translate in formal logic?

Question

Only if it doesn't rain will I go shopping tomorrow or will I not go shopping tomorrow.

If I go shopping tomorrow, then either I'll go shopping tomorrow, or I won't go shopping tomorrow.

Is any of these a valid sentence? Why?

Answer 1

"it rains" / p
"I (will) go shopping tomorrow" / q

1. (q v ~q) -> ~p
2. q -> (q w ~q)

("v" / inclusive disjunction, "w" / exclusive disjunction)

1. says that "not raining" is a necessary condition for my going / not going shopping, but not a sufficient one (in theory, it might be not raining, yet me doing something else). So the correct formalization is the way I've written it, not vice versa, nor with <->.

Now, the thing is that, therefore, if it DOES rain, then I should be doing something else than going shopping or not going shopping, which I obviously cannot. So sentence 1. is true in a possible world where it doesn't rain, and false in a world where it rains, so it's not valid. If we only look at the formalization, the antecedent is always true (tautology), while the consequent is a non-valid (or synthetic) sentence, so there is surely a "line" in the truth table where the implication is false. Actually there are two lines, those with p True.

2.'s consequent is a tautology, so the implication will be true in all possible worlds, so it's a valid sentence (no surprise since it's a consequence of a substitution instance of the second axiom of Principia Mathematica "q -> (p v q)")

edit: if I do go shopping, then of course that one and only one of the 2 variants of q w ~q will be true, namely q.

edit, 4 Jacquesh: of course you can't; but "p OR ~p" doesn't mean "p AND ~p". "I go shopping or I go swimming" means that I do at least one of those actions (in the case of sentence 1., I can do exactly one of the two actions), and "Either I go shopping, or I go swimming" means that I do one or the other of the actions, but not both, nor none. Both going and not going shopping isn't stipulated anyhere.

Answer 2

-R <--> (S v -S)
S --> (S v -S)

They are valid in that they are not inconsistent. However, both of these are tautologies. What you are essentially saying is that you will go shopping no matter what happens. The second one is even more pointless. You are saying you will either be shopping or not, if you go shopping; which is true, but completely repetitive.

EDIT: Actually, the first one, with the use of if, and only if means something different and a little strange. I just realized raining was also dependent on you shopping. You also get to predict the weather. It can be translated as whether or not I go shopping, it's not going to rain tomorrow and if that happens, I may or may not go shopping. (Because both side of the argument are equally dependent on one another).

Answer 3

Here's a transaltion in informal logic:

The subjects are confusing because we are trying to relate them when they are not related, and vice versa

In the first sentence the two actions are opposites, which confuses us, but the sentence is fine (awkward, but fine)
If you change the actions in the first sentence to things that "make more sense" you get:

Only if it doesn't rain will I 'have a picnic' or will I 'not watch a movie'

On the second sentence I wondering how one can go shopping and not go shopping. However maybe if you go shopping tomorrow, you may or may not go shopping the next day, which would be tomorrow once the IF is decided.

Just a thought...

Answer 4

I think the first sentence is missing a word. Grammatically speaking, it's not even a sentence because it's worded as a question.

"Only if it doesn't rain will I go shopping tomorrow.... or will I not go shopping tomorrow?" is how it should be worded as is.

With the extra wording, this is how it looks -
"Only if it does not rain will I got shopping tomorrow; otherwise, I will not go shopping tomorrow."

The second sentence makes sense because if he goes shopping tomorrow, then he is doing one of the following declarations of either going or not going shopping tomorrow.

"If I go shopping tomorrow, then EITHER I will go shopping tomorrow, OR I will not go shopping tomorrow." Since he's going shopping, he is doing one of the proposed actions.
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And this is why philosophers never get anywhere in real life besides being hermits living outta cardboard boxes. Wanna be a real philosopher? Go become a monk. Them Buddhist and Catholic Monestaries will appreciate all the deep existential thought you can provide.

Answer 5

The first one makes no sense because you say ONLY if it doesn't rain you will either shop or you won't shop. That means you can't do either if it rains, which is impossible.
The second makes no sense because you can't either go shopping tomorrow or not if the condition is that you ARE going shopping.

Answer 6

They're both wrong but the first one is trying to say you may or may not go shopping only if it doesnt rain - you have two options. But if it rains then you have your mind set to do only ONE thing, which wasnt mentioned, which would most probably be not going.

The second one doesnt depend on the rain (or anything else). Its basically saying, "If i go, I go."

.....But like i said, whoever wrote this could have done much better.

Answer 7

Neither one makes any sense at all.

Whether it rains or not, you are either going to go shopping or you're not going to go shopping.

And if you go shopping, you can't not go shopping.

It's an either/or proposition.

Answer 8

This question is a complete nonsense! You can't do and not do something at the same time!

Answer 9

Apparently, you will have to wait for the moment to arrive so you can see how the weather will affect your day.