![](bgc.png)
So, we have proved
/
0 |=
P
φ and φ is a tautology.
5 CONCLUSIONS
We have investigated the deduction problem of a for-
mula from a finite theory in the propositional Prod-
uct logic. The deduction problem has been solved
via translation of a formula to an equivalent satisfi-
able finite order clausal theory, consisting of order
clauses. An order clause is a finite set of order liter-
als of the form ε
1
ε
2
where ε
i
is either a conjunction
of propositional atoms or the propositional constant
0 (false) or 1 (true), and is a connective either P
or ≺. P and ≺ are interpreted by the equality and
standard strict order on [0, 1], respectively. The tri-
chotomy over order literals: either ε
1
≺ ε
2
or ε
1
P ε
2
or ε
2
≺ ε
1
, has naturally led to a variant of the DPLL
procedure with a trichotomy branching rule, which is
refutation sound and complete in the finite case.
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