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Brilliant

Logic II

via Brilliant

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Overview

Continue your adventures in logic with puzzles involving Venn diagrams and syllogisms, propositional logic, and first-order logic.

By the end of this course you'll have explored the deep foundations of truth as well as applied logic to AI expert systems and linguistic analysis.

Syllabus

  • Introduction: Survey the logical languages in the course, and solve some interesting puzzles on the way.
    • Fallacies: Deduce which arguments are valid or invalid.
    • Knights, Knaves, and Logic: Practice some classic knights and knaves puzzles.
    • Knights and Formal Logic: Use a knight and knaves puzzle to learn the beginnings of advanced logic.
  • Syllogisms and Sets: Visual tools for laying out logical thinking and avoiding fallacies.
    • Euler and Venn Diagram Basics: Using diagrams to relate categories is an essential tool for approaching syllogisms.
    • All, Some, and None: What do the statements "all", "some", and "none" look like visualized with Venn diagrams?
    • Spot the Fallacies!: Try to tell logic from illogic.
    • The Square of Opposition: See how different syllogism statements are related to each other.
    • And, Or, and Not: Use Venn diagrams to think through statements that combine "and", "or", and "not".
    • De Morgan's Laws: Apply your intuition to discover two fundamental laws of logic.
  • Logic Machines: Turn logic into arithmetic machines.
    • Truth Tables: Diagram out the meanings of each logical operator.
    • Basic Logic Gates: How do AND, OR, and NOT gates work?
    • Combinations: Combine gates together to make complex circuits.
    • Fredkin Gates: Take a peek at a gate used in biological and quantum computing.
    • Binary Numbers and Addition: How does computer arithmetic work?
    • Creating a Mechanical Adder: Use what you've learned about logic gates to create a circuit that adds numbers.
  • Arithmetic With Logic Gates: Build the functions of arithmetic using only logic gates.
    • Binary Refresher: Review the basics of binary.
    • Creating a Binary Comparator: When is one number larger than another?
    • Subtraction: Get into deeper complexity with binary subtraction.
    • Multiplication: Design the standard algorithm with logic circuits, then study an interesting shortcut.
    • Division: Finish the quadrilogy of arithmetic with one last design.
  • Propositional Logic: Turn logic puzzles into logic symbols.
    • Introduction to Formal Logic: Learn the basic terminology of formal logic.
    • Formal De Morgan: Prove De Morgan's Laws with formal logic tools.
    • Using Implication: Apply implication to break open new laws of thought!
    • Rules of Substitution: Which rules of substitution are logically valid?
    • Knights and Knaves Redux: Learn how to solve challenging knights and knaves puzzles using formal logic.
  • First-Order Logic: With a little extra logical power, describe the universe.
    • For All and There Exists: Practice using "all" and "some" formally and logically.
    • Formal Symbolization: Learn how to represent statements in first-order logic with formal notation.
    • Multiple Generality: Modify multiple logical variables rather than just one.
    • Duals and Prenex: Transform first-order logic statements using duals and learn about prenex form.
    • Quantifiers and Proof (I): Learn how to introduce and eliminate existential quantifiers.
    • Quantifiers and Proof (II): Derive complex proofs with universal generalizations and existential instantiations.
    • Functions and Identity: Apply logic to functions and arithmetic.

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