Talk:Obversion

Implication

Is the obverse of

$A\Rightarrow B$

just

$-B\Rightarrow -A$ ?

--Michael C. Price ^talk 10:37, 27 March 2009 (UTC)

Not the obverse. Your language is of conditional statements (hypotheticals, material implications). The rule of replacement in the language of your example involves the Transposition rule of replacement ([[1]]).

Obversion is a rule of class logic. The language in this article is of Aristotelian logic and used symbolically in set theory. For example: "All cats are animals" and "No cats are non-animals" is the rule of obversion in set theory. Your material implication above states that the obverse of "If it is a cat then it is an animal" is "If it is not a cat then it is not an animal", which is obviously false. One translation is "It is not the case that (If it is a cat then it is not an animal)", or "It is not the case that something is a cat and not an animal" Amerindianarts (talk) 14:37, 27 March 2009 (UTC)

Isn't the obverse of your example simply "if it is not an animal then it is not a cat", which is true, since the antecedent was swapped with the consequent?

Since the converse of

$A\Rightarrow B$

is just (I hope)

$B\Rightarrow A$,

I was hoping there would be simple formula for the obverse as well.

--Michael C. Price ^talk 15:31, 27 March 2009 (UTC)

I terms of class logic you are confusing the rules of obversion and contrapostion. Amerindianarts (talk) 19:43, 27 March 2009 (UTC)

I'll try again. Your proposition is the rule of replacement in methods of deduction called transposition. Its likely equivalent in class logic is contraposition, but in terms of class logic it is called the obverted contrapositive. This is because it is technically of the 'E' form of propositions and not the 'A' form. You can get from the obvert to the contrapositive to the obverted contrapositive in a manner of steps which in themselves are types of immediate inference applying the rules of distribution in class logic. Transposition as a rule of replacement preliminary to quantification logic accounts for all of these steps. In that sense it is equivalent. But in terms of distribution they are not. In the statement "All cats are animals" the subject is distributed and the predicate is not. In the statement "If it is not a cat then it is not an animal" the same rules of distribution do not apply. "All cats are animals" is a categorical statement. "If it is not a cat then it is not an animal" is conditional. There is a difference in form, distribution, and statement or proposition type. Amerindianarts (talk) 20:28, 27 March 2009 (UTC)

PS. The converse of "All cats are animals" is "Some animals are cats". In the language of conditionals this requires the use of existential quantifiers.Amerindianarts (talk) 20:57, 27 March 2009 (UTC)

But, according to mathworld, implications (or "if .. then .." statements, if your prefer) can be conversed:

Given the statement "if P, then Q," or P=>Q, the converse is "if Q, then P.

So my question is still, can they be obversed? --Michael C. Price ^talk 02:08, 28 March 2009 (UTC)

Yes. What you call obversion is the rule of replacement: Transposition. See Copi, Symbolic logic. It is a valid inference. As far as conversion, you can do it all you want, but, it IS NOT a valid inference, Amerindianarts (talk) 15:22, 28 March 2009 (UTC)

If by that you mean that an implication does not imply its converse, yes, I'm okay with that. Thanks for the Copi reference. The definition of transposition looks the same as contrapositive. Is that right? Are they synonyms? --Michael C. Price ^talk 16:29, 28 March 2009 (UTC)

In regard to prior conversations: I'm not a mathematician. In mathematics there may be some form of conversion where this is valid, but not in philosophical logic. From "all cats are animals" you can only deduce "some animals are cats". You cannot deduce "all animals are cats", or even "some animals are not cats" without at least an auxillary proposition such as "a dog is an animal". Thanks for your patience.

In regard to your last comment tranposition is actually the "obverted contrapositive". For this I refer you to Stebbing's Intro to logic, or some other good logic book that may cover the square of opposition, rules of distribution, etc. The contrapositive of "all cats are animals" is "No non-animal is a cat" and the obverted contrapositive is "All non-animals are non-cats" which is very clumsy. The last two examples show the problems of traditional logic and logic of natural language that quantification and conditionals have tried to remedy. For example, the double negative of ""No non-animal is a cat" ". This must be given a form other than a conditional or it is the equivalent of "~~q > p" or "q > p" ( or ~ q and ~p)which is not a valid inference. This can be done by eliminating the material implications of the hypothetical and converting to the negation of a conjunction. In traditional logic it follows the rules of subject and predicate distribution and the notion of "quantity". The rules of inference in symbolic follow this argument.

1) p > q ("all cats are animals")

2) ~(~q . p) ("Nothing is a non-animal and a cat")

3) by material implication, de morgens, and double negation: ~q > ~p (If it not an animal then it is not a cat)

There are many steps in between the propositions but I'd rather not risk natural language translations. Transposition presupposes itself in order to prove itself, more or less. Aristotle, despite the limitations of traditional logic, was a master of natural language arguments.

It should now be obvious that conversion of "p > q" to "q > p" is not valid in traditional logic, symbolic logic, and for the most part in natural language. The only way that conversion is valid is if the rule of equivocation is stipulated, i.e. " p > q if and only if q > p where both p and q are both the necessary and sufficient reasons (conditions) for each other.

I hope I have been clear and I hope it helps. You would probably do better reading the books I recommended. If you really want to get down to it, read Aristotle's Posterior Analytics Amerindianarts (talk) 18:34, 28 March 2009 (UTC)

Thanks. I've popped a few symbolic logic books into my shopping basket, so hopefully that should help clear a few things up. --Michael C. Price ^talk 23:57, 28 March 2009 (UTC)

I think the key here is that Obversion refers to categorization and sets, not to conditional statements. So, the proper way to represent it as a mathematical expression would be

"The obverse of A ∈ B is A ∉ -B."

Basically, if A is a subset of B, then it's not a subset of non-B.192.249.47.196 (talk) 17:08, 7 November 2011 (UTC)