YES

Problem:
 not(x) -> xor(x,true())
 implies(x,y) -> xor(and(x,y),xor(x,true()))
 or(x,y) -> xor(and(x,y),xor(x,y))
 =(x,y) -> xor(x,xor(y,true()))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [and](x0, x1) = x0 + x1,
   
   [true] = 0,
   
   [implies](x0, x1) = 6x0 + 2x1,
   
   [or](x0, x1) = 7x0 + 6x1 + 1,
   
   [not](x0) = 2x0,
   
   [=](x0, x1) = 4x0 + 4x1,
   
   [xor](x0, x1) = 2x0 + 2x1
  orientation:
   not(x) = 2x >= 2x = xor(x,true())
   
   implies(x,y) = 6x + 2y >= 6x + 2y = xor(and(x,y),xor(x,true()))
   
   or(x,y) = 7x + 6y + 1 >= 6x + 6y = xor(and(x,y),xor(x,y))
   
   =(x,y) = 4x + 4y >= 2x + 4y = xor(x,xor(y,true()))
  problem:
   not(x) -> xor(x,true())
   implies(x,y) -> xor(and(x,y),xor(x,true()))
   =(x,y) -> xor(x,xor(y,true()))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [and](x0, x1) = 4x0 + 4x1,
    
    [true] = 0,
    
    [implies](x0, x1) = 5x0 + 4x1,
    
    [not](x0) = x0,
    
    [=](x0, x1) = x0 + x1 + 1,
    
    [xor](x0, x1) = x0 + x1
   orientation:
    not(x) = x >= x = xor(x,true())
    
    implies(x,y) = 5x + 4y >= 5x + 4y = xor(and(x,y),xor(x,true()))
    
    =(x,y) = x + y + 1 >= x + y = xor(x,xor(y,true()))
   problem:
    not(x) -> xor(x,true())
    implies(x,y) -> xor(and(x,y),xor(x,true()))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [and](x0, x1) = 2x0 + x1,
     
     [true] = 0,
     
     [implies](x0, x1) = 6x0 + x1 + 1,
     
     [not](x0) = x0,
     
     [xor](x0, x1) = x0 + 4x1
    orientation:
     not(x) = x >= x = xor(x,true())
     
     implies(x,y) = 6x + y + 1 >= 6x + y = xor(and(x,y),xor(x,true()))
    problem:
     not(x) -> xor(x,true())
    Matrix Interpretation Processor: dim=3
     
     interpretation:
               [0]
      [true] = [1]
               [0],
      
                  [1 0 0]     [1]
      [not](x0) = [0 0 0]x0 + [0]
                  [0 0 0]     [1],
      
                      [1 0 0]     [1 0 0]  
      [xor](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                      [0 0 0]     [0 1 0]  
     orientation:
               [1 0 0]    [1]    [1 0 0]    [0]                
      not(x) = [0 0 0]x + [0] >= [0 0 0]x + [0] = xor(x,true())
               [0 0 0]    [1]    [0 0 0]    [1]                
     problem:
      
     Qed