# If yesterday were tomorrow, then today would be Friday.

(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?

• If yesterday were tomorrow, then you can do anything you set your mind to man. – fancynancy Feb 5 '15 at 6:01
• This belongs on Puzzles.SE. – apnorton Feb 6 '15 at 4:58
• Math.SE would love to have these sort of questions. :) – Sufyan Naeem Aug 13 '15 at 18:46

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.

• I'm not sure how 'yesterday was tomorrow' is a false assumption. You probably misunderstood. – servabat Feb 3 '15 at 6:33
• Perhaps my English grammar is holding me back from conveying the correct message... Surely, yesterday was not tomorrow. But isn't there a logic that can handle subjunctive possibilities? – Herr K. Feb 3 '15 at 6:59
• Old Yiddish saying: If Grandma had wheels, she would be a wagon. yiddishwit.com/gallery/wagon.html – Robert Israel Feb 3 '15 at 7:15
• +1, I guess that in my answer I've added more details to your good answer. – MphLee Feb 3 '15 at 17:13
• @RobertIsrael In Colombia, we have a different saying: If my aunt had wheels, she'd be a bus. – Miguelgondu Feb 3 '15 at 17:16

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.

• The two interpretations are: "if the date of yesterday was actually the date that it will be tomorrow" and "if the date of tomorrow was actually the date that it was yesterday". +1 for the algebraic interpretation. – Carl Mummert Feb 5 '15 at 20:36
• Thanks for your continued interest in the question. Could you illustrate, in your framework, how Sunday (as Robert Israel reasonably derived in his answer) may come about? Additionally, I'm still puzzled by your use of different $d$'s to translate the antecedent... – Herr K. Feb 6 '15 at 6:34
• @KevinC I think the tabular approach works better on this problem. From it, it seems more obvious that we are dealing with a kind of transformation from one "calendar" to another. In this case, we are adding add 2 to the date and day of the week. The date $x_0$ on the "old calendar" is a Wednesday. That day corresponds to the date $x_0+2$ in the "new calendar", which is a Friday. To get the Sunday result, you shift in the opposite direction: you subtract 2 to get the new date and day of the week. – Dan Christensen Feb 6 '15 at 15:13
• @KevinC I guess I have implicitly adopted the convention that if we have "If date-1 were date-2" then date-1 is on the new calendar and date_2 is the corresponding date on the old calendar. – Dan Christensen Feb 6 '15 at 15:24
• 1) how can you avoid sunday as possible solution? 2) What are the differences between your answer and my question? PS:I know that second question may sound...an attack but it is not. – MphLee May 20 '15 at 17:56

In my opinion the problem is not well defined:let the REAL today be $t_0$.

• The problem does not specify if we are in a situation where our hypothetical tomorrow (relative to a new "today" $t_1$) is the REAL yesterday (aka $t_0-1$)

(S) If [the REAL]yesterday($t_0-1$) were tomorrow($t_1+1$), then today($t_1$) would be Friday.

Question: What day is today($t_0$)?

$OR$

• if we have to imagine that our hypothetical yesterday (so a yesterday $t_1-1$ because is relative to $t_1$) is the REAL tomorrow ($t_0+1$)

(S) If yesterday($t_1-1$) were [the REAL]tomorrow($t_0+1$), then today($t_1$) would be Friday.

Question: What day is today($t_0$)?

So there can be two interpretations (Friday is $w_5$)

$1)$ if $t_0-1=t_1+1$ and $t_1=w_5$ find $t_0$

$2)$ if $t_1-1=t_0+1$ and $t_1=w_5$ find $t_0$

• By definition $t_1=\operatorname{Friday}$

• In the first interpretation $t_0-1=t_1+1$ so $t_0=t_1+2$

• In the second interpretation $t_1-1=t_0+1$ so $t_0=t_1-2$

so if Friday is $w_5$ and $\{w_i\}_{i\in\{1,2,...,7\}}$ are the days of the week then

$t_0=w_5+2=w_7$ that is Sunday

$t_0=w_5-2=w_3$ that is Wednesday

• Thanks for the answer. I understand the need to distinguish between the real today ($t_0$) and the hypothetical today ($t_1$). What I'm still confused about is why either "yesterday" or "tomorrow" in the antecedent should be hypothetical. That is, why does it have to be if $t_0-1=t_1+1$ or if $t_1-1=t_0+1$? Why can't it be translated as if $t_0-1=t_0+1$? – Herr K. Feb 3 '15 at 18:22
• BTW, I think there's a typo in "$t_1=w_3$ find $t_0$": should have been $w_5$. – Herr K. Feb 3 '15 at 18:24
• @KevinC thank you,, fixed the typo. In my opinion, the main reason is that the sentence is not "formal" and have not an unique interpretation, and imho is because it doesn't use a formal language. – MphLee Feb 3 '15 at 22:16

If you are interested in a logic which can handle this kind of conditionals, you might be interested in the field of Conditional Logic. These kinds of logics can be used to analyse conditionals with false antecedent. See

Enjoy!

Yesterday was some day. What if that day was tomorrow? If it was, today would be Friday. That means, find what day implies it would be Friday.

Yesterday was Saturday. If Saturday was tomorrow, today would be Friday.

• I thought this was the only solution as well, but think of it the other way. If tomorrow were yesterday, then today would be Friday. I'm convinced that this is the same as saying if yesterday were tomorrow then today would be Friday. In this way, Wednesday works too! Wednesdays tomorrow is Thursday, if Thursday were yesterday then today must be Friday! Thus, there are 2 solutions. – Paddling Ghost Feb 5 '15 at 20:04
• @user3440448 the answer is identical. If tomorrow were saturday, then today would be friday. In other words, today is sunday. – kηives Feb 7 '15 at 2:04

You are going back 2 days in the week. So, it would be a Wednesday.

In Hindi, there is one word for tomorrow and yesterday,so in this case no answer possible.

• @Mr Does that word really contribute here? In urdu, there is same situation as in hindi. The word used is "kal". But it doesn't participate in the relevant problem. – Sufyan Naeem Nov 28 '15 at 18:25

As mentioned by others, yesterday is never tomorrow, assuming a linear timeline. Thus, it has no bearing on the question.

In other words, the answer is the same as "Zip-A-Dee-Doo-Dah. What day is today?"

According to my phone, today is February 3rd. In general, the answer is "[insert today's date here]".

If yesterday was tomorrow (tomorrow being Saturday as today is Friday) then it was spoken on Sunday as yesterday was Saturday. It all hinges on the word "if" so don't read any more into it than that?

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