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 | \(((p \lor q) \land (\lnot p \lor r)) \rightarrow (q \lor r)\) |
| 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==