Supervenience, Entailment, and Impossible Objects

              
  
      



   
   
 


 
                      


        


              
  
      


  


 
  




            



       




      


        


              
  
      


     
     
            
 


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⇔ ∀x∀y(∀F (F ∈ A ⊃ (F (x) ≡ F (y))) ⊃ ∀G(G ∈ B ⊃ (G(x) ≡ G(y))))

4  x ≈A y    ∀F (F ∈ A ⊃ (F (x) ≡ F (y))!  (    %    %        (          (

⇔ ∀x∀y(x ≈A y ⊃ x ≈B y)

    


        


              
  
      


  



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⇔ ∀x∀y(∀F (F ∈ A ⊃ (F (x) ≡ F (y))) ⊃ ∀G(G ∈ B ⊃ (G(x) ≡ G(y))))

4  x ≈A y    ∀F (F ∈ A ⊃ (F (x) ≡ F (y))!  (    %    % (

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⇔ ∀x∀y(x ≈A y ⊃ x ≈B y)

    


        


              
  
      


 


 
 
  


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/ (#       #   !        ( /, '     3      !        +(-        + -!                    //  '     4  B = {P ∧ Q}! A = {P, Q}!      %    P (a), Q(b), ¬P (b), ¬Q(a) *   a   b  (#    %     #         


        


              
  
      


 


 
 
  


/ (#       #   !        ( 7    (!   (#         #    7, 8     B = {¬P }    A = {P }! % ¬P     P 

    


        


              
  
      





   


       . 9    +   

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  4             9  
       C1, C2 , ...Cn  *      A % %
 
    '     C1 ∧ C2 ∧ ... ∧ Cn 

4: Pi ∣¬Pi∣Pi ∧ Pi ∣Pi ∨ Pi      


        


              
  
      



   


*           

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$"  % (    & 
      
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!   1 : {α∣α ⊆ L}! %  +,- ∀Pi ∈  (Pi ∈ α  ¬Pi ∈ α) +/- ∀Pi ∈  (Pi ∉ α  ¬Pi ∉ α)     


        


              
  
      



   


9       d ∀kn ∈ N, d(kn ) ∈ W  ∣A(k)∣d = t(f ) A ∶= P ∣P (k)∣d = t ⇔ P ∈ d(k)" A ∶= ¬B ∣¬B∣d = t ⇔ ∣B∣d = f "

∣P (k)∣d = f ⇔ ¬P ∈ d(k) ∣¬B∣d = f ⇔ ∣B∣d = t

A ∶= B ∧ C ∣B ∧ C∣d = t ⇔ ∣B∣d = t   ∣B∣d = t" ∣B ∧ C∣d = f ⇔ ∣B∣d = f  ∣B∣d = f

    


        


              
  
      



   




A ⊧s B ⇔ ∀d ∀x ∀y (∣A(x)∣d = ∣A(y)∣d ⇒ ∣B(x)∣d = ∣B(y)∣d )

∀A A ⊧s B ∨ ¬B

+,+/-

∀Pi ∈ ∀Pi ∈

 (Pi ∈ α  ¬Pi ∈ α)  (Pi ∉ α  ¬Pi ∉ α)

    


        


              
  
      



   


 A ⊧s B ⇔ ∀d ∀x ∀y (∣A(x)∣d = ∣A(y)∣d ⇒ ∣B(x)∣d = ∣B(y)∣d ) *   ∀A A ⊧s B ∧ ¬B +,+/-

∀Pi ∈ ∀Pi ∈

 (Pi ∈ α  ¬Pi ∈ α)  (Pi ∉ α  ¬Pi ∉ α)

    


        


              
  
      



   


 A ⊧s B ⇔ ∀d ∀x ∀y (∣A(x)∣d = ∣A(y)∣d ⇒ ∣B(x)∣d = ∣B(y)∣d ) *   ∀A A ⊧s B ∧ ¬B ! 2 '  +,- ∀Pi ∈  +/- ∀Pi ∈ 

(Pi ∈ α  ¬Pi ∈ α) (Pi ∉ α  ¬Pi ∉ α)

    


        


              
  
      


 
  
  

  
  )  
 
  +;   -  )  
  + 

-   3   
 A/α

< A    α


A/α < A    α

    


        


              
  
      


 
  
  

  + -
  )  
 
  +;   -  )  
  + 

-   3   
 p/α ⇔ p ∈ α
p/α ⇔ ¬p ∈ α ¬A/α ⇔
A/α
¬A/α ⇔ A/α A ∧ B/α ⇔ A/α   B/α
A ∧ B/α ⇔
A/α 
B/α



A ⊧r B ⇔ ∀α( A/α ⇒

    

B/α )


        


              
  
      


 
  
  

          
   #
   A ⊧se B⇔ ∀α∀β((( A/α   A/β )  (
A/α  
A/β )) ⇒ (( B/α   B/β )  (
B/α  
B/β )))        2   ∀α(∣A(k)∣d = t   d(k) = α ⇔ A/α ∀α(∣A(k)∣d = f   d(k) = α ⇔
A/α    2 A ⊧r B ⇔ A ⊧se     


        


              
  
      


 
  
  

          
   #
   A ⊧se B⇔ ∀α∀β((( A/α   A/β )  (
A/α  
A/β )) ⇒ (( B/α   B/β )  (
B/α  
B/β )))        2   ∀α(∣A(k)∣d = t   d(k) = α ⇔ A/α ∀α(∣A(k)∣d = f   d(k) = α ⇔
A/α    2 A ⊧s B ⇔ A ⊧se B      


        


              
  
      


 
  
  

          
   #
   A ⊧se B⇔ ∀α∀β((( A/α   A/β )  (
A/α  
A/β )) ⇒ (( B/α   B/β )  (
B/α  
B/β )))        2   ∀α(∣A(k)∣d = t   d(k) = α ⇔ A/α ∀α(∣A(k)∣d = f   d(k) = α ⇔
A/α    2 A ⊧s B ⇔ A ⊧se B      


        


              
  
      


    

A ⊧se B ⇔ ∀α∀β ((( A/α   A/β )  (
A/α  
A/β )) ⇒ (( B/α   B/β )  (
B/α  
B/β ))) A ⊧se B ⇔ ∀α∀β(( A/α

 

A/β )⇒ (( B/α   B/β )  (
B/α  
B/β )))

∀α∀β((
A/α

 
A/β ) ⇒ (( B/α   B/β )  (
B/α  
B/β )))     

 


        


              
  
      


    



A ⊧pse B ⇔ ∀α∀β(( A/α   A/β ) ⇒ (( B/α   B/β )  (
B/α

 
B/β )))

8

A ⊧nse B ⇔ ∀α∀β((
A/α  
A/β ) ⇒ (( B/α   B/β )  (
B/α

 
B/β )))

A ⊧se B ⇔ A ⊧pse B

  A ⊧nse B

    


        


              
  
      


    



A ⊧pse B ⇔ ∀α∀β(( A/α   A/β ) ⇒ (( B/α   B/β )  (
B/α

 
B/β )))

8

A ⊧nse B ⇔ ∀α∀β((
A/α  
A/β ) ⇒ (( B/α   B/β )  (
B/α

 
B/β )))



A ⊧se B ⇔ A ⊧pse B

  A ⊧nse B

    


        


              
  
      






A ⊧pse B ⇔ ∀α∀β(( A/α   A/β ) ⇒ (( B/α   B/β )  (
B/α

   )

A ⊧pse B ⇔ A ⊧r B

    

 
B/β )))

 A ⊧r ¬B 


        


              
  
      




8

A ⊧nse B ⇔ ∀α∀β((
A/α  
A/β ) ⇒ (( B/α   B/β )  (
B/α

   ))

A ⊧nse B ⇔ ¬A ⊧r B

    

 
B/β )))

 ¬A ⊧r ¬B 


        


              
  
      


 
 


   )    ))

A ⊧pse B ⇔ A ⊧r B

 A ⊧r ¬B 

A ⊧nse B ⇔ ¬A ⊧r B

 ¬A ⊧r ¬B 

   

A ⊧se B ⇔ (A ⊧r B   B ⊧r A)  (¬A ⊧r B   B ⊧r ¬A)  (A ⊧r B   ¬A ⊧r B)  (B ⊧r ¬A   B ⊧r A)

    


        


              
  
      


    A ⊧se B ⇔ ⇔ ⇔ ⇔

+,+/+7+=-

(A ∣∣r B) (¬A ∣∣r B) (A ∨ ¬A ⊧r B) (B ⊧r A ∧ ¬A)

  



         '  +'  -!  ! 
   

    


        


              
  
      


    A ⊧se B ⇔ ⇔ ⇔ ⇔

+,+/+7+=-

(A ∣∣r B) (¬A ∣∣r B) (A ∨ ¬A ⊧r B) (B ⊧r A ∧ ¬A)

  



         '  +'  -!  ! 
   

    


        


              
  
      


    A ⊧se B ⇔ ⇔ ⇔ ⇔

+,- (A ∣∣r B)  +/- (¬A ∣∣r B)  +7- (A ∨ ¬A ⊧r B)  +=- (B ⊧r A ∧ ¬A)

  %! A ⊧se ¬A! ¬A ⊧se A

    


        


              
  
      


    A ⊧se B ⇔ ⇔ ⇔ ⇔

+,+/+7+=-

(A ∣∣r B) (¬A ∣∣r B) (A ∨ ¬A ⊧r B) (B ⊧r A ∧ ¬A)

  

    ∧   ∨)!     %      
9   C ∧ (A ∧ ¬A) ⊧se A ∧ ¬A

    


        


              
  
      


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