P -> Q

P

___________

 Q

Modus Tollens

P  ->  Q

~Q

__________

~P 

Hypothetical Syllogism 

P -> Q

Q -> R

__________

P -> R 

Disjunctive Syllogism 

P V Q

~P

_____

Q 

Constructive Dilemma

(P -> Q) & ( R -> S)

P V R

______________________

Q v S 

DeMorgans Rule 

~( P & Q).........................(~P v ~Q )

     ~( P V Q )........................( ~P & ~ Q ) 

Communtativity 

(P v Q)...... (Q V P)

(P & Q).......(Q & P) 

Associativity 

[P V (Q V R)] ......... [( P V Q) V R]

[P & (Q & R)]......... [( P & Q) & R] 

Distribution 

[P & ( Q V R)]........ [(P & Q) V (P & R)] 

same if & and v switched 

Transposition 

(P -> Q) ...... (~Q -> ~P) 

Material Implication 

P -> Q ....................... ~P v Q

Material Equivalence 

P=Q....[(P -> Q) & (Q->P)]

[(P & Q) v (~Q & ~P)] 

Exportation 

[(P&Q)-> R]......[P -> (R -> R)] 

Tautology 

PvP

P&P

=P