Wednesday, July 27, 2011

'Snow is white' is true if and only if snow is white

To: Joanne B.
From: Geoffrey Klempner
Subject: 'Snow is white' is true if and only if snow is white
Date: 23 January 2004 14:20

Dear Joanne,

Thank you for your essay of 13 January, with your second essay for the Metaphysics program, ''Snow is white' is true if and only if snow is white.'

I took the liberty of adding the missing quotation marks and 'is true'. Without these additions, there would not be much to write about!

By purest coincidence, my previous email - to Marcus, an US English professor from Tennessee who is taking the Possible World Machine program - concluded:

If asked to define truth, I would avoid a metaphysical answer. 'Is true' is the only predicate that can be substituted for X, without loss of truth value in all instances of the following formula:

'Snow is white' is X if and only if snow is white.

Put any sentence you can think of in place of 'snow is white', the result is always the same just in the one case where X=true. (Think of all the other things you can say about a sentence, that it is grammatical, poetic, has three words etc. etc. None of these can be substituted for X and still give the result we want.)

Marcus had offered his only definition of truth, based on the idea that 'the truth sets you free'. Well, we're not discussing that, but the point about metaphysics is that there are really two issues to consider. One issue is how we *identify* truth. The other issue is whether there is anything enlightening that the philosopher can *say about* truth besides simply identifying it.

I am going do to something I didn't do with Marcus, which is explain in as much detail as possible how the above formula works.

First, 'if and only if'. The statement 'P if and only if Q' asserts that P and Q have the same truth value. Either they are both true or they are both false. E.g. 'Paul is a bachelor if and only if Paul is unmarried', 'Geoffrey Klempner has three heads if and only if the number of Geoffrey Klempner's heads is between two and four.' Both those statements are true.

Second, why is the first occurrence of snow is white in quotes while the second isn't?

When we put a sentence in quotes we are not asserting it, but rather referring to it. If I say, 'Joanne wrote, 'Everyone sees snow as white even though under a microscope it is in fact red', I have said something true, regardless of whether this alleged 'fact' is true or not (it is certainly new to me!). I have not asserted that snow looks red under a microscope, but merely stated that you wrote this.

Putting these ideas together, if I say:

'Snow looks red under a microscope' is true if and only if snow looks red under a microscope,.

then I have said something true, according to the meaning of the phrase, 'if and only if'.

The statement is true if snow *does* look red under a microscope, and it is still true if snow *doesn't* look red under a microscope. Either the statements on the left hand side and the right hand side of 'if and only if' are both true or they are both false.

So what does this show?

If you asked me, 'What is truth?' then one possible answer would be to say, 'I can give you a formula which identifies truth, and only truth.' This is where the quote from my letter to Marcus comes in.

Imagine all the sentences anyone could ever composed arranged in a list. All the false sentences along with all the true ones. Then substitute each sentence, one by one, for 'Snow is white'.

In each case the resulting 'if and only if' statement is guaranteed true ONLY IN THE CASE WHEN the term 'is X' is understood as 'is true'. That is something to marvel at. How could it be? Why is that so?

Well, there are some complications. One question which you raise is who is using the terms 'snow' and 'white'? Alfred Tarski, the logician who first proposed the 'snow is white' formula talked about two 'languages', an object language and a meta-language. The object language is the language we are *talking about*. For example, suppose that we were talking about German instead of English. Then we would say,

'Schnee ist weiss' if and only if snow is white.

The meta-language, on the other hand, is the language we are using to talk about the object language, plus all the other things in the world. We are using the terms 'snow' and 'white' in the above formula to talk about snow and white, according to our understanding of what snow is and what white is.

There is a problem which you have identified, when the terms are vague. There are some sentences where we just don't want to talk about 'truth' or 'false'. For example, when we are describing a case which is right bang in the middle - e.g. a man gradually losing his hair. In the mid point of this process, we don't want to say that 'Fred is bald' is true but nor do we want to say that 'Fred is bald' is false.

What is the philosophical significance of this? I don't agree with you that the snow is white formula makes any philosophical claim about truth and perception. It is too general for that. But it does beg some important questions. It runs into the problem of vagueness. And it has nothing to say about the deep metaphysical issues surrounding the nature of truth.

All the best,

Geoffrey