Supervenience, Entailment, and Impossible Objects

! ! " # " $ ! ! !

% & ! '

%

! ! " # " $ !

% & ! '

%

( '

% & % # % & % (# ) *

#& % (#& +,- % . +/- +# - 0

( '

% & % # % & % (# ) *

#& % (#& +,- % . +/- +# - 0

) ▷ 1 2 ▷ 1 #( ▷ 1 $ ▷ $ ! 1 ! ) ▷ 1 ▷ 1 #3

(

(

⇔ ∀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)

(

(

⇔ ∀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)

3 %

!" # "

, . 5 ! ! # " / . 6 " 7 6 .

/ (# # ! ( /, ' 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 +

-

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

4: Pi ∣¬Pi∣Pi ∧ Pi ∣Pi ∨ Pi

*

$" % & "

''

$" % ( & " ( ) * + *,'" 4 8 # ! 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


B/β )))

8


B/β )))

A ⊧se B ⇔ A ⊧pse B

A ⊧nse B


)

A ⊧pse B ⇔ A ⊧r B

B/β )))

A ⊧r ¬B

8


))

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

-.! /012