I recently had an argument with my logic professor over the question of whether it is a truth of logic that nothing is colder than itself. I hold that it is a truth of logic, while he holds that it is not. I'm trying to figure out our precise point of disagreement.

To demonstrate his point, he wrote out in the language of first-order predicate calculus:

(x)(~Cxx) where c='1 is colder than 2'

He then declared that this is not a theorem.

I then pointed out that 'being colder than' is an irreflexive relation, which could be symbolized in the language of first-order predicate calculus. His rebuttal was: "But logic doesn't know that."

Who is right, me or my professor? Where does our point of disagreement lie? If one of us is wrong, what is the error we're making?

I have my own interpretation of this argument but I'd like to hear what others have to say first.

Re: Nothing is colder than itself: a truth of logic?

Benjamin, the answer is very simple:
Take the proposition: X1 is more Y than X2.
This is is constituted by and therefore implies: X1 is Y1, X2 is Y2, and Y1 is greater than Y2.
Now if X1 and X2 are one and the same concrete item, the predicates Y1 and Y2 must also be identical quantities, assuming that they are predicates that cannot co-exist in one and the same thing at the same time in the same respect.
Therefore, in such cases, X cannot be more Y than X.
This is true of the predicate "cold." Assuming that if something is cold to degree c1 it cannot at once be cold to degree c2, and vice versa, i.e. assuming c1 and c2 are mutually exclusive, it follows necessarily that:
X cannot be colder than X (itself).
To say otherwise would be a breach of the law of non-contradiction.

Something about you (optional) logician-philosopher

Re: Nothing is colder than itself: a truth of logic?

Yes it seems very obvious; but he seems to think there is a special kind of truth, a "truth of logic," which 'nothing is colder than itself' is not. Either:

(1) He is holding some nuanced epistemological viewpoint where that is plausible;

or

(2) He reasons: 'Logic' is identical to the predicate calculus. The predicate calculus does not contain instructions on how to analyze 'colder than'. Therefore it is not a "truth of logic" that nothing is colder than itself.

The second line of reasoning is an example of how a fixation with symbolic logic can lead one into absurdities. But before I accuse my professor of making such a fallacy, I want to make sure that (1) isn't the case. Could any viewpoint be entertained where 'nothing is colder than itself' is not a "truth of logic?"

Perhaps we should first define what a truth of logic is. How do you understand "truth of logic" as it is being used here?

Re: Nothing is colder than itself: a truth of logic?

X1 is more Y than X2 implies X1 is Y1 and X2 is Y2.
If X1 = X2 = X, say
and IFF Y1 and Y2 are incompatible, i.e. if Y1 then not Y2 and vice verrsa,
then X is Y1 and Y2 is impossible,
i.e. X is more Y than itself is impossible.
There is nothing more to say or that need be said.
Present this to your teacher with my compliments.
His formula (x)(~Cxx) where c='1 is colder than 2' is not where it is at.
The problem is not just "colder than" - it is "more Y than".
The solution is as I have said. I will not contribute further to this.

Something about you (optional) logician-philosopher

Re: Nothing is colder than itself: a truth of logic?

I know you said you won't contribute further to this, but here is a paraphrase of my professor's response in case anyone is interested:

"Logic must be defined as strictly as possible so as to prevent people from making metaphysical assumptions and calling them, and all that they imply, truths of logic. The best way to do that is to say simply that logic is the first-order predicate calculus with identity. I grant that X is more Y than itself is impossible, but it is not a truth of logic– it is a truth of the logic of quantitative relations, which is only an extension of logic proper.

In order to teach the above lesson to my students I give them arbitrary word problems to symbolize where they have to make the immediate inference that something is colder than itself. It is a kind of shock therapy."