"A is logical/logically possible/proven logically true"

Question

I am debating with 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 dissonance between the words we use speak 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 two 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 or 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: that maths is a subset of logic but logic is not in its entirety a subset of maths, hence all of maths is logic but not all of logic is maths.

Answer 1

In the academic study of logic, there is no such thing as a statement or argument being "logical".

There is the notion of satisfiability, which means that a statement is true in at least one conceivable situation, and validity, which means that a statement is true in all situations, i.e. a valid statement is one that can not ever become false. Every valid statement is also satisfiable, but not every satisfiable statement is valid. This perhaps comes closest to your "logically possible" and "logically true".

Note, though, that a statement may be true in the real world but not valid, i.e. not every true statement is a "logical truth". Likewise, just because a statement happens to be false in the real world does not mean it is "logically impossible". Using the above terminology, not every true statement is also a valid statement, and not every satisfiable statement is true in the real world. However, every statement that is valid must also be true in the current situation.

In modal logic there are also the terminologically closer concepts of possibility and necessity, but in standard modal logic without additional axioms, necessity does not imply possiblity nor does it imply truth in any actual situation.

We can also speak of a valid argument, which means that truth of the premises guarantees truth of the conclusion, in the sense that there can be no situation where all the premises are true but the conclusion false.
A sound argument is an argument which is valid and where in addition all premises are actually true. (The latter is not a requirement for a valid argument.) Soundness is what probably comes closest to people's intuition of an acceptable, "logical" argument.

However, a statement/argument being true or valid is not the same thing as being provable.
There are statements that are considered true in the real world/the intended situation but which can not be proven by the means of a certain proof system: Truth and provability diverge. This is, very roughly, what Gödel's first incompleteness theorem states.
Depending on the proof system, we may not even be able not prove all valid statements.
It is of course also possible to invent strange proof systems which are unsound, i.e. where a proof does not in fact guarantee the truth/validity of the statement. (Be careful that soundness of a proof system is a different "soundness" than the soundness of an argument mentioned above.) However, unless by accident, an unsound proof system will not typically be used in mathematical practice.
Finally, just because we know in theory that there exists a proof somewhere out there doesn't always mean we can find it in practice (with the available resources and within a human's life time).

TL;DR

• "A is logical" has no meaning in the formal study of logic.
• Not every statement that is satisfiable ("logically possible") is also valid ("logically true"), but every valid statement is satisfiable.
• Not every statement that is true is also valid (a "logical necessity"), but every valid statement is true in the real world.
• A statement/argument may be true/sound/valid but unprovable.

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The answer to your second question is no. Not all of math is logic, and a great deal of logic is more philosophy than math. See also

• Logic as a subset of mathematics and mathematics as a subset of logic
• Is all of mathematics based on logic?
• Difference between logic and mathematics
• Is logic part of philospohy or mathematics?

Answer 2

I will assume that by A being logical, you are tacitly implying that one can deduce A from certain established premises. When you say that B is proven to be logical, that is a dubious claim.

Let us first define when we mean as being 'logical' in this exigent. The proper language is 'sound'. Suppose there are certain true facts about the world (suppose). You can using arguments, logical or illogical, deduce new things that you think should be true about the world. When you use arguments which are logical in the sense that they preserve truth value (which is quite hard to characterize) you have a sound argument.

Now you say B is proven to be logical. This begs the question if it is sound or 'valid'. For something to be proven, I am making the assumption that you are acknowledging it to be true.

This is not now anything is done.

There are premises, there are arguments, and there are conclusions, and there are tests.

It can be the case that one is being illogical but the conclusions one has come across one's sloppy argument is consistent with what is observed.

The particular problem you raise is not simple. For instance I can, with my limited understanding, raise an analogous question. Does nature have a propensity for intrinsic geometry or embedded geometry. For instance we can study the general theory of relativity in four dimensions using null geodesics and declare that the geometric structure, as we imagine it, is that. I feel quite confused when I notice than the general theory of relativity can also be studied in five-dimensions as an embedded geometry. None of our math contradicts either views.

This is where I am going to opine. I am going to opine in the spirit of empiricism.

No matter how logical you were, if your deductions are inconsistent with present observations, you have to discard it.

No matter how illogical you think your friend was, if the present observations (and historical) are consistent, you cannot discard it. In fact you have to try to understand him.

We hope that truth always occurs as truth. If it does not, then it's temporal truth. Any other notion would lead to ineluctable epistemological despondency, particularly the notion of absolute truth.