Aristotle's On Interpretation Ch. 7. segment 17b17-17b26: Sketching out Aristotle's square of opposition
(17b17-17b26) of Ch. 7: Sketching out Aristotle's square of opposition
When we parse a thing as a particular in our speech, we convey that it is a single and indivisible unit. Conversely, a universal presents itself in our speech as a divisible cluster. This enables us to speak about a universal as a whole, i.e. universally, or in part, i.e. non universally. We may thereby assert, i.e. affirm or deny, something of (a) a particular (Camus is two-footed - Camus is not two-footed), (b) universally of a universal (every ostrich is two-footed - no ostrich is two-footed), or (c) non-universally (some ostrich is two-footed - not every ostrich is two-footed).
relations of opposition between assertions with a particular as subject
Now, with regards to the relations of assertions in which the subject is a particular, Aristotle has provided us with the basic groundwork in Cat. Ch. 10 13b12-13b36. There, the philosopher opposed the assertion “Socrates is healthy” to (i) “Socrates is sick” as contraries, and to (ii) “Socrates is not healthy” as affirmation to negation. He labelled the latter pair contradictories in On Int. Ch. 6.
(i) We may therefore produce contrary assertions about a particular either by affirming contrary things of it as subject (e.g. “Camus is healthy” - “Camus is sick”) or denying them (e.g. “Camus is not healthy” - “Camus is not sick”). (ii) Conversely, as Aristotle indicates in Ch. 6., to produce contradictory assertions we have to affirm and deny the same thing of the same thing (e.g. “Camus is healthy” - “Camus is not healthy”).
As such, two contradictory assertions necessarily oppose each other as affirmation to negation. With contrary assertions about particulars, on the other hand, this is not the case. As long as we assert contrary things of the same thing our assertions will be contrary even when they are both affirmations or both negations.
Furthermore, as Aristotle elaborates in Cat. Ch. 10, no matter the circumstances which underlie two contradictory assertions, one will always happen to be true and the other false. For contrary assertions, however, this is no necessary condition. Under given circumstances two contrary assertions may be both true or both false.
As we may observe in the table above, as long as we oppose an affirmation to a negation in which we assert the same thing, i.e. health or sickness, of the same thing, i.e. Camus, we will get a pair of assertions in which one will be true and the other false no matter the underlying circumstances. At first glance, this appears to be the case for contrary assertions as well. If Camus does not exist, however, he can neither be healthy nor sick. As such, there is at least one possible set of circumstances under which both contrary affirmations come up as false and both contrary negations as true.
relations of opposition between assertions with a universal as subject
In the present text, Aristotle charts out the relations of opposition between four assertions which assert the same thing of the same universal as subject. To begin with, he asserts one affirmation and one negation universally (e.g. every ostrich is fast - no ostrich is fast) and places them side by side. The former he calls a universal affirmation [uA] (καθόλου κατάφασις) and the latter a universal negation [uN] (καθόλου ἀπόφασις). Underneath both of them, he sets their opposites, i.e. a non-universal affirmation [nA] and a non-universal negation [nN] (e.g. some ostrich is fast - not every ostrich is fast).
At this point, the philosopher instructs us that each of the universal assertions contradicts with a non-universal one. Namely, (α) the universal affirmation with the non-universal negation (e.g. every ostrich is fast [uA] - not every ostrich is fast [nN]) and (β) the universal negation with the non-universal affirmation (e.g. no ostrich is fast [uN] - some ostrich is fast [nA]). Further, he holds that (γ) the universal affirmation is contrary to the universal negation and not contradictory and reasons that where both cannot be true together, (δ) their non-universal opposites can be.
By way of demonstration, we now take a closer look at each of the relations Aristotle describes as present in the above assertions and examine the properties which characterise each of these relations:
(I) (α) [uA] contradicts with [nN] and (β) [uN] contradicts with [nA]
In the first place, Aristotle identifies the pairs (α) and (β) as contradictory. We understand a pair of assertions to contradict when (i) one opposes the other as affirmation to negation, and (ii) the two assertions are never true or false at the same time, no matter the set of circumstances that underlies them. This, we may illustrate as follows:
(II) (γ) [uA] is contrary to [uN] and (δ) [nA] with [nN] are their opposites
In the second place, Aristotle recognises pair (γ) as contraries and (δ) as the opposites of those contraries. What foremost characterises the opposition in (γ) is that the two assertions are never both true at the same time, though there is one set of circumstances under which they are both false. Conversely, their opposites in (δ) can both be true together, yet never false. This, we may illustrate as follows:
This is what Aristotle communicates in this paragraph. As we conclude the present segment, we find that he has afforded us a basic structure and rules for what we may call a game of truth. As his students, we may thus choose to play this game and discover the way each of the four assertions relates to its three counterparts when its truth or falsity is known:
(i) when [uA] is true, then [nA] is also true. [uN] and [nN] are false.
That is, when it is true to claim that every ostrich is fast, then it also true to claim that some one ostrich is fast, yet to claim that not every ostrich is fast or none at all is false.
(ii) When [uA] is false, then [nN] is true. [nA] and [nN] are uncertain.
When it is false to claim that every ostrich is fast, then to claim that not everyone is fast is true. Yet, to claim that some ostrich is fast or that none at all remains uncertain.
(iii) when [uN] is true, then [nN] is also true. [uA] and [nA] are false.
When it is true to claim that no ostrich is fast, then to claim that not everyone is fast is also true. Yet, to claim that some or every ostrich is fast is false.
(iv) when [uN] is false, then [nA] is true. [uA] and [nN] are uncertain.
When it is false to claim that no ostrich is fast, then to claim that someone is fast is true. Yet, it remains uncertain whether every or not every ostrich is fast.
(v) when [nA] is true, then [uN] is false. [uA] and [nA] are uncertain.
When it is true to claim that some ostrich is fast, then to claim that noone is fast is false. Yet, it remains uncertain whether every or not every ostrich is fast.
(vi) when [nA] is false, then [uA] is also false. [uN] and [nN] are both true.
When it is false to claim that some ostrich is fast, then to claim that everyone is fast is also false. Yet, both not every ostrich is fast and none is fast are true.
(vii) when [nN] is true, then [uA] is false. [uN] and [nA] are uncertain.
When it is true to claim that not every ostrich is fast, then to claim that everyone is fast is false. Both claims that no ostrich is fast and some ostrich is fast remain uncertain.
(viii) when [nN] is false, then [uN] is also false. [nA] and [uA] are both true.
When it is false to claim that not every ostrich is fast, then to claim that none is fast is also false. Yet, both some ostrich is fast and everyone is fast are true.
We may thus take away the following from our observations above. (a) What most characterises the universal assertions is that when we know them to be true, we are also clear about which of the other three assertions are true or false. (b) Correspondingly, what we find characteristic of non-universal assertions is that when true they explicitly exclude their contradictories as false, yet leave room for uncertainty on whether the remaining two assertions are true or false.
Key points: (i) Aristotle charts out the relations of opposition between four assertions which assert the same thing of the same universal as subject. These are a universal affirmation, a universal negation, a non-universal affirmation and a non-universal negation (ii) The universal affirmation contradicts with the non-universal negation, the universal negation contradicts with the non-universal affirmation, the universal affirmation is contrary to the universal negation, the non-universal affirmation and negation are their opposites(iii) Contradictories are never true nor false together (iv) contraries are never true together but their opposites can be true together.