Semântica Proposicional Categórica

8/18/2019 Semântica Proposicional Categórica

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•

•

•

•

•


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→


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16/122

SE T →

SE T →

SE T →


17/122

SE T →

SE T

L


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L


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→


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¬A = A → ⊥ ¬A ⊥


25/122

∀ ∃


26/122

C

C

C

C

D

CD

f : a −→ b f : a, b

f

a −→ b D(f ) = a CD(f ) = b

◦ f : a −→ b

g : b −→ c h : c −→ d (h ◦ g) ◦ f = h ◦ (g ◦ f ) :

a −→ d

a f

g◦f

b

g

h◦g

c

h d ;


27/122

id

a

b

id : Obj(C ) −→ M or(C )

ida : a −→ a idb : b −→ b

f : a −→ b

a f

f

b

idb

a f b

b a

ida

f

idb ◦ f = f ◦ ida = f

SE T →

C Obj(C ), Mor(C ), D, CD, id, ◦

SE T →

SE T

Obj(SE T )

Mor(SE T )


28/122

Mor(SET ) −→ Obj(SE T )

D(f ) = A

CD(f ) = B

f : A −→ B

◦ : M or(SE T )2 −→ M or(SE T )

id : Obj(SE T ) −→ M or(SE T )

idA : A −→ A

idB : B −→ B

◦

f : A −→ B g : B −→ C h : C −→ D

a ∈ A

((h ◦ g) ◦ f )(a) = (h ◦ g)(f (a)) = h(g(f (a))) = h(g ◦ f (a)) = (h ◦ (g ◦ f )))(a) .

f : A −→ B a ∈ A b ∈ B

f (a) = b

(f ◦ idA)(a) = f (idA(a)) = f (a) = b = idB(b) = idB(f (a)) = (idB ◦ f )(a) .

SE T

= {0, 1, 2, 3, . . .}

SE T

m

n

n + m

n ◦ m = n + m


29/122

n

n+m

m

m + n

m + (n + k) = (m + n) + k

n

m

k

n

m◦n

m

k◦m

k

m ◦ (n ◦ k) = (m ◦ n) ◦ k = m + (n + k) = (m + n) + k

id

N

0

m

m

0

n

n

0 + m = m n + 0 = n

→

SE T →

Obj(SE T →)

Mor(SE T →)

h, k


30/122

A h

f

C

g

B

k D ,

g ◦ h = k ◦ f

Mor(SET →) −→ Obj(SE T →)

f : i, j −→ h, k D(f ) = i, j CD(f ) = h, k

◦ : Mor(SE T →)2 −→ Mor(SE T →)

Mor(SET →)

i, j ◦ h, k i ◦ h, j ◦ k

id : Obj(SET →) −→ M or(SE T →) idA, idB

◦ SE T →

A h

f

C i

g

N

l

B k D j M ,

i, j h, k i, j ◦ h, k

A i◦h

f

N

l

B j◦k M .


31/122

i, j ◦ h, k = i ◦ h, j ◦ k .

(i, j ◦ h, k) ◦ f, g = (i ◦ h) ◦ f, ( j ◦ k) ◦ g =

i ◦ (h ◦ f ), j ◦ (k ◦ g) = i, j ◦ h ◦ f, k ◦ g =

i, j ◦ (h, k ◦ f, g)

SET →

A idA

f

A

f

B

idB

B ,

A

f −→ B idf = idA, idB

A idA

f

A h

f

C

g

B

idB B

k D ,

h, k ◦ idA, idB = h ◦ idA, k ◦ idB = h, k

idA, idB ◦ h, k = idA ◦ h,idB ◦ k = h, k


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POSET

Obj(POSET )

Mor(POSET )

{a, b} ∈ a ≤ b f : A −→

B

f (a) ≤ f (b)

Mor(POSET ) −→ Obj(POSET ) f :

A −→ B D(f ) = A CD(f ) = B

◦ : M or(POSET )2 −→ Mor(POSET )

id : Obj(POSET ) −→ Mor(POSET )

idA : A −→ A

idB : B −→ B

◦

M, ≤

N, ≤ f (≤) : M, ≤ −→ N, ≤

f (≤) : M −→ N ≤

(m, n) ≤

f ◦ ≤ = ≤ ◦ f

f

m ≤ n

f

(f (m), f (n))

≤ f (m ≤ n) = f (m) ≤ f (n)


33/122

f (≤) : M, ≤ −→ N, ≤

f (m ≤ n) = f (m) ≤ f (n)

f : A −→ B g : B −→ C h : C −→ D

a ≤ b A f (a) ≤ f (b) B f : A −→ B

f (a) ≤ f (b) f (a) ≤ f (b) = f (a ≤ b)

g

h

((h ◦ g) ◦ f )(a ≤ b) = (h ◦ g)(f (a ≤ b)) = h(g(f (a ≤ b))) = h(g ◦ f (a ≤ b)) =

(h ◦ (g ◦ f )))(a ≤ b) .

f : A −→ B {a, b} ∈ A a ≤ b

f (a) ≤

f (b)

f (a ≤

b) = f (a) ≤

f (b)

f ◦ idA(a ≤ b) = f (idA(a ≤ b) = f (a ≤ b) = f (a) ≤ f (b) = idB(f (a) ≤ f (b)) =

idB(f (a ≤ b) = idB ◦ f (a ≤ b)

M, ⊕, e

M, ⊕

⊕ : A × A −→ A

{a,b,c} ∈ A


34/122

a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c

e ∈ M ⊕

a ∈ M

a ⊕ e = e ⊕ a = a

{a, b} ∈ M a ⊕ b = b ⊕ a

, +

, · 0 1

, +, 0 , ·, 1

M, ⊕, a N,,e {∗}

f (⊕) : M, ⊕, a −→

N,,e f (⊕) : M −→ N

M

M

M

Mor(M)

◦ : M ×M −→ M idM

M f ,g,h ∈ Obj(M) f ◦(g◦h) = (f ◦g)◦h = f ⊕(g⊕h) = (f ⊕g)⊕h idM M idM = e

f ◦ idM = idM ◦ f = f = f ⊕ e = e ⊕ f


35/122

(m, n) ⊕

f ◦ ⊕ = ◦ f

f

m ⊕ n

f

(f (m), f (n))

f (m ⊕ n) = f (m) f (n)

f (⊕) : M, ⊕, e −→ N,,d

f (m ⊕ n) = f (m) f (n)

{∗}a

e

M f

N

f : {a} −→ {e}

f (a) = e

f ◦ ⊕ = ◦ f f (⊕) = (f )

m n

f (⊕(m, n)) = (f (m, n))

f (⊕(m, n)) = (f (m), f (n)) f (m ⊕ n) = f (m) f (n)

, +, 0 A, ×, 1 A =

{2n | n ∈ } f : −→ A f (x) = 2x

, +, 0 A, ×, 1

0

1

f (0) = 20 = 1

a

b

f (a+b) =

2a+b

f (a + b) = 2a+b = 2a × 2b = f (a) × f (b)


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MON

Obj(MON )

M, ⊕, e N,, d

Mor(MON )

h : M, ⊕, e −→ N,, d

h : M −→ N

Mor(MON ) −→ Obj(MON ) f : M −→ N

D(f ) = M, ⊕, e

CD(f ) = N,, d

f : M, ⊕, e −→ N,, d

◦ : M or(MON → M ON )2 −→ M or(MON → M ON )

id : Obj(MON ) −→ M or(MON )

idM : M, ⊕, e −→ M, ⊕, e

idN : N,, d −→ N,, d

◦

M f

g◦f

N

g

h◦g

G

h H


37/122

M, ⊕, a N,,e G, , o

H, , u

{m, n} ∈ M f : M −→ N f (m ⊕ n) =

f (m) f (n) f (a) = e

{ p, q } ∈ N g : N −→ G g( p q ) = g( p) g(q )

g(e) = o

{r, s} ∈ G

h : G −→ H

h(r s) = h(r) h(s)

h(o) = u

{m, n} ∈ M g ◦ f : M −→ G g ◦ f (m ⊕ n) =

g ◦ f (m) g ◦ f (n) g ◦ f (a) = o

{ p, q } ∈ N h ◦ g : N −→ H h ◦ g( p q ) =

h ◦ g( p) h ◦ g(q ) h ◦ g(e) = u

{m, n} ∈ M (h ◦ g) ◦ f = h ◦ (g ◦ f ) : M −→ H

h ◦ (g ◦ f ) (m ⊕ n) = h ◦ (g ◦ f )(m) h ◦ (g ◦ f )(n) h ◦ (g ◦ f )(a) = u

=⇒ (h ◦ g) ◦ f (a) = h ◦ (g ◦ f )(a)

(h ◦ g) ◦ f (a) = h(o)

(h ◦ g)(e) = h(o) h(o) = u

(h ◦ g)(e) = u (h ◦ g) ◦ f = h ◦ (g ◦ f )

⇐= h◦(g◦f )(a) = u h◦(g◦f )(a) = h◦g(e)

h◦(g◦f )(a) = (h◦g)◦f (a)

(h ◦ g) ◦ f = h ◦ (g ◦ f )

f : M −→ N

a ∈ M

e ∈ N

f (a) = e


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(f ◦ idM )(a) = f (idM (a)) = f (a) = e = idN (e) = idN (f (e)) = (idN ◦ f )(a) .

G, , e,−1

G,

: A × A −→ A

{a,b,c} ∈ A

a (b c) = (a b) c

e ∈ G

a ∈ G

a e = e a = a

e ∈ G a−1 ∈ G

a ∈ G

a a−1 = a−1 a = e

a, b ∈ G a b = b a

, + , ·

0

1

−a ∈

1a

∈ a = 0

, +, −a

, ·, 1a

a a a a = a

a (a a) = (a a) a = e a = a


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G, , e,−1

H,, d,−1

{∗}

G

H

g() : G, , e,−1 −→

H,, d,−1 g() : G −→ H

G

(a, b)

∗

g ◦ = ◦ g

g

a b

g

(g(a), g(b))

g(a b) = g(a) g(b)

g() : G, , e −→ H, , d

g(a b) = g(a) g(b)

{∗}e

d

G g H

g : {e} −→ {d}

g(e) = d

g(a) g−1 g(g(a))−1

a

g

g−1◦ g

g(a−1) = g(g(a))−1


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a a−1 = e a−1 a = e

g(a a−1) = g(e)

g(a) g(a−1) =

g(a−1) g(a) = g(e) = d g(a−1) = (g(a))−1

GRU

Obj(GRU )

G, , e,−1 H,,d,−1

Mor(GRU )

h : G, , e,−1 −→ H,,

d,−1 h : G −→ H

M or(GRU ) −→ Obj(GRU ) f : G −→ H D(f ) =

G, , e,−1 CD(f ) = H,, d,−1 f

f : G, , e,−1 −→ H,, d,−1

◦ : M or(GRU → GRU )2 −→ M or(GRU → GRU )

id : Obj(GRU ) −→ Mor(GRU )

idG : G, , e,−1 −→

G, , e,−1 idH : H,, d,−1 −→ H,, d,−1

◦


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M f

g◦f

N

g

h◦g

G

h H

M, ⊕, a,−1 N, , e ,−1

G, , o,−1 H, , u,−1

{m, n} ∈ M f : M −→ N f (m ⊕ n) =

f (m) f (n) f (a) = e

f (m−1) = f (f (m))−1

{ p, q } ∈ N g : N −→ G g( p q ) = g( p) g(q )

g(e) = o

g( p−1) = g(g( p))−1

{r, s} ∈ G h : G −→ H h(r s) = h(r) h(s)

h(o) = u, e h(r−1) = h(h(r))−1

{m, n} ∈ M g ◦ f : M −→ G g ◦ f (m ⊕ n) =

g ◦ f (m) g ◦ f (n) g ◦ f (a) = o g ◦ f (m−1) = g ◦ f (g ◦ f (m))−1

{ p, q } ∈ N h ◦ g : N −→ H h ◦ g( p q ) =

h ◦ g( p) h ◦ g(q ) h ◦ g(e) = u h ◦ g( p−1) = h ◦ g(h ◦ g(m))−1

{m, n} ∈ M (h ◦ g) ◦ f = h ◦ (g ◦ f ) : M −→ H

h ◦ (g ◦ f ) (m ⊕ n) = h ◦ (g ◦ f )(m) h ◦ (g ◦ f )(n) h ◦ (g ◦ f )(a) = u

h ◦ (g ◦ f )(m−1) = h ◦ (g ◦ f )(h ◦ (g ◦ f )(m))−1

M, ⊕, a,−1 m⊕ m−1 = a

f (m⊕ m−1) = f (a)

f (a) = e

(1)

f (m ⊕ m−1) = e

(5) h ◦ g(e) = u h ◦ g(f (m ⊕ m−1)) = (h ◦ g) ◦ f (m ⊕ m−1)) = u

u = h ◦(g ◦f )(a) m⊕ m−1 = a u = h ◦(g ◦f )(m⊕ m−1)

(h ◦ g) ◦ f (m ⊕ m−1) = h ◦ (g ◦ f )(m ⊕ m−1)

(h ◦ g) ◦ f = h ◦ (g ◦ f )


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f : M −→ N

a ∈ M e ∈ N f (m ⊕ m−1) = f (a) = e

(f ◦ idM )(a) = f (idM (a)) = f (a) = e = idN (e) = idN (f (e)) = (idN ◦ f )(a) .


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f : A −→ B x1 x1 ∈ A

x2 x2 ∈ A x1 = x2 f (x1) = f (x2)

f : A −→ B

g, h : C

A f ◦ g = f ◦ h g = h

f


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A f B

C

h

g

f ◦h=f ◦g

C h

g

A

f

Af

B ,

=⇒ f : A −→ B g, h : C A

f ◦ g = f ◦ h g = h

x ∈ C g h C f ◦ g = f ◦ h

(f ◦ g)(x) = (f ◦ h)(x) f (g(x)) = f (h(x)) f

g(x) = h(x)

g = h

⇐= {x1, x2} ∈ A f (x1) = f (x2) x1 = x2

g

h

C

g, h : C

A

h

g

z 1

z 2

C

A

g(z 1) = x1 h(z 2) = x2 {x1, x2} ∈ A

f ◦ g = f ◦ h g h

C

f (g(z 1)) = f (h(z 2)) f

g(z 1) = h(z 2) x1 = x2

f : A −→ B C

g, h : C A f ◦ g = f ◦ h g = h

f : A B

SE T

m + n = m + p

n = p


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f : A −→ B y

y ∈ B x x ∈ A f (x) = y

f : A −→ B

g, h : B C g ◦ f = h ◦ f g = h f

A f

g◦f =h◦f

B

g

h

C

A f

f

B

g

B

h C ,

=⇒ f : A −→ B g, h : B C

g ◦ f = h ◦ f g = h

x ∈ A g ◦ f = h ◦ f g(f (x)) = h(f (x)) y ∈ B f

x ∈ A f (x) = y g(f (x)) = h(f (x))

g(y) = h(y)

g = h

⇐= g ◦ f = h ◦ f x ∈ A y ∈ B f (x) = y

g = h

y ∈ B f x ∈ A f (x) = y

g(y) = g(f (x)) = (g ◦ f )(x) (g ◦ f )(x) = (h ◦ f )(x) = h(f (x)) = h(y)

g(y) = h(y)

g = h


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f : A −→ B C

g, h : B

C g ◦ f = h ◦ f g = h

f

f : A B

SE T

n + m = p + m

n = p

f : A −→ B

y

y ∈ B x x ∈ A f (x) = y

x1 = x2 f (x1) = f (x2) {x1, x2} ∈ A

g = g

g

g ◦ f = idA f ◦ g = idB

y ∈ B y = f (x) g(y) = g(f (x)) = (g ◦ f )(x) =

idA(x) = x g(y) = g(f (x)) = (g ◦ f )(x) = idA(x) = x g = g

g = idA ◦ g

= (g ◦ f ) ◦ g = g ◦ (f ◦ g ) = g ◦ idB = g g

f

f −1

f −1

f −1 ◦ f = idA

f ◦ f −1 = idB

f : A −→ B

g : B −→ A f ◦ g = idB g ◦ f = idA


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A

id

f

B

g

id

A f

idA

B

g

A f B

A B

g

idB

g ◦ f = idA f ◦ g = idB ,

=⇒ f : A −→ B

x

x ∈ A y ∈ B f (x) = y

y

y ∈ B

x ∈ A g(y) = x

g(y) = x

f (x) = y

y ∈ B (f ◦ g)(y) = f (g(y)) = f (x) = y = idB(y) f ◦ g = idB

(g ◦ f )(x) = g(f (x)) = g(y) = x = idA g ◦ f = idA

⇐= f f

f

f ◦ g = f ◦ h g = h

f −1 ◦ (f ◦ g) = f −1 ◦ (f ◦ h) f −1

(f −1 ◦f )◦g = (f −1 ◦f )◦h

idA ◦ g = idA ◦ h


48/122

f

g ◦ f = h ◦ f g = h

(g ◦ f ) ◦ f −1 = (h ◦ f ) ◦ f −1 f −1

g ◦(f ◦f −1) = h ◦(f ◦f −1)

g ◦ idB = h ◦ idB

f

f : A −→ B C

f −1 : B −→ A

f −1 ◦ f = idA f ◦ f −1 = idB

f : A ∼= B f : A B

C 0 Obj(C ) 0

C a ∈ Obj(C )

!0 : 0 −→ a


49/122

A

A =

⇐= A = −→ B

=⇒ A = B

B = {x,y,...} x = y f : A −→ B

A

A = {x1,...} f, g : A −→ B f (z ) = x

g(z ) = y

z ∈ A f = g x = y A

B B , , , −, 0, 1

,

−

0

1


50/122

{x,y,z } ∈

AxB {x, y} ∈

x y = y x

AxB {x, y} ∈

x y = y x

AxB {x,y,z } ∈

x (y z ) = (x y) (x z )

AxB {x,y,z } ∈

x (y z ) = (x y) (x z )

AxB {x, 0} ∈

x 0 = 0 x =

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