4 ms·

Law of excluded middle. The two statements are equivalent.

by rcme 4y ago

maxbond 4y ago
I'd argue not for two reasons. 1. This isn't a two valued system, you can think it may be possible without believing it's possible. 2. Rhetorically these are very different statements, and I believe GP was referring to the rhetorical difference between them.
- rcme 4y ago
  If something is "not impossible", then it is possible. To say otherwise is to violate LEM, which is an essential part of all of our logical systems. The vast majority of mathematical proofs, many of which form the foundation of our physical understanding, would not hold without LEM. Stated another way, if something is "not impossible", then what is it?
  - maxbond 4y ago
    Sure but the claim is, "I believe X," not X. > I would probably have titled it "I think FTL travel might not be impossible". (Emphasis added) Belief isn't binary, ergo no LEM. LEM doesn't apply to all logical systems, it only applies to two-valued logic. In practice lots of logic is 3 valued or otherwise non-binary (such as partial equality or when we have a probability distribution over a set of outcomes). Some schools of mathematics (the constructivists and intuitionalists) actually reject the LEM. I personally, as a PhD in having internet opinions, wouldn't go so far as to reject it, but relying on the LEM or proof by contradiction is a risk. You've shown `~X`, and you're betting that a proof of `X` does not exist. But it could be that `X and ~X` is the paradox that reveals your axioms are flawed. But I digress. The criticism is that X isn't shown to be true, it's just not shown to be false. That would be sufficient if the only values were true and false, but that's not the case here. What the original commenter was saying was, even if you accept her argument in full, it moves the needle to "maybe".
    - rcme 4y ago
      The difference between X is true vs. I believe X is true is irrelevant. > I would probably have titled it "I think FTL travel might not be impossible". Again, what does “not impossible” mean? The only meaning, logically, is “not impossible = possible.” The fact that you’ve used the word “not” implies a two state system. LEM applies. The original title made sense. Your proposed change to the title is either: equivalent to the original title or logically inconsistent.
  - yarg 4y ago
    But a thing "not known to be impossible" is not necessarily possible.
    - rcme 4y ago
      “Not known to be impossible” is not the same as “not impossible.” I think original suggestion was trying to add some linguistic flair, but generally double negatives are hard to understand. It’s better to just state your point plainly.
  - throwawaymaths 4y ago
    Wrong, there is a middle: - known possible - not known to be possible or impossible - known impossible "Is possible" suggests case 1, if "is known" is implied (we are talking about science here, so that should be the case). "Is not known to be impossible" covers 2&3.
    - rcme 4y ago
      If "X is possible" implies "X is [known to be] possible" then shouldn't "X is not impossible" imply "X is [known to be] not impossible"?
      - maxbond 4y ago
        I think the misunderstanding here is that `X` is different from `X is known,` and `X` doesn't imply `X is known.` Consider if we are analyzing a program which does not halt (`~doesHalt`). The halting problem tells us that we can't actually prove this (`doesHalt is not known`). Can we flip the question around? Under what scenario would you accept the middle condition of, "we do not know whether X is or is not possible" to exist?
        rcme 4y ago
        There are plenty of things you can say "we don't know whether or not this exists." But that doesn't somehow invalidate LEM. Whether or not something is possible is a binary state, regardless of whether or not we know the answer. For instance, we don't know whether God exists or not (just a contrived example). Now consider the statement: "It's not impossible for God to exist." That's equivalent to saying "It's possible for God to exist." The two statements are totally equivalent. Sure, we don't know whether or not God exists, but that doesn't make the statement "It's not impossible for God to exist" fundamentally different from "It's possible for God to exist." Now consider the headline: "I think faster than light travel is possible." And the suggestion is to change this to "I think faster than light travel is not impossible." Again, the two statements are totally equivalent because whether or not something is possible is a binary state and the law of the excluded middle.
throwawaymaths 4y ago
logically, yes. As a matter of language, no, due to what the emphasis is on.
wowyallsuck 4y ago
I think the proposed rewrite better captures the author’s intention: > I was trying to explain why the arguments saying it's impossible are inconclusive and we shouldn't throw out the possibility too quickly Specifically the point is that the author believes that impossibility has not been adequately proven. Not that possibility has been proven.

Law of excluded middle. The two statements are equivalent.

by rcme 4y ago

Law of excluded middle. The two statements are equivalent.

I'd argue not for two reasons. 1. This isn't a two valued system, you can think it may be possible without believing it's possible. 2. Rhetorically these are very different statements, and I believe GP was referring to the rhetorical difference between them.

rcme 4y ago
If something is "not impossible", then it is possible. To say otherwise is to violate LEM, which is an essential part of all of our logical systems. The vast majority of mathematical proofs, many of which form the foundation of our physical understanding, would not hold without LEM. Stated another way, if something is "not impossible", then what is it?
- maxbond 4y ago
  Sure but the claim is, "I believe X," not X. > I would probably have titled it "I think FTL travel might not be impossible". (Emphasis added) Belief isn't binary, ergo no LEM. LEM doesn't apply to all logical systems, it only applies to two-valued logic. In practice lots of logic is 3 valued or otherwise non-binary (such as partial equality or when we have a probability distribution over a set of outcomes). Some schools of mathematics (the constructivists and intuitionalists) actually reject the LEM. I personally, as a PhD in having internet opinions, wouldn't go so far as to reject it, but relying on the LEM or proof by contradiction is a risk. You've shown `~X`, and you're betting that a proof of `X` does not exist. But it could be that `X and ~X` is the paradox that reveals your axioms are flawed. But I digress. The criticism is that X isn't shown to be true, it's just not shown to be false. That would be sufficient if the only values were true and false, but that's not the case here. What the original commenter was saying was, even if you accept her argument in full, it moves the needle to "maybe".
  - rcme 4y ago
    The difference between X is true vs. I believe X is true is irrelevant. > I would probably have titled it "I think FTL travel might not be impossible". Again, what does “not impossible” mean? The only meaning, logically, is “not impossible = possible.” The fact that you’ve used the word “not” implies a two state system. LEM applies. The original title made sense. Your proposed change to the title is either: equivalent to the original title or logically inconsistent.
- yarg 4y ago
  But a thing "not known to be impossible" is not necessarily possible.
  - rcme 4y ago
    “Not known to be impossible” is not the same as “not impossible.” I think original suggestion was trying to add some linguistic flair, but generally double negatives are hard to understand. It’s better to just state your point plainly.
- throwawaymaths 4y ago
  Wrong, there is a middle: - known possible - not known to be possible or impossible - known impossible "Is possible" suggests case 1, if "is known" is implied (we are talking about science here, so that should be the case). "Is not known to be impossible" covers 2&3.
  - rcme 4y ago
    If "X is possible" implies "X is [known to be] possible" then shouldn't "X is not impossible" imply "X is [known to be] not impossible"?
    - maxbond 4y ago
      I think the misunderstanding here is that `X` is different from `X is known,` and `X` doesn't imply `X is known.` Consider if we are analyzing a program which does not halt (`~doesHalt`). The halting problem tells us that we can't actually prove this (`doesHalt is not known`). Can we flip the question around? Under what scenario would you accept the middle condition of, "we do not know whether X is or is not possible" to exist?
      - rcme 4y ago
        There are plenty of things you can say "we don't know whether or not this exists." But that doesn't somehow invalidate LEM. Whether or not something is possible is a binary state, regardless of whether or not we know the answer. For instance, we don't know whether God exists or not (just a contrived example). Now consider the statement: "It's not impossible for God to exist." That's equivalent to saying "It's possible for God to exist." The two statements are totally equivalent. Sure, we don't know whether or not God exists, but that doesn't make the statement "It's not impossible for God to exist" fundamentally different from "It's possible for God to exist." Now consider the headline: "I think faster than light travel is possible." And the suggestion is to change this to "I think faster than light travel is not impossible." Again, the two statements are totally equivalent because whether or not something is possible is a binary state and the law of the excluded middle.

throwawaymaths 4y ago

logically, yes. As a matter of language, no, due to what the emphasis is on.

wowyallsuck 4y ago

I think the proposed rewrite better captures the author’s intention: > I was trying to explain why the arguments saying it's impossible are inconclusive and we shouldn't throw out the possibility too quickly Specifically the point is that the author believes that impossibility has not been adequately proven. Not that possibility has been proven.