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
Best Answer
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.