The Herbrand Universe: Making Infinite Domains Finite
How a 1930 theorem turns infinite first-order domains into finite propositional searches for every modern prover
How Temporal Logic Lets Computers Reason About Forever
LTL and CTL give algorithms the power to prove that systems behave correctly across infinite time.
The Halting Problem: Computation's Fundamental Limit
Turing proved no algorithm can universally decide if programs halt—permanently reshaping what AI can reason about
Why Proof Complexity Matters for SAT Solving
The mathematical theory explaining why some formulas defeat all solvers while others fall instantly.
How Automated Theorem Provers Discovered New Mathematics
Computer provers now settle open conjectures, find shorter proofs, and partner with mathematicians to discover truths humans alone couldn't reach.
Why Fuzzy Logic Isn't Just Probability in Disguise
Vagueness measures how much something satisfies a concept; probability measures uncertainty about crisp facts—different math for different problems.
How Knowledge Representation Shapes What AI Can Learn
Why your AI's representation language mathematically determines the concepts it can ever learn to express
Belief Revision: The Logic of Changing Your Mind Rationally
How rational agents should update beliefs when new information contradicts what they previously accepted.
How Epistemic Logic Models What Agents Know About Knowledge
The formal mathematics of what agents know about each other's knowledge, and why it determines what distributed systems can achieve.
The Lottery Paradox: When Rational Beliefs Contradict
When a million justified beliefs logically guarantee a contradiction, what gives?
Why Planning Is Hard: The Complexity of Thinking Ahead
PSPACE-completeness explains why planning is fundamentally hard, yet tractable structure makes real-world AI planning possible.
Why Description Logics Power the Semantic Web
How a carefully engineered fragment of logic enables decidable reasoning over web-scale knowledge.
Answer Set Programming: Logic Programming Meets NP-Complete Reasoning
Declare what solutions look like, let solvers figure out how to find them
Why Gödel's Incompleteness Theorems Don't Doom AI
Gödel's incompleteness constrains all reasoners equally—human and machine alike face the same mathematical horizon.
How Type Theory Revolutionized Program Verification
From logical foundations to production systems: how treating proofs as programs enabled mathematical guarantees for real-world software.
Why Default Reasoning Requires Abandoning Classical Logic
How defeasible inference forces a fundamental rethinking of logical consequence for intelligent systems
The Frame Problem: Why Commonsense Reasoning Defeats Simple Logic
How the challenge of representing what doesn't change revealed fundamental limits in AI's ability to reason about actions and their consequences.
Modal Logic: The Mathematics of Necessity and Possibility
Master the formal framework distinguishing what must be true from what merely might be—essential mathematics for verification, knowledge representation, and AI reasoning.
How Probabilistic Logic Programs Unite Uncertainty with Deduction
Discover how ProbLog and DeepProbLog merge logical deduction with probabilistic reasoning, enabling AI that handles uncertainty without sacrificing interpretability.
Why SAT Solvers Can Crack Problems That Exhaust Human Mathematicians
Discover how conflict-driven learning and proof certificates enable machines to solve logical problems billions of times faster than human reasoning allows.
The Chinese Room: Why Symbol Manipulation Might Not Be Understanding
Unpacking philosophy's most famous thought experiment and what it reveals about the gap between computation and comprehension