Understanding Pure Monadic Schemata

[disgradius]

My current understanding of schemata and structures in monadic logic is that, for example, if there are two predicate letters F and G that there would be (2*2)^the size of the universe of discourse possible structures as each object in that universe of discourse could take on any of the 4 possible truth assignments (TT, TF, FT, FF) from F and G.

I'm having some issues understanding how it's possible to find the number of pure monadic schemata possible independent of the size of the universe of discourse as it seems that the number possible would depend on the number of structures possible which in turns requires a size for the universe of discourse. My thinking is that there would be 2^# of structures possible for potential schemata till equivalence since we can combine them like truth assignments can be combined to form those schemata without quantifiers found in simpler truth functional logic.

Sorry if this is really confusing, in a nutshell what I'm asking is:
how can I find the number of pure monadic predicate letters that can be created from a given number of monadic predicate letters? Do I have to know the size of the universe of discourse and if not how would I go about doing this?

Apologies if this should go in the hw&coursework section, wasn't really sure if I needed a specific problem to post there.

Thanks!

 

[Preno]

If I understand correctly, your question, properly phrased, is: how many different open monadic formulas are there up to equivalence in a language which consists of the monadic predicates F₁, ..., F_n and a single variable?

Well, each element x is fully characterized by some formula of the form F₁(x) & ~F₂(x) & ... & F_n(x). There are 2ⁿ of these formulas. Each property of x then bijectively corresponds to some set of these. Thus, there are 2⁽²ⁿ⁾ different open formulas containing only x up to equivalence. Likewise, the number of different open formulas over m variables should be 2^(2mn).

 

[disgradius]

Right, sorry if it was confusing but I actually need the combinations of those open formulas which normally I would assume to be just another 2^(previous # of formulas) except that I don't know what they correspond to in general terms. The way I thought of it before was to assume there was an expression of each structure with a set universe of discourse of n such that I would get 2^[(2^(#predicate letters))^n] possible. Except, while I can follow these general steps to find the total number possible, because I don't know what the new conception of the intermediate step that I used to call structures is, I can't manipulate it to consider only subsets of the population satisfying certain conditions or eliminate subsets etc.

So if I were asked a modification of the above question, say what are the number of possible schematas till equivalence but excluding those implied by a certain formula, how would I approach this? At the moment I only have a general notion of how these different formulas are combined and the combinations of those combinations etc without knowing what it is I'm combining which is what gives me trouble.

Apologies for my terrible explanation, this is still rather fuzzy in my head. I'll try to illustrate with an example

The previous structure I knew of had these elements:
we'll say there are two predicate letters Fx and Gx with only one variable x
there are 2^2 different combinations of True or False possible with only these two predicate letters
within a universe of discourse (1, 2, 3) there are 4^3 possible structures where each number (1, 2, 3) is a general object that can take on any of the possible true/false combinations.
if there is a schemata that is true only of one particular structure and I combine these schemata, I get 2^(64) possible schemata.
Before, since I knew what the structures were, I could manipulate the quantity of schemata I could get with certain conditions etc.
Now, the system I'm using doesn't have a universe of discourse, only a general concept of the number of possible schemata. In this general situation, how do I include or exclude certain groups? Let's say if I want to exclude those schemata that are implied by (Ex) Fx (There is a structure where Fx is true) where imply means that every structure that is true for the schemata I mention is true for another arbitrary schemata which I want to exclude.

I'm guessing I need to know what the middle layer is composed of the one you mentioned above, modify that with whatever conditions I want then take 2^(the remaining formulae). I just don't know what that middle layer is.

Thanks again!