Talk:Converse (logic)

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[edit] Converse or inverse

This article should be merged with inverse (logic)? Gene.arboit 02:18, 11 September 2005 (UTC)

I don't think the two articles should be merged. The converse is a much more imoprtant and complex idea in mathematics than the logical inverse is. For example, consider the following proposition:
A → (B → C)
A naïve understanding of the converse says that the converse of this is:
(B → C) → A
But in fact, as mathematicains and logicians understand and use the word "converse", the original proposition is understood to be equivalent to :
(A ∧ B) → C
and so its converse is:
C → (A ∧ B).
and in general, one wants equivalent statements to have equivalent converses, which is not true of the simple definition of the converse of (A→B) as being (B→A). -- Dominus 04:06, 12 September 2005 (UTC)
Agreed. Amerindianarts 09:48, 13 September 2005 (UTC)
I think I see the point. (B → C) → A, i.e. (B ∧ ¬ C) ∨ A, is not equivalent to C → (A ∧ B), i.e. ¬ C ∨ (A ∧ B). Should this be mentioned in the converse article, perhaps? Gene.arboit 02:57, 14 September 2005 (UTC)
I don't think it should be in the article. It seems superfluous really. I might be wrong, I'm a logician and not a mathematician, and there are differences, I know. Thus, I'm a little confused about conversion as a process of immediate inference being illustrated in terms of hypotheticals and material implication symbolically. We use it as a simple rule in traditional logic as a means in contraposition, and it is used in set theory. But the rule of exportation used above [A → (B → C) iff (A ∧ B) → C] is a rule of inference in deductive systems that don't use the rule of conversion.Amerindianarts 06:35, 14 September 2005 (UTC)
I did not mention the rule of exportation as a central issue here, only as an example that when mathematicians (and logicians) talk about the converse of a statement of the form (B → C), they do not always mean (C → B). They might mean one of several different things, depending on the context. My point is that "converse" is a more complex concept than is suggested by the article, and that if the information in the article is considered to be a complete description of the notion of the "converse", then the article is wrong.
For example, consider the theorem that says:
If a, b, and c are positive real numbers, and ac > bc, then a > b.
Is the converse of this theorem also true? I think all mathematicians will agree that it is.
The theorem is clearly a statement of the form (B → C). If one takes the article at face value, and applies its definition, one will conclude that the converse of the theorem is:
If a > b, then a, b, and c are positive real numbers, and ac > bc.
This is not only false, but absurd. But mathematicians nevertheless will say that the converse of the theorem is true, not false.
The only thing wrong is that the converse of the statement is not what the article says it is. So the article gives an incorrect and incomplete understanding of its subject, and that's why I think it should be changed. -- Dominus 13:15, 14 September 2005 (UTC)

I agree with you, I think, if I'm understanding what you say, but I know nothing about the meaning of "converse" in mathematics. In traditional logic "converse" is limited to the results of the process of conversion. Other than that it becomes a dictionary definition or a mathematical concept. The converse stated in the article is wrong, I agree. The converse in traditional logic is more complex and the correct converse as accepted by traditional logicians is not stated in the article. Also, It has nothing to do with truth tables because its validity is measured by the rules of distribution. I think your notion of converse is much larger than mine and I have tried to limit it per definition within traditional logic. I started a new article at Conversion (logic) to try to fix this. Inversion is also a much more complex issue and I think because of all the controversy on the validity of Inversion (in traditional logic) that it would require an article of its own which would probably be essay for the most part. Amerindianarts 16:13, 14 September 2005 (UTC) I think that the point I was trying to make is that within the logic I know, "converse" has a limited function as a technical term or application. It is not used in truth-functional logic, quantification theory, or modal logic. "Conversion" is used in set theory, but I’m not knowledgeable about its application. This is not to say that the term "converse" does not have its uses because just about everything has its converse, and I don’t know how the term is used in mathematical logic. But we have to be careful about what it is we are trying to define, e.g. some broader application, or as a technical term. What is the scope and the range of "converse"? Another example is "contrapositive", or "contraposition". As a process within logic it is limited to traditional logic and requires a reasoning process involving obversion, conversion, and obversion again. But in truth functional logic this type of inference is replaced with the rule of transposition. You can call transposition "contraposition" if you like, but that is not what it is.Amerindianarts 17:09, 14 September 2005 (UTC)

@Dominus: In mathematics, the formation of the converse depends on the choice of a subset of the hypotheses which corresponds to some sort of "setting", like "a, b, c are positive reals" in the above case. Once this choice has been made, the theorem can be written in the form
S → (B → C),
where S is the "setting" and B is the remaining hypotheses. Then, the "correct" converse is
S → (C → B).
--Gwaihir 00:03, 15 September 2005 (UTC)
Indeed. But that is not what the article says, is it? -- Dominus 14:24, 15 September 2005 (UTC)

Could you explain the difference (if there is one) between converse and inverse? 134.250.72.141

First, don't confuse the method (conversion or inversion) with the product of the method (converse or inverse). In conversion the subject of the original proposition is the predicate of the inferred proposition and the predicate of the original proposition is the subject of the inferred proposition, with the quality of the original proposition retained. It is valid only for E and I propositions, and A propositions with limitations. The process of inversion is where the subject of the inferred proposition is the contradictory of the original proposition, and requires successive conversion and obversion. It is valid only for A and E propositions. The inverse of I and O propositions cannot be obtained because attempts to derive a non-S subject yield an O proposition which cannot be converted. "Inversion" is barely recognized by logicians and is a term invented by Keynes. Amerindianarts 18:01, 8 November 2005 (UTC)

[edit] Conversion and converse

If this receives no objections within a reasonable period of time I am going to redirect "converse" to the article "Conversion". The reason is that the article "converse" deals with the product of the method of inference "conversion". The two are not the same, conversion being a process which may require other rules, and the converse simply the product of conversion. The article "converse" also deals with truth tables and implication, which conversion is not. Conversion is a rule of categorical statements, not implication, and validity for categorical statements of traditional logic utilizes Venn diagrams and not truth tables. This is in reference to philosophical logic. Anything other than this may require an article on the converse (mathematics).Amerindianarts 01:21, 1 December 2005 (UTC)