MANY VALUES TRUTH TABLES
We still want to be able to represent the truth values of sentences and their
logical properties. And we can still do this using truth tables. The only
difference is that there are now three possible truth values for each
sentence, i.e. true, false, or indeterminate (T, F, I). So now, when we write
all the possible different truth value combinations in a truth table, we need
to represent more possible combinations, 3n, where n is the # of variables.
THREE VALUED CONNECTIVES
To be able to make truth tables we have to start by redefining the sentence
connectives, adding in the third possible truth value of “indeterminate.” It is
important to know that although we have added a new truth value, this
does not keep sentence connectives from being functions. Sentence
connectives are still functions, only now they are functions from {T I F} into
{T I F} instead of functions from {T F} to {T F}.
Most many-valued or normal logics usually agree with the truth table results
for the values of truth and falsity. For example, in classical logic, only one
conjunct has to be false for the whole sentence to be false. As a result, 6
out of the 9 positions in a many-valued truth table are already filled in a
manner identical to in a truth table in bivalent logic. How we fill in the
remaining slots depends on how we regard indeterminate sentences. And
the different ways that different logicians use regarding indeterminate truth
values of each of the various connectives distinguishes their respective
theories.
NEGATION
-A is true iff A is false
A
T
I
F
|
|
|
|
~A
F
I
T
Both Kleene and chief competitors agree on this analysis of negation.
NOTE: If you cannot determine whether the resulting truth value
is T or F, it must be I.
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CONJUNCTION
CONJUNCTION
A
A&B
T
B
I
T
I
F
T
I
F
I
I
F
F
F
F
F
 a conjunction is true if both conjuncts are true
 a conjunction is false if either conjunct are false
Example: The sentence “Holmes lived in London as a child and Holmes
took a bus to school.” is indeterminate.
TEST: If a compound has the same truth value whether a component is T
or F, it also has that same value if that component is Indeterminate. In
contrast, if the truth value of the compound depends on the truth values of
both components, then whenever one component is indeterminate, so is
the truth value of the compound (whole) sentence.
For example, if A is T and we have A & B, whether the whole sentence is
true depends on whether B is true or not. So the conjunction fails the test,
and the remaining slots of components with indeterminate truth values are
indeterminate.
Again, both Kleene and chief competitors agree on this analysis of
conjunctions.
DISJUNCTION
A
AvB
T
B
I
T
I
F
T
T
T
T
I
I
F
T
I
F
 If either disjunct is true, the compound sentence is true.
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 If both disjuncts are false, the compound sentence is false.
 We apply the test to fill in the values not already given by classical logic.
For example, if A is T and we have A v B, the whole sentence is true no
matter whether B is true or false. Therefore, the sentence would be true
if B were indeterminate. Otherwise, the value is indeterminate.
Example: The sentence “Holmes lived in London as a child or Holmes
took a bus to school.” is indeterminate.
Both Kleene and chief competitors agree on this analysis of disjunctions.
CONDITIONALS
There are three different approaches to analyzing the truth tables of
conditional statements. As a result, there are three different logics:
Kleene’s logic, Lukasiewicz’s logic, and Godel’s logic.
KLEENE’S APPROACH
CONDITIONAL (K3)
B
AB
T
A
T
I
F
T
T
T
I
F
I
I
T
F
I
T
 If the antecedent of a conditional is false, the compound sentence is
true.
 If the antecedent of a conditional is true and the consequent is false,
then the compound sentence is false.
 If the consequent is true, the whole sentence is true.
 Makes the three remaining positions indeterminate.
RESULT: An indeterminate that implies an indeterminate, an
indeterminate.
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Consequence: p  p is not a tautology. Furthermore, there are no
tautologies in Kleene’s logic.
Example: The sentence “If Holmes lived in London as a child, then
Holmes lived in London as a child.” is indeterminate, rather than
true.
LUKASIEWICZ’S APPROACH
In reaction against this, Lukasiewicz makes the truth value of a conditional
with two indeterminates true.
CONDITIONAL (L3)
B
AB
T
A
T
I
F
T
T
T
I
F
I
T
T
F
I
T
For Lukawiecz, if the truth values of the antecedent and consequent are
indeterminate, then the truth value of the whole is true.
Consequence: In L3, p p is a logical truth, true even when the
antecedent and the consequent are both indeterminate.
Example: The sentence “Holmes lived in London as a child, then Holmes
lived in London as a child.” is true.
GODEL’S APPROACH
Godel’s modification is to make a conditional with an indeterminate
antecedent and a false consequent false.
CONDITIONAL (G3)
A
AB
B
T
I
F
T
I
F
T
T
T
I
T
T
F
F
T
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Consequence: A sentence with an indeterminate antecedent and a false
consequent is false. Also, the sentence p  p is a tautology.
Example: The sentence, “If Holme’s lived in London as a child, then New
Mexico is the capital of Pittsburgh.” is false.
BICONDITIONAL (K3)
A
A  B
B
T
I
F
T
I
F
T
I
F
I
I
I
F
I
T
The biconditional is indeterminate whenever any component is
indeterminate.
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