Re: inverse/obverse

Hi Mike & Ken:

[Oof. This Google groups page is definitely a beta. I just hit the
'tab' key and lost my entire post. Hope you don't get it twice.
Apologies in advance, in that sorry case.]

I don't think that we've answered Mike's original questions. Here, I'd
like to clarify the difference between the obverse and the inverse.
Originally--and not without reason, given the coverage in his two
textbooks--Mike thought that obversion and inversion were different
names for the same transformations of categorical logic propositions. I
believe, however, that they are distinct.

Inversion isn't explained very well, except in research monographs. For
example, P.F. Stawson, in his classic, "Introduction to Logical
Theory," London: Methuen, 1963, explains _obversion_ completely (p.
157), but then follows up with contraposition and inversion, only to
explicate _contraposition_ and ignore _inversion_! Why is this? Perhaps
because obversion enjoys complete validity and inversion is
problematic.

In fact, _obversion_ is different from _inversion_, even though the
names of the two are kind of similar. You have to pay attention to the
technical meaning of the terms, as Ken pointed out earlier. It's a
terminological mess, and the logical issues are not completely resolved
to this very day, I think. And it could be that that's why Mike's
professor balked at a more succinct answer.

To answer the question: Obversion and inversion are not the same, if
I'm not mistaken.

(1) In obversion, one negates the predicate and changes the quality of
the categorical proposition. For example, if I first say "All human
beings are animals", I can conclude, validly, that "No human beings are
non-animals". In categorical logic, this means that

(1a) sAp -> sEp', where p' is the negation of the predicate p.

and obversion is valid for the other four types of categorical
proposition:

(1b) sEp -> sAp'; "no human being is an animal" implies "all human
beings are non-animals"
(1c) sIp -> sOp'; "some human being is an animal" implies "some human
being is not a non-animal"
(1d) sOp -> sIp'; "some human being is not an animal" implies "some
human being is a non-animal".

So that's obversion.

Inversion is different, however, and it does not enjoy such general
validity in transforming categorical propositions. Indeed, there are
two case of inversion: (2) partial inversion and (3) full inversion.

(2) In partial inversion, one negates the subject and changes the
quantity and quality of the proposition:

(2a) sAp -> s'Op; "all human beings are animals" implies "some
non-human being is not an animal"; [This is not valid unless there is
something that is not an animal.]

(2b) sEp -> s'Ip; "no human being is an animal" implies "some non-human
being is an animal"; [This is not valid unless there is something that
is an animal.]

(3) In full inversion, one negates the subject and the predicate and
changes the quantity of the proposition:

(3a) sAp -> s'Ip'; "all human beings are animals" implies "some
non-human being is a non-animal"; [This is not valid unless there is
something that is not an animal.]

(3b) sEp -> s'Op'; "no human being is an animal" implies "some
non-human being is not a non-animal"; [This is not valid unless there
is something that is an animal.]

Well, I hope I've got these correctly defined. Online, you might want
to have a look at Terence Parson's page about Aristotle's "Square of
Opposition" in the Stanford Encyclopedia of Philosophy:
http://plato.stanford.edu/entries/square/

Also, there is a new Google discussion group on Aristotle's logic,
http://groups.google.com/group/aristotle-logic?lnk=li&hl=en

which has started with the "Topics" and "Categories." Aristotle's
"Categories," of course, is about what kinds of things can be the
predicate of a categorical proposition. The "Topics" is a lot easier to
read than the "Categories," however, and it explains how the breakdown
of things into their categories (or predicates) is useful. Soon enough,
I'll initiate a topic in aristotle-logic on Google Groups that covers
"On Interpretation." There, in the 2nd and 3d books, Aristotle gets
right into the negation of predicate terms.

Just to tantalize you about "obversion":

(1) Parsons says in his online article that "Aristotle discussed some
instances of obversion in 'De Interpretatione'."

(2) But Gunther Patzig says that "...if we appeal to the laws of
obversion and contraposition--which Aristotle never uses and never
mentions...." ("Aristotle's Theory of the Syllogism," Dordrecht: D.
Reidel, 1968).

I hope to get to these issues in the coming months over at the
aristotle-logic group
(http://groups.google.com/group/aristotle-logic?lnk=li&hl=en).

Thanks!
--Ron Allen

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