Paradox

In an logical Space S(A) based on the Axiom Set A if we find a statement x s.t.  x  =  ¬  x
or  x  ⇔ ¬  x then  x is called a paradox.

There are so many paradoxes :-

1. Self-Reference : x =  ¬ x s.t "This statement is false." :)
2. Vicious circularity, or infinite regress : 
         "What happens when Pinocchio says, 'My nose will grow now'?" [from Wikipedia ]

3. Russell paradox: S = {x| x ∉ S} is there any element in S . 

4. Proof that Almighty doesn't exist : Let A =Almighty exist and it has all power .
                                                           So it must have power P defined as who have P he will be 
                                                           destroyed .
                                                           Thus A is a paradox.

Principal Of Exclusion : We should define Our Logical Space / Language s.t. Paradox does not exist. So if you have a Language  where some statement can be a Paradox then exclude such statements from definition and you have to reduce your Language space such that no other statement can conclude by logical consequence in a paradox . 

A language with a paradox is not a logical language and it is something which can't lead us to truth . So we must see some logical language . 


Now one question is there can we describe "Paradox" in a logical language . :) 
Can we describe the illogical language in such logical language say L .
If we bind our self by logic can we tell that there is a out of logic universe or universe of error . 

Abstract Algebra Glimpse

As we know magma , semigroup , Monoid , Group and Ring , field  from definition let's see some good features and examples of them .

Group example : 1. (Zn,+) Zn = {0,1,...(n-1)} , '+' is addition modulo n
             2. (Un,.)  Un = {0< x < n | x ⊥ n ( relatively prime )} , '.' is multiplication modulo n
           
An algebraic structure G = (A,P)| Γ have substructure H = (B,P)| Γ  ⇔  B ⊆ A ∧ Γ(B,P) = true .
i.e according to above definition of G,H : (H ≤ G)  ⇔  (B ⊆ A ∧ Γ(B,P) = true)  

If (G,+) , (H,+) are groups then (H ≤ G) ⇔  ∀ x,y ∈ H x+(-y)  ∈ H

left and right cosets of H ≤ G are xH,Hx respectively where x   ∈ G 

1. Two left/right cosets are either equal or disjoint .
2. If xH = Hx ( H,x are as previous ) then H ◅ G (Normal Subgroup)

Algebra Glimpse

An algebra is (Ω , θ , Γ )  defined as :-
1. Ω  :  List of Symbol/Variable
2. θ   :  List of Operation/mapping from E1,E2,... ⊆ Ω* to F1,F2,... ⊆ Ω* [ '*' is Kleene closure  
            A* = ( ∅  ∪ A  ∪ (A^2) ... )] . 
3. Γ   : List of properties that algebra follows

Sometimes we express an algebra  by (Ω , θ) then mention the Γ  i.e algebra A =  (Ω , θ)| Γ  
Now in abstract algebra deals with some mathematical models and definitions  :-

Definition : A definition is a statement describing an object exactly as it is . 
Set : An order-independent collection of well defined distinct objects .
Multiset : An order-independent collection of well defined [duplication is possible ] objects.
Tuple : An ordered list of elements .

Set operations :  ∈ , ⊆ , ∪ , ∩ , ()' [ Complement ] ,2^() [Power Set]
                         
1. x ∈ A means x is an element of set A
2. A ⊆ B means A is subset of B
3. A ∪ B = {x | x ∈ A ∨ x ∈ B }
4. A ∩ B = {x | x ∈ A ∧ x ∈ B }
5. A' = {x | x ∉ A}
6. 2^A = {x | x ⊆ A}

Product of Sets : A × B = {(x,y)| x ∈ A ∧ y ∈ B } # of element = |A|.|B|
Relation : R  ⊆ A × B # of relation 2^|A|.|B|
Mapping :  φ | A→ B  is representing a map from A to B i.e related elements of A to elements of B
Function :  φ | A→ B  is representing a function from A to B where for one element x of A function 
                  φ map to y i.e. only one elements of B . 
                 
    
Category of functions :   
Let a function  φ | A→ B        
1. one-one : ∀ x ∈ A φ(x) = φ(y)  ⇒ x = y # of one-one functions |B|!/(|B| - |A|)!
2. onto :  ∀ y ∈ B y = φ(x) for some x ∈ A # of onto functions |B|^|A|
3. bijection : one-one onto function # of bijections |A|! = |B|!

Group Theory : 
1. magma (set G, operator +) : ∀ x,y ∈ G  (x+y)  ∈ G 
2. semigroup(magma G, operator +): ∀ x,y,z ∈ G  (x+y)+z = x+ (y+z)  
3. Monoid ( semigroup G, operator +): ∀ x ∈ G ∃ 0 ∈ G s.t x + 0 = 0 + x = x (o is the identity)
4. Group (semigroup G, operator +): ∀ x ∈ G ∃ (-x) ∈ G s.t x + (-x) = (-x) + x =  0 ( (-x) is the inverse 
                                                                                                                                      of x )
5. Ring (G ,+ , .) :  1. (G,+) is commutative Group and  (G , .) is semigroup 
                               2. ∀ x,y,z ∈ G (x+y)z = xz + yz  and
                                                        z(x+y) = zx + zy   (distributive law)

6. Field (G, + , .) : 1. (G , +) and (G - {0} , .) are abelian Groups
                              2. (G,+ , .) is a ring   

Boolean Algebra ( B , +, . ,()') :
1. (B,+) and (B, .) are commutative Monoids with  identity 0,1 respectively 
2. x + x' = 1 and x.x' = 0 
3. (B , + , .) and (B , . , +) are identity rings . 

There are some theorems on boolean algebra :
1. x + x = x.x = x 
2. x + 1 = 1 and x.0 = 0
3. (x')' = x 
4. x + xy = x(x+y) = x (Absorption Law)
5. (xy)' = x' + y' and (x+y)' = x'y' (De'Morgan's  Law)
6. Duality of '+' and '.' : If we change '+' and '.' in a valid boolean identity the new identity will also be
    valid.


S is a Set then (2^S, ∪ , ∩ , ()')  ;  P is the set of logical atoms then (P,∨,∧ ,¬  ) are all boolean algebra .

One more thing we can have such  Boolean Algebra  say B as Kleene Algebra  s.t we can  find a partial order in B and arrange elements of B as (1, x1, x2 , ... ) where 1 ≤ x1 ≤ x2 ... where 
x ≤ y  ⇔ x + y = y