My observations. 1. Regarding 1029b13 "kai proton eipomen enia peri autou logikos." Here I have a comment rather than question about Burnyeat's interpretation.
Ross: logikos suggests plausibility rather than truth, a reference to abstract considerations (logoi) rather than to the precise nature of the facts in question.
Bostock: three accounts about logikos (a) often means definition, it would be quite nature to suppose that Aristotle is proposing to begin by defining his notion of essence. (b)"Logical remarks" concern language, or "how to express oneself." (c) In contrast with "physical remarks," one remains general and abstract, whereas the other goes deeply into the nature of the objects concerned.
Burnyeat: aims at persuading readers of Zeta that Simplicius' third meaning and Andronicus' sense of "logical" are so closely connected and can be treated as one. (25) Burnyeat goes by (a) Simplicius' third meaning for the use of the word "logical" at Physics III 3.202a21-22: the puzzle proceeds form generalities rather than from principles peculiar to the subject; and (b) Alexander’s contrast: (i) the argument from opposites for the immortality of the soul in Plato’s Phaedo (70c-72e) is logical: it proceeds from the premise that everything comes to be from its opposite, and this is a premise which applies to lots of different kinds of things. (ii) The argument from the definition of soul as self-mover in Plato’s Phadedurs (245c-246a) is not logical: it proves immortality form the essential nature of its subject, the soul. And then Burnyeat concludes his discussion of logikos with two qualitifications: (a) a "logical" approach need not start from outside the science altogether. It may start from a more general level within the science. (b) The principles that a "logical" discussion abstracts from need not be causal.
My observation: Burnyeat does not make it clear about what exactly he means by the above two qualifications. It seems to me that Burnyeat is trying to say that the constrast between logical and non-logical is quite independent of the constrast between linguistic and non-linguistic. So, although Z4 begins by a "logical" approach, it does not mean a "linguistic" approach, rather, a more general, not specific to the subject in question. But, on the other hand, Burnyeat holds that "with each of the four independent sections there is a definite movement from logical discourse to metaphysical discoures. I don't see how Burnyeat makes it clear on his contrast between "logical" and "metaphysical." I feel even more puzzled when it comes to Burnyeat's refutation of Ross' interpretation at 1030a27, where Ross suggests a move from the linguistic question "how one ought to express" to "how things are."
2. about 1029b19-20 en hoi ara me enestai logi auto, legonti auto, outos ho logos tou ti en eniai hekastoi...
a formula (logos) is the essence of a thing iff (a) this formula expresses/defines (legonti) the thing, and (b) it is not circular by including that thing itself.
My observation: it seems to me that Bostock is adopting a linguistic approach to construe this passage (and even the whole Z4), since he defines logos as "any meaningful series of words is counted as a formula" and orismos as "only a formula of a certain kind will be a definition." How do you construe this passge? I think the key point may rest on how to translate "legonti." Expresses? or Defines?
3. about 1030a2-6 hoper gar ti esti to ti en einai; otan d' allo kat' allou legetai, ouk estin hoper tode ti, eiper to tode tais ousiais huparxei monon.
Aristotle seems to switch the statement "the essence is precisely what a thing is" to a different line of argument: "an essence must be a this," whereas a compound such as "pale man" is not a this. Aristotle says in Z3 that only a substance is a this, and compounds are not substances.
My observation: Bostock that Aristotle cannot be entitled to calim that "only a substance is a this," because Aristotle is wrong to assert "an essence must be a this." But, wait, what does Aristotle mean by a this? (this relates to the discussion of this in the end of Z3)
-a particular as opposed to a universal?
-the category of substance as opposed to other categories?
-a form or essence? a form in contrast to privation? a form in contrast to matter? a form in contrast to the coincidental attribute?
4. about 1030a10-12 ...all' ean proton tivos ei; toiauta d' estin hosa legetai me toi allo kat' allou legesthai. ouk estai ara oudeni ton me genous eidwn huparxon to ti hn einai, alla toutois monon.
"There is a formula of something primary, i.e. of something which is expressed without predicating one thing of another. A what-being-is, then, will belong to nothing but what is a form of a genus."
My question: Both Bostock and Burnyeat construe "a form of genus" as "species." Bostock says "if Aristotle is aware of any potential ambiguity in the word, arising from these two different contrasts, then clearly he is indicating here that it is form in the sense of 'species' that he intends." So it is about secondary substances?
At 30a27-b7 Aristotle proceeds to add an explanation of how "primarily" and its opposite are to be understood. Owen (1960) has it called "focal meaning." For instance, the word "medical" applies primarily to the art, skill, or knowledge of medicine. Other things are called "medical" by reference to this, e.g. a patient is called a medical patient because she is being treated by the art of medicine. So, perhaps, I can say, it is a formula of something "primary" in the sense of its species, i.e. secondary substance, whereas a medical patient, you and me, is to individualize such secondary substance?
5. about 1030b7-13
In addition to "of something primary" or "a form of a genus," Aristotle seems to add more restriction on the formulae that can be taken as expression definitions: such a formula must be a formula "of one thing." So perhaps only a substance is a "one" or "unity"?