The Anti-LEM Manifesto

What is LEM?

A proposition is a sentence or formula that can be evaluated as having a truth value such as true or false. The Law of Excluded Middle (LEM) states that for any proposition, either the proposition hold or its negation holds. Formally, it is written for arbitrary proposition $P$, $$\vdash P \lor \lnot P$$. It looks reasonable, maybe even seemingly valid and holds in all context. After all, everything statement that you can say has to either hold or not hold. WRONG!

Where does it go wrong?

Just as a person has bad intuitions about reasoning [example], one can have bad intuitions in our world. My vendetta against LEM is on the following basis:

LEM doesn't actually align with our intuitions.
Accepting LEM leads to unintuitive claims.
Not only it is possible to do away with LEM, but it is more useful.
There are many other ways to denotee excluded middle that doesn't rely on LEM.

Knowing the enemy
1. History of LEM
2. LEM in Mathematics
3. LEM in Computer Science
4. Constructive and Non-construction
5. Defining OR and defining NOT
Relationships with other Logical statements
1. Proof by contradiction
2. Disjunctive Syllogism
3. Double Negation Elimination
4. Axiom of Choice
Alternatives to LEM
1. Double negated LEM
2. Intuitionistic Logic
Another sub-heading

The Anti-LEM Manifesto

What is LEM?

Where does it go wrong?

Table of Contents