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
<table border = "1" cellpadding = "5"><tr><td>1. Modus Ponensp --> qp:. q</td><td>2. Modus Tollensp --> q~q:. ~p</td><td>3. Hypothetical Syllogismp --> qq --> r:. p --> r</td></tr><tr><td>4. Disjunctive Syllogismp v q~p:. q</td><td>5. Conjunctionpq:. p & q</td><td>6. Constructive Dilemma(p --> q) & (r --> s)p v r:. q v s</td></tr><tr><td>7. Simplificationp & q:. p</td><td>8. Absorptionp --> q:. p --> (p & q)</td><td>9. Additionp:. p v q</td></tr></table>

Rules of Replacement
<table border = "1" cellpadding = "5"><tr><td>10. De Morgan's Theorems~(p & q) = ~p v ~q~(p v q) = ~p & ~q</td><td>11. Commutationp v q = q v pp & q = q & p</td><td>12. Associationp v (q v r) = (p v q) v rp & (q & r) = (p & q) & r</td></tr><tr><td>13. Distributionp & (q v r) = (p & q) v (p & r)p v (q & r) = (p v q) & (p v r)</td><td>14. Double Negationp = ~(~p)</td><td>15. Transpositionp --> q = ~q --> ~p</td></tr><tr><td>16. Material Implicationp --> q = ~p v q</td><td>17. Material Equivalencep <--> q = (p --> q) & (q --> p)p <--> q = (p & q) v (~p & ~q)</td><td>18. Exportation(p & q) --> r = p --> (q --> r)</td></tr><tr><td>19. Tautologyp = p & pp = p v p</td><td></td><td></td></tr></table>
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
<a href="https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson">Truth Tables</a>