but eventually one comes to ultimate premises that can only be asserted, not argued.

Argument cannot free us of assertion since every argument has premises and they must be asserted if one is making an argument as opposed to merely entertaining one.  One who makes an argument is not merely asserting its conclusion; he is asserting its conclusion on the basis of premises that function as reasons for the assertion; and yet the premises  themselves are merely asserted.  There is no escaping the need to make assertions.

If you refuse to accept ultimate premisses, then you are bound for a vicious infinite regress or a vicious circle, between which there is  nothing to choose.    (The viciousness of a logical circle is not mitigated by increasing its 'diameter.') This shows the limited value of argument and discursive rationality. One cannot avoid the immediate taking of something for true.  For example, I immediately take it to be true, on the basis of sense perception, that a couple of  black cats are lounging on my desk:

IMG_0863

 

A Wittgenstein Paradox

Ludwig Wittgenstein had no respect for academic philosophy and steered his students away from academic careers.  For example, he advised Norman Malcolm to become a rancher, a piece of advice Malcolm wisely ignored.  And yet it stung his vanity to find his ideas recycled and discussed in the philosophy journals.  Wittgenstein felt that when the academic hacks weren't plagiarizing his ideas they were misrepresenting them.

The paradox is that his writing can speak only to professional philosophers, the very people he despised.  Ordinary folk, even educated ordinary folk, find the stuff gibberish. When people ask me what of Wittgenstein they should read, I tell them to read first a good biography like that of Ray Monk, and then, if they are still interested, read the aphorisms and observations contained in Culture and Value (Vermischte Bemerkungen).

Only professional philosophers take seriously the puzzles that Wittgenstein was concerned to dissolve.  And only a professional philosopher will be exercised by the meta-problem of the origin and status of philosophical problems.  So we have the paradox of a man who wrote for an audience he despised.

"There is less of a paradox that you think.  Wittgenstein was writing mainly for himself; his was a therapeutic conception of philosophy.  His writing was a form of self-therapy.  He was tormented by the problems.  His writing was mainly in exorcism of his demons." 

This connects with the fly and fly bottle remark in the Philosophical Investigations.

Why does the bug need to be shown the way out? Pop the cork and he's gone.

Why did Wittgenstein feel the need to philosophize his way out of philosophy? He should have known that metaphilosophy and anti-philosophy are just more philosophy with all that that entails: inconclusiveness, endlessness . . . . He should have just walked away from it.

If the room is too smoky, there is no necessity that you remain in it. You are free to go, the door is unlocked. This figure's from Epictetus and he had the quitting of life in view. But the same holds for the quitting of philosophy. Just do it, if that's what you want. It can be done.

What cannot be done, however, is to justify one's exit. (That would be like copulating your way to chastity.) For any justification proffered, perforce and willy-nilly, will be just more philosophy. You cannot have it both ways. You either walk away or stay.

On Translating ‘Some Individual Exists’ Fressellianly

An astute reader comments:

You write:

2. But can this presupposition be expressed (said) in this logic? Here is a little challenge for you Fressellians: translate 'Something exists' into standard logical notion. You will discover that it cannot be done. Briefly, if existence is instantiation, which property is it whose instantiation is the existence of something? Same problem with 'Nothing exists.' If existence is instantiation, which property is it whose non-instantiation is the nonexistence of anything? Similarly with 'Everthing exists' and 'Something does not exist.'

But couldn't we translate those expressions this way (assuming  we have only two properties: a, b)?
1. "something exists" -> "there is an x that instantiates either a or b or ab"
2. "everything exists" -> "there is an x that instantiates a and there is a y that instantiates b and there is a z that instantiates ab"
3. "nothing exists" -> 1 is false
4. "something doesn't exist" -> 2 is false

I am afraid that doesn't work.   We need focus only on on 'Some individual exists.'  The reader's proposal could be put as follows.  Given the properties F-ness and G-ness,

What 'Some individual exists' says is exactly what 'Either F-ness is instantiated or G-ness is instantiated' says.

I would insist however that they do not say the same thing, i.e., do not have the same meaning.  The expression on the left says that some individual or other, nature unspecified, exists.  The expression on the right, however, makes specific reference to the 'natures' F-ness and G-ness.  Surely, 'Some individual exists' could be true even if there are are no individuals that are either Fs or Gs. 

Note that it is not a matter of logic what properties there are.  This is an extralogical question.

On the Frege-Russell treatment of existence, 'exist(s)' is a second-level predicate, a predicate of concepts, properties, propositional functions and cognate items.  It is never an admissible  predicate of individuals.  Thus in this logic every affirmation of existence must say of some specified concept or property that it is instantiated, and every denial of existence must say of some specified concept or property that it fails of instantiation.

This approach runs into trouble when it comes to the perfectly meaningful and true 'Something exists' and 'Some individual exists.'  For in these instances  no concept or property can be specified whose instantiation is the existence of things or the existence of individuals.  To head off an objection: self-identity won't work.

That there are individuals is a necessary presupposition of the Frege-Russell logic in that without it one cannot validly move from 'F-ness is instantiated' to 'Fs exist.'  But it is a necessary presupposition that cannot be stated in the terms of the system.  This fact, I believe, is one of the motivations for Wittgenstein's distinction between the sayable and the showable.  What cannot be said, e.g., that there are individuals, is shown by the use of such individual variables as 'x.'

The paradox, I take it, is obvious.  One cannot say  that 'There are individuals' is inexpressible without saying 'There are individuals.'  When Wittgenstein assures us that there is the Inexpressible, das Unaussprechliche,  he leaves himself open to the retort: What is inexpressible? If he replies, 'That there are individuals,' then he is hoist by his own petard.

Surely it is true that there are individuals and therefore expressible, because just now expressed.

"The suicide of a thesis," says Peter Geach (Logic Matters, p. 265), "might be called Ludwig's self-mate . . . . "  Here we may have an instance of it.

‘Blacklisted’ Blacklisted

Here: "POLICE chiefs have banned IT staff from using the word blacklist over fears it is RACIST." (Via VFR)

This sort of thing is insane, of course. And so I suspect that to argue against it is foolish: it only lends credibility to a view that ought to be mocked and derided. 

But I do argue it out here.  One late-night comic lampooned the 'crispy critter' tanning lady (who brought her child into the tanning booth with her) by saying that the she is so dark it's racist!  That's the way to go.  You PeeCee liberals are so stupid it's racist! What is the antecedent of the last two occurrences of 'it'?  Don't worry, we be in PeeCee land now.  We don't need to talk no sense.

Cosmologists are going to have to be careful what with their talk of black holes.  Someone might take that as 'code' for 'black ho' a phrase that in PeeCee logic (and no, I'm not talking about the propositional calculus) implies that all black females are whores.

Tanning%20Lady

Wittgenstein and Rejectionism

I characterized Rejectionism with respect to the question why there is anything at all as follows:  "The rejectionist rejects the question as ill-formed, as senseless."  London Ed suggests that Wittgenstein may be lumped in with the rejectionists.  He has a point, though I do insist on the distinction between taking 'Why is there anything at all?' as an explanation-seeking why-question and taking it as a mere expression of wonder at the sheer existence of things.  We know that Wittgenstein was struck with wonder at the sheer existence of things.  What is now to be discussed is whether Wittgenstein can be read as making a rejectionist response to the ultimate explanation-seeking why-question.

 Ed quotes from Anthony Kenny's book, Wittgenstein:

Logic depends on there being something in existence and there being facts; it is independent of what the facts are, of things being thus and so. That there are facts is not something which can be expressed in a proposition. If one wants to call there being facts a matter of experience, then one can say logic is empirical. But when we say something is empirical we mean that it can be imagined otherwise; in this sense every proposition with sense is a contingent proposition. And in this sense the existence of the world is not an empirical fact, because we cannot think it otherwise.

This passage cries out for commentary.

1. Does logic depend on there being something in existence?  Yes, if we are talking about the Frege-Russell logic that young Ludwig cut his teeth on.    In 'Fressellian' logic, existence is instantiation.  To say that cats exist is to say that something is a cat.  (The concept cat is instantiated.)  To say that dragons do not exist is to say that nothing is a dragon. (The concept dragon is not instantiated.)  This works nicely – but only on the assumption that individuals exist.    So Kenny is surely right that (Frege-Russell) logic requires that something exists, in particular that individuals exist.

2. But can this presupposition be expressed (said) in this logic? Here is a little challenge for you Fressellians: translate 'Something exists' into standard logical notion.  You will discover that it cannot be done.  Briefly, if existence is instantiation, which property is it whose instantiation is the existence of something?  Same problem with 'Nothing exists.'  If existence is instantiation, which property is it whose non-instantiation is the nonexistence of anything?  Similarly with 'Everthing exists' and 'Something does not exist.'

3. I surmise that this is one of the motivations for Wittgenstein's infamous and paradoxical saying/showing distinction. What can be said can be said clearly.  But not everything can be said.  It cannot be said that there are beings or that there are objects or that there are individuals.  For again, how does one express (say) that there are beings (existents) in Frege-Russell logic?  This system of logic rests on presuppositions that cannot be expressed within the system.  The presuppositions cannot be said but thay can be shown by the use of variables such as the individual variable 'x.'  That is the Tractarian line.

4. Kenny also says that logic depends on there being facts.  That's not clear.  Near the beginning of the Tractatus, LW affirms the existence of facts.  He tells us that the world is the totality of facts (Tatsachen) not of things (Dinge).  But does the Frege-Russell logic require that there be facts?  Not as far as I can see.  The mature Frege certainly did not posit facts.  Be that as it may.

5. Is Wittgenstein a rejectionist?  Does he reject the question 'Why is there anything at all?' as senseless or ill-formed? The case can be made that he does or at least could within his framework.

When I raise the question why anything at all exists, I begin with the seemingly empirical fact that things exist: me, my cat, mountains, clouds . . . .   I then entertain the thought that there might have been nothing at all.  I then demand an explanation as to why there is something given (a) that there is something and (b) that there might not have been anything.

A Wittgensteinian rejection of the question might take the following form. "First of all, your starting point is inexpressible: it cannot be said that things exist.  That is a nonsensical pseudo-proposition. You can say, sensibly, that cats exist, but not that things exist. That things exist is an unsayable presupposition of all thinking.  As such, we cannot think it away.  And so one cannot ask why anything exists."

6. This form of rejectionism is as dubious as what it rests upon, namely, the Frege-Russell theory of existence and the saying/showing distinction.

Obama’s Historical Howlers

Here.  The funniest is Obama's reference to the construction of the "Intercontinental Railroad" in the 19th century.  That would be something, a railroad that crossed oceans.  Think of all the pontoons that would be needed to float the tracks on. 

The article documents Obama and his gang's unconcern with truth — as if we needed more evidence of that.

Cabinets Gone Wild

Another outstanding column by Victor Davis Hanson.  Excerpt:

Attorney General Eric Holder dropped charges against the New Black Panther Party for voter intimidation. That may explain why he said nothing when the same group put out a dead-or-alive bounty poster on George Zimmerman in the Trayvon Martin shooting case. Holder's department is suing the state of Arizona for passing a law to enforce the largely unenforced federal immigration law. Holder suggested that the Arizona law was racially inspired even as he admitted that he had never read it. Holder has praised the race-baiting Al Sharpton for his "partnership" and called the country "cowards" for not holding a national conversation on race on his terms. The attorney general has referred to African-Americans as "my people," and he has characterized congressional oversight of his office's failure to rein in the Fast and Furious scandal as racially motivated attacks on himself.

Some Inaccurate Negative Stereotypes About Stereotypes

• People ascribe a stereotype to everybody in the subject group. "All Germans are efficient." "All English people have bad teeth." In fact, these researchers were not able to locate anybody who believes that a stereotype is true of all members of the stereotyped group. Stereotypes are probabilistic tools, and even the most dull-witted human beings seem to know this. People who believe that Mexicans are lazy or that the French don't wash, understand perfectly well that there are lots of industrious Mexicans and fragrant Frenchmen.

• Stereotypes exaggerate group characteristics. No, they don't. Much more often, the opposite is true. For example, the racial stereotypes that white Americans hold of black Americans are generally accurate; and where they are inaccurate, they always under-estimate a negative characteristic. The percentage of black American families headed by a female, for example, was 21 at the time of one survey (1978): the whites whose stereotypes were being investigated offered estimates of from 8 to 12 per cent. It is not true that stereotypes generally exaggerate group differences. As in this example, they are much more likely to downplay them.

• Stereotypes blind us to individual characteristics. Nope. It is not the case that when we pass from a situation where we have nothing to go on but a stereotype (cab driver being hailed by young black male) to one where a person's individuality comes into play (interviewing a black job applicant), our stereotypes blind us to "individuating traits." On the contrary, researchers have found that the individuating traits are seized on for attention, and stereotypes discarded, with rather more enthusiasm than the accuracy of stereotypes would justify. Teachers' judgments about their students, for example, rest almost entirely on student differences in performance, hardly at all on race, class or gender stereotypes. This is as one would wish, but not as one would expect if the denigrators of stereotyping were to be believed.

• The real function of stereotypes is to bolster our own self-esteem. Wrong again. This is not a factor in most stereotyping. The scientific evidence is that the primary function of stereotypes is what researchers very prettily call "the reality function." That is, stereotypes are useful tools for dealing with the world. Confronted with a snake or a faun, our immediate behavior is determined by generalized beliefs — stereotypes — about snakes and fauns. Stereotypes are, in fact, merely one aspect of the mind's ability to make generalizations, without which science and mathematics, not to mention much of everyday life, would be impossible. Researcher Clark R. McCauley:

Standing next to the bus driver, we are more likely to ask about traffic patterns than about the latest foreign film. On the highway, we try to squeeze into the exit lane in front of the man driving a 10-year-old station wagon rather than trying to pull in on the man driving a new Corvette. Looking for the school janitor, we are more likely to approach a young man in overalls than a young woman in overalls. This kind of discrimination on the basis of group differences can go wrong, but most of us probably feel that we are doing ourselves and others a favor when we respond to whatever cues and regularities our social environment affords us.

Taken verbatim from John Derbyshire, Stereotypes Aren't So Bad