Re: axiom of cardinality in boolean algebra?

On Feb 3, 4:07 pm, Tegiri Nenashi <TegiriNena...@xxxxxxxxx> wrote:
On Feb 3, 3:31 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

In predicate logic, aside from 0-ary predicate letters, i.e.,
sentential letters (if your predicate language even includes
sentential letters),

There are always sentential formulas (e.g. Ax:P(x)), so the matter if
my predicate language allows sentential letters is not important?

Some people call those 'prime formulas'. As to importance, yes, my
remark was actually in agreement with what you're saying.

My question arose in the context of algebraic logic. (For the
record, it is neither Kleene, nor cylindric algebra). I have axiomatic
algebraic system corresponding to predicate calculus, so my question
is how do I interpret the models.

I don't know what you mean by interpret the models. Ordinarily, what
we interpret are languages. By the way, I know about (not details
though) cylindric algebra vis-a-vis predicate logic, but where do I
find out about Kleene's algebra for this?

The elements sure have to be
relations, and I identified some 0-ary ones. As expected all 0-ary
relatons are be proven top satisfy boolean algebra axioms. I was
wondering if I have to postulate that there are only two such
elements.

Hmm, I guess I'm starting to see what you're getting at, but I'm not
really sure. But, in a 2-valued logic, what 0-ary relations are there
other than the two?

But let me see whether we're on the same page. Is the following
correct?:

You have a certain 1st order axiomatization. And the models of the
theory thus axiomatized are a certain kind of algebra that in some
sense corresponds to first order logic itself (in some way similar to
Lindenbaum-Tarski algebras corresponding to sentential logic and
cylindric algebras corresponding to1st order logic).

Some further investigation (with Mace4), however, revealed that the
arity of the predicate can't be established algebraically. What I
initially thought to be zilliary relations, can as well be interpreted
as unary relations. Then there can be more than two values in the
lattice between lattice top and bottom element...

I don't know what you mean by establishing a predicate algebraically.
Perhaps this is material that I haven't studied yet. I have Halmos's
book on algebraic logic. Do know if this matter is addressed in that
book?

MoeBlee

.

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