Statements and truth
Six units on the basic tools of logic: statements and truth values, the words and, or and not, conditionals and the contrapositive, truth tables, validity and soundness, and the syllogism. Each unit has diagrams of truth tables or regions, flashcards, and 10 or 11 practice questions with the reasoning shown.
About this course
In 1966 the psychologist Peter Wason showed people four cards, E, K, 4 and 7, and said each card had a letter on one side and a number on the other. The rule to test was: if a card has a vowel on one side, it has an even number on the other. Most people turned over E alone, or E and 4. The right answer is E and 7. That small puzzle is the kind of thinking this course trains, one careful reading at a time.
The six units begin with the line between statements and everything else. Aristotle wrote in On Interpretation that a prayer is a sentence but neither true nor false, so "Close the door" cannot be true or false while "The door is closed" can. And, or and not come next, with the rule that "p or q" stays true when both are true. Conditionals and the contrapositive follow, and then truth tables, where three letters give 8 rows. The tables are often credited to Wittgenstein's 1921 Tractatus and to Emil Post, but the history turned out to be older. Validity and soundness separate a good argument from true sentences: "All fish can fly. A salmon is a fish. So a salmon can fly" is valid and has a false conclusion. The last unit covers syllogisms such as "All whales are mammals", which medieval students called Barbara. Each unit has diagrams of truth tables or regions, flashcards, and 10 or 11 practice questions with the reasoning shown.
Before you start
No mathematics is required beyond counting. Truth tables grow by doubling, so three letters give 2 × 2 × 2 = 8 rows, and the course explains that as it goes. What helps most is patience with careful reading, since many questions turn on a single word such as or, only if or unless.
When you finish
You will be able to tell a statement from a command or an opinion, negate a claim correctly, and use De Morgan's laws. You can identify the one case that breaks an if-then sentence, form the converse, inverse and contrapositive, fill in an 8-row truth table, explain how an argument can be valid but unsound, and test a syllogism for an undistributed middle.
- 1Statements and truth values“Close the door” can be obeyed or ignored, but it cannot be true or false. “The door is closed” can, and that one sentence of difference is where logic starts.
- 2And, or, notOn a menu, “soup or salad” usually means one of the two. In logic, “p or q” stays true when both are true, and that single rule settles most arguments about the word or.
- 3Conditionals and the contrapositive“If it rains, the match is cancelled” can be broken in only one way: it rains and the match goes ahead.
- 4Truth tablesThree letters give 2 × 2 × 2 = 8 rows, and a truth table writes out every one of them, so no case is left to guesswork.
- 5Validity and soundness“All fish can fly. A salmon is a fish. So a salmon can fly.” The conclusion is false, yet the argument is valid, and logic has a separate word for what went wrong.
- 6Syllogisms“All whales are mammals. All mammals breathe air. So all whales breathe air.” Medieval students called this pattern Barbara, and its three a’s are a code for the three sentences.
Sources for this course (9)
- P. D. Magnus, Tim Button, Aaron Thomas-Bolduc, Richard Zach, forall x: Calgary , chapters 1 to 6 on sentences, arguments, validity, soundness, connectives and truth tables (forallx.openlogicproject.org/html/)
- Stanford Encyclopedia of Philosophy, Robin Smith, "Aristotle's Logic" (plato.stanford.edu/entries/aristotle-logic/)
- definition of deduction from Prior Analytics I.2 24b18-20, the three figures and the medieval names Barbara and Celarent
- Aristotle, On Interpretation 17a, E. M. Edghill translation (classics.mit.edu/Aristotle/interpretation.1.1.html)
- "a prayer is a sentence, but is neither true nor false"
- Irving H. Anellis, "Peirce's Truth-functional Analysis and the Origin of Truth Tables" (arxiv.org/abs/1108.2429): Peirce manuscript of 1893, Shosky's 1997 find on a 1912 Russell typescript
- Emil Post, "Introduction to a general theory of elementary propositions", American Journal of Mathematics 43 (1921)
- Wittgenstein, Tractatus Logico-Philosophicus (1921, English 1922)
- 4.31. Everyday example sentences are made up for teaching.
Checked: 2026-10-17