> Here are some examples in FOL:
>
> "Mary is female"
> female(mary)
Could be
Inheritance Mary female
or
Evaluation female mary
(the latter being equivalent to female(mary) )
but none of these has an uncertain truth value attached...
> This is a [production] "rule": (not to be confused with an inference rule)
> "A female child is called a daughter"
> daughter(X) <- child(X) & female(X)
> where universal quantification is assumed.
You could say
ForAll $X
ExtensionalImplication
And
Evaluation child ($X)
Evaluation female ($X)
Evaluation daughter($X)
which is equivalent to the pred logic formulation
you've given.
But it will often be more useful to say
Implication
And
Evaluation child ($X)
Evaluation female ($X)
Evaluation daughter($X
which leaves the variable unbound, and which replaces the purely
extensional implication with an Implication that is mixed extensional
and intensional.
And one will normally want to attach an uncertain TV like an
indefinite probability to an expression like this, rather than leaving
it with a crisp TV.
The definition of
IntensionalImplication A B
is
ExtensionalImplication Prop(A) Prop(B)
where Prop(X) is the fuzzy set of properties of X
The definition of Implication is a weighted average of extensional and
intensional implication
I guess that gives a flavor of the difference
> *** bonus question ***
> Can you give an example of something expressed in PLN that is very
> hard or impossible to express in FOL?
FOL can express anything, as can combinatory logic and a load of other
Turing-complete formalisms.
However, expressing uncertainty is awkward and inefficient in FOL, as
opposed to if one uses a specific mechanism like indefinite truth
values.
Similarly, expressing intensional relationships is awkward and
inefficient in FOL as there is no built in notion of fuzzy sets of
properties
And there is no notion of assigning a truth value to a formula with
unbound variables in FOL, but one can work around this by using
variables that are universally bound to a context that is then itself
variable (again, more complex and awkward)
-- ben
-------------------------------------------
agi
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