# Is categorical logic the same as predicate logic

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.

• Note that there's a terminology overload here which I've seen lead to confusion before: "categorical logic" (or sometimes "categorial logic") can also refer to approaches to logic via category theory. (For that matter, "categorical" also has a separate meaning in model theory.) Aug 1, 2021 at 17:33
• 'Categorical syllogism' is the right name; 'categorical logic' only adds confusion if used in that sense -few tend to that rubbish. Aug 1, 2021 at 20:56
• No they are NOT IDENTICAL. One form must be translated into the other. Also the purposes are distinct. LOGIC as almost all young teens call it these days 98 percent of the time means MATHEMATICAL LOGIC aka discrete mathematics. There are real world cases where those techniques will fail. Categorical syllogisms are designed in a specific wording structure to eliminate rhetoric & emotions. Math will tell you logic is about validity. Philosophy focuses on Soundness. Validity alone often fails in the real world. Aug 1, 2021 at 21:47
• It is a subset of; see Monadic predicate calculus Aug 2, 2021 at 8:24

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.

• @Logikal, first, It is not a "problem" if my answer doesn't mention a special rule followed by a tiny minority of writers on the topic, and your use of the word "ignorant" was offensive. I don't know what set you off, but please keep the personalities out of it. Second, I did mention chains of syllogisms in my answer. Third, your comments about mathematical logic make no sense. Maybe you are using some peculiar terminology of your own, but if so, you are going to have to explain it if you want people to understand you. Aug 2, 2021 at 0:43
• @Logikal, I did not edit it. My original answer included the part in parenthesis which you left out in your quote. The statement is not false. A syllogism can only have two premises. A syllogistic argument can chain syllogisms together as I noted in the part you did not quote. If a premise is hidden, then you don't have a syllogism. You seem to be complaining that my answer was not a complete encyclopedic entry on Aristotelian logic. Aug 2, 2021 at 1:54
• @Logikal, As to your comment on math, what didn't make sense is saying that mathematical logic "works with rhetorical phrases" (not according to the normal meaning of "rhetorical"), that "people seem to word premises any kind of way" (not true, there are formal rules), that "they leave the window open for a convenient switch in the context of a premise" (not according to the usual meaning of "context"). Also, you seem to equivocate between "mathematical logic" as formal logic studied mathematically and "mathematical logic" as the logic used in mathematics. Two different topics. Aug 2, 2021 at 2:00
• @Logikal, Do you know what the word "equivocate" means? Please read my comment again. I did not deny that there is a distinct field called mathematical logic. It is a topic I am rather familiar with. What I said is that you seem to sometimes mean one thing by the phrase and sometimes something else. Aug 2, 2021 at 2:12
• @Logikal, If by "the new set of philosophy people" you mean modern philosophy which began with Descartes), then yes, I'm a student of modern philosophy. So are roughly 99.9% of all English speakers who are students of philosophy. If you want to make a point about Classical or Medieval philosophy, then it is incumbent on you to express yourself in a way that the 99.9% will understand you. Aug 2, 2021 at 2:17

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.

• N.B. There are studies in this direction. The best known is Łukasiewicz, Jan (1951, 1957). Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, 2nd edition. Oxford: Clarendon Press. Aug 2, 2021 at 13:35
• @TankutBeygu Yes, and not just him. There are more recent works, for example John Corcoran's. But, yes, while all these people say interesting things, they don't actually produce any actual relational syllogistic. Aug 2, 2021 at 16:32
• How about researchgate.net/publication/…? Aug 2, 2021 at 19:36
• @DavidGudeman Thanks, I'll try to look at it, but we would already know if he had succeeded. Aug 3, 2021 at 13:52