I have some trouble with translating certain sentences into a statement of propositional logic. It is homework, so I will also be happy with some hints.
Please keep in mind that I translated these sentences from dutch into english, so there can be mistakes. But the keywords are the same.
The sentences are:
a. To pass philosophy it is not necessary to make notes every week.
Keyword: not necessary
p := pass philosophy, m := make notes every week
My first intuition was $\neg(p \to m)$ but I couldn't find a proof to that. Can someone tell me if this is right or wrong (and if wrong explain why? Maybe give a hint to the right solution?)
EDIT
I woke up today and was thinking that I could eventually say:
$(s \land \neg m) \to s$
Intuitively this makes sense. "Passing philosopy and not making notes every week, implies passing philosophy". I also found that this is a tautology so I assume its proper. Any notes on this one?
b. There is still some soup in the fridge if you want.
s := there is soup in the fridge, l := you want soup
This is confusing. The soup will be in the fridge if I will like some or not. So there is in my opinion no implication made. I came up with
$s \lor l$
but I am very sure that this is wrong. I can't think of anything better. Actually I would want to just drop $l$ completely because I think it is not necessary. Any help?
Thinking about it , if I just go for the keywords (if in this case) than it should be
$l \to s$ right? If you like some than there is some soup in the fridge. But that feels so wrong…
EDIT
I settled here now with just s. Arguing that "if you like" is not necessary to be evaluated. There is soup or not.
c. To eat at mc donalds is equivalent to destroying the rainbow forest
e := eat at mc donalds
d := destroy the rainbow forest
The keyword equivalent here leads me to $\leftrightarrow$ but I don't think this is right. Because $\leftrightarrow$ means $(e \to d) \land (d \to e)$ and the last implication is definitely wrong.
I settled now with $e \to d$. But I don't think this is right.
EDIT
I more and more think that it should be $e \to d$. If I eat at mc donalds than I destroy the rainbow forest. Which makes sense right? ;o)
I seriously would appreciate any help. Also tips on how to study those kinds of questions. I digged two days through my lecture notes plus the silabus but the material is very bad. And my fellow studends also have no clue 🙂 So hopefully I get some answers here.
Best Answer
It looks like all three are trick questions, and the best answer to each of them might be "this meaning cannot be expressed in propositional logic".
Sentence (a) speaks about necessity. Your suggestion $\neg(p\to m)$ is logically equivalent to $p\land \neg m$, in other words "I will pass philosophy, and by the way I'm not taking notes". That is something quite different from saying that notes are not necessary for passing. Propositional logic cannot in itself speak about necessity -- I've gone on at length about that in an earlier answer.
In sentence (b) you have found the problem yourself -- the naked truth of the entire sentence doesn't at all depend on whether you want soup or not. The only slightly defensible propositional rendering would be simply $s$ itself, but that entirely fails to encode the real content of the sentence, namely "... and you're welcome to eat it". Propositional logic cannot express permission either.
Sentence (c) is just using "equivalent to" in a casual, decidedly non-logical way. In ordinary conversation, the meaning of this sentence is a value judgement, namely that the moral desirability of eating at McD is no higher than the moral desirability of destroying rainforests (not "rainbow forest", I think). Propositional logic is unable to express moral judgements or desirability.
It is possible that the point of the exercise is to let you discover for yourself some problems that modal logics attempt to address (especially if there's modal logic later in your course). Or it may simply be to make you aware of the dangers of translating natural language to logic by thoughtless pattern-matching.