Set

1. P (P Q) ----- addition
2. (P Q) P ----- simplification
3. [P (P Q] Q ----- modus ponens
4. [(P Q) Q] P ----- modus tollens
5. [ P (P Q] Q ----- disjunctive syllogism
6. [(P Q) (Q R)] (P R) ----- hypothetical syllogism
7. (P Q) [(Q R) (P R)]
8. [(P Q) (R S)] [(P R) (Q S)]
9. [(P Q) (Q R)] (P R)

1. P (P P) ----- idempotence of
2. P (P P) ----- idempotence of
3. (P Q) (Q P) ----- commutativity of
4. (P Q) (Q P) ----- commutativity of
5. [(P Q) R] [P (Q R)] ----- associativity of
6. [(P Q) R] [P (Q R)] ----- associativity of
7. (P Q) ( P Q) ----- DeMorgan's Law
8. (P Q) ( P Q) ----- DeMorgan's Law
9. [P (Q R] [(P Q) (P R)] ----- distributivity of over
10. [P (Q R] [(P Q) (P R)] ----- distributivity of over
11. (P True) True
12. (P False) False
13. (P False) P
14. (P True) P
15. (P P) True
16. (P P) False
17. P ( P) ----- double negation
18. (P Q) ( P Q) ----- implication
19. (P Q) [(P Q) (Q P)]----- equivalence
20. [(P Q) R] [P (Q R)] ----- exportation
21. [(P Q) (P Q)] P ----- absurdity
22. (P Q) (Q P) ----- contrapositive

1. Universal Instantiation:
   
   x P(x)
   -------
   P(c)

   where c is some arbitrary element of the universe.


Go to Universal Instantiation for further explanations and examples.


2. Universal Generalization:
   
   P(c)
   ----------
   x P(x)

   where P(c) holds for every element c of the universe of discourse.


Go to Universal Generalization for further explanations and examples.


3. Existential Instantiation:
   
   x P(x)
   -------
   P(c)

   where c is some element of the universe of discourse. It is not arbitrary but must be one for which P(c ) is true.


Go to Existential Instantiation for further explanations and examples.


4. Existential Generalization:
   
   P(c)
   ----------
   x P(x)

   where c is an element of the universe.


Go to Existential Generalization for further explanations and examples.

1. x [ P(x) Q(x) ] [ x P(x) x Q(x) ]
2. [ x P(x) x Q(x) ] x [ P(x) Q(x) ]
3. x [ P(x) Q(x) ] [ x P(x) x Q(x) ]
4. x [ P(x) Q(x) ] [ x P(x) x Q(x) ]