How to interpret “$P\,$ but $Q\,$” when symbolizing natural language statements?

If it is wednesday then I won't study, but if it rains then I will study and watch TV

Let's make that into propositions:

$P:$ It is wednesday

$Q:$ I will study

$R:$ It rains

$S:$ I will watch TV

Now let's get the premises in that sentence... here is my problem: I'm not sure what to do with the "but". Would this be right?

$P \implies \lnot Q$

$R \implies Q \land S$

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Yes that is correct. "But" translates as "and", as there is no logical operator exclusively for "but". –  Anonymous Sep 27 '13 at 22:54
In English, "but" is just "and" with a subtextual judgement that what follows contrasts with what precedes. Such interpretations have no value in logic. –  Joshua Ciappara Sep 27 '13 at 22:56
@Anonymous: Could you check on Greg's answer? He says that it isn't correct, so now I don't know what's going on :( –  Zol Tun Kul Sep 27 '13 at 23:00
@Omega I don't think Greg is right. It seems that he is confused. –  Anonymous Sep 27 '13 at 23:02
@Anonymous I think you're confused =p. "But" certainly can't mean "and" in this context. If Omega's guess in his OP is correct, then a plausible event (P and R, it's Wednesday and raining) implies a logical contradiction (he both studies and doesn't study). So his original guess can't be correct. –  GMB Sep 27 '13 at 23:05

What you have isn't quite right. As a hint, a more complete way of expressing your sentence is:

"If it is Wednesday and it doesn't rain then I won't study, but if it rains then I will study and watch TV."

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So it should be like $P \land \lnot R \implies Q$ and then $R \implies Q \land S$? –  Zol Tun Kul Sep 27 '13 at 22:56
I think you want "$\lnot Q$" in that first expression, not "$Q$". But otherwise, you're right. –  GMB Sep 27 '13 at 22:58
Ah, I said $Q$ because your quote said "then I will study". –  Zol Tun Kul Sep 27 '13 at 23:10
Oh, oops, you're right. Edited. –  GMB Sep 27 '13 at 23:11
@Omega: The sentence probably should read "..., but if it is Wednesday and it rains then I will study and watch TV." So your second phrase should really be $P \land R \implies Q \land S$. –  GMB Sep 27 '13 at 23:24

The actual sentence you are analyzing is somewhat ambiguous, because it is not entirely clear whether it says anything about what happens when it's not Wednesday. Getting to the heart of your question, in natural language, "but" is used in place of "and" to reflect one of at least two situations:

1. If $a$ then $b$, but if $\neg a$ then $c$. (Contrasting what happens if a certain proposition does or does not hold)

2. If $a$ but $\neg b$ then $c$. (Generally, "but not" is preferred to "and not", but most particularly when one would commonly think of $b$ happening when $a$ happens: "If a man falls six stories but survives, he will likely be permanently disabled.")

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We typically use "and": $\quad p$ but $q \iff p \land q$.

If we are going to go Michael's route and interpret what might be meant, then I suggest we need

$$P \rightarrow \Big((\lnot R \rightarrow \lnot Q) \land (R \rightarrow (Q \land S))\Big)$$

If it's Wednesday, then $\Big($if it's not raining then I won't study, but if it's raining, then I'll both study and watch t.v.$\Big)$ since we are discussing what we'll do on Wednesday, not any other day.

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This isn't right in the context that he is using "but." In his context of "but," $P$ and $R$ aren't mutually exclusive events. –  GMB Sep 27 '13 at 22:57
Well then, @Greg, I've updated, but now, yours isn't quite right. –  amWhy Sep 27 '13 at 23:10
I still disagree with your first expression. In the context of OP's question, $P$ and $R$ can both happen (it can be both Wednesday and raining). According to your expression, that implies both $Q \land \lnot Q$. –  GMB Sep 27 '13 at 23:15
@Greg: I do understand what you're saying. But I do suggest that we are discussing alternatives for Wednesday, not anytime it rains, since it would be quite unlikely to state "(Whenever it rains) if it rains, then I'll study and watch t.v"! I'm just saying...The premise: "If it's Wednesay" encompasses the entire statement. –  amWhy Sep 27 '13 at 23:18
@GregBodwin Maybe we need to know more about the context and perhaps even particular speaking patterns of the speaker to fully clarify what this sentence in terms of propositional logic. –  Doug Spoonwood Sep 28 '13 at 0:58

The distinction between 'but' and 'and' in English is not very mathematical. We use 'but' instead of 'and' when the second statement is seen to be somehow contradictory to the first.

In mathematics, such a distinction is useless, so you can just use 'and' for both. Obviously, when you are writing a logical statement in English, use 'but' as you see fit, and how you feel you would naturally use it.

Bad (although not incorrect) English:

The sphere has volume $V$, and when we cut it up and put it back together again in this way, it has volume $2V$.

The Weierstrass function is continuous everywhere and differentiable nowhere.

Better, more idiomatic English:

The sphere has volume $V$, but when we cut it up and put it back together again in this way, it has volume $2V$ - better English.

The Weierstrass function is continuous everywhere and differentiable nowhere.

Both are equally valid and equivalent mathematical statements, however.

[Note: of course, this applies to a great many languages besides English.]

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