Great IdeasKnowing and thinking
Logic and argument
Reasoning well: proofs, arguments and the mistakes to avoid.
The question
What makes an argument good? How can we reason step by step to conclusions we can trust?
How the books answer
Oldest first. Each stop is our own short summary of what the author argued; follow the link to the rung for the whole lesson.
c. 390s–380s BC
Plato, Protagoras and Gorgias (with Charmides, Laches, Lysis, Euthydemus and Ion)
Socrates tests people's beliefs by asking questions. He gets someone to state a view, then shows that it clashes with other things they believe. The method, called dialectic, teaches that a good argument must hang together.
“For I certainly think that I and you and every man do really believe, that to do is a greater evil than to suffer injustice: and not to be punished than to be punished.”
c. 350–322 BC
Aristotle, Categories, On Interpretation, Prior and Posterior Analytics, Topics, Sophistical Refutations
Aristotle was the first to study the forms of argument. A syllogism is an argument where, once some things are granted, something else must follow. 'All people are mortal; Socrates is a person; so Socrates is mortal' is valid whatever the subject.
“A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.”
c. 300 BC
Euclid, The Elements, Books I–IV
Euclid shows logic building a whole science. He starts with a few definitions and postulates and proves one proposition after another, each resting on earlier ones. For two thousand years his Elements was the model of how to prove something.
1265–1274 (unfinished)
Thomas Aquinas, Summa Theologica, First Part
Aquinas writes in the form of disputed questions. For each question he first lists the best objections to his own answer, then gives his answer and his reasons, and then replies to each objection. It is a model of taking the other side seriously.
“The existence of God can be proved in five ways.”
1637
René Descartes, Discourse on the Method (with Rules for the Direction of the Mind)
Descartes gives himself four rules: accept nothing as true that you don't clearly know, divide each problem into parts, go from the simplest to the hardest, and check that nothing has been left out. He wanted to reason in every field the way mathematicians do.
1677 (published after his death)
Baruch Spinoza, Ethics, Demonstrated in Geometrical Order
Spinoza writes his Ethics like Euclid, with definitions, axioms and numbered proofs. He believed that truths about God, the mind and happiness could be shown as firmly as truths about triangles. Readers admire the rigor even when they doubt the starting points.
“By God, I mean a being absolutely infinite—that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.”
1748
David Hume, An Enquiry Concerning Human Understanding
Hume says logic and math can prove relations of ideas, but no amount of logic can prove a matter of fact, such as what will happen tomorrow. For that we rely on experience. His point still guides how people think about proof and evidence.
“All these suppositions are consistent and conceivable.”
Go to the rung: Hume: An Enquiry Concerning Human Understanding
1953 (published after his death)
Ludwig Wittgenstein, Philosophical Investigations
Wittgenstein, who had once tried to build a perfect logical language, changed his mind. He came to think that words get their meaning from how they are used in many different activities, which he called language games. Many puzzles in philosophy, he argued, come from taking words out of their ordinary use.
For young readers
Logic is the art of thinking clearly. If you know all cats have whiskers, and Tiger is a cat, then you know Tiger has whiskers. That's a logical step.
Logic also helps you spot mistakes. If someone says 'my friend got sick after eating pizza, so pizza makes people sick,' you can ask: did anything else happen? Good thinkers check each step.
Talk about it
- Make up your own 'All... Some... So...' argument. Is it valid?
- Aquinas listed the best arguments against his view first. Why is that a good habit?
- Descartes says to break a big problem into small parts. Try it on a chore or a puzzle.
- Can an argument be logical and still have a false conclusion? How?
Where else on the site
- The Parts of a Syllogism
- The Valid Forms and Their Names
- Premises and Conclusions
- What a Fallacy Is
- Writing a Proof
- Validity
Every rung tagged with this idea:
These pages are our own short synthesis, written for families. They were inspired by Mortimer J. Adler's Syntopicon, the index of great ideas in volumes 1 and 2 of Great Books of the Western World, but the questions, summaries and groupings are ours, not the Syntopicon's. Short excerpts come from public-domain translations, credited where they appear.