(S) If yesterday were tomorrow, then today would be Friday.

Question: What day is today?

This seems to be an old puzzle, and depending on the interpretations, the answers are Wednesday or Sunday (or perhaps Friday as well?). I would like to understand the logic required to analyze and answer the above question.

The following is my attempt in formalizing the analysis. Please let me know if (and where) I err.

Model

Let the actual "today" be $t$, so that "yesterday" in the antecedent of (S) is $t-1$ and "tomorrow" in the antecedent is $t+1$.

Let $D(\tau)$ denote the day of week of date $\tau$.

The subjunctive "today" in the consequent of (S) can be formalize in two ways: as (i) "the yesterday of tomorrow", or (ii) "the tomorrow of yesterday". The two interpretations thus lead to two ways to translate (S): $$(t-1)=(t+1) \quad\Rightarrow\quad D(t+1)-1=\text{Friday}\tag{1}$$ $$(t-1)=(t+1) \quad\Rightarrow\quad D(t-1)+1=\text{Friday}\tag{2}$$

From $(1)$, we have $D(t-1)=\text{Saturday}\Rightarrow D(t)=\text{Sunday}$.
From $(2)$, we have $D(t+1)=\text{Thursday}\Rightarrow D(t)=\text{Wednesday}$.

But both interpretations also seem to suggest Friday as a solution, which is implausible (?) and indicates that the model I've proposed has flaws.

What's wrong, and how can we improve the model?

Solution 1:

Yesterday was not tomorrow. From a false assumption, any conclusion is possible.

But one interpretation of the puzzle goes like this. The only day of the week $x$ for which is would be correct to say "If yesterday was $x$, then today would be Friday" is Thursday. So on that interpretation, $x = $Thursday, which actually happens to be tomorrow, so today is Wednesday.

Alternatively, "If $y$ was tomorrow, then today would be Friday" is true if $y$ is Saturday. On that interpretation, Saturday is actually yesterday instead of tomorrow, and today is Sunday.

Solution 2:

If yesterday were tomorrow, then today would be Friday. What day is today?

(Deleted my previous answer and starting over.)

Let $d$ be any date, past or present or future, expressed as an integer, with the natural ordering of days.

Let $d_0$ be the supposed current date.

Let $D(x)$ be the day of the week for date $x$ (Monday, Tuesday, etc.).

We are given:

$d-1=d_0+1$ and $D(d)=Friday$

Can there be any other meaningful interpretations?

Then the day of the week for the supposed current date would be $D(d_0)$ where

$D(d_0)=D(d-2)=Wednesday$

Edit

A tabular approach...

CURRENT DATES AND DAYS:

..............................................Date...........Day..............

Today..............................$x_0$...............Wednesday.

Tomorrow.....................$x_0+1$........Thursday.......

Day After Tomorrow..$x_0+2$........Friday...........

NEW DATES AND DAYS:

..............................................Date...........Day..............

Yesterday.......................$x_0+1$.......Thursday.......

Today...............................$x_0+2$......Friday...........

First, fill in the current dates with today's date of $x_0$. Then fill in the new dates with yesterday's date of $x_0+1$. Then fill in the new days with today's day of Friday. Finally, fill in the current days with day-after-tomorrow's day of Friday.