| RavenBlog |
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| Comments on Sunday 29 April 2007: |
| Why is it that "this sentence is true" hardly ever gets any attention, but logical paradox people love "this sentence is false"? Is it that hard to recognise that they are equally broken? In fact, "this sentence is true" has an advantage over "this sentence is false", in that one of the more elegant solutions resolves the latter but not the former; the solution suggests that all statements are contractions of "it is true that (statement)", which is itself a contraction of "this statement is true: (statement)". Inserting our much-loved paradox, we get "this statement is true: this statement is false", which is logically equivalent to "A and not A", which always resolves to false. But try that with "this sentence is true" and it resolves only to "A and A", whose truth value remains dependent on whether A was true to begin with. This topic came up in conversation the other night, and most of the usual arguments were made, including splitting the sentence into two and constructing similar self-referencing-but-non-paradox sentences to show why it's difficult to produce a cohesive rule for what sentences you can designate "indeterminate" and move on. Two things didn't come up - one was what I've just said, that "this sentence is true" is not an example of a similar self-referential sentence that is okay (in fact, it was proposed as a logically acceptable sentence, and I didn't catch it), and the other was the clarification I woke up thinking this morning. The trick is that it's not the sentence that can't be self-referential; "this sentence contains more than two words" is fine, for example. It's the truth value of the sentence that can't be self-referenced. I don't think you can construct a logical paradox that doesn't reference an 'output' truth value within the paradox. It is possible to construct a non-paradox that does reference a truth value within itself, but only if you allow a logical 'indeterminate' to exist during the resolution, eg. "this sentence is false, and one equals zero." {Indeterminate AND false} equals false, so the sentence is false, so it's {true AND false}, which still equals false (I don't think you can produce a paradox flip-flop with indeterminates, only indeterminates that may later resolve in one direction). My tentative conclusion; all logical paradoxes can be resolved by the addition of an 'indeterminate' state, which is the result whenever an unassigned truth value within the paradox is referenced. {Indeterminate OR true} is true, {Indeterminate AND false} is false, and all other logical operators with an indeterminate parameter are indeterminate. Can you construct me a paradox that doesn't match these rules, or a non-paradox that resolves to indeterminate by these rules? [07:18] |
| Lambert |
| Mm, sorry.. I'm stumped. |
| Melange |
| What are you missing in your life that means you spend time thinking about these kinds of things? It's fascinating. Like reading Murakami. |
| RavenBlack |
| I don't know what I'm missing - maybe menage-a-quatres? |
| Lusus |
| Why is it that I feel as though my soul has been kicked in its metaphorical face everytime I read your blog? (very tired)lol<\/A>/ |
| Leon |
| Recently my little brother got a fortune cookie with a message that read "ignore the previous cookie", he mused over the paradox of the previous cookie reading "ignore the next cookie". What would one do in this situation and can it be modelled with a Turing machine? |
| RavenBlack |
| I don't see a problem with the "ignore the cookies" paradox, unless you have a standing directive to obey all cookies. Even then, since the cookies are in a consecutive order, "ignore the next cookie" would be the first directive received and, obeying it, you would never even see the second - unless the third was "now obey the previous cookie", which is where it really gets complicated since, if you were really ignoring the second cookie, then, in your mind, the previous cookie would be the first one, which you already obeyed. But in fact, if you are genuinely ignoring the "next" cookie then you will never know you're supposed to stop ignoring, because ignoring it would include not ending that directive based on it. |
| Leon |
| Yes, but then you would be thinking in C. In reality the third cookie would be the next cookie and you would have to ignore it, as you said. I think the paradox comes in when you have read both and have already ignored the first directive while at the same time obeying it by obeying the second. |
| Leon |
| "In reality the third cookie would be the next cookie and you would have to ignore it" - I should clarify. I mean if you skipped the second cookie then the message in the third cookie would be the one you read next and should therefore ignore. I'm very confused now. |
| Leon |
| I've figured it out - it's no paradox, you're right. It's just that the outcome is always the set of all possible outcomes. Whether you ignore both or only one or neither, you have always ignored both, obeyed both and ignored one and simultaneously the other whilst obeying them both as well. Whether or not you choose to ignore or obey cookies in the first place, there is no scope to avoid the set of all possible outcomes, no matter what you do or don't do. |
| Leon |
| Sorry I've just finished my exams and have a little too much time on my hands as you can tell. |
| RavenBlack |
| I suppose it becomes a clearer issue if the cookies aren't "ignore" but are instead "do not obey" or even "disobey". That way it's pretty much the same paradox as "the next sentence is false / the previous sentence is true". Apart from the aspect where cookie-directives don't have to be obeyed, whereas logic-based statements (once fully interpreted) do have to have a truth value. Kind of. |
| knut |
| I was thinking to myself "why does everyone worry about "this sentence is false", and not about "this sentence is true"?" So I googled "this sentence is true". And found this. Cool |