Monday, November 8, 2010

Fuzzy Logic

Dictionary
[A] is the truth value of A
|[A]| is the absolute value of the truth value of A

DEFINING SENTENCE CONNECTIVES

NEGATION
·        If A is true (1) then ~A is false (0).
·        The sum of the truth values of A and ~A should always be 1.
·        [~A] = 1 – [A]

NEGATION
A       |         ~A
1       |         0
0.75  |         0.25
0.5    |         0.5
0.25  |         0.75
0       |         1

CONJUNCTION
·        a conjunction is as true as the least true of the conjuncts
·        a conjunction is false if either conjunct are false
·        a conjunction is completely true if both conjuncts are completely true
·        [A & B] = Min([A], [B])

CONJUNCTION
B
A & B         1       .75    .5      .25    0

A       1                 1       .75    .5      .25    0
.75              .75    .75    .5      .25    0
.5                .5      .5      .5      .25    0
.25              .25    .25    .25    .25    0
0                 0       0       0       0       0

DISJUNCTION
·        A disjunct is as good as its strongest disjunct.
·        [A v B] = Max([A], [B])

DISJUNCTION
B
A v B                   1       .75    .5      .25    0

A       1                 1       1       1       1       1
.75              1       .75    .75    .75    .75
.5                1       .75    .5      .5      .5
.25              1       .75    .5      .25    .25
0                 1       .75    .5      .25    0

CONDITIONAL
·    If the consequent is at least as true as the antecedent, then the conditional is absolutely true (has a truth value of 1).
·    If the consequent is not as true as the antecedent, then the conditional is less than absolutely true (truth value less than 1).
·    If the consequent is less true than the antecedent, the amount that it misses the complete truth by is the absolute value of the difference between the truth of the antecedent and the consequent.
·        If [A] <= [B]                                       [A à B] = 1
·        If [A] > [B]                                         [A à B] = 1 – |[A] – [B]|

CONDITIONAL
B
A à B        1       .75    .5      .25    0

A       1                 1       .75    .5      .25    0
.75              1       1       .75    .5      .25
.5                1       1       1       .75    .5
.25              1       1       1       1       .75
0                 1       1       1       1       1

Fuzzy Logic Results

The principle of noncontradiction is absolutely true whenever the truth of the sentence is absolutely true or false.  It can never be more than .5 true.

p      |       ~p    |       p & ~p
1      |       0      |           0
.75   |       .25   |           .25
.5     |       .5     |           .5
.25   |       .75   |           .25
0      |       1      |

Harry is wiry.           Harry is not wiry.                Harry is wiry and Harry is not wiry.
1                                  0                                              0
.75                               .25                                           .25
.5                                 .5                                             .5
.25                               .75                                           .25

LAW OF THE EXCLUDED MIDDLE

The law of the excluded middle is absolutely true whenever the truth value of the sentence is absolute, and it is never less than .5 true.

p      |       ~p    |       p v ~ p
1      |       0      |               1
.75   |       .25   |           .75
.5     |       .5     |           .5
.25   |       .75   |           .75
0      |       1      |           1

Mike is short. Mike is not short.                 Mike is tall or Mike is not tall.
1                                  0                                              1
.75                               .25                                           .75
.5                                 .5                                             .5
.25                               .75                                           .75