# Is the categorical syllogistic a formal system?

I attended a recent presentation of a doctoral thesis in which the candidate described the categorical syllogistic as a formal system - making reference to Jan Łukasiewicz.

The reasons given for this claim included:

1. one can give a satisfactory semantics by simply employing an extensional semantics to represent the meaning of sentences and to decide whether a syllogistic inference is valid;

2. one can convert the categorical syllogistic into symbolic script (i.e., Boolean Algebra);

3. one can represent the categorical syllogistic in mathematical terms even though Aristotle does not define it as such.

Are these sufficient reasons to make the claim that the categorical syllogistic is (or at least can be transformed into) a formal system? I for one would not describe the categorical syllogistic as a formal system based upon the definition of a formal system as I know it.

• Sorry...why wouldn't you consider it a formal system? Aristotle's syllogistic theory has been formulated and reformulated many times, Lukasiewicz being a notable example. Jun 18 '14 at 23:21
• Depends on what definition one would use for a "formal system." Many modern definitions require that a formal system employ a fully symbolic language. That is clearly not the case among the classical systems. So, if one goes with that definition, they are proto-formal systems. Jun 19 '14 at 9:59
• That's a silly requirement. Generally, if a system is genuinely formal, then a symbolic language can easily be adapted as Lukasiewicz did. For example, Professor Larry Moss (logicforlanguage.blogspot.com/2010/08/…) symbolizes "All X are Y" as ∀(X,Y). Many of his papers then introduce a deductive system for this logic. Check out his papers on natural logic for further information. Jun 19 '14 at 21:08
• For an in depth reading into the question see "What Does it Mean to Say that Logic is Formal?" by John MacFarland. Intro to the paper as well as link to the PDF at johnmacfarlane.net/books.html . Jun 19 '14 at 21:17

Those are indeed sufficient to justify the claim that the categorical syllogistic can be transformed into a formal system. A formal system (or a proof system) can be thought of as a triple (L, Γ, ⊢), where:

1. L is a formal language,
2. Γ is a set of axioms, and
3. ⊢ is a direct derivability relation.

I have described the categorical syllogistic both from the semantical and the axiomatic points of view in an earlier post. Not to repeat myself I would like to direct you to (Definition 1) there for the semantics of the categorical syllogistic; and to (Definition 4) and (Definition 5) respectively for an adequate set of axioms and a derivability relation that characterize the categorical syllogistic. The axioms that (Definition 4) contains are taken from Łukasiewicz (1951); the definition of derivability from Vlasits (forthcoming).

Those definitions (for Γ and ⊢) and the very important completeness and soundness proofs that you can find in all standard expositions of Aristotelian metalogic should be sufficient to convince yourself that Aristotle's categorical syllogistic 'can be transformed into a formal system'. Perhaps what's missing, while not so important, is an explicit characterization of (1), that is, the formal language of the categorical syllogistic. But that's a pretty easy thing to characterize and there are lots of different ways we can go about doing it. For consistency with that earlier post, I'll stick to Vlasits' characterization:

Definition 6. (Language) Given a set of terms T = {a,b,...}, a set syllogistic relation symbols R = {a,e,i,o}, the language of categorical syllogistic is defined as L = {XrY : x,y ∈ T ∧ x ≠ y ∧ r ∈ R}.

Now that we have a precise characterization of the language (Definition 6), the set of axioms (Definition 4), and the syllogistic derivability relation (Definition 5), if the above description of a formal system as consisting of those three components is correct, then it would be reasonable to conclude that the categorical syllogistic can be (or has been) transformed into a formal system.

As regards reasons (1) and (2) that you listed. (1) is true, witness our extensional semantics given in (Definition 1) of that linked post. Checking whether a syllogistic inference is valid with that semantics reduces to pretty easy set-theoretic proofs (e.g. (Fact 2) in the earlier post shows an easy proof of Bocardo), which are often accompanied by Venn diagrams in elementary courses that discuss the syllogistic. (2) is false, I think, since the algebraic 'profile' of the Łukasiewicz axioms is far from forming a Boolean Algebra.

References

"Earlier Post" (2014) Reply to Crab Bucket: "Aristotle's Syllogistic and Proofs by Contradiction".
Łukasiewicz, J. (1951) Aristotle's Syllogistic: From the Standpoint of Modern Formal Logic, 2nd Edition.
Vlasits, J. (forthcoming) "Divisional Semantics for Aristotle's Assertoric Syllogistic", Aug. 22, 2012.

• Thanks for this. But, in your conclusion, you say 2) is false based on Łukasiewicz. But, you do not mean to imply 2) is necessarily false, do you? If so, why? Jun 19 '14 at 10:04
• @user155194 I do not. Since I haven't seen the work of the doctoral candidate, I wanted to look at some concrete implementation of the categorical syllogistic and see whether the three points are true for that axiomatization. (2) is not true for the Łukasiewicz axioms. (Now, Aristotle's logics are not propositional but term logics, so even the idea of a logical connective in the ordinary sense is absent. I wonder how could one axiomatize any of his logics in such a way that there are logical connectives in it obeying the laws of Boolean Algebra. That might be a question worth researching..) Jun 19 '14 at 15:25
• Yes - the reference to Boolean Algebra was made in reference to more complex forms of syllogism from the later ancient and medieval periods (ex. those which involved either or etc.) not the standard Aristotelian categorical syllogisms. Jun 19 '14 at 21:14

Categorical syllogisms can be done entirely in first-order logic (a formal system) using the following translations:

"All A are B" translates: For all x, A(x) implies B(x)

"Some A are B" translates: There exists x such that A(x) and B(x)

"No A are B" translates: For all x, A(x) implies not B(x)

"Some A are not B" translates: There exists x such that A(x) and not B(x)

where A and B are logical predicates.

Example: Translate "Some animals are not mammals."

Answer: There exists x such that Animal(x) and not Mammal(x)

where Animal and Mammal are the obvious predicates.

Once you get used to it, the predicate notation and rules of inference really simplify things. You can play around with these translations using my proof software available free at my website.

See examples.