Using symbolic logic in mathematical proof?

[Terrell]

is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct.

For example, I was asked to prove the statement "If 0<a<b, then a^2 < b^2"
I: 0<a<b
A: multiply the given inequality by a
B: multiply the given inequality by b
C: 0 < a^2 < ab
D: 0 < ab < b^2
E: a^2 < b^2
(I ^ A) --> C
(I ^ B) --> D
(C ^ D) --> E
therefore, (I ^ E) --> Q

am i doing this right? and what i really want to know is how do i do it the most efficient way? or is proving math theorems simply a different animal compared to symbolic logic proofs? is practicing with symbolic logic proofs mainly a way of getting my brain to think in a certain way? thank you all.

 

[micromass]

This is not a good way of proving theorems. Nobody does this kind of symbolic proof. Nobody wants to read symbolic proofs nowadays. So forget about symbolic logic proofs and learn the actual way of proving theorems the way mathematicians do it. I understand that proof books encourage these kind of proofs, and if you find a proof book that does, stay far away from it.

 

[Terrell]

This is not a good way of proving theorems. Nobody does this kind of symbolic proof. Nobody wants to read symbolic proofs nowadays. So forget about symbolic logic proofs and learn the actual way of proving theorems the way mathematicians do it. I understand that proof books encourage these kind of proofs, and if you find a proof book that does, stay far away from it.

noted. but what if i where to translate it back from symbols to words and math notation? would that still be a bad way? and are you implying that this symbolic proofs are simply a way to get the reader into thinking some sort of way to segue to the way mathematicians really do it? thanks! also what books would you recommend?

 

[Stephen Tashi]

noted. but what if i where to translate it back from symbols to words and math notation? would that still be a bad way?

The goodness or badness of how a proof is presented is a matter of tradition and cultural conventions. The style you are proposing resembles the abbreviated way that homework problems are traditionally presented. This abbreviated style has more to do with making the work easy to grade than with presenting a coherent proof. (No grader wants to read a essay in order to check work that is mainly calculation.)

What you are calling "symbolic logic" is not symbolic logic in the mathematical sense. For example, the style you propose is missing the use of logical quantifiers such "$\forall$" and "$\exists$". There is a field of study called Symbolic Logic and it can involve elaborate notation that allows steps in proofs to be written in an abbreviated form. However, in courses on other subjects (e.g. Calculus, Linear Algebra, etc) the elaborate symbolic machinery of symbolic logic is not used in writing proofs because it would make conceptually simple proofs extremely long. ( Furthermore, many students who take Calculus or Linear Algebra have never taken a course in symbolic logic.)

Particular instructors have particular standards about how they want homework type proofs written. The style that is easiest to grade is the "step-reason" format. It is a series of "steps", each annotated by a definition, theorem or assumption that justifies the step. To understand the "goodness" or "badness" of a proof with respect to pleasing an instructor, you have to learn the peculiarities of that instructor's tastes. There have been instructors that forbade the step-reason format and required proofs to be written using complete sentences and in the form of an essay.

Proofs in textbooks and mathematical journals, are usually presented with a mixture of the essay style and a symbolic style. They may contain short passages that have the "step-reason" format, but rarely would the entire content be presented in that style. To understand what's good or bad style with respect to publications, you have to read such material to learn the accepted conventions.

 

[FactChecker]

There is a practical use of what you are trying to do. If you want to develop a computer automated "theorem prover", I think this might be what you would want to do.
But this is not an approach that a human would want to use. Your example illustrates how it makes even the simplest proofs more difficult to follow. A computer might like it, but a human would not.

That being said, what do you think is wrong with your proof? (Except the last line confuses me. Why isn't it just I ---> E?)