description/proof of that proposition 1 or proposition 2 iff if not proposition 2, proposition 1
Topics
About: logic
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
- 4: Note
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that any proposition 1 or any proposition 2 if and only if if not the proposition 2, the proposition 1.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(p_1\): \(\in \{\text{ the propositions }\}\)
\(p_2\): \(\in \{\text{ the propositions }\}\)
//
Statements:
\(p_1 \lor p_2\)
\(\iff\)
\(\lnot p_2 \implies p_1\)
//
2: Natural Language Description
For any propositions, \(p_1\) and \(p_2\), \(p_1 \lor p_2\) if and only if \(\lnot p_2 \implies p_1\).
3: Proof
Whole Strategy: Step 1: suppose that \(p_1 \lor p_2\) is true and prove that \(\lnot p_2 \implies p_1\) is true; Step 2: suppose that \(\lnot p_2 \implies p_1\) is true and prove that \(p_1 \lor p_2\) is true.
Step 1:
Let us suppose that \(p_1 \lor p_2\) is true.
If \(\lnot p_2\) is true, \(p_2\) will be false, so, \(p_1\) will be true (because \(p_1 \lor p_2\) is true), and so, \(\lnot p_2 \implies p_1\) is true.
Step 2:
Let us suppose that \(\lnot p_2 \implies p_1\) is true.
If \(p_2\) is true, \(p_1 \lor p_2\) is true. If \(p_2\) is false, \(\lnot p_2\) is true, and \(p_1\) is true (because \(\lnot p_2 \implies p_1\) is true), and so, \(p_1 \lor p_2\) is true. So, \(p_1 \lor p_2\) is true anyway.
4: Note
Of course, also \(\lnot p_1 \implies p_2\) is equivalent, because \(p_1 \lor p_2 \iff p_2 \lor p_1\) and also because \(\lnot p_1 \implies p_2\) is the contraposition of \(\lnot p_2 \implies p_1\).
