What are the main inference rules used in propositional logic?

In propositional logic, there are several main inference rules that are used to derive new logical statements from existing ones. These rules include:

1. Modus Ponens: This rule states that if we have a statement of the form "If A, then B" and we also have A, then we can infer B. Symbolically, it can be represented as (A → B) and A, therefore B.

2. Modus Tollens: This rule states that if we have a statement of the form "If A, then B" and we also have ¬B (not B), then we can infer ¬A (not A). Symbolically, it can be represented as (A → B) and ¬B, therefore ¬A.

3. Disjunctive Syllogism: This rule states that if we have a statement of the form "A or B" and we also have ¬A (not A), then we can infer B. Symbolically, it can be represented as (A ∨ B) and ¬A, therefore B.

4. Conjunction: This rule states that if we have two statements A and B, then we can infer the statement "A and B". Symbolically, it can be represented as A and B, therefore (A ∧ B).

5. Simplification: This rule states that if we have a statement of the form "A and B", then we can infer A or B individually. Symbolically, it can be represented as (A ∧ B), therefore A or B.

6. Addition: This rule states that if we have a statement A, then we can infer the statement "A or B" for any statement B. Symbolically, it can be represented as A, therefore (A ∨ B).

7. Contradiction: This rule states that if we have a statement and its negation, then we can infer any statement. Symbolically, it can be represented as A and ¬A, therefore B (where B can be any statement).

These inference rules form the foundation of propositional logic and are used to derive new logical statements based on given premises.