7 ms·

sigh hello downvoters... There is no coherent, constant theory of truth for any language which has the word "truth." This was proven by Tarki ages ago. The

by rhelz 2y ago

sigh hello downvoters... There is no coherent, constant theory of truth for any language which has the word "truth." This was proven by Tarki ages ago. The reason is the Liar's paradox. It blows up any such theory of truth. For example, the correspondence theory of truth is blown up with this version of the liar sentence: "This sentence does not correspond to reality". Ok, does that sentence correspond with reality, or not? :-) This is a genuine problem, which nobody has ever found a solution for. See the Stanford encyclopedia article on the liar paradox. The good news is that we don't need to define truth at all. Just like we don't have to define "point" or "line" in geometry, or "set" in mathematics. These are, as it were, "axiomatic" concepts, which we use to define everything else with. They cannot be defined in terms of anything more basic or simpler, because they are the most basic, simple concepts we have.

dredmorbius 2y ago
Are there examples of self-contradictory statements which aren't strictly self-referential? That is, a statement which expresses a statement about something other than itself, or a small set of closed relations. Forms I've encountered of the liar's paradox or similar constructions are not deterministically true or false, yes, but they're also not particularly useful. This is problematic if you're trying to come up with a closed, consistent, axiomatic logical system (as Gödel and Russell were each attempting, independently AFAIU), but not so much if you're simply looking to assess statements about the real world, or even a simulation.
- rhelz 2y ago
  The paradox doesn't depend on self reference at all, e.g.: Alen: What Bob is about to say is true. Bob: What Allen just said is false.
  - sickofparadox 2y ago
    The paradox is the entire block after your colon, and it is still referencing itself.
    - rhelz 2y ago
      chuckle love the handle, and its an interesting point. Let's think about this. When you say "the entire block after your colon", you are reifying a new object, and then observing that that object is self-referential. For sake of argument, let's say that's a valid move. But what are the rules to this game? I mean, can I just say "I'm lying" and then, by fiat, construct a new object consisting of me and you, and claim that this new object is self-referential and therefore you are participating in a paradox too? What about the object which consists of the totality of what the human race has written thus far? Since one of the things we've written is the liar sentence, does that condemn the human race, for all eternity, into hopeless paradox?
  - dredmorbius 2y ago
    That would be the "small set of closed relations" case I'd mentioned above. The system of two statements is self-referential, in that it references nothing outside itself, that is, it only concerns what Allen (presuming "Alen" is a typo) and Bob say, but no other larger-world context. A somewhat trivial and contrived counterexample might be: Allen: What Bob is about to say is true and water is wet. Bob: What Allen just said is false and water is not wet. Here, we'd have to consider the truth values of both assertions, whether or not what the other party said is true, and whether or not water is wet. Both assertions would have to be true for the full statement to be true, either assertion being false would falsify the combined statement. And it is possible that both statements are false.
- AnimalMuppet 2y ago
  > Are there examples of self-contradictory statements which aren't strictly self-referential? Depends on how you define "strictly" ;-) "This statement is false" is strictly self-referentially false. "All absolute statements are false" is kind of indirectly self-referentially false, because it refutes its category, not itself as a statement. "We can only trust the results pure logic, direct sense impressions, and repeatable experiments" is a statement that is not any of those categories, and so it tells you, not that it's false, but at least that you can't trust that it's true. So that's kind of indirect, first because it's using the category, and second because it's not directly saying it's false. But, depending on your definitions, none of those may get outside your bounds of "strictly"...
  - dredmorbius 2y ago
    The difference between your two examples is that "This statement is false" is undecidable, as it's false if true and true if false. "All absolute statements are false" includes itself, but also other statements. It's more akin to the notion of "the set of all sets that do not include themselves". That is to some extent solipsistic and tautological, in that the truth of the statement is defined by the statement itself, rather than on some exogenous state of the world. Much as "non-lactating mammal" or "spineless vertebrate" would also be dependent on definitions (mammals are lactating animals, vertebrates are animals having a spinal column).
carapace 2y ago
This paradox in all it forms relies on the convention that a part can stand for the whole, which concept is itself trivially false (or merely conventional.) The only case where the "paradox" is not symbolic mummery is in the case of "yourself", when a human makes a statement regarding that human's essential identity. (E.g. "I am" or just "I", is the simplest form.) In that case the Buddhists and others have determined experientially that there is no essential self, it is an illusion, if "you" go searching for it there is nothing there. This leaves us with the "mystery" of subjective awareness. Fortunately there is nothing meaningful to say about subjective awareness (including "this" statement) whatsoever, so it offers no challenge to the logician or semanticist.
- rhelz 2y ago
  // there is no essential self, it is an illusion, // Who is is being fooled by this illusion?
  - carapace 2y ago
    Exactly. Operationally, dwelling on that question is the short, wide path to Enlightenment. https://en.wikipedia.org/wiki/Self-enquiry https://en.wikipedia.org/wiki/Self-enquiry

by rhelz 2y ago

dredmorbius 2y ago

Are there examples of self-contradictory statements which aren't strictly self-referential? That is, a statement which expresses a statement about something other than itself, or a small set of closed relations. Forms I've encountered of the liar's paradox or similar constructions are not deterministically true or false, yes, but they're also not particularly useful. This is problematic if you're trying to come up with a closed, consistent, axiomatic logical system (as Gödel and Russell were each attempting, independently AFAIU), but not so much if you're simply looking to assess statements about the real world, or even a simulation.

rhelz 2y ago
The paradox doesn't depend on self reference at all, e.g.: Alen: What Bob is about to say is true. Bob: What Allen just said is false.
- sickofparadox 2y ago
  The paradox is the entire block after your colon, and it is still referencing itself.
  - rhelz 2y ago
    chuckle love the handle, and its an interesting point. Let's think about this. When you say "the entire block after your colon", you are reifying a new object, and then observing that that object is self-referential. For sake of argument, let's say that's a valid move. But what are the rules to this game? I mean, can I just say "I'm lying" and then, by fiat, construct a new object consisting of me and you, and claim that this new object is self-referential and therefore you are participating in a paradox too? What about the object which consists of the totality of what the human race has written thus far? Since one of the things we've written is the liar sentence, does that condemn the human race, for all eternity, into hopeless paradox?
- dredmorbius 2y ago
  That would be the "small set of closed relations" case I'd mentioned above. The system of two statements is self-referential, in that it references nothing outside itself, that is, it only concerns what Allen (presuming "Alen" is a typo) and Bob say, but no other larger-world context. A somewhat trivial and contrived counterexample might be: Allen: What Bob is about to say is true and water is wet. Bob: What Allen just said is false and water is not wet. Here, we'd have to consider the truth values of both assertions, whether or not what the other party said is true, and whether or not water is wet. Both assertions would have to be true for the full statement to be true, either assertion being false would falsify the combined statement. And it is possible that both statements are false.
AnimalMuppet 2y ago
> Are there examples of self-contradictory statements which aren't strictly self-referential? Depends on how you define "strictly" ;-) "This statement is false" is strictly self-referentially false. "All absolute statements are false" is kind of indirectly self-referentially false, because it refutes its category, not itself as a statement. "We can only trust the results pure logic, direct sense impressions, and repeatable experiments" is a statement that is not any of those categories, and so it tells you, not that it's false, but at least that you can't trust that it's true. So that's kind of indirect, first because it's using the category, and second because it's not directly saying it's false. But, depending on your definitions, none of those may get outside your bounds of "strictly"...
- dredmorbius 2y ago
  The difference between your two examples is that "This statement is false" is undecidable, as it's false if true and true if false. "All absolute statements are false" includes itself, but also other statements. It's more akin to the notion of "the set of all sets that do not include themselves". That is to some extent solipsistic and tautological, in that the truth of the statement is defined by the statement itself, rather than on some exogenous state of the world. Much as "non-lactating mammal" or "spineless vertebrate" would also be dependent on definitions (mammals are lactating animals, vertebrates are animals having a spinal column).

carapace 2y ago

This paradox in all it forms relies on the convention that a part can stand for the whole, which concept is itself trivially false (or merely conventional.) The only case where the "paradox" is not symbolic mummery is in the case of "yourself", when a human makes a statement regarding that human's essential identity. (E.g. "I am" or just "I", is the simplest form.) In that case the Buddhists and others have determined experientially that there is no essential self, it is an illusion, if "you" go searching for it there is nothing there. This leaves us with the "mystery" of subjective awareness. Fortunately there is nothing meaningful to say about subjective awareness (including "this" statement) whatsoever, so it offers no challenge to the logician or semanticist.

rhelz 2y ago
// there is no essential self, it is an illusion, // Who is is being fooled by this illusion?
- carapace 2y ago
  Exactly. Operationally, dwelling on that question is the short, wide path to Enlightenment. https://en.wikipedia.org/wiki/Self-enquiry https://en.wikipedia.org/wiki/Self-enquiry