VazScep wrote:What does this abstract description buy you? I mean, why talk about integral domains at all? I understand that once you eliminate quantifiers, you are effectively propositional, and that classical propositional logic can be understood as the logic of two objects, and that AND and XOR are minimal connectives. But how does the algebra of integral domains do heavy work here?

I think it´s useful if you you want to look at problems of computing. Whether it´s useful to go to ID isn´t that important, I think that any two element ring satisfies the other axioms. You get a progression there from unary logic, which is the trivial ring, to binary logic as the minimal non-trivial ring.

VazScep wrote:By comparison, the algebraic description for intuitionistic logic says that truth values form a bounded lattice with the Heyting property: for any truth values P and Q, there must be a uniquely largest truth value, called P → Q such that (P → Q) ∧ P is less than Q. Classical propositional logic is then the lattice containing just T and F, where F < T. But the full algebraic description is necessary since intuitionistic logic is the set of formulas with maximal truth value over any bounded lattice with the Heyting property.

An issue I see there is that you define this using 3 binary operations. I´m not sure it´s a minimal statement of ZOL, because it is defined with expansions to higher order logics in mind.

VazScep wrote:I took you to be saying that you can formalise the notion of consistency within in ZOL and derive a theorem saying "I am consistent."

Fair enough, but that would indeed take quantifiers. What you can show is that you can not prove 1=0 whithin ZOL, basically following from the completeness of the operations.

ここにいる皆は俺の宝石のような単語を値するか。

Goathead wrote:ここにいる皆は俺の宝石のような単語を値するか。

Translation please?