Worked proofs
Each page proves one argument with a semantic tableau: the premises assumed true and the conclusion false, decomposed until every branch contradicts itself or an assignment survives. How semantic tableaux work →
Valid arguments
- Modus ponens
p→q, p ⊨ qIf p → q holds and p is true, q follows. Modus ponens is the rule most proofs lean on, and the tableau below closes on every branch. - Modus tollens
p→q, ¬q ⊨ ¬pIf p → q holds and q is false, p must be false too: anything that made p true would make q true. Denying the consequent denies the antecedent. - Hypothetical syllogism
p→q, q→r ⊨ p→rTwo conditionals sharing a middle formula chain into one: from p → q and q → r you get p → r. This is how short steps build a long argument. - Disjunctive syllogism
p∨q, ¬p ⊨ qA disjunction needs at least one true side, so p ∨ q with ¬p leaves q. Ruling out one disjunct leaves the other standing. - Constructive dilemma
p∨q, p→r, q→r ⊨ rWhichever of p or q holds, r follows — so r follows outright. A disjunction plus a conditional from each side gives the shared conclusion. - Contraposition
p→q ⊨ ¬q→¬pp → q and ¬q → ¬p say the same thing: nothing can make the antecedent true while the consequent fails. A conditional equals its contrapositive. - De Morgan's law
¬(p∧q) ⊨ ¬p∨¬q¬(p ∧ q) does not say which conjunct fails, only that they cannot both hold — which is exactly ¬p ∨ ¬q. Denying a conjunction spreads the negation. - Material implication
p→q ⊨ ¬p∨qp → q holds in exactly the rows where ¬p ∨ q holds: either the antecedent fails, or the consequent holds. A conditional is a disjunction in disguise. - Double negation
¬¬p ⊨ p¬¬p and p are true in the same rows, so a doubled negation can be dropped wherever it stands. Two negations cancel. - Biconditional elimination
p↔q, p ⊨ qThe two sides of p ↔ q always carry the same truth value, so p gives you q — and q would give you p. A biconditional works in both directions. - Law of excluded middle
⊨ p∨¬pp ∨ ¬p is true in every row and needs no premises at all: every proposition is either true or false, with no third option. - Principle of explosion
p, ¬p ⊨ qFrom p and ¬p anything follows: no assignment makes both premises true, so none can make them true while the conclusion fails.
Invalid arguments
- Affirming the consequent
p→q, q ⊨ pA fallacy: p → q with q leaves p open, because q can hold for reasons that have nothing to do with p. The open branch below gives the countermodel. - Denying the antecedent
p→q, ¬p ⊨ ¬qA fallacy: p → q says nothing about what happens when p fails, so ¬p leaves q open. The open branch below shows q true while p is false. - Affirming a disjunct
p∨q, p ⊨ ¬qA fallacy: p ∨ q is inclusive, so both sides may hold at once. Knowing p tells you nothing about q, as the open branch below shows.