Nine Rules of Logic

Below are the nine rules of inference required to carry out the reasoning governed by sentential or propositional logic.  This is the most basic level of logic which deals with inferences based on sentential connectives like "if..., then," "or" and "and."

Rule #1:  modus ponens

1. P implies Q
2. P
_____________              
3. Q

Example:

1. If John studies hard, then he will get a good grade in logic.
2. John studies hard
_____________
3. He will get a good grade in logic.

Rule #2:  modus tollens

1. P implies Q
2. Not Q
_____________        
3. Not P

Example:

1. If Joan has been working out, then she can run the 5 K race.
2. She cannot run the 5 K race.
_____________
3. Joan has not been working out.

Rule #3:  Hypothetical Syllogism

1. P implies Q
2. Q implies R
_____________
3. P implies R

Example:

1. If it is Valentine's Day, Guillaume will invite Jeanette to dine at a fine restaurant.
2. If Guillaume will invite Jeanette to dine at a fine restaurant, then they will dine at L'Auberge St. Pierre.
_____________
3. If it is Valentine's Day, the Guillaume and Jeanette will dine at L'Auberge St. Pierre.

Rule #4:  Conjunction

1. P
2. Q
_____________
3. P & Q

Example:

1. Charity is playing the piano.
2. Jimmy is trying to play the piano.
_____________
3. Charity is playing the piano and Jimmy is trying to play the piano.

Rule #5:  Simplification

1. P & Q                                                  1. P & Q
_____________                                       _______________
2. P                                                          2. Q

Example:

1. Bill is bagging groceries, and James is stocking the shelves.
_____________
2. James is stocking the shelves.

Rule #6:  Absorption

1. P implies Q
_____________
2. P implies (P & Q)

Example:

1. If Allison goes shopping, she will buy a new top.
_____________
2. If Allison goes shopping, then she will go shopping and buy a new top.

Rule #7:  Addition

1. P
_____________
2. P or Q

Example:

1. Mallory will carefully work on decorating their new apartment.
_____________
2. Either Mallory will carefully work on decorating their new apartment, or she will allow it to degenerate into a pigsty.

Rule #8:  Disjunctive Syllogism

1. P or Q                                                1. P or Q
2. Not P                                                 2. Not Q
_____________                                     ________________
3. Q                                                       3. P

Example:

1. Either Mary will grade the exams herself or she will enlist Jason's aid.
2. She will not grade the exams herself.
_____________
3. She will enlist Jason's aid.

Rule #9:  Constructive Dilemma

1. (P implies Q) & (R implies S)
2. P or R
_____________
3. Q or S

Example;

1. If Jennifer buys dwarf fruit trees, she can make peach pies; and if she plants flowers, the yard will look colorful.
2. Either Jennifer buys dwarf fruit trees or she plants flowers.
_____________
3. Either Jennifer can make peace pies or the yard will look colorful.

Stand firm in Christ,
Chase

Footnote:
Craig, William Lane. Moreland, J.P. Philosophical Foundations for a Christian Worldview. Pages 30-39.