Formal Logic Basics
The slow, deliberate machinery of valid reasoning. Five symbols, four rules, and a truth table — almost every "gotcha" argument can be checked mechanically.
Why bother with symbols?
Natural language is ambiguous. "If you study, you'll pass" reads as cause-and-effect, but logic treats it as a one-way conditional — and conditional reasoning trips even smart people. Symbols force precision.
A century-old result by Frege, Russell and Tarski showed that almost all valid mathematical proof can be reduced to manipulating these symbols by mechanical rules. You don't need that machinery for daily life, but the rules themselves are worth a sharp Saturday morning.
Valid ≠ true ≠ sound
An argument is valid if accepting the premises forces accepting the conclusion. The premises can be false — the argument is still valid.
An argument is sound if it is valid and the premises are true. "All cats meow; my dog is a cat; ∴ my dog meows" is valid but unsound (premise 2 is false).
The five connectives
NOT, ~, !
AND, &
OR, |
->, IF…THEN
<->, IFF
Truth table builder
Type a propositional expression using variables (single uppercase letters: P, Q, R) and connectives. The truth table updates live.
Valid argument forms
Four you actually use
- Modus ponens — "If P then Q. P. ∴ Q." (Affirming the antecedent — valid.)p → q ⊢ q when p
- Modus tollens — "If P then Q. Not Q. ∴ Not P." (Denying the consequent — valid.)p → q, ¬q ⊢ ¬p
- Disjunctive syllogism — "P or Q. Not P. ∴ Q." (Either-or, one ruled out — valid.)p ∨ q, ¬p ⊢ q
- Hypothetical syllogism — "If P then Q. If Q then R. ∴ If P then R." (Transitivity of implication — valid.)p → q, q → r ⊢ p → r
Invalid look-alikes (commit a fallacy): "If P then Q. Q. ∴ P." is affirming the consequent. "If P then Q. Not P. ∴ Not Q." is denying the antecedent. Try both in the builder above — neither is a tautology.
Argument validity checker
Enter premises and a conclusion. We build the truth table of (premises) → conclusion and check whether it's a tautology (= valid).
Natural deduction — build a proof step by step
Apply one inference rule at a time. Derive the goal.
Pick a puzzle. You start with the premises. Select 1 or 2 numbered lines as inputs, pick an inference rule, hit Apply, and the next derived line appears. Reach the goal to clear the puzzle.
— 📖 How to use it
P→Q AND P → derives QMT: tick lines like
P→Q AND ¬Q → derives ¬P∧I: tick two lines
P, Q → derives P∧Q∧E: tick one line
P∧Q, type P or Q in Extra → derives that side∨I: tick one line
P, type any formula Q in Extra → derives P∨QDS: tick
P∨Q AND ¬P → derives Q (or vice versa)HS: tick
P→Q AND Q→R → derives P→RDN: tick
¬¬P → derives P
Categorical syllogisms
Aristotle's specialty. Each syllogism uses two premises about categories ("All A are B," "No A are B," "Some A are B") to derive a conclusion. Step through the classics:
De Morgan's laws
How negation distributes over and / or
Two equivalences that look small but rewire most logical puzzles:
¬(P ∧ Q) ≡ ¬P ∨ ¬Q ¬(P ∨ Q) ≡ ¬P ∧ ¬QIn English: "not (both A and B)" is the same as "not A, or not B (or both)." Negation flips ∧ to ∨ and vice versa. Try them in the truth-table builder — identical columns.
Practical use: turning hard-to-reason negations into easier disjunctions. "I'm not going if both of them go" ≡ "I'm going if at least one of them doesn't go."
A peek at predicate logic
Quantifiers: ∀ and ∃
Propositional logic treats whole statements as atoms. Predicate logic opens them up: P(x) means "x has property P."
- ∀x P(x) — "for all x, P(x)." Universal.
- ∃x P(x) — "there exists an x such that P(x)." Existential.
- ¬∀x P(x) ≡ ∃x ¬P(x) — "not everything is P" = "something is not P." The quantifier flips.
- ¬∃x P(x) ≡ ∀x ¬P(x) — "nothing is P" = "everything fails to be P."
Quantifier order matters: "Every student has a teacher" (∀s ∃t T(s,t)) ≠ "There is one teacher every student has" (∃t ∀s T(s,t)). Swapping ∀ and ∃ usually changes meaning.
Connect the dots
Quiz
15 questions on formal logic.
Flashcards
Twenty essential logic terms and rules.
Teacher mode
Lesson outline, quick-reference card, and a printable worksheet with answer key.
Lesson outline (45 min)
- 5 min · Hook — Read aloud: "If you study, you'll pass. You didn't pass. Did you study?" Most class will guess incorrectly. Reveal modus tollens.
- 10 min · Concept — Propositions, connectives, valid vs. sound. Build the AND, OR, → truth tables with the class on the board.
- 10 min · Builder — Live-type the four valid forms (modus ponens, tollens, disjunctive, hypothetical) into the truth-table builder. Show they're tautologies.
- 15 min · Practice — Pairs translate three short English arguments into symbols and check validity. Report back invalid examples.
- 5 min · Wrap — Brief intro to ∀ and ∃ as a teaser. Homework: find one ad or political statement and translate it.