I'm wondering if someone can offer an intuition of what is meant when mathematicians talk of recasting first-order logic statements as algebraic expressions. e.g. Halmos, Barnes, or Mack?
Here is a simple example to illustrate this question, adapted from forall x: Calgary.
If we had two variables "lover" and "loved" that ranged over the domain {"Imre", "Juan", "Karl"}, and a predicate $L$ (for Lovers) which was true for the following values of lover and loved.
| lover | loved |
|---|---|
| Imre | Juan |
| Juan | Karl |
| Imre | Karl |
| Karl | Imre |
then it is true that everyone loves someone $\forall lover \exists loved L(lover, loved)$
If we had an algebraization of FOL:
- Would we have defined a set closed under some list of operators?
- Could such an algebraization read like one of these two forms?
$forall(exists(variable(loved, variable(lover,L(lover,loved)))))$
$L(lover,loved).variable(lover).variable(loved).exists().forall()$
Or is an algebraization of FOL something else?
Thanks
David