25.4 The Chart
The relationship between tautologies, FO validities and logical truths can be represented in an Euler diagram like this.
This is a just like the chart you’ve seen before, showing the relation between tautologies and logical truths. Only now there’s a new area for FO validities.
Let’s see if you can place some sentences in here.
Identity truths, like p=p, use to be our paradigmatic example of a sentence that is a logical truth but not a tautology.
We aren’t changing that: they are still logically true, and they are still not tautologies.
But now we’ve added a logical system, FOL, that includes identity. So p=p does not work as an example of something that is a logical truth but no an FO validity, since it is an FO validity.
What we need is a new paradigmatic example. Recall that = is our only predicate given a special symbol and meaning in FOL. But there are other predicates with logical significance, like less-than, <.
Like identity, less-than is written infix: we write a<b and 3<5.
It is a fact about the logic of the < relation that two things can’t be less than each other. So if a<b, we know ~(b<a). That means this sentence:
is a logical truth, but not an FO validity. So it goes in region C.
You might be wondering how we could ever place something in region D, contingent truths.
If we just give you an atomic sentence like H(a), we can’t know whether it goes in region D or E without knowing two things:
- What it actually means, like Alberto is hungry.
- What the facts in the world are: when did Alberto have lunch?
So sometimes you will need additional information to place sentences. Namely, you need to know the intended interpretation of the symbols as well as the common knowledge about the world that the interpretation presupposes.
Of course, if a sentence is necessarily false, you can place it in region E regardless of what the interpretation is.