Sic et non

What do the words ‘yes’ and ‘no’ mean? Well, I can explain how their meanings add up. If I add the word ‘yes’ to the question ‘is it raining’ I get something with the same meaning as the assertion ‘it is raining’

“Is it raining? – yes” = ‘it is raining’

Similarly adding the word ‘no’ to the same question gives the corresponding denial:

“Is it raining? – no” = ‘it is not raining’

Does this shed any light on the question about negation and the principles of contradiction and excluded middle? I think so. For I suggested that one who questions either of these principles hasn’t really understood the meaning (or rather, the use) of words like ‘not’. Certainly, one who thinks that both ‘yes’ and ‘no’ are simultaneous replies to the same question, hasn’t understood the meaning of these interjections. For example, suppose someone utters ‘no’ then ‘yes’ in close succession.

“Is it raining? – no… yes” = ‘it is raining’

We would naturally take him or her to have changed their mind. I.e. At the time of uttering ‘no’ they were saying it was not raining, and then retracted it by uttering ‘yes’. 

That deals with the principle of contradiction, namely is that it is impossible meaningfully to assert and deny the same thing.  As for excluded middle, that follows from the fact that nothing else apart from ‘yes’ and ‘no’ counts as a reply to a question. Note that ‘not sure’ is not a relevant reply, for it is really the reply ‘no’ to the question ‘are you sure that it is raining?’, rather to the question 'is it raining?'.  I.e.

"Is it raining? - not sure" = "are you sure it is raining? - no" = "I am not sure it is raining"

Failing to answer is failing to answer. Bullshitting is also failing to answer the question, but by means of asserting all sorts of other irrelevant things, or just outright nonsense.

What the meaning of 'not' is not

There were (or was?) a flurry of comments on my earlier post about the concept of negation, and the meaning of the word ‘not’. I would say these are quite separate things. Indeed, there are many separate things. Negation is not the same as the concept of negation. For the expression “the concept of negation” is a noun phrase referring to the concept of negation, whereas ‘negation’ refers to negation. As for the word ‘not’, it is an adverb not a noun, and while it clearly has a meaning, it cannot be the same as the meaning of the word ‘negation’. Otherwise we could replace the word ‘negation’ with the word ‘not’ without change of meaning. But ‘snow is not black’ does not mean the same as ‘snow is negation black’. Note also (and I find this a bit puzzling) that the ‘the meaning of the word “not”’ is a noun phrase that refers to the meaning of the word ‘not’. Yet we can’t replace ‘the meaning of the word “not”’ with the word ‘not’ without change of meaning.

There is also the verb ‘negate’. This is derived from the Latin ‘nego’ meaning to deny, and the meaning is still pretty much the same. Negating is denying. This is what the word ‘not’ does when we attach it to a sentence. So the word ‘not’ achieves what the verb ‘deny’ does, and so it has the effect of a verb, yet it is an adverb!

All very confusing. I shall talk about the interjections ‘yes’ and ‘no’ tomorrow.

Was Wittgenstein a rejectionist?

The Phoenician Maverick helps me out of the mid-week posting famine with this post on whether Wittgenstein was a 'rejectionist' with respect to the question why there is anything at all.

Just a point on the quote: it is Kenny paraphrasing Waismann's note of what Wittgenstein may have said, i.e. it is Wittgenstein's voice here, not Kenny', as Bill seems to suggest in his point #4. Otherwise I agree with him that Wittgenstein's position rests on the saying/showing distinction which is in turn closely connected with the Frege-Russell account of existence. Whether I agree with that position is something which I decline to talk about for the moment. Altissimum enim est huiusmodi negotium et maioris egens inquisitionis.

Wittgenstein on negation

Plato suggests the following problem in the Theaetetus:

In judging one judges something; in judging something, one judges something real; so in judging something unreal, one judges nothing; but judging nothing, one is not judging at all.

According to Anscombe* Wittgenstein returned to this problem again and again throughout his life. It presents a formidable challenge to his picture theory of language. He thought, in his early work, that in a proposition we supposedly put together a picture of the world just as in the law-courts of Paris of the early twentieth century, a car accident is represented by means of dolls. So how do we get a picture of negation?

What any picture, of whatever form, must have in common with reality, in order to be able to depict it - correctly or incorrectly - in any way at all, is logical form, i.e. the form of reality. So what reality does a negative proposition represent? What is the reality represented by 'snow is not black'? According to Wittgenstein, the negation operator 'not' does not make a picture at all, but simply performs a truth functional operation on the picture given by the corresponding affirmation. The picture he drew in his early Notebooks (above) shows this clearly**. And in the Tractatus he writes (my emphasis)

4.0621 But it is important that the signs 'p' and '-p' can say the same thing. For it shows that nothing in reality corresponds to the sign '-'.

Determinatio negatio est. Determination is negation. By drawing a circle in the sand we delimit all the sand on the beach outside the circle as well as all which is inside.

Does this help us to understand how the concept of negation is learned?  Is negation a concept at all?

*G.E.M. Anscombe, An Introduction to Wittgenstein's Tractatus (London 1971) p.13
**Taken from a nice paper by Robert Pippin here about this subject.

Meno and negation

Jason suggested in a comment on the last post that my problem about how we learn to apply negation is related to the ‘Meno question’.

A couple of comments. First, there are two Meno questions or Meno problems. The first is to explain how knowledge is more valuable than mere true belief, and that is the Meno problem as commonly understood. The second problem, which is clearly related to the first, is to explain how we learn anything at all. If knowledge is different from true belief, how can we possibly acquire knowledge by teaching? If teaching is the mere repetition of true propositions (‘The battle of Hastings was in 1066’), and if learning is the acceptance of those propositions as true, on account of the authority of the teacher, or for whatever reason, and if knowledge is more than simple acceptance of the truth, it logically follows that we can’t acquire knowledge from teaching, as so defined. So how do we acquire knowledge? Is it already there, and does teaching in the proper sense require uncovering what lay hidden?  – the Latin root of ‘educate’ means ‘drawing out’. Or does it come from without?

The second problem clearly underlies the difficulty about negation, but there is a further difficulty. The medieval philosophers, following Boethius*, divided discourse into three types, namely written, spoken and conceptual. Written discourse signifies spoken discourse, by convention. It is a convention, e.g., that the written word ‘dog’ signifies the noise that comes out of my mouth when I utter ‘dog’. In turn, spoken discourse signifies mental discourse, also by convention. It is by convention that the spoken word ‘dog’ signifies the idea of a dog. French people use the word ‘chien’ to signify that same idea, Germans use the word ‘Hund’ (I think). Ockham, though he was English, would have used the Latin word canis. However, mental discourse signifies not by convention, but ‘naturally’. The English and French and German and Latin words for dog all signify exactly the same thing, namely the idea of a dog. For otherwise we could not communicate, unless we could signify the same idea in another person’s mind as the one we want to convey. But we cannot teach the signification of the idea itself – we don’t have it available to match up with the thing signified, for ideas are private. So ideas or ‘mental terms’ signify naturally. Ockham explains this right at the beginning of his Summa Logicae.

Now the word ‘not’ is what Ockham would have called a syncategorematic term. He explains the distinction in chapter 4, although it does not originate with him. A syncategorematic term is one that does not signify on its own (in the way that the term ‘dog’ on its own signifies or ‘supposits for’ all dogs), but signifies by making other words signify. For example, ‘every’, ‘some’, ‘only’ and of course ‘not’.

If Ockham is right, there is a sense in which we cannot learn the meaning of the word ‘not’. Of course we learn that the English word ‘not’, the spoken word, corresponds to the mental ‘not’, and in that sense we learn the meaning. But in another sense we cannot learn or acquire the mental term itself. It must be already there.

Thus, if he is right, we don’t learn the meaning of negation. It is already there. But is he right?

*See his commentary on Aristotle’s Perihermenias ed. 2a, I, Patrologia Latina 64, col. 407B.

Negation, denial and the Principle of Contradiction

In learning negation I asked whether we learn the principle of contradiction, and the concept of negation, by observation and experience, or whether it is somehow 'hard wired' into our consciousness. I didn't spell it out, but I was implicitly making two claims. One, that negation is 'hard wired'. Two, that the principle of contradiction follows directly from our concept of negation, i.e. anyone who insists on the possibility that "Socrates is white and Socrates is not white" simply cannot have understood the meaning of 'not'.

Taking the first. It is absurd that the concept of negation is anything we could learn. How, e.g. could you see that something is not white without understanding what negation was, even if you hadn't learned the word 'not' which corresponds to it. Understanding that something is not the case is no less fundamental than understanding that it is the case (presumably those who believe we learn the concept of negation would not defend the learning of affirmation – how would we learn the idea of being the case?). Ergo, the concept of negation is hard-wired.

The second point is harder to prove. I would like to argue that it is a consequence of the position which I have frequently defended here, namely that to assert that p is true is simply to assert p, and that to say that p is false is simply to deny that p. I.e. truth and falsity reduce to affirmation and denial.  Does the principle of contradiction follow from this?

Learning negation

I have a further question about this discussion and generally about any argument that we learn logical laws by experience, observation or induction or anything like that. Suppose it is argued that I learn that no x is white and not white by observing particular x's and noting of each one that it is either white, or not white. I then generalise this to 'no x is white and not white', and further generalise (by substituting other predicates like 'round', 'soft', 'large' etc) to 'no x is F and not F'.

I ask, how did I learn the meaning of the negation 'not'? Is this a sign whose meaning I understood correctly before all these observations? Or as part of the process of observation that led to the general conclusion? Surely not the first. Could anyone who thought it was possible that 'Socrates is white and Socrates is not white' was true, really understand the meaning of the word 'not'? It means negation, and negation means denial, and how could you assert and deny the same thing at the same time? So not the first.

But if the second, that means we learn the concept of negation by observation. Perhaps by your teachers pointing to different things and saying 'not white' when they were not white, and 'white' if the things were white. But that doesn't tell me whether the predicate 'not white' also applies to the white things. To do that, my teachers would have to say 'not not white' when pointing to the white things. And that still doesn't of itself tell me how to use the negation operator for I still haven't been taught that 'not not not white' applies to the not white things, and so on ad infinitum. To understand negation properly, I would have to understand its basic properties before all this took place. But if I understood that, the first point would apply, i.e. I would have to understand that 'x is white and x is not white' can never be true, on account of the meaning of the negation sign.

On the point attributed to Tim Crane, namely that one can perceive something 'as A and not-A' but rejects it through giving greater weight to the principle of contradiction, I'm not sure we can perceive something as A and not-A. Rather, it may seem that it is A and not A, but our understanding of the meaning of the word 'not' assures us that it is not the case that it is A and not A.

More about non-men

Anthony has asked for some more about indefinite (or 'infinite') negation.  Well he will have to wait for our book to come out (working title: Time and Existence: Duns Scotus' Questions on the Perihermenias) if he wants much more. Opus II Book I Q4 ("Does an Indefinite Name Posit Something") and Opus II Book II Q2 (Does ‘This Is Not Just; Therefore, This Is Non-Just’ Follow?) refer. Until that happy day, here is Thomas Aquinas on the same subject, in his commmentary on the Perihermenias, Book II lecture 2:

Latin	English*
Sicut ipse dicit,enunciatio aliqua virtute se habet ad illud, de quo totum id quod in enunciatione significatur vere praedicari potest: sicut haec enunciatio, homo est iustus, se habet ad omnia illa, de quorum quolibet vere potest dici quod est homo iustus; et similiter haec enunciatio, homo non est iustus, se habet ad omnia illa, de quorum quolibet vere dici potest quod non est homo iustus.	It must be noted that, as Aristotle himself says, the enunciation, by some power, is related to that of which the whole of what is signified in the enunciation can be truly predicated. The enunciation, "Man is just,” for example, is related to all those of which in any way "is a just man” can be truly said.So, too, the enunciation "Man is not just” is related to all those of which in any way "is not a just man” can be truly said.
Secundum ergo hunc modum loquendi, manifestum est quod simplex negativa in plus est quam affirmativa infinita, quae ei correspondet. Nam, quod sit homo non iustus, vere potest dici de quolibet homine, qui non habet habitum iustitiae; sed quod non sit homo iustus, potest dici non solum de homine non habente habitum iustitiae, sed etiam de eo qui penitus non est homo: haec enim est vera, lignum non est homo iustus; tamen haec est falsa, lignum est homo non iustus.	According to this mode of speaking it is evident, then, that the simple negative is wider than the infinite affirmative which corresponds to it. Thus, "is a non-just man” can truly be said of any man who does not have the habit of justice; but "is not a just man” can be said not only of a man not having the habit of justice, but also of what is not a man at all. For example, it is true to say "Wood is not a just man,” but false to say, "Wood is a non-just man.”
Et ita negativa simplex est in plus quam affirmativa infinita; sicut etiam animal est in plus quam homo, quia de pluribus verificatur. Simili etiam ratione, negativa simplex est in plus quam affirmativa privativa: quia de eo quod non est homo non potest dici quod sit homo iniustus. Sed affirmativa infinita est in plus quam affirmativa privativa: potest enim dici de puero et de quocumque homine nondum habente habitum virtutis aut vitii quod sit homo non iustus, non tamen de aliquo eorum vere dici potest quod sit homo iniustus.	The simple negative, then, is wider than the infinite affirmative-just as animal is wider than man, since it is verified of more. For a similar reason the simple negative is wider than the privative affirmative, for "is an unjust man” cannot be said of what is not man. But the infinite affirmative is wider than the private affirmative, for "is a non-just man” can be truly said of a boy or of any man not yet having a habit of virtue or vice, but "is an unjust man” cannot.
Affirmativa vero simplex in minus est quam negativa infinita: quia quod non sit homo non iustus potest dici non solum de homine iusto, sed etiam de eo quod penitus non est homo. Similiter etiam negativa privativa in plus est quam negativa infinita. Nam, quod non sit homo iniustus, potest dici non solum de homine habente habitum iustitiae, sed de eo quod penitus non est homo, de quorum quolibet potest dici quod non sit homo non iustus: sed ulterius potest dici de omnibus hominibus, qui nec habent habitum iustitiae neque habent habitum iniustitiae.	And the simple affirmative is narrower than the infinite negative, for "is not a non-just man” can be said not only of a just man, but also of what is not man at all. Similarly, the privative negative is wider than the infinite negative. For "is not an unjust man” can be said not only of a man having the habit of justice and of what is not man at all—of which "is not a non-just man” can be said—but over and beyond this can be said about all men who neither have the habit of justice nor the habit of injustice.

Note that 'man is just' etc is better translated as 'a [or the] man is just'.  According to Aristotle and the scholastics, the apparently negative 'is non-just' is really something positive or affirmative said about anyone who has a determinate nature such as a man or an animal.  Thus is not as wide as definite negation, because not being a just man can apply to anything you like, so long as it is not just, or is not a man.

That's not to say the matter is any clearer, really.

*Translated by Jean T. Oesterle Milwaukee: Marquette University Press, 1962