# Possible vs may (or impossible vs cannot) in Aristotle's Prior Analytics

In chapter 13 of the Prior Analytics, it is written that "it is evident if it is possible for A to apply to B, it is also possible for it not to apply". Which naively speaking makes sense, because unless something is clearly impossible, both "it is possible A is B" and "it is possible A is not B" are true at the same time. Do you agree with that?

Therefore, until chapter 16, I was quite convinced that every time we have a possible premise and there is a syllogism, that the conclusion will also neither be assertoric nor apodictic. However, reading chapter 16, I was surprised to find the following example:

If A cannot apply to any B, and B may apply to some C, it must follow that A does not apply to some C. For if A applies to all C, and cannot apply to any B, B too cannot apply to any A; and so if A applies to all C, B cannot apply to any C. But it was assumed that it may apply to some.

In other words, the proof by contradiction says that if the opposite was true, then because "B does not apply to any A" and "A applies to all C" we conclude that "B cannot apply to any C", which contradicts the second premise, which makes the conclusion apodictic. It also makes some sense.

Would the same be true if instead of "B may apply to some C", the second premise was "It is possible for B to apply to some C"? If yes, could you please clarify how I could reunite that with the previous treatment of "it is possible" premises?

Thanks!

• Re "possibility", see A's Modal Syllogism: "Modern modal logic treats necessity and possibility as interdefinable. A gives these same equivalences in On Int. However, in Prior An, he makes a distinction between two notions of possibility. On the first, which he takes as his preferred notion, “possibly P” is equivalent to “not necessarily P and not necessarily not P”. He then acknowledges an alternative definition of possibility according to the modern equivalence, but this plays only a secondary role in his system." Nov 21, 2022 at 10:21
• @MauroALLEGRANZA Thanks! Why not make a full answers, so I can close this question?
– S11n
Nov 21, 2022 at 11:40

Yes, "B may apply to some C", is "it is possible that (some C is B)" (I've have written this way to show the interplay of modal operators and (modern) quantifiers).

Regarding "possibility" in Aristotle, see Aristotle's Modal Syllogism:

"Modern modal logic treats necessity and possibility as interdefinable. [...] Aristotle gives these same equivalences in De Int. However, in Prior Ana, he makes a distinction between two notions of possibility. On the first, which he takes as his preferred notion, “possibly P” is equivalent to “not necessarily P and not necessarily not P”. He then acknowledges an alternative definition of possibility according to the modern equivalence, but this plays only a secondary role in his system."

And see the para on Modal Conversions for difficulties in interpreting it.

A good place to start with is modern edition of Prior Ana by Robin Smith (with intro and notes); but see page XXVI:

In A 8-22, Aristotle extends his theory to include deductions involving modally qualified categorical sentences. In contrast to the account of assertoric deductions this theory is problematic in the extyreme.

For recent accounts: Adriane Rini, Aristotle's modal proofs: Prior Analytics A8-22 in predicate logic (Springer, 2011) as well as Marko Malink, Aristotle's Modal Syllogistic (Harvard UP, 2013).

Especially interesting is Rini's approach in Part III (page 119-on) "about the problematic syllogistic – the syllogistic involving a new sense of possibility. Initially Aristotle offers what appears to be a new definition of possibility as contingency, according to which what is possible is what is neither necessary nor impossible. Aristotle explains that anything that is contingently phi is also contingently not phi."

And see Table 22b (page 172) for the (possible) reading of the quoted text above (36a34):

∀x(Bx → L~Ax) and ∃x(Cx & QBx); therefore ∃x(Cx & ~Ax)

where L is necessary, M is possible and Q symbolizes the third modality (provisionally interpreted as contingent) introduced to formalize what is neither necessary nor impossible:

I use the expressions ‘to be possible (endechestai)’ and ‘what is possible (to endechomenon)’ in application to something if it is not necessary but nothing impossible (A13, 32a19-22).

Re: Would the same be true if instead of "B may apply to some C", the second premise was "It is possible for B to apply to some C"?

According to Rini, "B may apply to some C" is ∃x(Cx & QBx) while "It is possible for B to apply to some C" is "M(∃x(Cx & Bx)).

Compare with The De Re/De Dicto Distinction.