🧮 Formal Logic

What you’ll learn

1. Arguments, Not OpinionsDefine what logic studies — the relation between premises and conclusions — and separate validity from truth.Logic is the study of what follows from what: an argument is a set of premises offered in support of a conclusion, and logic evaluates the support, not the subject matter. Validity is a property of an argument's form — if the premises were true, the conclusion would have to be — while truth is a property of individual statements. A sound argument is the gold standard: valid form plus actually true premises.
2. Twenty-Three Centuries of RigorTrace formal logic from Aristotle and the Stoics through Boole and Frege to the modern symbolic era.Aristotle founded logic in the 4th century BC with the syllogism — arguments valid by their form — and his Organon dominated the curriculum for two millennia. The Stoic Chrysippus built the rival system we now call propositional logic, reasoning with whole statements joined by 'if,' 'and,' and 'or.' The modern era arrived when George Boole turned logic into algebra (1854) and Gottlob Frege built the first complete formal system (1879), the foundation of both modern mathematics and computing.
3. Propositions and ConnectivesLearn the atoms of propositional logic — simple statements — and the connectives NOT, AND, OR.Propositional logic reduces language to propositions — statements that are either true or false — and a handful of connectives: ¬ (not), ∧ (and), ∨ (or), → (if…then), ↔ (if and only if). Simple propositions get letters (p, q, r); connectives build compounds whose truth depends only on the truth of the parts. Note one convention: logic's ∨ is inclusive or — 'p or q or both' — unlike the exclusive 'or' English often intends.
4. Truth TablesEvaluate any compound proposition mechanically by tabulating every combination of truth values.A truth table lists every possible assignment of true/false to a formula's letters — 2 to the power n rows for n letters — and computes the compound's value in each row. The tables for ∧, ∨, and ¬ define those connectives completely, and the method decides any question in propositional logic by brute force. The tabular method was made standard in the early 1920s by Emil Post and Ludwig Wittgenstein, realizing a dream older than either: reasoning checked by pure routine.
5. The ConditionalMaster the material conditional p → q, its one false case, and the difference between a conditional and its converse.The conditional p → q ('if p then q') is false in exactly one case: p true and q false — a promise is broken only when the condition happens and the payoff doesn't. This makes conditionals with false antecedents 'vacuously true,' a convention that surprises newcomers but keeps logic consistent. The vital distinctions: the converse (q → p) and inverse (¬p → ¬q) do NOT follow from p → q, but the contrapositive (¬q → ¬p) is exactly equivalent to it.
6. Valid Forms and Formal FallaciesRecognize modus ponens and modus tollens, and the two look-alike fallacies that mimic them.Two argument forms carry most deductive reasoning: modus ponens (p → q; p; therefore q) and modus tollens (p → q; ¬q; therefore ¬p). Each has an invalid twin: affirming the consequent (p → q; q; therefore p) and denying the antecedent (p → q; ¬p; therefore ¬q) — fallacies that feel right because they run the true conditional backward or inside out. Telling the four apart at a glance is the most practically useful skill in formal logic.
7. Chains of InferenceBuild multi-step deductions from valid rules, and see what a fully formal proof looks like.Real proofs chain small valid steps: hypothetical syllogism links conditionals (p → q, q → r, so p → r), disjunctive syllogism eliminates options (p ∨ q, ¬p, so q), and each derived line becomes a premise for the next move. Frege's Begriffsschrift (1879) first made proofs fully formal — every step an instance of an explicit rule — and Whitehead and Russell's Principia Mathematica pushed the program to its famous extreme, taking hundreds of pages to reach 1 + 1 = 2.
8. Equivalences and De Morgan's LawsUse logical equivalences — especially De Morgan's laws — to rewrite and simplify statements, and classify formulas as tautologies, contradictions, or contingencies.Two formulas are equivalent when their truth-table columns match, which licenses substituting one for the other anywhere. The workhorses: double negation (¬¬p = p) and De Morgan's laws — ¬(p ∧ q) is equivalent to ¬p ∨ ¬q, and ¬(p ∨ q) to ¬p ∧ ¬q: negation flips 'and' into 'or.' Formulas sort into tautologies (true in every row), contradictions (false in every row), and contingencies (varies) — a classification that matters everywhere from proofs to database queries.
9. Categorical Logic and Venn DiagramsWork with 'all,' 'no,' and 'some' statements and test syllogisms visually with Venn diagrams.Aristotle's categorical logic deals in four sentence shapes: All S are P, No S are P, Some S are P, Some S are not P — statements about classes rather than whole propositions. A three-circle Venn diagram tests any classical syllogism: diagram the premises, then check whether the conclusion is already forced. The diagrams are named for John Venn, who introduced them in 1880, and they expose invalid syllogisms that sound perfectly persuasive aloud.
10. Logic Becomes ComputingSee how Boolean logic became physical circuitry — and why logic is the foundation of every computer.In his 1937 MIT master's thesis — often called the most influential ever written — Claude Shannon showed that relay switching circuits obey Boole's algebra: wire switches in series for AND, in parallel for OR, and any Boolean function becomes hardware. Logic gates implementing NOT, AND, and OR combine into adders and processors; a modern chip is billions of them. Formal logic thus completed a strange journey: from Athenian debate tool to the physical substrate of civilization's machines.

Questions this course answers

Grounded in trusted sources

• Irving Copi, Carl Cohen & Victor Rodych, Introduction to Logic (15th ed., Routledge)
• Stanford Encyclopedia of Philosophy — Classical Logic; Aristotle's Logic; Frege
• George Boole, An Investigation of the Laws of Thought (1854)
• Gottlob Frege, Begriffsschrift (1879)
• Whitehead & Russell, Principia Mathematica (1910–13)
• Claude Shannon, A Symbolic Analysis of Relay and Switching Circuits (MIT, 1937)

Every Wunder lesson is built from real, reputable sources — never invented.

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