Can element of a set be a logical sentence?

Question

Let $A$ be a set. Is there a set $B$ whose elements are these elements $x \in A$ which are logical sentences?

Such a set would be constructed using the axiom of specification, provided that $$\phi(x) : x \ \text{is a logical statement}$$ is a propositional function. In the textbook I am studying now (Kazimierz Kuratowski - Introduction to Set Theory) the concept of propositional function is defined so vaguely that I can't decide if this is true.

Answer 1

A propositional function is what we get from a sentence of our language, like e.g. "Socrates is a philosopher" removing the name Socrates and using instead a "place holder" : a variable like $x$.

The resulting expression : "$x$ is a philosopher", is like a mathematical function: assigning to $x$ a "value" : Plato, Napoleon, what we get is a sentence : either true (for Plato) or false (for Napoleon).

In predicate logic, a propositional function $\phi(x)$ is called : open formula.

See : Kazimierz Kuratowski & Andrzej Mostowski, Set theory, North Holland (1968), page 45 :

we shall consider propositional functions. They are expressions which contain
variables. If each variable is replaced by the name of an arbitrary element, then the propositional function becomes a sentence. For instance,

$x> 0, x^2 < 5, X$ is a non-empty set

are examples of propositional functions. By substitution we obtain, e.g., the following sentences:

$1 > 0, 25 < 5$, the set of prime numbers is a non-empty set.

An open formula $\phi(x)$ is what is used in the set-builder notation :

$\{ x \mid \phi(x) \}$

to define a set: the set of all and only those objects such that $\phi(x)$ holds of them.

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Regarding propositional calculus, we satrt with a language (or alphabet), i.e. a set of propositional symbols : $p_0,p_1,\ldots$ and the connectives : $\lor, \land, \lnot$.

We define expressions as finite strings of symbols.

Finally, we have :

The set $\text {Prop}$ of propositional formulas [also : well-formed formulas] is the smallest set $X$ with the properties :

(i) $p_i ∈ X$,

(ii) if $ϕ,ψ ∈ X$, then $(ϕ ∨ ψ), (ϕ \land ψ) ∈ X$,

(iii) if $ϕ ∈ X$, then $(¬ϕ) ∈ X$.