Whose Logic is Three-Valued Logic?

(1)

Whose Logic is Three-Valued Logic?

U

U / ∈ U ι

D ( ι ) = U ∪ { U } Θ( ι ) = U

o

I = W × T W T

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(2)

3 = {0 , 1 , 2} Θ( o ) = 2 p

w t

ι o

o ( ι )

f : D ( o ) −→ D ( ι ) f ( U ) = 2 2

σ ( P ) : D ( ι ) −→ 3

| A |

vi

= A i = w, t ∈

W × T v i d ( i ) ⊆ D ( ι )

i w

t x ι v ( x )

D ( ι ) v ( x ) ∈ / d ( i ) | x |

vi

= Θ( ι )

ι w t

w t

F o ( ι )

I F F |I F |

vi

= u

_i

{ u ∈ d ( i ) : | F |

vi

( u ) = 1} = { u

_i

}

|I F |

vi

= Θ( ι ) F w

t |I F |

vi

F

|I F |

vi

o A B

α

| A = B |

vi

=

⎧ ⎪

⎨

⎪ ⎩

2 | A |

vi

= Θ( α ) | B |

vi

= Θ( α ) 1 | A |

vi

= | B |

vi

= Θ( α )

0

2 =

λ =

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(3)

p ∼ p 1 0 0 1 2 2

p ∧ q 1 0 2 1 1 0 2 0 0 0 2 2 2 2 2

p ∨ q 1 0 2 1 1 1 2 0 1 0 2 2 2 2 2

= o

p ⊃ q 1 0 2 1 1 0 2 0 1 1 2 2 2 2 2

p ≡ q 1 0 2 1 1 0 2 0 0 1 2 2 2 2 2

p ¬ p 1 0 0 1 2 2

p & q 1 0 2 1 1 0 2 0 0 0 0 2 2 0 2

p ∨ q 1 0 2 1 1 1 1 0 1 0 2 2 1 2 2 p ⊃ q 1 0 2

1 1 0 2 0 1 1 1 2 1 2 2

p ≡ q 1 0 2 1 1 0 2 0 0 1 2 2 2 2 2

p p ∨ q 1

q

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(4)

p p

p p p p p p

p 10

¹⁰¹⁰¹⁰

π − 3

A α B β |A|

vi

= Θ(α) ⇒

|B|

vi

= Θ(β)

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(5)

p

¬ p p

p p

¬ p

¬ p ¬¬ p

π

n 7

ⁿ

F

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(6)

F

n n

F

n F ( n )

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(7)

π

F

F ( n ) n

F ( n )

n F n

F n

F (0)

F (1) F ( n )

F (0)

F(1) F (2)

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(8)

p

n F ( n )

n

^∗

∗

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