How is categorical syllogism defined as a formal logic system?

Question

Is Aristotelian categorical syllogism (as in Copi's Introduction to Logic) a special case of the first-order logic? (This is my thought after making efforts.)

Is it exactly the specialization of the first-order logic without any function symbol (but only predicate symbols)? (This is my thought after making efforts.)

If it is still a specialization of the above specialization, what kind of specialization is it exactly?

How would you define it as a formal logic system?

Thanks.

Answer 1

"Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification."

Those are the opening words of a classic paper by Timothy Smiley in the Journal of Symbolic Logic in 1962, "Syllogism and Quantification". (Smiley and John Corcoran, whom Mauro Allegranza rightly mentions, independently got to similar places, but I think this particular piece by Smiley is the most illuminating of all.)

Smiley continues "It is now more than twenty years since the invention of the requisite framework [for regimenting syllogistic logic using modern quantifiers], the logic of many-sorted quantification." And he proceeds to show how we get a natural version of Aristotelian syllogistic in first-order many-sorted logic.

So we start from what you might call a basic many-sorted calculus (just lots of single-sorted logics, with different sorts of variables, pasted together!). And if for example a variable of the sort $a$ is designated as running over the class of men, statements like All men are F and Some men are F can get regimented by $(\forall a)Fa$ and $(\exists a)Fa$. And off we go ...

However, we will want to introduce some unary "sortal" predicates, one for each sort of variable, with the plan that each new predicate is true of exactly those individuals which constitute the range of variables of the corresponding sort. So, for example if any variable $a$'s sort ranges over the class of men, we'll want a corresponding sortal predicate $A$ which is interpreted as 'man' and is true of just the things that $a$ ranges over. And we need a story about how to augment basic many-sorted logic with sortal predicates so everything behaves well, and about how we get e.g. the traditional conversion of $E$-propositions ('No A are B' implies No B are A'). In other words we need to be able to show, in our many sorted logic with sortals, that $(\forall a)\neg Ba$ implies $(\forall b)\neg Ab$. And so on. Smiley spells all this out very elegantly.

Download the paper and read it, it is only fifteen pages. Jstor: https://www.jstor.org/stable/pdf/2963679.pdf

Answer 2

It’s important to distinguish between Categorical/Syllogistic Logic and Categorical Syllogism. Syllogistic logic uses argument forms known as syllogisms, while Categorical Syllogism is a specific argument form, namely “Every A is B; every B is C; therefore, every A is C.”

As others have said, Aristotle’s logic is roughly a many-sorted monadic quantified logic. However, Aristotle’s logic does not contain the Principle of Explosion and does not permit the use of predicates that cannot be satisfied by anything in the actual world. For example, Aristotle’s syllogistic logic does not validate $\forall x\in \emptyset . \phi (x)$ for an arbitrary first-order formula $\phi(x)$. As a consequence, Aristotle’s logic validates $\forall x \in A. \phi(x) \to \exists x \in A. \phi(x)$, which Classical FOL does not.