My own view is that this argument is as convincing as an argument for any philosophically interesting position, and so should be taken seriously. Trivialism should not be treated as a special case in this regard. Philosophers have committed to claims on the basis of a lot less.
How does trivialism differ from assuming the existence of a Tegmark IV universe?
Tegmark IV is the space of all computable mathematical structures. You can make true and false statements about this space, and there is nothing about it that implies a contradiction. You may think that any coherent empirical claim is true in Tegmark IV, in that anything we say about the world is true of some world. But being true in some world does not make it true in this world. If I say that the sky is green, I am implicitly referring to the sky that I experience, which is blue. That is, I am saying that the sky which is blue is green. So I’m contradicting myself, and the statement is false. You don’t even need to think of alternate universes to reason through this. After all, some planet in our galaxy surely has a green sky.
A spectral argument given in defense of trivialism in the dissertation runs like this...
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic. In particular, the aspects of natural language that make it inconsistent probably can’t be translated into classical logic. After all, that’s why we invented classical logic in the first place.
Level IV refers to parallel worlds in distinct mathematical structures, which may have fundamentally different laws of physics.
Trivialism induces a mathematical structure, and so is contained in the level IV multiverse. I think there’s some meta-level confusion in the rest of the first part of your comment.
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic.
It’s not clear to me how this claim affects the argument. Asserting the negation of the converse of (c) doesn’t imply anything about (c).
Did these points come up in the dissertation?
The argument is not central to the dissertation. He reports it from a trivialist to establish the existence of at least one trivialist.
How does trivialism differ from assuming the existence of a Tegmark IV universe?
Because even if we assume the existence of every mathematical structure, we are still assuming that they are coherent. Mind you, there are consistent models of some paraconsistent logic (even in set theory), but there is no model of the theory of all sentences. This is pretty standar model theory: the class of models of the total theory is empty (viceversa: the theory of the class of all models is empty). Anyway, assuming trivialism is uninteresting (as the name correctly imply ;)): we still can play a formal game that mimics the difference between truth and falsity.
This is pretty standard model theory: the class of models of the total theory is empty (viceversa: the theory of the class of all models is empty).
I’m not sure why level IV would restrict itself to standard model theory. In a tri-valued logic (i.e., all propositions are either true, false, or both), there are non-trivial models of trivialism.
I read this sentence differently than its author intended, I think:
It never ceases to astound me that Graham Priest was willing to take this project seriously enough to act as its supervisor – a clear demonstration of the philosophical spirit.
It makes sense; as he lays out in the first section, it isn’t clear why dialetheism is different from trivialism. If they weren’t different, then a good part of his advisor’s field would become trivial! Taking on a willing grad student to devote time to separating the two is just good politics.
it isn’t clear why dialetheism is different from trivialism.
Imagine all well-formed logical statements, stretching out in an infinite list.
Each of these statements are to be marked “true” or “false.” For each possible marking, there is a shortest set of rules that generates that marking. Those rules are “rules of logic” you’d be following if that was how all the statements were to be marked true or false.
Trivialism is a particularly simple rule: “mark all true.” Dialetheism points to a category of markings, where both A and not-A are true for some A—and thus points to a category of rules that generate such patterns.
Trivialism is a particularly simple rule: “mark all true.”
This is one form of trivialism; the dissertation also uses it to mean something like “whatever marking you place on the list, every item is marked true (but also possibly marked something else).”
The Kabay dissertation is interesting in a bizarre way.
Heh.
If you leave out the phrase “and so should be taken seriously”, I’d agree with that.
It was definitely worth skimming through. Two… well, not really questions, but thoughts:
How does trivialism differ from assuming the existence of a Tegmark IV universe?
A spectral argument given in defense of trivialism in the dissertation runs like this:
a. Natural language is inconsistent.
b. Therefore, by explosion, every sentence in natural language is true.
c. Every classical proposition may be interpreted in natural language.
d. Therefore, classical logic is inconsistent.
The error in the argument is actually quite subtle!
Tegmark IV is the space of all computable mathematical structures. You can make true and false statements about this space, and there is nothing about it that implies a contradiction. You may think that any coherent empirical claim is true in Tegmark IV, in that anything we say about the world is true of some world. But being true in some world does not make it true in this world. If I say that the sky is green, I am implicitly referring to the sky that I experience, which is blue. That is, I am saying that the sky which is blue is green. So I’m contradicting myself, and the statement is false. You don’t even need to think of alternate universes to reason through this. After all, some planet in our galaxy surely has a green sky.
It all looks shaky, but most obviously, just because every classical proposition may be interpreted in natural language doesn’t mean that every natural language proposition may be interpreted in classical logic. In particular, the aspects of natural language that make it inconsistent probably can’t be translated into classical logic. After all, that’s why we invented classical logic in the first place.
Did these points come up in the dissertation?
From one of Tegmark’s pop sci papers:
Trivialism induces a mathematical structure, and so is contained in the level IV multiverse. I think there’s some meta-level confusion in the rest of the first part of your comment.
It’s not clear to me how this claim affects the argument. Asserting the negation of the converse of (c) doesn’t imply anything about (c).
The argument is not central to the dissertation. He reports it from a trivialist to establish the existence of at least one trivialist.
Because even if we assume the existence of every mathematical structure, we are still assuming that they are coherent. Mind you, there are consistent models of some paraconsistent logic (even in set theory), but there is no model of the theory of all sentences. This is pretty standar model theory: the class of models of the total theory is empty (viceversa: the theory of the class of all models is empty).
Anyway, assuming trivialism is uninteresting (as the name correctly imply ;)): we still can play a formal game that mimics the difference between truth and falsity.
I’m not sure why level IV would restrict itself to standard model theory. In a tri-valued logic (i.e., all propositions are either true, false, or both), there are non-trivial models of trivialism.
Trivialism would not respect Tegmark IV’s subsections which comply with our model of logic.
I read this sentence differently than its author intended, I think:
It makes sense; as he lays out in the first section, it isn’t clear why dialetheism is different from trivialism. If they weren’t different, then a good part of his advisor’s field would become trivial! Taking on a willing grad student to devote time to separating the two is just good politics.
Imagine all well-formed logical statements, stretching out in an infinite list.
Each of these statements are to be marked “true” or “false.” For each possible marking, there is a shortest set of rules that generates that marking. Those rules are “rules of logic” you’d be following if that was how all the statements were to be marked true or false.
Trivialism is a particularly simple rule: “mark all true.” Dialetheism points to a category of markings, where both A and not-A are true for some A—and thus points to a category of rules that generate such patterns.
This is one form of trivialism; the dissertation also uses it to mean something like “whatever marking you place on the list, every item is marked true (but also possibly marked something else).”
I wonder … when he signed the declaration on page 4, what was he asserting?
Wow. I’ve actually used all but one of those arguments (the principle of sufficient reason one) as reductios against various things.