Lesson Logic Rules of Inference and Replacement

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This is a reference page about the Rules of Inference and Replacement.
The explanation of each rule is omitted.
A similar reference sheet should be found in the back of your logic textbook.

Rules of Inference

1. Modus Ponens p --> q p :. q	2. Modus Tollens p --> q ~q :. ~p	3. Hypothetical Syllogism p --> q q --> r :. p --> r
4. Disjunctive Syllogism p v q ~p :. q	5. Conjunction p q :. p & q	6. Constructive Dilemma (p --> q) & (r --> s) p v r :. q v s
7. Simplification p & q :. p	8. Absorption p --> q :. p --> (p & q)	9. Addition p :. p v q

Rules of Replacement

10. De Morgan's Theorems ~(p & q) = ~p v ~q ~(p v q) = ~p & ~q	11. Commutation p v q = q v p p & q = q & p	12. Association p v (q v r) = (p v q) v r p & (q & r) = (p & q) & r
13. Distribution p & (q v r) = (p & q) v (p & r) p v (q & r) = (p v q) & (p v r)	14. Double Negation p = ~(~p)	15. Transposition p --> q = ~q --> ~p
16. Material Implication p --> q = ~p v q	17. Material Equivalence p <--> q = (p --> q) & (q --> p) p <--> q = (p & q) v (~p & ~q)	18. Exportation (p & q) --> r = p --> (q --> r)
19. Tautology p = p & p p = p v p

For any given line, a logical expression marked in red is the same as the logical expression marked in blue.
For instance, ~(p & q) is the same as ~p v ~q due to De Morgan's Theorem.

There are 19 rules in total.
9 rules of inference and 10 rules of replacement. The order of the rules shown doesn't matter.

A rule of inference MUST be used on a whole line only.
In contrast, a rule of replacement can be used for a portion of a line.

The rules of inference must flow in the direction shown above.
Example: We can go from p --> q to p --> (p & q) because of the absorption rule. However we cannot go from p --> (p & q) back to p --> q

For any rule of replacement, we can flow in either direction from red to blue, or vice versa.

Related Topic
Truth Tables

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