Rules of Inference

The rules of valid inference are a set of laws by which the syntax of statements in a system of logic, such as propositional or first-order logic, may be manipulated.

Rules of inference are, in themselves, simple valid argument forms. They may be divided into basic rules, which are fundamental to logic and cannot be eliminated without losing the ability to express some valid argument forms, and derived rules, which can be proven by the basic rules and serve as shortcuts for logicians when constructing a logical argument or proof.

Basic Rules

There are, essentially, two basic rules for each basic logical operator of propositional logic (¬, ∧, ∨, →, and ↔)—one for introduction (I) of the operator and one for elimination (E) of the operator. In first-order logic, we add the quantifier rules (for ∀ and ∃). In modal logic, we add rules for the modal operators (□ and ◊)

Conjunction (∧) rules

• Conjunction introduction (Adjunction, ∧I)
• Conjunction elimination (Simplificiation, ∧E)

Disjunction (∨) rules

• Disjunction introduction (Addition, ∨I)
• Disjunction elimination (Separation of cases, ∨E)

Negation (¬) rules

• Negation introduction (Proof by contradiction, reductio ad absurdum, ¬I)
• Negation elimination (Proof by contradiction, reductio ad absurdum, ¬E)

Conditional (→) rules

• Conditional introduction (Deduction theorem, →I)
• Conditional elimination (modus ponens, →E)

Biconditional (↔) rules

• Biconditional introduction (↔I)
• Biconditional elimination (↔E)

Quantifier (∀ and ∃) rules

• Universal introduction (∀I)
• Universal elimination (∀E)
• Existential introduction (∃I)
• Existential elimination (∃E)

Derived Rules

The derived rules are the rules that can be inferred from the basic rules. All of the valid argument forms can be derived from the basic rules, but the term "rules" is normally used for those that are especially simple, and for those that are uncommon in normal spoken arguments but are rather used when formulating and proving the validity of arguments. The following is a list of common derived rules:

• Associative laws
• Conditional exchange rule
• Commutative laws
• Contraposition rule
• Double negation rule
• De Morgan's laws
• Exportation rule
• Hypothetical syllogism
• Negated conditional rule
• Modus tollens