Undergrad Question regarding writing proofs

[Munnu]

TL;DR

A few questions about proof writing.

I have a couple general questions regarding writing proofs:

1. Do proofs typically fall into being one out of all of the rules of inference (page 6-7 on this pdf)
2. or is it that generally, most proofs may categorically qualify within a very small subset of the rules of inference (say “many common proofs are generally modus ponens or hypothetical syllogism”)
3. or is it possible that many proofs may not use any rules at all?

And if yes to 1 and/or 2, is it important for a newcomer proof writer to begin by always decomposing into symbolic logical statements (akin to the format seen on: pg 6-7 middle column “tautology”) and then consciously apply a rule of inference (like they do on page 20-21 on this pdf)? https://www.site.uottawa.ca/~lucia/courses/2101-10/lecturenotes/04InferenceRulesProofMethods.pdf

I'm trying to understand are 1 and 2 generally implicit in proofs or is it that 1 and 2 are typically techniques used for propositional and predicate logic and might not even apply depending on the discipline?

Thank you for any help.

 

[jedishrfu]

I don't think you can categorize proof in that way. Of course, when you first learn about geometric proofs you are taught the rules of inference and various methods of proof that can be used as the proof basis ie the rules of inference are the atomic building blocks of proofs and the methods are the cookbook recipes used to structure the proof.

However, as proofs get more and more complex you find that they are broken down into smaller theorems and lemmas that are proved independently and are then used to prove the bigger statement.

These smaller proofs may be divided into still smaller ones until you have the smallest ones will use those proof strategies you first learned.

https://en.wikipedia.org/wiki/Mathematical_proof

I don't think anyone has ever categorized proofs in the way you are thinking. Erdos was fond of saying that there is a book kept by GOd with all the most elegant proofs. Some of his colleagues put together a book with the Erdos title.

The Book

https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

and this article on writing proofs:

https://deopurkar.github.io/teaching/algebra1/cheng.pdf

 

[valenumr]

There is also proof by induction and proof by contradiction to name a couple. Sometimes axioms can just be directly applied, but that is not always the case.

 

[PeroK]

Summary:: A few questions about proof writing.

I have a couple general questions regarding writing proofs:

1. Do proofs typically fall into being one out of all of the rules of inference (page 6-7 on this pdf)
2. or is it that generally, most proofs may categorically qualify within a very small subset of the rules of inference (say “many common proofs are generally modus ponens or hypothetical syllogism”)
3. or is it possible that many proofs may not use any rules at all?

And if yes to 1 and/or 2, is it important for a newcomer proof writer to begin by always decomposing into symbolic logical statements (akin to the format seen on: pg 6-7 middle column “tautology”) and then consciously apply a rule of inference (like they do on page 20-21 on this pdf)? https://www.site.uottawa.ca/~lucia/courses/2101-10/lecturenotes/04InferenceRulesProofMethods.pdf

I'm trying to understand are 1 and 2 generally implicit in proofs or is it that 1 and 2 are typically techniques used for propositional and predicate logic and might not even apply depending on the discipline?

Thank you for any help.

Just an opinion, but this formal approach is much harder than informal (natural) proofs. Unless you are familiar with informal mathematical proofs, then material will be hard to digest.

It's like the difference between numerical algebra and abstract algebra. It's a lot easier to grasp group theory and the theory of rings if you are already familiar with the algebra of numbers and functions and trigonometry.

 

[Munnu]

As a follow-up, I have a question regarding Rules of Inference in Propositional Logic.

In referencing pg 6-7 of this link: (https://www.site.uottawa.ca/~lucia/courses/2101-10/lecturenotes/04InferenceRulesProofMethods.pdf), I’ve come to qualify Modus Ponens and Modus Tollens as “kinds of proofs” (direct and contrapositive), and then categorize Hypothetical Syllogism as a methodology or tool in order to prove a proposition that’s in the form of one of the two above proof types.

I don’t know what to categorize Disjunctive Syllogism, Addition, Simplification, Conjunction, and Resolution as (pg 6-7). I’m unsure if these would fall under one of those two categories: “proof types” vs “method within a proof to help prove a proof” or if they fall under a separate category.

Thank you for any help.