To introduce the basics of logic, one ordinarily begins with propositional logic and then proceeds to predicate logic. Unfortunately, the examples of propositions typically either use "real-world sentences" (e.g., "George Washington lived in Mt. Vernon") or, when strictly mathematical, actually use quantifiers implicitly (e.g., "$6$ is an even number" -- which of course really means "there exists an integer $k$ for which $6 = 2 k$; or "an integer that is a multiple of $6$ is even" which similarly involves quantifiers implicitly).
For a very brief introduction to logic that I'm writing -- part of a textbook that is not about logic or set theory -- I'm looking for simple examples of propositions that deal only with mathematical objects, not real-world entities and that do not implicitly involve quantifiers. And I want to use these in order to provide instances of conjunction, disjunction, implication, and equivalence.
I am willing for the examples to involve terms such as "integer", "natural number" along with $=$, $<$, etc. Thus I take as understood such propositions as "$0 = 1$" and "$3 < 4$". But not expressions such as "$2 \in \mathbb{N}$" that explicitly involve sets of objects.
Can you provide any more such examples (other than just changing the particular numbers in those)?
I know this is asking a lot!
I think "6 is an even number" works just fine as a propositional logic claim ... to treat it as an existential seems unnecessarily complicated. And you can still represent it using something like $Even(6)$ ... that involves a predicate and a constant, which we typically only introduce in predicate logic, but it has no quantifiers. And, you can do propositional logic with such claims just fine.