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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.
Where LEM fails
Imagine you are a mathematician and you want to show to the mathematical community that a propositino is a theorem. To do this, you supply a proof to show that your proposition holds. Now you have this insight: your proposition has an associated proof. A proof is a kind of data that you can manipulate to
Relation between LEM and other statements
Ex Falso Quodlibet
Double Negation Elimination
Double negation elmination simply states that if you
LEM isn't true, but it's not not true.