Rules of inference

An argument is a sequence of propositions. This sequence ends with a conclusion that is prefixed by one or more premises (or hypothesis). The sequence is termed a valid argument if the truth of all it’s premises implies the truth of the conclusion.

The rules of inference provide base structures on which to build incrementally complex valid arguments.

Name Tautology
Modus ponens (law of detachment) \((p \land (p \rightarrow q)) \rightarrow q\)
Modus tollens (law of the contrapositive) \((\lnot q \land (p \rightarrow q)) \rightarrow \lnot p\)
Conjunction \(((p) \land (q)) \rightarrow (p \land q)\)
Simplification \((p \land q) \rightarrow p\)
Addition \(p \rightarrow (p \lor q)\)
Hypothetical syllogism \(((p \rightarrow q) \land (q \rightarrow r)) \rightarrow (p \rightarrow r)\)
Disjunctive syllogism \(((p \lor q) \land \lnot p) \rightarrow q\)
Resolution \(((p \lor q) \land (\lnot p \lor r)) \rightarrow (q \lor r)\)

Rules of inference with quantifiers

Name Expression
universal instantiation \(\forall x P(x) \therefore P(c)\)
universal generalisation \(P(c) \therefore \forall x P(x)\)
existential instantiation \(\exists x P(x) \therefore P(c)\)
existential generalisation \(P(c) \therefore \exists x P(x)\)
universal modus ponens \((\forall x P(x) \rightarrow Q(x)) \land P(a) \therefore Q(a)\)
universal modus tollens \((\forall xP(x) \rightarrow Q(x)) \land \lnot Q(a) \therefore \lnot P(a)\)

==todo: add section on fallacies==