Hello I am debating my friend and want to quote some statements I disagreed with him about. Please try to give an unbiased answer because I feel like there is a lot of misunderstanding or a dissonance between the words we use speaking to each other and the definition that we both imagine them to have. I might be right, he might be right and we are both confident therefore I suggested we take it to people who have studied philosophy and mathematics (or either).
"A is logical B is proven to be logical
Are both A and B correct?"
This is what my friend says
What I tell him is that A being logically possible doesn't make it true whereas B being proven logically true makes it true.
What I meant by that (and I might be wrong so again please be unbiased in telling us who is right) was that saying something is logical isn't enough. You have to say whether you mean it is logically possible or logically proven to be true since I argued that these 2 things are NOT the same thing.
He says “no I said A is logical not A is logically possible” which confuses me since it is an unfinished sentence/statement (according to my perspective on things of course).
Second quick question regarding the academic fields of logic compared to mathematics. I told my friend "all of maths is logic but not all of logic is maths” and my friend disagreed saying “all of logic is maths and all of maths is logic”. I don't want to comment on what my friend meant but what I meant by my statement is that maths is a subset of logic but logic is not in its entirety a subset of maths and hence why all of maths is logic but not all of logic is maths.
It sounds as if you and your friend are discussing the distinction between syntax and semantics in formal logic (although your terminology is non-standard). In a formal language a sentence (a string of symbols) is syntactically correct if its structure follows the rules of the language - I think this corresponds to your term "logically possible". But it is only semantically correct (or "true") if it can be derived from a set of axioms following the rules of inference or transformation in the language - I think this corresponds to your term "logically proven".
In his book Godel, Escher, Bach, Douglas Hofstadter uses the MU puzzle to illustrate the difference between syntax and semantics. The sentence $MU$ is syntactically correct in the formal system $MIU$ (which allows any sentence that contains only the symbols $M$, $I$ or $U$). But $MU$ cannot be derived following the transformation rules in $MIU$ starting from the sentence $MI$, which is the single axiom of $MIU$. So it is not true in $MIU$.