allzermalmer

Truth suffers from too much analysis

3 Value Logic

Posted by allzermalmer on May 10, 2013

I am going to use Polish Notation in expressing these truth tables of 3 value logic and 2 value logic. Lower case letters are variables: x, y, z, …
Capital Letters represent logical operators: N, A, K, E, C

Nx = ~x
Axy = x v y
Kxy = x & y
Cxy = x → y
Exy = x ↔ y

This notation is explicated in the text book Formal Logic by logican A.N. Prior in the late 1950’s to early 1960’s. The notationw as used by Polish logican Jan Lukasiewicz. Lukasiewicz was one of the first logicans to formally organize a three value logic. The logical matrix or logical matrices of both 2 value logic and 3 value logic are presented.

2 value logic uses 1 and 0.
3 value uses 1, 1/2, and 0.

1 stand for true.
1/2 stands for indeterminate.
0 stands for false.

I have put in bold those portions of truth tables in 3 value logic that do not have a similar truth table in 2 value.

Affirmation & Negation (x & Nx) : 2 value

  • (1) If x = 1 then Nx = 0.
    (2) If x = 0 then Nx = 1.

Affirmation & Negation (x & Nx) : 3 value

  • (1) If x = 1 then Nx = 0.
  • (2) If x = 1/2 then Nx = 1/2
  • (3) If x = 0 then Nx = 1

Conditional (Cxy) : 2 value

  • (1) If x = 1 and y = 1, then Cxy = 1
  • (2) If x = 1 and y = 0, then Cxy = 0
  • (3) If x = 0 and y = 1, then Cxy = 1
  • (4) If x =0 and y = 0, then Cxy = 1

Conditional (Cxy) : 3  value

  • (1) If x = 1 and y = 1, then Cxy = 1
  • (2) If x = 1 and y = 1/2, then Cxy = 1/2
  • (3) If x = 1 and y = 0, then Cxy = 0
  • (4) If x = 1/2 and y = 1, then Cxy = 1
  • (5) If x = 1/2 and y = 1/2, then Cxy = 1
  • (6) If x = 1/2 and y = 0, then Cxy = 1/2
  • (7) If x = 0 and y = 1, then Cxy = 1
  • (8) If x = 0 and y = 1/2, then Cxy = 1
  • (9) If x = 0 and y = 0, then Cxy = 1

Conjunction (Kxy) : 2 value

  • (1) x = 1 and y = 1, then Kxy = 1
  • (2) x = 1 and y = 0, then Kxy = 0
  • (3) x = 0 and y = 1, then Kxy = 0
  • (4) x = 0 and y = 0, then Kxy = 0

Conjunction (Kxy) : 3 value

  • (1) If x = 1 and y = 1, then Kxy = 1
  • (2) If x = 1 and y = 1/2, then Kxy = 1/2
  • (3) If x = 1 and y = 0, then Kxy = 0
  • (4) If x = 1/2 and y = 1, then Kxy = 1/2
  • (5) If x = 1/2 and y = 1/2, then Kxy = 1/2
  • (6) If x = 1/2 and y = 0, then Kxy = 1/2
  • (7) If x = 0 and y = 1, then Kxy = 0
  • (8) If x = 0 and y = 1/2, then Kxy = 0
  • (9) If x = 0 and y = 0, then Kxy = 0

Disjunction (Axy) : 2 value

  • (1) If x = 1 and y = 1, then Axy = 1
  • (2) If x = 1 and y = 0, then Axy = 1
  • (3) If x = 0 and y = 1, then Axy = 1
  • (4) If x = 0 and y = 0, then Axy = 0

Disjunction (Axy) : 3 value

  • (1) If x = 1 and y = 1, then Axy = 1
  • (2) If x = 1 and y = 1/2, then Axy = 1
  • (3) If x = 1 and y = 0, then Axy = 0
  • (4) If x = 1/2 and y = 1, then Axy = 1
  • (5) If x = 1/2 and y = 1/2, then Axy = 1/2
  • (6) If x = 1/2 and y = 0, then Axy = 1/2
  • (7) If x = 0 and y = 1, then Axy = 1
  • (8) If x = 0 and y = 1/2, then Axy = 1/2
  • (9) If x = 0 and y = 0, then Axy = 0

Biconditional (Exy) : 2 value

  • (1) If x = 1 and y = 1, then Exy = 1
  • (2) If x = 1 and y = 0, then Exy = 0
  • (3) If x = 0 and y = 1, then Exy = 0
  • (4) If x = 0 and y = 0, then Exy = 1

Biconditional (Exy) : 3 value

  • (1) If x = 1 and y = 1, then Exy = 1
    (2) If x = 1 and y = 1/2, then Exy = 1/2
    (3) If x = 1 and y = 0, then Exy = 0
    (4) If x = 1/2 and y = 1, then Exy = 1/2
    (5) If x = 1/2 and y = 1/2, then Exy = 1
    (6) If x = 1/2 and y = 0, then Exy = 1/2
    (7) If x = 0 and y = 1, then Exy = 0
    (8) If x = 0 and y = 1/2, then Exy = 1/2
    (9) If x = 0 and y = 0, then Exy = 1
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