Talk:Transposition (logic)/Archive 1

Same as Contrapositive?

I can't see the difference between transposition (logic) and contrapositive (logic). Should we merge? --Michael C. Price ^talk 09:01, 29 March 2009 (UTC)

No. If you can't see the difference then all our previous discussions has been for naught. You need to pop a little more than books on symbolic logic into your shopping cart. Amerindianarts (talk) 12:49, 29 March 2009 (UTC)

Can Amerindianarts please clarify -- is contraposition the application of transposition to the extremes? What I mean is that if A is the statement that "P implies Q", then the transposition of A is "not Q implies not P" and there will then be the contrapositive statements "All P are Q" and "No not-Q is P". i.e. I'm saying that transposition deals with the general logic, while contraposition is only dealing with the All-or-Nothing condition. Is that true? If it is, then we can say that contraposition is a special case of transposition and this may merit some form of merging (because my last contribution to the contrapositive page actually then applies to transposition). If not, then there should be a clear and definite statement that defines the differences, because I really cannot understand the differences. Asking us to buy books on symbolic logic will not be a good answer if they apply the same form of argument that both Price and me cannot understand. So, please offer another explanation (preferably in concrete examples). Thank you. Burning.flamer (talk) 03:41, 30 March 2010 (UTC)

Oops, forgot to check for even more cases. There seems to be also the talk of Obversion between the 2 of you, so what is that about too? Burning.flamer (talk) 15:34, 30 March 2010 (UTC)

As the articles are currently written, this is the only difference:

• Contraposition: The contrapositive of an implicational statement is another implicational statement
• Transposition (logic): Transposition is a rule of inference that says that from an implicational statement you may infer the contrapositive of the statement
• Contraposition (traditional logic) is not about contrapositives in the modern sense, but about syllogistic logic.

So the first one is about a syntactic transformation of statements, while the second is about a rule of inference and the third is about syllogistic logic. The source of confusion here seems to be the focus on syllogistic logic without clear warnings that this is what's going on. — Carl (CBM · talk) 15:49, 30 March 2010 (UTC)

Dear Carl, it seems that there is, yet again, abstract reasoning without concrete examples. Do you mean that transposition is the rule that states that contrapositives are the same as each other (hence usable in a logical discussion) while contraposition is the act of finding the contrapositive? i.e. The statement that "All cats are animals" has a contrapositive "No cat isn't an animal". Contraposition is to find this contrapositive statement from the original, and transposition is the rule stating that you can always do so without changing the meaning, so both statements can be used in a discussion to mean the exact same thing. If this is the case, then my section on the simple proof is more applicable to transposition than contraposition. Burning.flamer (talk) 17:22, 30 March 2010 (UTC)

I was just describing how the articles appear to be written. My background is in mathematical logic, not in syllogisms, which have only historical interest at this point.

In general terms, I think that "transposition is the rule that states that contrapositives are the same as each other (hence usable in a logical discussion) while contraposition is the act of finding the contrapositive" matches the structure of our articles.

There are two different meanings of "contrapositive", it seems. One is for syllogisms, while the other is for implicational statements. The article contraposition (traditional logic) is about the syllogistic ones, while contraposition is about the implicational ones. So examples such as "All cats are animals" are only relevant to contraposition (traditional logic), since they are not implicational statements.

In contemporary terms, one has to start by translating statements such as "No A is B" and "Every B is not A" into implicational formulas, before it even makes sense to talk about contrapositives. Moreover, according to contraposition (traditional logic), not every syllogistic phrase even has a contrapositive in the sense of syllogistic logic. — Carl (CBM · talk) 18:17, 30 March 2010 (UTC)

Ah, that is all fine and good, and I thank you for trying to clarify. I seriously would rather that you are definitive (which you say you are not, despite being versed in mathematical logic -- I got here from the page on mathematical proof, anyway). However, your explanation is yet again full of jargon and abstractions that are simply just not conducive to understanding. What do "syllogisms" and "implicational statements" mean? The relevant wiki entries are just as vague as these are.

From what I have read, syllogisms seem to mean "to use prior statements to deduce further implications" i.e. the method of using axioms to directly proof further facts. If my reading is correct, then what is the use of differentiating between the use of contraposition for syllogisms and for implicational statements? I mean, "No A is B" is always translatable into "All A is not B" which is equivalent to "A implies not B" which is now an "implicational statement" (I read it this way, unless you have a more precise definition that says I am wrong).

Anyway, I really just mean to understand why the extreme sensitivity employed. Why differentiate between these cases if you cannot even give concrete examples as to its utility? But I digress. Thank you for helping to clear the doubts, and also to confirm that, as per the current writing, I am not baseless. So, some higher-level being please enlighten us asap. (I mean, dear Carl, please do continue the discourse too. Thank you too!) Burning.flamer (talk) 00:19, 31 March 2010 (UTC)

By syllogistic logic, I mean traditional logic. The individual statements in this logic are phrases of these four forms:

Every A is a B

No A is a B

Some A is a B

Some A is not a B

A syllogism is a particular type of argument that deduces a phrase of one of these forms from two other phrases of these forms. All these phrases are called "propositions" in traditional logic. Some, but not all, propositions have a "contrapositive" in traditional logic, as explained at contraposition (traditional logic). The ones that don't have contrapositives are of the form "Some A is B".

In modern logic, that sort of traditional logic is replaced by formal logical systems such as first order logic. Some statements in these logics are in the form of an implication: $A\to B$. Every such implicational formula has a contrapositive, as described at contraposition. The contrapositive of $A\to B$ is $\lnot B\to \lnot A$. This is a different meaning of "contrapositive" than in syllogistic logic.

So the sensitivity is that not every "proposition" of syllogistic logic has a "contrapositve" in the sense of syllogistic logic, while every implicational formula of first-order logic has a "contrapositive" in the sense of first-order logic. — Carl (CBM · talk) 01:09, 31 March 2010 (UTC)

Ah! Thank you! It now seems to me that what you mean by traditional syllogistic logic is simply just a form of logic that allows the "Some A is B", which is murkier than we would accept in modern logic. Because implicational formulas are absolute i.e. they can only be used to convey the "All A are B" or the "No A is B" cases, they will always have contrapositives and hence having the differentiation.

Now, I would venture to say that this form of differentiation would then be controversial. I state this because we do not really have to separate the 2 concepts, as long as we keep them clear. For example, in geometry, we learn that 4 sided figures are quadrilaterals, with some special cases. They are the trapezium, the parallelogram, the rectangle, the rhombus and the square. Each special case has more and more restrictions and these restrictions simplify the analysis of problems involving these special cases. Similarly, vectors and matrices are intimately related, and the matrix product exists for both. Under computer science, both are represented as lists. In Wolfram Mathematica, the matrix product is simply the dot product. What happened here is operator overloading -- if the operation is meaningless in the context, it will simply return an error. Or even in mathematics, the cross product is only valid in 3D space. It still does not hinder its use in analysing a problem, until we leave 3D space.

What this means is that contraposition is invalid on the general syllogism space, and we simply need to define the subset that we know as first order logic that has the restriction of being only A or E type statements, instead of all 4 AEIO. Then, we can define contraposition to only work in this subset with no loss of generality. Moreover, attaching this very discussion will help to clarify the rational behind this restriction and help readers understand why such a scheme is useful. (Hence, maybe a small scale merge is in order. Or at least a huge and noticeable disclaimer about why the current state of events are as it is.) Burning.flamer (talk) 17:54, 31 March 2010 (UTC)

There are also more general things that can be stated in first order logic but not in syllogistic logic. For example: (*) "for every person x there is another person y such that y likes every person that x does not like". Syllogistic logic only admits propositions of the four special forms listed above, and so it is only able to reason about a few types of logical relationships. — Carl (CBM · talk) 18:38, 31 March 2010 (UTC)

Wow, so you mean first order logic is not a subset of syllogistic logic. Okay, I don't even see how to simply find the contrapositive of that statement you have just posted. What do you suggest we do about this situation now? Clearly, leaving it as it is is not a good idea because all the abstractions are hiding such important implications. Also, regarding the simple venn diagram proof I had posted on contraposition, should it be copied over to transposition too? Burning.flamer (talk) 23:47, 31 March 2010 (UTC)

The statement I posted above as (*) is not an implication so it does not have a contrapositive. As for improving the articles, I don't have a lot of suggestions. I don't know why they are written with such a focus on syllogistic logic. I was just explaining what is going on, in case someone else sees a way to improve things. Maybe Amerindianarts has an idea. — Carl (CBM · talk) 00:10, 1 April 2010 (UTC)