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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.

Captain HD
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  • What you call the "academic field of logic" is a branch of philosophy born with Aristotle. See this ; if you open a book of logic written by a philosopher, as you say it in your last paragraph, it's far from being entirely mathematized. In the context of mathematics, here is, among many others, a question with valuable answers: https://math.stackexchange.com/q/3536681 – Jean Marie Oct 10 '20 at 09:42
  • It might be better asking this on philosophy stack exchange. There are a number of people over there who are well versed in mathematical logic too. – user400188 Oct 10 '20 at 09:47
  • The reason I hesitated to go on philosophy stack exchange (although I did) was because my friend believes that maths is enough to answer philosophical questions like tackling questions on existence (actually after a debate about the existence of God specifically he went to mathematicians and I was surprised telling him that its like going to biologists to ask about physics). I could think of better analogies but whatever. Do you agree in what I have said (that he should have taken it to philosophers)? – Captain HD Oct 10 '20 at 09:51
  • @JeanMarie Thank you for your reply. Unfortunately my friend still doesn't believe he is wrong. Could you help us out by making a clear-cut conclusion of the questions in my post? – Captain HD Oct 10 '20 at 10:04
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    What does it mean”A is logical”? – Mauro ALLEGRANZA Oct 10 '20 at 10:17
  • The question does not seem to be neither about Logic nor about Philosophy – Mauro ALLEGRANZA Oct 10 '20 at 10:18
  • No idea. I told him he has to specify whether it is logically possible or logically proven to be correct but he insists that what I am saying doesnt make sense. Lets ignore what I wrote. Do you agree with me in saying that something being logical is DIFFERENT from something being logically proven? That which is logically proven MUST be a fact of reality. If I prove the existence of the metaphysical using Aristotelian logic (meaning "logically prove it") then it must exist. Of course I am ASSUMING that I have logically proven it not saying I did but would you agree with me? – Captain HD Oct 10 '20 at 10:25
  • Saying $x$ is prime is different from proving $x$ is prime. – vvg Oct 10 '20 at 10:31
  • @vvgiri I mean I gave him the example that there's a difference between saying "a triangle is logical" and saying "a triangle is logically proven to exist" but your example works too (although they're slightly different). Can you make your answers to each question clear-cut? I am asking for this because I want to avoid any excuses or anything like that if I turn out to be right because my friend was too hot-headed about this. I genuinely want to help him and I think the first step in that direction is to try humble him or make him more open to me being right about these discussions on logic. – Captain HD Oct 10 '20 at 10:43
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    @captainhd: What does a triangle is logical mean? If you say a triangle is made by joining three points with line segments between pairs, it is logical, but not always true. For eg: when the three points are collinear, you get only a line passing through three points. One could argue it s a triangle with area zero, but then you need to define the universe of what you consider to be valid triangles. – vvg Oct 10 '20 at 10:56
  • I told that the statement is unfinished and must specify whether it is saying that triangles are POSSIBLE or PROVEN LOGICALLY but he told me "I did not say logically posssible". He is seriously confused. "A is logical"? I interpreted his words as meaning "A is a logical entity in the sense that it could exist without contradicting the principles of logic" but whenever I interpret it like that by saying "logically possible" he gives me a strange response and told me "no I did not say A is logically possible" and I therefore was told to just type "A is logical" as it is. I really dk what to do! – Captain HD Oct 10 '20 at 11:00
  • A statement '$P \rightarrow A$ is true' is logical, but not necessary to be true. $P$ is true if and only if $A$ is true. You may believe $P$ to be true, but that is very different from $P$ actually being true. It looks like you are saying $P$ is always true because it is logical and your friend is saying, 'no, that is not the case, It is logical, but not proven true'. If you agree with this characterization of the scenario, your friend is right. – vvg Oct 10 '20 at 11:21
  • @vvgiri It is the other way around. I am the one saying that something being “logical” (which I assume to mean logically possible although still unclear to me) is not necessarily true. However if something has been logically proven then it IS true. This was my position and I copied and pasted only what he wanted me to copy and paste in quotations saying “A is logical, B is logically proven. Are they both correct?” And I told him that I wanted to add “A is logically possible” and not leave it unfinished as just “A is logical” but he told me not to. Why he told me no to is unclear to me. – Captain HD Oct 10 '20 at 12:30

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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 $MIU$.

gandalf61
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  • Would you mind elaborating more on things? To give a bit more detail I was essentially saying that something being logical doesn’t make sense to leave at that without explaining further. Something is either logically impossible, logically possible or logically necessary/true. This is why I was confused by his simple statement of just “A is logical but being logical doesn’t make it true” and I got confused and thought he meant A is logically possible since the example he initially gave and that I took was that of a triangle. He said a triangle “is logical but doesn’t (or doesn’t have to) exist” – Captain HD Oct 10 '20 at 12:42