Timeline for Boolean-Valued Models vs. the Infinite-valued Logic of Lukasiewicz and set theory
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14 events
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| Jan 29, 2012 at 9:49 | comment | added | Thomas Benjamin | By the way, Chang in his paper had the closed interval [0,1] as the truth-values for the infinite-valued Lukasiewicz logic. It seems rather telling that the elements of [0,1] can be put in a 1-1 correspondence to a complete Boolean algebra. | |
| Jan 28, 2012 at 13:36 | comment | added | Thomas Benjamin | Emil: I'm looking at the article "Boolean-valued Model" from Springer's Online Encyclopedia of Mathematics (www.encyclopediaofmath.org/index.php/Boolean-valued_model) and found an interesting statement..."If a Boolean algebra B is a two-element algebra (i.e. B={0,1} then the B-model M is the classical two-valued model." This seems to suggest that if B had an infinite number of elements, then the B-model M would essentially have an infinite-valued logic, which under certain conditions could be the infinite-valued logic of Lukasiewicz? | |
| Jan 25, 2012 at 11:34 | comment | added | Emil Jeřábek | ... ZFC-style forcing is quite tied to the particular set theory it is done in. There is no correspondingly developed set theory over Łukasiewicz logic, and if you pick such a theory (such as the theory with full comprehension), it is quite unclear whether one can adapt forcing to it. There is some work (Takeuti, Titani) on forcing in a ZFC-like set theory over Gödel–Dummett logic, but that’s in a different ballpark from Łukasiewicz logic (the lack of contraction makes a huge difference). | |
| Jan 25, 2012 at 11:26 | comment | added | Emil Jeřábek | I was not referring to any quotient, but to the fact that the logic of an algebra-valued model is determined by the (quasi-)equational theory of the algebra (at least the propositional logic, it gets more complicated for first-order logic). For example, Boolean algebras satisfy the equation $x\lor- x=1$, which means that any Boolean-valued model satisfies the law of excluded middle $\varphi\lor\neg\varphi$. See plato.stanford.edu/entries/consequence-algebraic for a quite thorough explanation of the correspondence (they unfortunately do not treat first-order models). As for forcing, ... | |
| Jan 24, 2012 at 9:52 | comment | added | Thomas Benjamin | Emil: Thanks for the comment. Very helpful. What I would like to know is how Chow could come to be so misguided. What was the source of his confusion, in your opinion? Also, when you say, "the number of truth values as such is mostly irrelevant, what matters most is what equations hold in the algebra of truth values," are you referring to taking the quotient M^B/U where U is an ultrafilter to form an actual model(in Chow's case, a model of ZFC)? Also, can one actually do forcing in the infinite-valued predicate logic of Lukasiewicz? Thanks for your help. | |
| Jan 23, 2012 at 11:45 | comment | added | Emil Jeřábek | Chow’s claim is thoroughly misguided, just forget it. Fuzzy set theory is quite a different thing from forcing. An infinite Boolean-valued logic is still exactly the same classical logic as the two-valued one, no matter how large a Boolean algebra you take (which is the reason why forcing in ZFC using Boolean-valued models can work in the first place). The number of truth values as such is mostly irrelevant, what matters is what equations hold in the algebra of truth values. The comprehension axiom is inconsistent with classical logic, hence it cannot hold in any Boolean-valued model. | |
| Jan 23, 2012 at 10:07 | comment | added | Thomas Benjamin | To Emil Jerabek: Also, I understand one can do Forcing in the infinite-valued predicate logic of Lukasiewicz. Is that sort of forcing essentially similar to Forcing with Boolean-valued models? | |
| Jan 23, 2012 at 10:02 | comment | added | Thomas Benjamin | To Emil Jerabek: I guess what I expect to get is a return to Naive Set Theory, since in the infinite-valued logic of Lukasiewicz, at least Chang may have correctly proven that the Axiom of Comprehension without parameters is consistent. My understanding of boolean-valued models comes from Timothy Chow's "A Beginner's Guide To Forcing" in which he claims that Boolean-valued Models are a type of 'fuzzy set theory'. Can't a Boolean-valued model based on a Boolean algebra with an infinite number of elements be construed as an infinite-valued logic? | |
| Jan 20, 2012 at 12:51 | comment | added | Emil Jeřábek | I see: Chang indeed constructs a model valued in the standard MV-algebra, but White’s argument is proof-theoretic. | |
| Jan 20, 2012 at 12:13 | comment | added | Emil Jeřábek | Ah, OK, thanks. Anyway, @Thomas: I’m not sure what do you expect to get. Boolean-valued models are models of classical logic, because Boolean algebras are the equivalent algebraic semantics of classical logic. In order to have a model satisfying only Łukasiewicz logic, it needs to have values in an MV-algebra instead. I didn’t read Chang’s or White’s paper, but I’d be surprised if that’s not what they do. | |
| Jan 20, 2012 at 11:59 | comment | added | boumol | Emil, I would say you remember "half of the story". It is true that some claims in White paper are indeed false, this is proved in Appendix A of Petr Hájek's paper "On White's Expansion of Lukasiewicz Logic". | |
| Jan 20, 2012 at 11:43 | comment | added | Emil Jeřábek | I may misremember it, but I was under the impression that White’s paper is considered faulty among the fuzzy logic people. | |
| Jan 20, 2012 at 10:16 | comment | added | boumol | No, as far as I am aware. Some recent papers following this old research line are due to Petr Hájek and Shunsuke Yatabe. | |
| Jan 20, 2012 at 0:25 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |