# Propositional Logic for variable tautologies?

My textbook states that x = x+1 is not a proposition. Why? I thought about domains but no matter the domain this statement is always false because 0 is not equal to 1.

My textbook defines a proposition in the following way:

A boolean proposition (or simply proposition) is a statement which is either true or false (sometimes abbreviated as T or F). We call this the A boolean proposition (or simply proposition) is a statement which is either true or false (sometimes abbreviated as T or F). We call this the truth value of the proposition.

• $x =x + 1$ is true if you are evaluating modulo $1$. Commented Feb 13 at 23:33
• The expression above is a formula of predicate logic with equality: a formula is called sentence when there are no free occurrences of variableds. Commented Feb 14 at 6:44
• But it is not a formula of propositional logic. Commented Feb 14 at 7:12
• As has been said, correctly as far as I can say, your expression is of course not a formula of propositional logic (in a narrower sense). But of course it is a 'proposition' or 'sentence' in the usual, common sense of these words. I am not sure why your book says otherwise. As others indicated, it may be that the authors mean a more narrower sense of some kind of more formal language, then it seems correct. But then still I think it should and could be more clear on this. Your book may on the other hand just not been very carefully written in this regard, and so to some extend incorrect. Commented Feb 17 at 9:50

Your book means to say that while there exists an x such that x = x+1 is a (false) proposition, x = x+1 contains a free variable, so it does not have a definite truth value, so it is not a (Boolean) proposition.