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# 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 itâ€™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==