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Apr 30:

Aristotelean contraposition consists of obverting an initial proposition, converting this obversion by switching the places of the first and second propositions (keeping any negation), and then obverting the conversion.

Obversion is often explained in this way:

A-type proposition: "All S are P" becomes "No S are non-P".

E-type proposition: "No S are P" becomes "All S are non-P".

I-type proposition: "Some S are P" becomes "Some S are not non-P".

O-type proposition: "Some S are not P" becomes "Some S are non-P".

I see two different unary operators "not" and "non-". I am informed that they are not equivalent, as "not" defines a negative proposition type (E or O) while "non-" is considered to define a positive proposition type.

My question is whether there is a meaningful distinction between these two operators. If there is a meaningful distinction, I see no explanation or interpretation of how "not non-P" differs from "P", and the list of examples (that seems to be repeated in multiple sources) seems woefully incomplete. If there is not a meaningful distinction coming from Aristotle, is there a meaningful distinction for modern logics (particularly paraconsistent logics)?

May 4:

Having read and considered the one answer to date and various comments and spent a good deal of time reading SEP (Aristotle's Logic, and The Traditional Square of Opposition), I am ready to start responding and following up.

Based on the answer and comments, it seems that my Apr 30 comparison of "not non-P" to "P" was malformed - it is effectively comparing one-and-a-half components of a proposition with one component - and I should instead compare "non-P" to "P" or "is-not non-P" to "is P". So part of my initial misunderstanding is resolved, I believe.

I like the replacement of "All" with "Every" that Mauro ALLEGANZA has been systematically using. The change prevents the need for a difference in number agreement (is/are) in the propositions, which reduces the amount of interpretation of what is changing from one proposition to another; and this change effectively eliminates the suggestion of a whole having properties that its parts may not share. ("The whole is greater than the sum of its parts" is something that Aristotle made famous.)

I also like Mauro ALLEGANZA's replacement of the complicated construction that operationally pairs a quantifier with "is" and infixes the subject. Using the "belongs to" relation makes proposition construction and proposition transformations both simpler.

What I do not see in any of the responses here or in the common presentations of obversion and contraposition is any treatment of existential import (i.e., which quantifiers should evaluate as true/false when one entity or the other or both is empty). SEP makes a good argument that "every" necessarily has existential import (i.e., "every" is not satisfied when the subject is empty) and that the O-proposition must not have existential import.

The arguments that SEP makes about existential import depend on the "contradictory" and "contrary" relationships between proposition types and on what is logically necessary to make Aristotle's syllogistic examples correct.

None of these necessary clarifications of the meaning of the proposition types is apparent in the examples or definitions of obversion that are currently presented on Wikipedia, for example. So, the common presentations of this aspect of so-called Aristotelean logic are indeed woefully incomplete, just as I worried on Apr 30. (I say so-called because I understand from SEP that Aristotle did not endorse contraposition and wrote about a few forms of obversion, but mainly these transformations were added between the 12th and 14th centuries CE and were then taught as part of "Aristotelean logic".)

Regarding paraconsistent logics, if the logical system evaluates correctly when empty entities are included, then it is compatible with a Kleene logic. Kleene logics and Priest logics are operationally identical but have different interpretations: Kleene nulls are unknown or undetermined, while Priest nulls are contradictory or overdetermined. A Priest logic need not be explosive and would therefore be paraconsistent. Consequently, a logic that is valid in the presence of empty entities could be paraconsistent. However, I was really wondering if there might be a more substantial difference between "not" and "non-" that might have led to something more interesting; but, obviously, that wonder is forlorn.

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  • The difference is merely pedantic, "not P" applies negation to a predicate, "non-P" creates a new negated predicate. The main uses of this are to uphold the distinction between affirmative and negative propositions, and to make the double negation "not non-P" sound more "naturally". However, Aristotle is not an intuitionist, so "not non-P" reduces to P, and, according to Oxford reference, "there is no good logical or philosophical way of making the distinction. Modern logic allows that there is frequently an equivalence between positive-looking and negative looking statements". – Conifold Apr 30 '20 at 18:28
  • Are you asking what's the difference between privation and negation? – Geremia May 6 '20 at 3:17
  • @Geremia - I don't know that I'm asking that question, but if you think it is relevant, then perhaps it is. (I understand set-wise vs logical complementation, and I understand the linguistic property of privation - that, for example, "unalive" is not usually acceptable language while "toothless" is acceptable for something that is expected to have teeth. I do not really know what Aristotle thought was expected of logical utterances. – lambdakiki May 6 '20 at 4:25
  • You are likely under the impression that "all logic is logic" and that it is all the same thing. You would be wrong about that. Different logic systems may have different rules that do not carry over. Only in Mathematical logic does existiental import arise. It would never arise in Aristotelian logic. The non is a prefix to a noun. The not is a word by itself & makes the proposition negative. The non prefix keeps the proposition POSITIVE. There are distinctions between non & not. They do not always mean the same thing. Sometimes people take them to be equal but this is not reliable. – Logikal Sep 6 '20 at 7:54
  • Take for example, the I type proposition some s are non-p. This is distinct from the O type proposition some s is NOT. I can swap the subject & predicate terms and still have truth preserved. Do that with an O type proposition & you will see it is not 100% accurate. It fails at times. Then we have distinct terms such as symmetrical, asymmetrical (the not case) & non-symmetrical. Another set is transitive, intransitive ( the not case) & non transitive. These are distinct terms that don't overlap. The non prefix usually indicates a condition where the truth value can alternate depending on info. – Logikal Sep 6 '20 at 8:01
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Part of the explanation is with translation: the original ancient Greek text can be translated in different ways.

The issue is: what is the best way to translate the negation used to negate a statement:

"every S is P", whose negation is "not every S is P",

compared with the assertion of a "privative" predicate:

"every S is not-P" ?

See e.g.Prior An., 25a14-25a27 [translated by A.J. Jenkinson, from J.Barnes edition of A's Complete Works]:

take a universal negative with the terms A and B. Now if A belongs to no B, B will not belong to any A.

Slightly different into R.Smith's translation:

let premise AB be universally privative. Now, if A belongs to none of the Bs, then neither will B belong to any of the As.

Having said that, there is no place in A's treatment of negation supporting a paraconsistent reading.


At a little more detailed level of discussion, we have to consider that what we today translate with quantifiers are not exactly what Aristotle meant.

The four forms of categorical propositions may be translated as follows:

PaS: ‘P belongs to every S’,

PeS: ‘P belongs to no S’,

PiS: ‘P belongs to some S’,

PoS: ‘P does not belong to every S’.

We may note that we have here three "quantifiers" acting on the subject: "every", "none" and "some".

We have only one relation: "belong to" and we have a standard negation acting on statements.

See De Int, 18a34:

For if every affirmation or negation is true or false it is necessary for either to be the case or not to be the case.

[...] every affirmation will contain either a name and a verb or an indefinite name and a verb. Without a verb there will be no affirmation or negation. [...] when ‘is’ is predicated additionally as a third thing, there are two ways of expressing opposition. Because of this there will here be four cases [...] I mean that ‘is’ will be added either to ‘just’ or to ‘not-just’, and so, too, will the negation. Thus there will be four cases. What is meant should be clear from the following diagram:

(a) ‘a man is just’

(b) ‘a man is not just’. This is the negation of (a).

(c) ‘a man is not-just’

(d) ‘a man is not not-just’. This is the negation of (c).

And see Prio An., 51b5:

the expressions ‘not to be this’ and ‘to be not-this’ [...] do not mean the same thing. [...] Therefore it is clear that ‘it is not-good’ is not the denial of ‘it is good’. If then of every single thing either the affirmation or the negation is true if it is not a negation clearly it must in a sense be an affirmation. But every affirmation has a corresponding negation. The negation then of this is ‘it is not not-good’.

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  • As I understand it for propositions Aristotle stipulates the negation must be of the copula. So 'Every S is P' would become 'Every S is-not P'. Is this not correct? Perhaps the rule is different where the quantifiers 'Every' and 'Some' are used. – user20253 Apr 30 '20 at 10:44
  • @Peter J, no I would say only particular propositions have the negation through the copula. Universals are negated at the quantifier. Every s is p is properly expressed as All s is p. To negate All s is p all I need to do is change the QUANTIFIER. The copula has not changed. There is no such thing as EVERY S IS NOT P. Propositions are not literally sentences. So it doesn't matter how it reads whatsoever. What is being expressed is what is important. All s is not p is best interpreted as Some s in not p. Any other issue is the problem of the communicator. The speaker has to be clear. – Logikal Apr 30 '20 at 14:02
  • @Logikal - Thanks. I wasn't sure about universals and quantifiers. . – user20253 Apr 30 '20 at 14:25
  • @Peter J, no problem at all. If I want to negate SOME S IS P I would now add NOT to the quantifier. I would like to be clear that adding NON does not attach to a copula. NON will attach to a subject or a predicate. Also notice Some s is not p is a particular NEGATIVE. If I have SOME S IS NON P that is not a NEGATIVE proposition. It happens to be an affirmative particular. So the OP has a point. A non p is indeed distinct from a IS NOT P type proposition. All nons are not to be translated as negations. However math people do this all the time in Mathematical logic. Thus why the confusion. – Logikal Apr 30 '20 at 14:55
  • @Logikal - I think I see the point and it's an interesting one. – user20253 Apr 30 '20 at 16:24

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