# Is it possible to apply a predicate to a proposition?

I have a logical question for you which I am not really sure how to phrase, but here it goes: Is it possible to apply a predicate to a proposition? And do any of you know of some sub-logic which allows this or a paper which deals with this topic?

Example: In predicate logic, the proposition "the cat is on the mat" may be formalized as follows:

("C": is a cat; "M": is on the mat)

∃x(Cx ∧ Mx)

Or in propositional logic it may be just called "p":

"p": "the cat is on the mat"

Is it then possible to apply a predicate (e.g. "S": is a sentence) to this proposition in its entirety? I'm not sure how to formalize it, but maybe something like "Sp" or "S(∃x(Cx ∧ Mx))" which is supposed to say "'The cat is on the mat' is a sentence".

• I second order and higher language your examples are perfectly reasonable.
– user9166
Feb 22, 2018 at 0:38
• As a picky detail, this actually says "a cat is on the mat" or "there is at least one cat on the mat". Also, any time you use "there exists", you formally need a bounding set, eg "there exists x in the set of all living creatures S such that..."
– user935
Feb 23, 2018 at 16:53

In FOL (i.e. predicate logic) no.

We have variables, constants, function and predicate symbols.

With the first three we build terms.

With terms and predicate we build atoms (or atomic formulas).

Example: x=0 and E(x,y) are atoms.

With connectives and quantifiers we build "complex" formulas from atomic ones, like your example: ∃x(Cx ∧ Mx).

Some versions allow also the use of 0-ary predicate symbols Pi, that are simply the propositional letters of propositional calculus.

See Dirk van Dalen, Logic and Structure, page 56-on.

Your approach is more akin to "formalizing" the meta-language.

We may define (in the meta-tehory) the class Sent of sentences and then we can write e.g.

"∃x(Cx ∧ Mx)" ∈ Sent.

A more "subtle" approach is via arithmetization.

For a gentle approach to coding, see Ch.4: M.Fitting, Russell’s Paradox, Gödel’s Theorem, of Melvin Fitting & Brian Rayman (editors), Raymond Smullyan on Self Reference, Springer (2017).