# Translation to FOPL

As we have seen, the notation is different from the predicate logic one, but
semantically, (some of) the description logics are subsets of first order
predicate logic. A concept corresponds to an unary predicate while a role
corresponds to a binary predicate. More formally, a concept *C*
corresponds to a predicate logic formula *ϕ _{C(x)}* with one
free variable

*x*such that for every interpretation

*I*the set of elements of

*Δ*satisfying

^{I}*ϕ*is exactly

_{C(x)}*C*. An atomic concept

^{I}*A*is translated into the formula

*A(x)*, the constructors intersection, union, and negation are translated into logical conjunction, disjunction, and negation. If

*C*is translated to

*ϕ*and

_{C(x)}*R*is an atomic role, then ∃

*R.C*and ∀

*R.C*are translated as

where *y* is a new variable. For further details of relationship
between first order predicate logic and description logics see The Description Logic Handbook.

(c) Marek Obitko, 2007 - Terms of use