Topics 141a - 142b

In the previous post we left open the question: what does it mean for Aristotle to predicate something of something according to essence ?  And can a subject, as in modern mathematics, have more than one valid definition ?  Topics 141a provides a clue. Apparently a characteristic of essential predication is using 'anterior and better-known terms' (see 141b for an interesting discussion involving absolute anteriority and relative, according to us, better-known-ness).  Anyhow the uniqueness of definition is assumed as used by Aristotle as argument against definitions using terms which are not anterior and better-known.  For given such a definition we could replace the lesser-known term by a better-known term. The result would be a different definition. So there would be two definitions of the same subject which  contradicts the uniqueness postulate. Interestingly Aristotle speaks of the 'expression', the definition being equal to the 'expression'.  This is clearly not an expression in our modern syntactic sense. It must be its sense, its Fregean Sinn.  Aristotle rejects definitions using terms merely better known to us, rather than being in themselves anterior - or  absolutely better known. Such definitions would not express the essence of the subject.  It is curious how in this passage a point is seen as anterior to a line but a line better known to us than a point (this does not seem to be consistent with the later theory of the continuum in the Physics).  Aristotle states that if we know the subject then we know its genus and difference but not vice-versa. 142a is a strikingly insightful anticipation of Mates' puzzle, indeed the problem of the relativity of definitions. It shows the consequences if we take anteriority according only to a person or group of people. 

We conclude that the better-known relation $\triangleleft$ is not something odd, an intrusion of sociological factors, but one of the deepest and most purely logical concepts in the Topics. We might call it the relation of logical precedence. In 142a we get the rules: $\sim A \triangleleft A^\circ$ and $A,B < C$ implies that $\sim A \triangleleft B$ and $\sim B \triangleleft A$.

The discussion in 142b is one of the most important in all the Topics. It deals with the non-circularity of definitions.  A definition is proposed $\Sigma = \Gamma\Delta$. But $\Delta$ involves a term $\Sigma'$ whose definition involves $\Sigma$.  Aristotle proposed that we substitute in the definition of $\Sigma$ such a term by its definition to detect the circularity.  But what if there is no circularity. Is the resulting expression still a definition ? Does this contradict the uniqueness of definitions ? This seems contradicted by 143a when Aristotle seems fine both with a definition involving proximate genus and difference and a definition involving a further genus and all the necessary differences. For instance if $\Sigma = \Gamma\Delta$ and $\Gamma = \Gamma'\Delta'$ then Aristotle apparently accepts also the definition $\Sigma = \Gamma'\Delta'\Delta$.