Some genera 'admit degree' and some do not. Take an adjective admitting degree like 'desirable'. Of course this must be specified to desirable to a certain kind. It might seem that an adjective admitting degree particularized to a given kind can be represented as a set $X$ and relation $<$ expressing comparison. Then the superlative (if it exists) is unique (152a). For this to be the case we might need the anti-symmetry property $x < y\,\&\, y < x \rightarrow x = y$. We define a superlative of $X$ relative to $<$ , to be an element $s \in X$ such that $\forall x \in X. x\neq s \rightarrow x < s$. The problem with Aristotle's discussion is the expression 'the superlative', or rather for instance 'the most desirable'. Does this definite article already imply that there is only one superlative (if at all), so that the mentioned argument of Xenocrates is just a case of the topic stated further ahead, that involving two things equal to a third being equal amongst themselves. The only way to make sense of this argument is to cast it in the form: $x$ is a superlative and $y$ is a superlative (of $X$) therefore $x = y$. Notice that in category theory we also use the expression 'the product of $A$ and $B$' when in reality we are speaking not of a unique object (and pair of arrows) but of an equivalence class. Anyhow we have that the superlative is a case of category-theoretic construction, it is the terminal object in the induced category of the pre-order $(X,<)$.
Now Aristotle's furnishes us with an interesting discussion of superlatives applied to pluralities (which might be seen to answer the objection above as well as present an early example of brilliant linguistic analysis, pointing out the misuse of the definite article and the assumptions implied). For instance the 'Spartans' and 'Peloponnesians' are both said to be the bravest of the Greeks. What do such plural superlatives mean ? How are we to make sense of Aristotle's argument that the Spartans being the bravest and the Peloponnesians being the bravest only means that one term is contained in the other ? The definition of $A$ being the supremum in $X$, for $A \subset X$ (let us write this as a predicate $Sup_X(A)$) seems to be stated explicitly to be $\forall x \in X. \forall y \in A. x\notin A \rightarrow y > x$. S
Suppose that $Sup_X(A)$ and $Sup_X(B)$ and that $A$ and $B$ were not contained in each other. Then there is an $a\in A$ such that $a \notin B$ and a $b\in B$ such that $b \notin A$. But then by definition $a > b$ and $b < a$ from which we derive a contradiction. Note the rather complex logical steps and rules for dealing with multiple implicitly required in Aristotle's argument. But if for instance $A \subset B$ there is no contradiction. It only follows that the $A$ are the best within $B$.
Question: what exactly does Aristotle mean by a predicate 'expressing the essence' of a subject ? And is it possible for Aristotle for there to be two different valid definitions of the same subject ?
In 153b we get an anomalous genus-difference constructions: justice is the virtue of the soul, where 'of the soul' is the difference. The context is as follows. Given a definition $\Sigma = \Gamma\Delta$ how does this relate to the definition of $\Sigma^\circ$ ? How do $\Gamma^\circ$ and $\Delta^\circ$ come in ? Sometimes the same $\Delta$ is used. Justice and injustice are contraries, yet their definitions are virtue of the soul and vice of the soul. So it is $\Gamma^\circ$ that enters. Aristotle concludes that if we now the definition of $\Sigma^\circ = \Gamma' \Delta'$ then we know the definition of $\Sigma = \Gamma\Delta$, there being three cases. i) $\Gamma = \Gamma'$ and $\Delta' = \Delta^\circ$, ii) $\Gamma' = \Gamma^\circ$ and $\Delta' = \Delta$ and iii) $\Gamma' = \Gamma^\circ$ and $\Delta' = \Delta^\circ$. Here we assume that $X^{\circ\circ} = X$.
It seems that modifiers such a difference in the Topics are perhaps not quite correctly expressioned as conjunctions of properties. In 155a Aristotle gives some basic quantifier rules. Harder to refute a definition through reasoning that to construct one ! Property is closest to definition. Accident is just plain predication. Book VII seems to be part of an independent treatise.
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