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Ryan Collison

I write about cool science and share it here.

Mathematical Induction: Infinite Dominos

Mathematical Induction: Infinite Dominos

Mathematical induction lets you prove a statement for every positive integer with just two steps: prove the base case, then prove the chain rule. It's the domino argument made rigorous — and it works for infinitely many cases.

Methods of Proof: Direct Contradiction Induction

Methods of Proof: Direct Contradiction Induction

Every mathematical proof uses one of a handful of logical moves. Knowing which weapon to reach for — direct proof, contradiction, or induction — is most of what separates elegant proofs from ugly ones.

Quantifiers: For All and There Exists

Quantifiers: For All and There Exists

The symbols ∀ (for all) and ∃ (there exists) carry the weight of precise logical thought. Getting them right is the foundation of proofs, type theory, and formal verification.

Common Logical Fallacies: Where Reasoning Goes Wrong

Common Logical Fallacies: Where Reasoning Goes Wrong

A good fallacy doesn't announce itself—it sounds convincing. This guide maps the most common logical traps, from straw men to appeals to authority, so you can spot them before they mislead you.

Arguments and Validity: When Conclusions Follow

Arguments and Validity: When Conclusions Follow

You can build a valid argument from completely false premises. Validity is about structure, not content — and that distinction is what makes logic useful for detecting bad reasoning even when you can't check the facts.

Tautologies and Contradictions: Always True and Never True

Tautologies and Contradictions: Always True and Never True

"It will either rain or not rain" is vacuously true — a tautology that carries zero information. "It will rain and not rain" is a contradiction that breaks logic. Both extremes reveal where meaningful statements actually live.

Logical Equivalence: Different Statements Same Meaning

Logical Equivalence: Different Statements Same Meaning

"If it rains, I'll bring an umbrella" and "If I have no umbrella, it didn't rain" are logically identical. Understanding equivalence is the key to elegant, efficient reasoning.