=============================================================================
notation: '==' is logical equivalence; '->' is implication; '^' is AND; 
'v' is OR; '~' is NOT; '<->' is biconditional; XOR is exclusive or
Precedence Order (highest to lowest): () NOT AND OR -> <-> 
=============================================================================
 identity laws         p ^ T == p              p v F == p
-----------------------------------------------------------------------------
 domination laws       p v T = T               p ^ F = F
-----------------------------------------------------------------------------
 negation laws         p v ~p == T             p ^ ~p == F 
-----------------------------------------------------------------------------
 idempotent laws       p v p == p              p ^ p == p 
-----------------------------------------------------------------------------
 commutativity       (p ^ q) == (q ^ p)        (p v q) == (q v p) 
------------------------------------------------------------------------------
 associativity  (p ^ q) ^ r == p ^ (q ^ r)     (p v q) v r == p v (q v r)
------------------------------------------------------------------------------
 double negation elimination     ~(~p) == p 
------------------------------------------------------------------------------
 contraposition                  p -> q == ~q -> ~p 
------------------------------------------------------------------------------
 implication elimination         p -> q == ~p v q
-------------------------------------------------------------------------------
 biconditional elimination       p <-> q == (p -> q) ^ (q -> p) 
-------------------------------------------------------------------------------
 De Morgan    ~( p ^ q) == ~p v ~q       ~( p v q) == ~p ^ ~ q
-------------------------------------------------------------------------------
 distributivity of ^ over v and v over ^
   p ^ (q v r) == (p ^ q) v ( p ^ r)
   p v (q ^ r) == (p v q) ^ (p v r) 
-------------------------------------------------------------------------------
 absorption                     p v (p ^ q ) == p           p ^ (p v q) == p
-------------------------------------------------------------------------------
            Rules of Inference for Propositional and Predicate Logic
 ':' is therefore; 'v' is "OR"; '~' is NOT; '^' is AND; '->' is implication
-------------------------------------------------------------------------------
   p  
   q               Conjunction (Conj.) 	
-----------
: p ^ q
  	
   p -> q          Modus Ponens (M.P.)  
   p  
----------
:  q
  	
   p -> q          Modus Tollens (M.T.)  
   ~q  
----------
: ~p
  	
  p -> q           Hypothetical Syllogism (H.S.) 	
  q -> r  
----------
: p -> r
  	
   p v q           Disjunctive Syllogism (D.S.) 	
   ~ p  
----------
:  q
  	
 (p -> q) ^ (r -> s)  Constructive Dilemma (C.D.) 	
  p v r  
---------------------
: q v s
  	
   p -> q            Absorption (Abs.) 	
---------------
:  p -> (p ^ q)  
 
   p ^ q             Simplification (Simp.) 	
-----------
: p
  	
   p                 Addition (Add.) 	
-----------
: p v q
  	
    p v q            Resolution (Res.)
   ~p v r
-----------
:   q v r

Fallacy of affirming the consequence (or conclusion)    
p -> q
q
-------
p        

Fallacy of denying the antecedent (or hypothesis) 
p -> q
~p
-------
~q        
                Rules of Inference for Quantified Statements 
-------------------------------------------------------------------------------
Given Universe of Discourse = U and c member of U
Universal Instantiation  
   Ax P(x)
----------- 
:  P(c) for some element c

Universal Generalization 
   P(c) for an arbitrary c
----------
:  Ax P(x) 

Existential Instantiation
   Ex P(x)
-----------
:  P(c) for some element c

Existential Generalization 
   P(c) for an arbitrary c
----------
:  Ex P(x)