01Overview
Logic began as part of philosophy. Aristotle’s Organon gives the syllogism: from “all men are mortal” and “Socrates is a man” it derives “Socrates is mortal”. What matters is not whether the conclusion is true but whether the form is valid — already a sketch of symbol manipulation.
In the nineteenth century Boole turned logic into algebra: and, or and not became multiplication, addition and complement. Frege then added quantifiers and predicates to build first-order logic. By the twentieth century logic had become the foundational language of mathematics — and the starting point of computer science: circuits, programs, database queries and theorem provers all manipulate formal statements.
02History
- 350 BC
Aristotle, Organon
Systematised the syllogism and introduced the idea of formal validity.
- 1854
Boole, The Laws of Thought
Described propositional logic in algebra: truth values 0/1, with and, or, not as multiply, add, complement.
- 1879
Frege, Begriffsschrift
Introduced quantifiers and predicates, building the first complete first-order logic.
- 1930
Gödel's completeness theorem
Semantic entailment and formal provability in first-order logic coincide exactly.
03Core concepts
Propositional logic
Combines atomic propositions with truth-functional connectives; it is decidable.
Predicate logic
Adds objects, predicates and quantifiers — enough to express almost all of mathematics.
Inference rules
Mechanical rules that derive conclusions from premises, such as modus ponens and resolution.
04Hands-on
Do it once — it beats reading it ten times
Verify with a truth table that (p→q) ∧ p → q is a tautology, then write a 20-line truth-table printer
05Self-check
Say it in your own words first, then open the answer. Whatever you cannot explain is what to review next.
Can you explain "Propositional logic"?
Combines atomic propositions with truth-functional connectives; it is decidable.
Can you explain "Predicate logic"?
Adds objects, predicates and quantifiers — enough to express almost all of mathematics.
Can you explain "Inference rules"?
Mechanical rules that derive conclusions from premises, such as modus ponens and resolution.
What problem did it solve, and why was there nothing better at the time?
Go back to the history section: how was this done before, and what did it cost?
If you had to build it today, how would you break it down?
Use the hands-on task above: write a plan of at most three steps before you start.
06Key people
Aristotle
384 – 322 BC
Built the syllogism and the first system of formal logic.
George Boole
1815 – 1864
Turned logical operations into algebra; Boolean algebra carries his name.
Gottlob Frege
1848 – 1925
Founder of modern mathematical logic.
07Further reading
- Organon
- Begriffsschrift