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I looked through the nice paper by Tarski On the Calculus of Relations. In the beginning he touched a motivation behind Theory of Relations but this part was not clear to me (page 1, very beginning):

De Morgan clearly realized the inadequacy of traditional logic for justification... witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal

As far as I understood from wikipedia, the problem with Aristotle logic is that it does not contain singular terms and does not allow to talk about a singular object. It allows to talk about multiple objects such as 'all Socrates', 'some Socrates', 'no Socrates' and sounds awkward, but does not allow to talk about one particular singular 'Socrat'. 1) Is this right?

2) Did Aristotle logic restriction motivate Theory of Relations?? To me it seems unlikely, since I don't see the connection (unfortunately). So what did motivate development of Theory of Relations?

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up vote 6 down vote accepted

Put $Rxy\ $ for $y$ is the head of $x$, $Hx\ $ for $x$ is a horse, $Ax\ $ for $x$ is an animal. Then the inference De Morgan mentions -- "A horse is an animal, therefore any horse's head is an animal's head" -- gets regimented into modern quantificational logic as $$\forall x(Hx \to Ax) \therefore \forall y(\exists x(Hx \land Rxy) \to \exists x(Ax \land Rxy))$$ which is easily proved valid in a standard first-order logic (try it! -- I used to set variants as a standard intro logic exam question!).

However, this sort of inference involving quantifiers embedded in the scope of other quantifiers cannot be well-handled by "traditional" logic. Indeed it is a standard textbook example to show that the post-Frege treatment of generality does better than Aristotelian logic. And the "traditional" logic's shortcomings over multiple generality stem from its inadequacy at handling relations like $R$. There's a wonderfully illuminating discussion of this in the second chapter of Michael Dummett's magisterial Frege: Philosophy of Language.

(By contrast, traditionalists can fudge up treatments of singular terms; but that's another story.)

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Nice reply. Perhaps you could emphasize the point that Aristotelian logic based on syllogisms was too weak, and Frege is generally thought to be the one to have introduced quantifiers (a.k.a. relational logic). – Mikhail Katz Feb 13 '14 at 13:12

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