# Natural Deduction - AvB |- BvA

1. Jan 4, 2009

### temp_tsun

Introduction

Hello, I have some problems proving the following problem: "AvB |- BvA". I would like to prove this by using natural deduction only. (I know it is possible proving it using truth tables).

Syntaxis

To avoid confusion I will use the following syntaxis to show my problem.
- Capital letters are used for variables.
- Instead of the mathematically symbols like "v" which means OR, I will use OR (all capital to substitute the mathematical symbols (few examples: (A AND B), (NOT(A))).
- I use something that I would like to call a diagram proof and hopefully you can explain how to do it using my familiar proof technique. I will demonstrate some below.
- I use |- to represent the mathematical symbol. It means: "When a conclusion C is derived from a set of assumptions {A1, A2, A3, ... , An} we write {A1, A2, A3, ... , An} |- C".
- When discharging a assumption I prefer to use [A] arround the proposition.

Diagrams

Note: I will use dots to represent spacings.
Note: I will use as much brackets as I can to make everything clear.

{A AND B} |- (B AND A)

(A.AND.B)..........(A.AND.B.)...........(Assumption)
----------..........---------
....B........................A................ (by AND elimination)
--------------------------
.............B.AND.A.........................(by AND introduction)

Problem Statement

Prove using natural deduction: (A OR B) |- (B OR A)

The attempt at a solution

.....................................................................[NOT(B.OR.A)]......(Assumption)
..................................................................-------------------
AvB.........[NOT(A)].................(Assumption)........NOT(B).AND.NOT(A).(Rewrite)
-----------------..........................................--------------------
.B.......................................................................NOT(B)..............(AND elimination)
-------------------------------------------------------------------
...................................(B.AND.NOT(B))..........................................(AND introduction)
--------------------------------------------------------------------
.......................................FALSE...................................................
--------------------------------------------------------------------

Is this correct? Because I've the feeling I'm missing something.

Questions

A few questions about natural deduction itself.
- Can I use every assumption as long I discharge them later in the proof? (except for those that are already given in the excersize?

Thanks for explaining everything.

Greetings TKM