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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.

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  • $\begingroup$ $x =x + 1$ is true if you are evaluating modulo $1$. $\endgroup$
    – fleablood
    Commented Feb 13 at 23:33
  • $\begingroup$ The expression above is a formula of predicate logic with equality: a formula is called sentence when there are no free occurrences of variableds. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 14 at 6:44
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    $\begingroup$ But it is not a formula of propositional logic. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 14 at 7:12
  • $\begingroup$ 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. $\endgroup$
    – Ettore
    Commented Feb 17 at 9:50

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At first it might be tempting to consider "x = x+1" as a proposition since it's obviously false in under normal arithmetic, it has no solution since there is no real number that satisfies the equation.

BUT such interpretation is outside the domain of propositional logic, which deals with declarative statements and their truth values, not mathematical equations and their solutions. So the statement "x = x+1" isn't a boolean proposition, it is not asserting a fact that can be true or false. It's expressing a mathematical relationship that, in the algebra case, does not hold for any real number x.

"x = x+1" could be seen as an assignment/increment statement in programming, where the value of "x+1" is being assigned to the variable "x", which is not a proposition either, but rather an operation or command.

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  • $\begingroup$ The formula above can be interpreted in a domain with only one element called "one". $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 14 at 6:45
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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.

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