Could we use predicates instead of propositions in the definition of axiomatic system?

Question

An axiomatic system is a finite sequence of propositions a_1,a_2..,a_N which are called axioms 56:23

In the whole lectures, two kind of logics are introduced:

Proposition: A variable which is either true or false. It is also remarked, you can force a proposition to be either true or false.
Predicate: A proposition valued function of some variable.

In the whole lecture, I see that proposition logic is used to define both a predicate and an axiom system. My question is why is it that we is that propositions are used over predicates in the definition in an axiomatic system? Would there be any issues in doing mathematics if we use predicates instead of proposition for the definition of axiom set?

Excuse me if the question is very stupid.

This is like my sixth rewatch of this same lecture and I still feel I don't understand anything. — tryst with freedom, Mar 08 '22 at 23:38
For what it's worth I'm not a fan of Schuller's treatment of foundations; see e.g. here or here. — Noah Schweber, Mar 09 '22 at 00:37
Yes, I had gone through your post about a few months back. I had even commented on it. Very enlightening. But, I still believe there is much to be learned from what he is saying even if it has some incorrect parts @NoahSchweber. Also knowing why the wrong is wrong is instructive as well — tryst with freedom, Mar 09 '22 at 00:39
See also What is the difference between Deductive system and Logical system? — Mauro ALLEGRANZA, Mar 09 '22 at 13:11
Hmm he gives an exact definition of the proposition earlier though @MauroALLEGRANZA — tryst with freedom, Mar 09 '22 at 13:49
Well formed means quantified predicates, so the issue with using predicates I guess is relating somethign to their quantification? @MauroALLEGRANZA — tryst with freedom, Mar 09 '22 at 14:11
@Buraian - not exactly. Well-formed formula (WFF) means an expression (string of symbols) that match the syntactical specification of the language. In prop logic $p \to q$ is a WFF while $p \to \land q$ is not. In predicate logic both $P(x)$ and $\forall x P(x)$ are formulas correctly written: the first one is an "open" formula (it has free var) while the second one is a sentence (no free var). In the language of set theory $(x \in y)$ is a correct formula (with free var) as well as the sentence $\forall x \lnot (x \in \emptyset)$. — Mauro ALLEGRANZA, Mar 10 '22 at 07:32

score 1 · Answer 1 · answered Mar 09 '22 at 03:47

Proposition in Maths

A proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's Elements (c. 300 BCE), all theorems and geometric constructions were called "propositions" regardless of their importance.

Wikipedia

Proposition in Philosophy

In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning.[1] Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false.

Wikipedia

As the person in the video is talking about propositions in relation to the ZFC axioms, I think they mean "an elementary theorem that can be stated without proof" or perhaps just "true logical statement". They're almost certainly not talking about "proposition" as it's used in propositional logic as the ZFC axioms are predicate logic formulas — i.e., the ZFC axioms are quantified.

Mauro ALLEGRANZA · Accepted Answer · 2022-03-09T14:51:45.853

See also the post linked above referring to Schuller's lectures; what the author calls proposition (in the formal sense) is a formula of the language without free variable, like e.g. $\forall x \forall y (x \in y)$ in the language of set theory.

In the Lectures, the author starts with propositional calculus where a proposition is symbolized by a propositional variable $p_i$.

Then he introduces predicate logic, with predicates symbols: $P(x)$.

See 45:25 where the author states that a quantifier turns a predicate $P(x)$ of a single variable into a proposition.

Usually, we call this a sentence, i.e. a formula with no free variables; the issue is that it is like a proposition of propositional calculus because - having no free variables - it has a definite truth value.

Could we use predicates instead of propositions in the definition of axiomatic system?

No, because we want that axioms have a definite truth value.

In light of this, the statement above can be rewritten as:

"An axiomatic system is a finite sequence of sentences (formulas without free variables) $a_1,a_2, \ldots, a_n$ which are called axioms."

See examples regarding arithmetic and geometry.

Could we use predicates instead of propositions in the definition of axiomatic system?

2 Answers2

Proposition in Maths

Proposition in Philosophy