Difference between (propositions used in proofs) and (propositions of propositional logic)

Question

In the most recent revision of the Theorem wikipedia article, it says:

A theorem is a statement […] 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. [emphasis mine]

What is the difference then, between a proposition in math and a proposition in propositional logic?

Clicking through the links on said article leads me to the same Proposition article, and it seems to me (in other instances) “statements” and “(logical) propositions” are synonyms (i.e. things which can have a true/false value).

So what am i missing? What nuance is there between the two concepts?

Thanks

The distinction above is vague: both are statement proved in thecontext of a mathematical theory. — Mauro ALLEGRANZA, Jan 29 '23 at 16:57
For comparison, consider lemma and corollary. These aren't logically distinct things from any other theorem. Their names provide useful context about how they relate to a body of work. "Proposition" has a similar role in that lexicon. — Alex K, Jan 29 '23 at 19:52

score 0 · Accepted Answer · answered Jan 29 '23 at 22:48

A proposition in propositional logic is just a formula that has a truth value; it may or may not be true in a given situation, and may even be contradictory. A proposition in informal mathematics, on the other hand, is a statement thought to be true in reality.

In addition, a proposition in propositional logic is a rigorously defined formal string of symbols (like $(p \lor q) \to r$), whereas the other kind of proposition is typically formulated in natural language (like "The number of brackets in an arithmetic expression is even"), and in fact is usually thought of more in terms of its conceptual content rather than a concrete formulation.

@conatus To add to paragraph one: for example, the if-then symbol in a mathematical lemma/theorem, unlike in formal logic, is not the material conditional, because its context is specifically mathematics, which is on a lower level of abstraction than formal logic. — ryang, Jan 30 '23 at 10:48

Difference between (propositions used in proofs) and (propositions of propositional logic)

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