I'm doing some practice problems for my Methods of Proof final, and one of the questions asks whether "Today is Presidents' Day" is a proposition, propositional function, or neither. In my class, we define a propositional function $P(n)$ as a proposition whose truth value depends on the value of $n$. I was thinking that "today is Presidents' Day" may be a propositional function because the truth value depends on what day it is. For example, today is March 29th, 2022, so it is not Presidents' Day, meaning the statement is false. However, the truth value of the statement is true if it is Presidents' Day. Therefore, the truth value of the statement depends on what day it is, so the statement $P(n):$ "Today is Presidents' Day" is a propositional function with the domain $D:=\{n:n \,\text{is a calendar date\}}$. Is this line of reasoning correct? The reason why I ask is that the solution manual said this was a proposition rather than a propositional function.
EDIT: To be more precise, our definition of a propositional function is:
Let $P(x)$ be a statement involving the variable $x$, and let $D$ be a set. If for all $x$ in $D$, $P(x)$ is a proposition, then $P(x)$ is a propositional function. The set $D$ is the domain of the variable.
I still think the statement is a propositional function based on this definition. Is that true?
Best Answer
What counts as a proposition is always a bit of a controversial matter, but I agree with the book on this one. Sure, the truth-value depends on the context, but that is so with just about any claim, e.g. 'My shirt is red' is true one day, and false the next.... but I would not consider that a propositional function. I would say a function has a much more explicit reference to a 'variable', e.g. '$x$ is red'. This of course makes most sense in the kinds of domains where we typically apply logic, i.e. math, where it is really important to distinguish between expressions like '$x$ is even' and '$2$ is even'. That is, this whole distinction is really with the purpose of making a distinction between expressions like $Even(x)$ as it may occur in $\forall x \ Even(x)$ or $\exists x \ Even(x)$, as opposed to expressions like $Even(2)$.