# Predicate logic: Aristotle vs Frege

Consider the following argument.

1) People who write novels are more sensitive than people who play soccer. 2) Alf writes novels. 3) Brian plays soccer.

Conclusion: 4) Alf is more sensitive than Brian.

Here is how you formalise this argument using predicate logic (many thanks to Mauro Allegranza):

1) ∀x∀y((Nx & Sy) ⊃ Mxy) 2) Na 3) Sb Conclusion: 4) Mab

M= more sensitive than N= writes novels S= plays soccer a= Alf b= Brian

Here is my question: am I right in thinking that you can't formalise this argument using Aristotle's logic (i.e., it would be an invalid argument in it)? The reason is that in Aristotle's logic you can't infer from the fact that people who write novels are more sensitive than people who play soccer that people who play soccer are less sensitive than people who write novels. As a result, you can't infer that Alf is more sensitive than Brian. In order for this argument to be valid you need a language where you can express relations between singular terms. Am I right?

Thank you very much for your help!

Fisher

• Exactly; Aristotle's syllogism is "translated" into modern symbolism in monadic predicate logic; i.e. all predicates must be unary: no binary relations, like "x is more sensitive than y" are expressible. – Mauro ALLEGRANZA Feb 9 '17 at 14:07
• What if Alf is a novellist who plays soccer? – zoli Feb 9 '17 at 14:44
• This specific limit of A's syllogistic logic was clearly identified by Augustus De Morgan; see Augustus De Morgan and the Logic of Relations. – Mauro ALLEGRANZA Feb 9 '17 at 14:45
• @zoli - in that case, the first premise: ∀x∀y((Nx & Sy) ⊃ Mxy) is false. – Mauro ALLEGRANZA Feb 9 '17 at 15:02
• There are at least two unstated properties of $Mxy$ involved here. One is that a comparison "more sensitive than" is anti-symmetric and transitive. Another is that the comparison applies to individuals rather than to populations. – hardmath Feb 9 '17 at 15:28

## 1 Answer

am I right in thinking that you can't formalise this argument using Aristotle's logic?

Exactly; Aristotle's syllogism is translated into modern symbolism in monadic predicate logic; i.e. all predicates must be unary, that means that a binary relation, like "$x$ is more sensitive than $y$", is not expressible in it.

This specific limit of A's syllogistic logic was clearly identified by Augustus De Morgan; see Augustus De Morgan and the Logic of Relations.