# Boolean-Valued Models vs. the Infinite-valued Logic of Lukasiewicz and set theory

Is anyone familiar with an old paper of C.C. Chang entitled "The Axiom of Comprehension in Infinite-Valued Logic" which shows that the Axiom of Comprehension without parameters is consistent in the infinite-valued logic of Lukasiewicz and a newer paper by Richard White ("The Consistency of the Axiom of Comprehension in the Infinite-Valued Predicate Logic of Lukasiewicz") in which he shows that the full Axiom of Comprehension is consistent in the infinite-valued predicate logic of Lukasiewicz? I ask because I am interested if finding out if anyone has found a way construe this infinite-valued predicate logic of Lukasiewicz as a Boolean-valued model (or contrariwise, can the full Axiom of Comprehension be shown to be consistent in some Boolean-valued model)? If so, please provide for me the reference(s).

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No, as far as I am aware. Some recent papers following this old research line are due to Petr Hájek and Shunsuke Yatabe. – boumol Jan 20 '12 at 10:16
I may misremember it, but I was under the impression that White’s paper is considered faulty among the fuzzy logic people. – Emil Jeřábek Jan 20 '12 at 11:43
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. – Emil Jeřábek Jan 20 '12 at 12:13
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. – Emil Jeřábek Jan 23 '12 at 11:45
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, ... – Emil Jeřábek Jan 25 '12 at 11:26