Law of Identity: F(a) → F(a)

Law of Non-contradiction: ¬ (p & ¬ p)

Law of Excluded Middle: p ∨ ¬ p

These are sentences are necessary truths and have as their terms all possible sentences. The power of these Laws is the highest of any Laws in the universe. They seem weak, boring, and stupid. But there are no other sentences with such God-like dominion. Indeed, God himself must follow these laws. There are no laws higher than these!

Leibniz adds two

Leibniz formulated two additional principles that are sometimes counted as laws of thought:

• Principle of Sufficient Reason
• Identity of Indiscernibles

Modern update

Let the formal language P be the formal language for truth-functional propositional logic. Let PS be the formal system that is the deductive apparatus for P. Then these are the axioms of PS:

[PS1] A → (B→A)

[PS2] (A → (B→C)) → ((A→B) → (A→C))

[PS3] (¬A → ¬B) → (B→A)

Did you notice that A → A is not an axiom? That is because it can be derived from [PS1] and [PS2], above.

1. A → ((A→A) → A) [axiom, by PS1]
2. (A → ((A→A) → A)) → ((A → (A → A)) → (A → A)) [axiom, by PS2]
3. ((A → (A→A)) → (A → A)) [by MP of 1 and 2]
4. A → (A → A) [axiom, by PS1]
5. A → A [by MP of 3 and 4]