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the truth value
•
and
⊥
for
◦
(later these symbols are seen as abbreviations for
specific formulas).
But classical logic cannot handle inconsistency since an explosion occurs.
In order to handle inconsistency we allow additional truth values and the first
question is:
1. How many additional values do we need?
It seems reasonable to consider countably infinitely many additional truth
values - one for each proper constant we might introduce in the theory for the
knowledge base. Each proper constant (a proposition, a property or a relation)
can be inconsistent “independently” of other proper constants. We are inspired
by the notion of indeterminacy as discussed by Evans [11]. Hence in addition
to the determinate truth values ∆ =
{•, ◦}
we also consider the indeterminate
∇
{
,
,
,...}
truth values
to be used in case of inconsistencies. We refer
to the determinate and indeterminate truth values ∆
=
∪∇
as the truth codes.
Wecanthenuse,say,(∆
∪∇
)
\ {•}
as substitutes for the natural numbers
ω
.
The second question is:
2. How are we going to define the connectives?
One way to proceed is as follows. First we want De Morgan laws to holds;
hence
=
{
0
,
1
,
2
,
3
,...}
ϕ ∨ ψ ≡¬
(
¬ϕ ∧¬ψ
). For implication we have the classically acceptable
ϕ → ψ ≡ ϕ ↔ ϕ ∧ ψ
and vice versa,
leaving the other values unchanged (after all, we want the double negation law
ϕ ↔¬¬ϕ
. For negation we propose to map
•
to
◦
to hold for all formulas
ϕ
). For conjunction we want the idempotent
law to hold and
•
should to be neutral, and
◦
is the default result. For biimpli-
cation we want reflexivity and
•
should to be neutral,
◦
should be negation, and
again
is the default result. The universal quantification is defined using the
same principles as a kind of generalized conjunction and the existential quantifi-
cation follows from a generalized De Morgan law.
We do not consider a separate notion of entailment - we simply say that
◦
ϕ
entails
ψ
iff
ϕ → ψ
holds. While it is true that
ϕ ∧¬ϕ
does not entail arbitrary
ψ
, hence we do not have a relevant logic [1] in
general (but only for so-called first degree entailment). Our logic validates clear
“fallacies of relevance” like the one just noted, or like the inference from
we do have that
¬ϕ
entails
ϕ → ψ
ϕ
to
ψ → ψ
, but these do not seem problematic for the applications discussed above.
Our logic is a generalization of Lukasiewicz's three-valued logic (originally
proposed 1920-30), with the intermediate value duplicated many times and or-
dered such that none of the copies of this value imply other ones, but it differs
from Lukasiewicz's many-valued logics as well as from logics based on bilattices
[8, 12, 6, 13].
4
Conjunction, Disjunction, and Negation
The motivation for our logical operators is to be found in the key equalities
shown to the right of the following semantic clauses (the basic semantic clause
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