Logic & Reason

Aristotle's syllogistic logic — developed in the Prior Analytics — was the dominant framework for 2,000 years. A syllogism consists of two premises and a conclusion, with the validity determined by the "mood" and "figure" of the terms. "All men are mortal; Socrates is a man; therefore Socrates is mortal" is valid because of its form, not its content. Aristotle catalogued 256 possible syllogistic forms and identified the 24 that are valid.

In the late 19th century, Gottlob Frege revolutionized logic by developing predicate logic — a far more powerful system that could express claims about all members of a class ("All triangles have three sides"), about particular members ("Some philosophers are logicians"), and about relations between objects ("Socrates is wiser than Callias"). Predicate logic proved essential for the mathematical foundations program that Russell and Whitehead pursued in Principia Mathematica (1910–1913), which attempted to derive all of mathematics from logical axioms.

This program ran into Gödel's incompleteness theorems (1931) — perhaps the most stunning results in 20th-century mathematics. Gödel proved that any consistent formal system powerful enough to express basic arithmetic contains true statements that cannot be proved within the system. No formal system can be both complete (able to prove all truths) and consistent (free from contradiction). This result deflated the logicist program and raised deep questions about the nature of mathematical truth and the limits of formal reasoning.

Formal logic studies argument structure with mathematical precision. Informal logic studies the reasoning we actually use in everyday life — in persuasion, decision-making, scientific reasoning, and political argument — where the precision of formal logic is often unavailable or inappropriate. At the center of informal logic is the catalogue of fallacies: systematic errors in reasoning that are persuasive without being valid.

The most common fallacies include ad hominem (attacking the person rather than the argument), straw man (misrepresenting an opponent's argument to make it easier to attack), false dilemma (presenting only two options when more exist), begging the question (assuming what you're trying to prove), and appeal to authority (accepting a claim because an authority figure endorses it, regardless of whether they have relevant expertise). Recognizing these patterns is one of the most practically useful skills philosophy offers.

One of the most fertile connections in 20th-century philosophy was between logic and the philosophy of language. Frege's logic required a precise account of what it is for a name to refer and for a sentence to be true. His distinction between Sinn (sense) and Bedeutung (reference) — between the meaning of an expression and what it picks out — became the foundation of analytic philosophy of language.

Bertrand Russell's theory of descriptions used logical analysis to solve puzzles about non-referring expressions. "The present king of France is bald" seems to be a meaningful sentence — we understand it — but France has no king, so what is the sentence about? Russell showed that definite descriptions ("the present king of France") are not names but quantified expressions that make existential claims: "There exists exactly one thing that is currently king of France, and it is bald." Since the existential claim is false, the whole sentence is false — not meaningless, not a reference failure, but simply false.