# Is "today is Presidents' Day" a proposition, propositional function, or neither?

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?

• People seem to have an irrepressible desire for a notation that conflates a function with a value of the function. Saying "If for all $x$ in $D$, $P(x)$ is a proposition, then $P(x)$ is a propositional function" is calling the same thing ($P(x)$) both a proposition and a propositional function (and the two almost as close together as possible), which is particularly awkward in the context of this definition. (Note also that $P(x)$ was called a statement in the sentence before.) What it should say instead is "If $P(x)$ is a proposition for all $x$ in $D$, then $P$ is a propositional function". Commented Mar 30, 2022 at 11:58
• In my view it all depends whether you choose to consider "today" as a single, fixed, but unknown value, or as a variable into which you can substitute any given day. So you might say if you ask it once on a certain day, it's a proposition. But if you ask repeatedly on different days and want to take a view of the statement across a distribution of days, it's a propositional function. Commented Apr 7, 2022 at 8:52

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 $$Even(2)$$.
Today is what philosophers call an indexical, so the sentence will mean a different proposition if it's written tomorrow. But if we regard it as a function, it's a function of "today" rather than a date we get to specify freely. For a given value of today, one proposition has been stated. Your solution manual will be for a course that probably expects "Day $$n$$ is Presidents' Day" to be the appropriate format for a propositional function, so $$n$$ can be chosen arbitrarily forever.