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Tuesday, March 23, 2010

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A question: given that LNC has something to do with the meaning of the word 'not', what actually is the meaning of this word? For example, if it means 'certainly' then

Snow is white and it is *not* the case that snow is white

is 'certainly' true! But it certainly does not mean 'certainly'. What does it mean then? I'm asking because we sometimes explain the meaning of words by giving examples of true and sentences which contain the word. Perhaps we could explain the meaning of 'not' by saying that

p and not-p

is never true? Just a thought.

That should read "by giving examples of true and *false* sentences which contain the word.

I just wanted to chime in and mention how nice it was to read the following:

"Why then do you genuflect before the authority of scientists when they spout gibberish?"

That is a sentiment not uttered or written often enough.

One thing I'm curious of, if Bill (Valicella, not Hill) is willing to reply: You said that you recognize reality may in fact not conform to our logic, but at the same time rational discourse requires (if you wouldn't put it this way, let me know) that certain ground rules, like the LNC, are true. Does that mean you're saying that, if the LNC doesn't hold, we can't know this or will never know this (barring perhaps something like divine revelation)?

William,

You ask what the meaning of 'not' is. And you seem to be suggesting that 'not' can be explained in terms of 'never.' But 'never' means 'not at any time' or 'not ever' so we move in a circle if we try to define 'not' in terms of 'never.'

Some notions are so basic that they cannot be defined in other, more basic, terms. But although we cannot provide a synonym or an analysis, we can show how to use 'not.' If proposition p is true, then not-p is false, and if p is false, then not-p is true. So if there are just two truth values, we can explain 'not' in this way.

You are right that LNC and 'not' are closely connected: a proposition p and its negation (not p) cannot both be true.

Note that 'not' cannot be define din tewrms of 'ralse' since in a 2-valued logic, 'false' just means 'not true' -- and so we move in a circle.

'Not,' if it denotes anything, denotes an operator on propositions, a negation operator. But 'not' can also function as a universal quantifier. Reflect on the ambiguity of 'All men are not rich.' That could mean: 'not(all men are rich)' or 'No men are rich.'

>>But 'never' means 'not at any time' or 'not ever' so we move in a circle if we try to define 'not' in terms of 'never.'
Good point. Perhaps we could start with excluded middle. Whatever we put in the placeholder 'p', (it is the case that) p or not-p . Not asserting any fundamental truth, rather just explaining a rule of use. A further rule of use is that not-not-p = p. Apply the negation operator twice, and you are back to the original proposition. From this we get LNC, as follows.

1. p or not- p (excluded middle)
2. not (not-p and not-not-p) (1, De Morgan)
3. p = not-not-p (meaning of double negation)
4. not(not-p and p) (2, 3, substitution)

Perhaps there is still a problem with 'always', or of 'following a rule. Does this involve some prior concept of negation? If something is always the case, you have to grasp there are *no* exceptions.

On your second point 'if p, then not-p is false'. I think this is fundamentally the same thing. Assuming 'p is false' means the same as 'not p', this gives 'if p, then not-not-p'. This in turn means the same as 'not (p and not-not-not-p)', using an appropriate definition of 'if… then'. Assuming double negation (p = not-not-p), this gives 'not(p and not-p)' which is LNC again.

This takes us some way from quantum mechanics, though if it is correct that these fundamental principles are merely semantic laws, it follows that these explanations of quantum mechanics are flawed in some way: they would indicate a fundamental misunderstanding of *syncategoremata*.

Conversely, and using the valid principle of modus tollens (negatio antecedentis sequitur ad negationem consequentis) if the quantum mechanics are not flawed, it would follow that these fundamental principles are not merely semantic laws.

William,

I just realized who you are, our old friend 'ocham.' You really love to cover your tracks lest the U.K. thought police come after you for heterodox opinions in logic.

Your derivation (1)-(4) is correct, but I am not sure what you are after. What was traditionally called the Three Laws of Thought are interdefinable. For example:

1. p = p (Law of Identity. Read '=' as triple bar.)
2. (p --> p)& (p --> p) From 1 by defn of material equivalence.
3. p --> p From 2 by Simplification
4. ~p v p From 3 by defn of mat'l implication
5. p v ~p (LEM) From 4 by Commutation

Yes it is I, Ocham. How did you guess? I was merely using my first name (which is the same as yours).

Are you sure they are interdefinable. E.g. intuitionist logic holds that p implies not-not-p, but not conversely, and so accepts the LNC, but not LEM. That was why I mentioned double-negation.

In my adherence to classical logic I am certainly not heterodox!

I have some news for you Bil, is your email still good?

I identified you from your e-mail address which is included in an e-mail I receive from Typepad whenever a comment is left at this site. I am also informed of the IP # of the computer the commenter is using. This allows me to block undesirables. Yes, my e-mail is still good.

I was joking about your 'heterodox opinions.' You are right about intuitionism.

Does 'p is false' means the same as 'not p'? That's not clear to me for the same reason that it is not clear to me that 'p is true' means the same as 'p.' I do grant, though, that the truth predicate adds nothing to the sense of 'p.' But it does seem to put that sense into relation to something external to it. You will recall Frege's claim that declarative sentences have both Sinn and Bedeutung, the latter being either the True or the False -- whatever the hell they are! But I know what he is getting at: the truth predicate is redundant in one way, but not in another.

There are a host of tricky questions here. I am still not sure what your concern is, or your thesis, if you have one.

I must go now. I was merely wondering how we could derive the laws from basic semantic principles.

On 'p is false', as you know I subscribe to a form of the redundancy theory of truth, which we have argued about before.

I will email you. I am thinking about giving up lens-grinding for a time, and take a sabbatical for philosophical purposes. On the other hand, I am not sure I could stand it.

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