In philosophical logic, categorical logic is the logic that deals with the logical relationship between categorical statements. I wonder if categorical logic is considered the same as predicate logic (first-order logic) in formal logic.
As @TankutBeygu notes in his comment, there is a terminological difficulty here. "Categorical logic" sounds like a general class of logical systems that deals with categories, but the only example of categorical logic is the logic of the syllogism, which is a specific formalism invented by Aristotle.
A syllogism is made of three statements, each of which has one of the following four forms of categorical sentence.
- All A is B.
- No A is B.
- Some A is B.
- Some A is not B.
In English, you have to modify the sentences a bit to have them sound right (I don't know if this is necessary in Greek). For example, you wouldn't say "All blue is colored", you would say "Everything blue is colored". With that, here is a syllogism:
Everything blue is colored.
Everything colored is extended in space.
Everything blue is extended in space.
This is a syllogistic proof. The first two statements are the premises and the third is the conclusion. You can see that this is a very limited form of proof. It can't handle an argument with more than two premises (except by chaining them in pairs), it can't handle logical connectives such as "A and B", "A or B", or "A implies B", and it can't handle general relations. You can translate every categorical sentence into predicate calculus like this:
All A is B --> (exists x.A(x)) and (forall x.A(x)->B(x))
No A is B --> (exists x.A(x)) and (forall x.A(x)->not B(x))
Some A is B --> exists x.A(x) and B(x)
Some A is not B --> exists x.A(x) and not B(x)
However, there is no way to translate the following predicate logic sentences into categorical sentences:
forall x.A(x) -> (B(x) or C(x))
forall x,y. A(x,y) -> B(x)
Therefore, the syllogism is strictly less powerful than predicate logic. One of the motivations for inventing predicate logic is that syllogisms are not powerful enough to do mathematics.
Predicate logic is usually understood these days as what is called "predicate logic" in mathematical logic. As such, it is certainly not identical with Aristotle's syllogistic. It is literally antinomic with it.
I believe there would be no problem in principle working out a formal model of Aristotle's syllogistic, although I think nobody has done it yet.
One interesting question would be how you extend predicate logic, with typical expressions such as Socrates is mortal, to relational logic, for example Romeo loves Juliet.