Useless Poll: How would you interpret this sentence?

In My Humble Opinion
1

I’m taking a Discrete Math course and we’re starting with propositional logic. I think of off-the-wall scenarios too much. Now, the teacher doesn’t use tricky “unfair” questions, so I’m not in any danger of doing poorly on a test because of this since it wouldn’t actually show up on a test, but thought it’d be interesting to see the responses. The poll is about the statement “s”.

p: Billy can run quickly.
q: Billy can only run quickly. [Alternatively: Billy can not run slowly.]
r: Billy can not run fast.
s: Billy can not not run quickly.
–If you don’t want my reasoning to pollute you, take the poll now. Statement r is simply there for flow, it’s not in the poll.–

I personally think that despite the fact that it has two nots, they don’t fully cancel each other. Specifically, s is equivilent to q, not p. The first not modifies “not run fast,” he can not “not run quickly,” extrapolating that since the denial is specific he can still operate in the exclusion of that statement. In other words, he can only run quickly. My teacher disagrees (and on a test I’d probably err on this side as well due to the nature of the course), and says that the not not cancel each other, and it simply means “he can run quickly” with no other underlying statement.

Of course, even though I’d agree with him on a propositional logic test, simply because of all the double negation finagling we have to learn, this poll is mostly about what the natural conclusion you’d immediately come to would be. For me, that’s still the notion that s and q are more or less the same statement because of the specific denial. Obviously even though this quiz is about “natural English” and your conclusions it’s still a clumsy, theoretical statement, but I do what I can.

Remember that it takes a couple minutes to make the poll and the hamsters will show the thread before that so hold your horses.

2

I think it’s that your interpretation follows the accepted conventions of English grammar, while your teacher is interpreting the NOTs in formal mathematical terms.

3

If you say someone can only do something, you have eliminated everything else - that is the only thing he can do. Therefore, statement s equals q - there is only one thing Billy can do - run quickly.

4

As a matter of grammar (as opposed to math) I think you were correct that s and q are the same.

Billy can not [not run quickly]. Hence, Billy can only run quickly. But it is such an awkward wording that (while technically grammatically correct) it could be confusing.

5

I think the "not"s cancel each other and S and P are identical.

6

S = Billy can’t help but run quickly when he runs. It doesn’t mean he always runs though.

7

I’m surprised at the number of “S = Q” responses. I think S = P, and the reason is the difference between “cannot” and “can not”.

If I cannot eat apples, then it is impossible for me to eat an apple. If I can not eat applies, then it is possible for me not to eat an apple. Whether I can in fact eat an apple is unknown.

If Billy cannot run quickly, it is impossible for him to run quickly. If Billy can not run quickly, then it is possible for him to run slowly. Whether he can in fact run quickly is unknown.

Thus, if Billy can not NOT run quickly, then it is possible for him to fail to run slowly. Whether he can in fact run some way other than quickly is unknown…but, the statement that it is possible for him to fail to run slowly is equivalent to the statement that it is possible for him to run quickly, therefore S = P. (This does assume that slowly and quickly are the only possible ways to run. If there are three or more possibilities, then that last statement doesn’t follow, but “S = Q” wouldn’t be correct in that case either.)

I now sit back and eagerly await being proven completely wrong. :slight_smile:

8

I see different possible interpretations here.

P (“Billy can run quickly.”) seems pretty straight forward. It obviously says that Billy can run quickly but it sets no limits on other ways Billy can run or other non-running actions he can do.

But Q (“Billy can only run quickly.”) is open to interpretation. Is the meaning “There are a number of different ways to run but, when he chooses to run, Billy can only run in one manner - quickly.”? Or does it mean “There are a number of different actions that a normal person can do. But Billy can only do one thing - run quickly.”?

The alternative statement Q (“Billy can not run slowly.”) would seem to say the first interpretation is the correct one.

Leaving R aside, we come to S (“Billy can not not run quickly.”) which also has two interpretations. One interpretation of this is that the two nots cancel each other out and the statement is equivalent to “Billy can run quickly.” But that seems unlikely - the nots were presumedly put in the statement for a reason. So the second interpretation seems correct here; that S means “Most people have the option of running quickly or not running quickly. Billy does not have that option because he cannot choose to not run quickly.”

So I didn’t vote “S and P are the same” because I felt that while they might possibly be the same, that would be the less likely option.

And I didn’t vote “S and Q are the same” because I again I felt it was possible they were the same but the interpretations of the statements that would make them equivlaent were not the likeliest combination.

Compare Q1 (“There are a number of different ways to run but, when he chooses to run, Billy can only run in one manner - quickly.”) and S1 (“Billy can run quickly.”): S1 doesn’t prohibit the option of Billy running slowly but Q1 does.

Compare Q1 and S2 (“Most people have the option of running quickly or not running quickly. Billy does not have that option because he cannot choose to not run quickly.”): S2 says that Billy must run quickly - he cannot do anything else. Q1 only prohibits Billy from running in a non-quick manner - but he can choose not to run at all.

Compare Q2 (“There are a number of different actions that a normal person can do. But Billy can only do one thing - run quickly.”) and S1. Q2 places restrictions on Billy’s actions while S1 does not.

Now compare Q2 and S2. They both mandate Billy to continuously run quickly. They are the only combination which would make S and Q equivalent but, as I stated above, I feel that Q1 and S2 are the likeliest combination.

I also didn’t vote “S is not in any way the same as either P or Q” because of the absolute nature of this conclusion. I think that my interpretations are the reasonable ones but I am open to the possibility that the other interpretations might be correct. So while I conclude that S is probably not the same as P or Q, I cannot state it as a proven fact.

So I voted for the last option.

I will concede I may be overthinking this.

9

S is not in any way the same as either P or Q:

Billy can not not run quickly ≠ Billing can only run quickly.
The latter statement could be reworded as "Whenever Billy runs, Billy runs quickly: or as “Billy cannot run non-quickly”.

Both of those also apply to the FIRST statement as well; but the FIRST statement could also be reworded as “Billy cannot refrain from running, NOR can Billy run non-quickly”.

The latter statement, in contrast, makes no assertions about Billy’s ability to NOT RUN AT ALL.

{This post posted before reading other folks’ comments}

EDIT: Hmm, it’s an exaggeration to say S is not in any way the same as P or Q. It is the same in some ways. Sorry, I suppose I really should have opted for Yo, dude, you forgot _____ as the best answer…

10

Apologies, the former was intended, but I was trying to keep the statements as simple as possible and missed that interpretation. However, given the nature of the poll, that sort of overthinking is perfectly valid.

11

You know what “not running quickly” is? Billy fails at that. He must, therefore, run quickly. It doesn’t matter if you apply the “only” to the verb (The only action for Billy is running quickly) or to the adverb (running can only be quick), as it’s present in both statements. Whatever you meant in S, you also mean in Q.

Given the choices of A, B, and C:

  1. Billy can not choose C = A or B
  2. Billy can not not choose C = C

S translates as 2. Thus, if Billy can not not run quickly, then Billy must run quickly. I should vote for both P and Q, as Q being true necessarily means P is true too.
What I think your teacher (and Roland) miss is that “to cancel out” means that the “can” should be deleted too. It’s part of the not. So if he cancels out the two nots, it should yield:

Billy runs quickly.

You can either interpret that as “He always runs quickly” or “whenever he runs, it’s quickly.” It makes no difference.

12

Well, I didn’t reason using the “not not cancels itself out” proposition, but even if I did, the “can” would not be cancelled out…again, because there is a difference between “cannot not” and “can not not”.

13

According to S, when Billy runs, he’s incapable of running any speed except fast; he cannot run at a non-fast speed. This is the same as Q.

14

No, according to S, when Billy runs, he is capable of not running at a speed that is not fast. It’s a slight distinction, but here it makes all the difference.

Look at it this way: “cannot” establishes the impossibility of a positive case, whereas “can not” establishes the possibility of a negative case. The difference is that the latter says nothing about the possibility of a positive case. With “can not”, either positive OR negative can still be true. (It’s the same distinction that causes atheists fits when talking about the difference between NOT believing in God and DISbelieving in God.)

Once you know that, it’s easy to see even without going through the derivation that a possibility that starts with “Billy can” cannot be equivalent to a statement which asserts that ONLY the positive or the negative can be true. Q asserts that only the positive is true, so no matter how many “nots” you throw in there after the “can” in S, you will never arrive at a statement logically equivalent to Q.

ETA: To the P = Q supporters, would you consider the two statements “I cannot eat apples” and “I can not eat apples” to mean the same thing? If you wouldn’t, then I believe my reasoning holds (though I’m happy to be proven wrong if you think I am).

15

I can see where you’re coming from, however, in a pinch I would say those two sentences are the same without some rewording. It is slightly ambiguous, I’ll give you that, but without changing sentence 2 to something like “I can choose not to eat apples” I’d probably be more likely to say the two sentences are the same.

16

My immediate impression is that s=q. However, there is certainly room for s=p, for the reason stated by Roland Orzabal. That being the case, the answer is “4.”

17

Damn, I misread it. I missed the second “not” in S … voted for the wrong option…

18

I’m with you, Roland. Your first post was more or less exactly how I reasoned it out. It’s surprising to me that others would intuitively see “I cannot X” and “I can not X” as having the same meaning.

For instance, let’s say I’m planning to go to the movies on Saturday, and invite my kid along. We might have the following conversation:

Kid: Do I *have *to go?
Me: Nope. You can go, or you **can *not ***go; it’s up to you.
Kid: Okay, I’ll think about it.
[Later, after school]
Kid: Hey, I can’t (cannot) go to the movies. We just got assigned a ton of homework that’s due Monday.

To me, there’s a clear difference. If someone “can not” X, they can choose not to X, or refuse to X. If they “cannot” X, they’re unable to X, whether they want to or not.

So what “Billy can not not run quickly” means, then, is:
Billy can choose not to choose not to run quickly.
or
Billy can refuse to refuse to run quickly.

Imagine that Billy’s and his friends are in gym class. Some of his friends don’t like the gym teacher, so to annoy him, they refuse to run quickly. (Note that we don’t know what they’re doing instead. Maybe they’re running slowly, maybe they’re taking a nap. As long as it’s not “running quickly”, it qualifies.) And of course, they’re able to refuse - they teacher can’t force them. So to put it another way, Billy’s friends can not run quickly. Billy’s friends encourage him to join their boycott, but he chooses not to. And of course, they can’t force him, either. So, Billy can not not run quickly. Now, if Billy weren’t *able *to run quickly, he couldn’t defy the boycott, even if he wanted to. So if he is able to defy the boycott, if **he can not not run quickly, then that must mean that Billy can run quickly.

For another example, consider the statement:
“I’m not not licking toads.”
We all understand this to mean that Homer is, in fact, licking toads. We know that he is not X, where X is “not licking toads”. So whatever he is doing can’t be categorized as “not licking toads”. Therefore, he is licking toads.

And if Marge said, “Homer, I order you to not lick toads!” he could respond, “You’re not the boss of me! I can not not lick toads if I want to!” Clearly, this means that he can lick toads. He can X, or he can not X. He can “not lick toads”, or he can not “not lick toads”. It’s his choice.

19

In the Englishlanguage, cannot and “can not” are the samewords.

20

Cite for common-usage English accepted, CS, but not in the context of a qualified propositional logic statement*, which is what the OP is. Granted, I should have phrased my question about the apple sentences as “are these two logically equivalent” rather than “do these two mean the same thing”, to emphasize that difference.

Of course, on re-reading the OP just now, I see that Jragon did say that the poll was targeted at “natural English”, which validates the numerous S = Q responses (as well as reaffirming my faith in the Dope in general ;)). He just didn’t say that until AFTER the part where he instructed us to stop reading so as not to be influenced by his own reasoning…which I did.

Short version, logically S = P, conversationally S = Q. Still, in a discrete math class, I’d assume the former (which the OP also noted).

*You can infer that you’re talking about qualified prop logic because the question implies a difference between P and Q (that is, there is a difference between “Billy can [sometimes] run quickly” and “Billy can only run quickly”). S still equals P in non-qualified prop logic, but the existence of the qualifiers means that “not not” doesn’t automatically cancel itself, so you actually have to do the derivation.