# Showing/rewriting logical validity

Hey guys, just a bit confused with this question. I'm not sure exactly how the answer should be given:

Question:

Express the following argument in symbolic form using logical connectives. Be careful to define any notation you introduce:

"1) If I earn some money then I will go for a holiday this summer.
2) I will either go for a holiday or work this summer.

3) Therefore, if I don't go for a holiday this summer then I will not have earned any money and will be working"

What I got so far:

let p = "I earn some money"
and q = "I go away for holiday this summer"
and r = "I work this summer"

then by the first statement: $(p\rightarrow q)$
by the second statement: $(q\lor r) \land \lnot(q \land r)$ (exclusive or)

I've yet to fully work out the third statement, but my question is the answer given by three different statements in symbollic form or am I supposed to somehow combine it?

edit: What would be the third statement? would it be $\lnot(q \rightarrow p)\land r$?

• Yes, three different statements. Also, no mention of "exclusive or" so don't add it... just $$q \vee r$$ – GEdgar Apr 27 '11 at 12:49
• but $(q\lor r)$ means q OR r. the statement states either q or r, but not both hence exlusive or – Arvin Apr 27 '11 at 12:57

The inclusive vs. exclusive or issue is often one translating natural language to symbolic logic. You have to decide which "or" the natural language means. I think in this case I would agree with exclusive, but that is not a mathematical problem. You have represented the exclusive or correctly. For the third statement, I would see it as $\lnot q \rightarrow (\lnot p\land r)$, seeing the "then" as grouping the last two phrases. I'm sure both of the translation decisions can be argued both ways.