Talk:Inverse (logic)/Archive

Old stuff

Seems wrong, because: In predicate logic, the contrapositive of the statement "p implies q" is "not-q implies not-p." These are logically equivalent. Patrick 01:00 Dec 9, 2002 (UTC)

Agreed. Removing " In general, the truth or falsity of S says nothing about the truth or falsity of its inverse." -- Tarquin 11:47 Dec 9, 2002 (UTC)

Why would that be removed? The inverse is the contrapositive of the converse, so if the inverse is true, the converse is true. Ergo, if both a hypothetical proposition "P ⇒ Q" (equivalent to "¬Q ⇒ ¬P") and its converse "Q ⇒ P" (equivalent to "¬P ⇒ ¬Q", the inverse) is true, then the antecedent and the consequent are logically equivalent, i.e., P ⇔ Q. The truth or falsity of a hypothetical proposition S says nothing about the truth or falsity of its converse S′, and thus says nothing about the truth or falsity about the converse's logical equivalent – the inverse – either. See the truth table for reference. – 80.203.113.135 01:06, 8 June 2006 (UTC)

The article is absolutely fine the way it is. I don't know anything about "obversion" and "contraposition", but in mathematics and symbolic logic one can talk about the converse, inverse, and contrapositive of a given statement "p => q". The converse is "q => p", the inverse is "~p => ~q", and the contrapositive is "~q => ~p". The statement and its contrapositive are logically equivalent, as are the converse and inverse. -- kier07 16:27, 19 July 2006 (UTC)

No, the article isn't fine the way it is. I refer you to your basic college texts for logic and you need to read the section below here, re: "tags". In symbolic logic "q => p" is not a valid equivalent of "p => q". You can call it the converse if you like, but it is not logically equivalent. To infer "q => p" from "p => q" commits the fallcy of Affirming_the_consequent. Pretty simple stuff, really. From "p => q" you can infer that there is a "q" which implies "p", but it is a fallacy to assert that "q => p" can be universally implied from "p => q". In order to prove "p => q" and "q => p" are logically equivalent the proof requires auxillary premises which would state a bi-conditional relationship between the propositions. This is not conversion, nor has anything to do with inversion. Amerindianarts 16:52, 19 July 2006 (UTC)

I have read the re: "tags" section, and I guess I understand what the issue is. As Awis noted, converse, inverse, and contrapositive have a certain meaning in propositional logic. In propositional logic, it's a lot simpler than what you're describing. I will give you an example. The original statement might be "if a man is a politician, then he is corrupt." Then the converse would be "if a man is corrupt, then he is a politician", the inverse "if it is not the case that a man is a politician, then it is not the case that he is corrupt", and the contrapositive "if it is not the case that a man is corrupt, then it is not the case that he is a politician". You can look this up in any dictionary and many math books. Perhaps there should be two articles: inverse (propositional logic) and inverse (traditional logic), because these concepts seem quite different. And, if this article were "inverse (propositional logic)", then it WOULD be absolutely fine the way it is. -- kier07 21:47, 19 July 2006 (UTC)

You may be right (I ask for sources the degree of which I supply and not general references to books) but Copi makes no reference to inversion as a rule of inference in Symbolic Logic. True, if "The original statement might be "if a man is a politician, then he is corrupt." Then the converse would be "if a man is corrupt, then he is a politician", but it is not an equivalent inference, which is what the rules of inference in symbolic logic are based upon. The valid or equivalent conversion of "if a man is a politician, then he is corrupt" is "if a man is corrupt then he might possibly be a politician". One reason that inversion disappears from logic books after traditional logic is because it really has no practical application, and an equivalent is a possibility only in remote situations. A universal affirmative proposition has no inverse which is logically equivalent except with a change in quantity, but the possibility of a valid inversion remains. From a dictionary of philosophy I have taken the following definition of "Inversion"-- "In traditional formal logic, replacement of a proposition by a logically equivalent one, its inverse, having as subject the negation of the original subject, e.g. 'Some non-cats are non-black' is the inverse of 'All cats are black'. This corresponds to Stebbing' and Copi's essays in college text books, and the definition given in the 'Glossary of Logical Term's in Mcmillan's Encyclopeida of Philosophy. We are dealing with logical equivalence and a universal affirmative has no logical equivalent inverse, and "if a man is a politician, then he is corrupt." Then the converse would be "if a man is corrupt, then he is a politician", 'is not a logically equivalent converse.Amerindianarts 22:32, 19 July 2006 (UTC)

See for example moonbase.wwc.edu/~aabyan/Articles/ProofStyles.pdf; the usage of the term "inverse" that I am familiar with is mentioned there. --kier07 1:12, 20 July 2006

Bad link, check it. Amerindianarts 01:34, 20 July 2006 (UTC)

I think the URL works. If not, try moonbase.wwc.edu/~aabyan/Articles, and go to the ProofStyles one. I see that his articles are probably not of the published kind; they're his casual writings. Still, he is a computer science professor at a university. But I know that the definition of inverse, converse, contrapositive that I'm familiar with is well known. See also Merriam Webster's online dictionary, for example. --kier07 16:22, 20 July 2006 (UTC)

Dictionary definitions have no bearing here. Nix that. See Conversion (logic), Transposition (logic), Contraposition, and Obversion

Correct page for your reference is here

The proof given for conversion in the article you refer (actually given as a rule of replacement, and not a proof) is

A → B ≡ ~A v B, BUT;

by rule of replacement "Commutation" in symbolism you get

A → B ≡ B v ~A

then A → B ≡ ~B → ~A again by implication as a rule of replacement which proves the rule of transposition (or the law of contraposition), and not conversion. In order to get A → B ≡ B → A you must use inversion, which as a rule and according to the current article here at WIki, is not a logical equivocation. It would seem that the entire definition in the article is based upon a fallacy. Thus, I'm really confused by this definition and there may be some distinction between computer language and the philosopher's symbolic logic of natural language. But that definition of conversion is a fallacy in college texts on formal logic and I don't know how they get around that, and that doesn't change the fact that the current article would then assume too much (or not enough) about inversion in logic, and still needs a rewrite explaining these differences. So, the current article is not "fine". Even if there are separate articles on the different renditions, the article needs to show a little more expertise and explanation. You need to check the books for formal logic I have cited and you will see that the whole story is being neglected here. It needs to explain the differences in definition which the traditional logician would experience in reading the article. If the traditional definition refers to only a change in the quality of the subject with no reference to a change in quality of the predicate, then there needs to be an explanation of how the truth tables can apply.--Amerindianarts 18:03, 20 July 2006 (UTC)

First of all, can you tell me why dictionary definitions do not apply here? Seems to me we were looking for published sources that explain different possible meanings of these words. A dictionary seems like a great idea in that case. Is there some Wikipedia page you can point me to that explains why dictionary definitions are not considered?

Some observations. The articles conversion (logic), contraposition (logic), transposition (logic), and so on are all in the realm of traditional logic. There should definitely be an article "inversion (logic)", and perhaps you should start it; seems like you know enough about traditional logic to do that. In fact, when I was poking around for information on this, I found "full inversion" and "partial inversion"; the distinction should probably be explained.

In terms of propositional logic, like I said, there's nothing incorrect in the article. There are no fallacies, and it does not contradict itself. The truth table thing, for instance, seems fine... all it's showing is that an implication and the inverse of an implication are logically distinct. The article is not exactly inexpert, but obviously it is incomplete and confusing, because you found it confusing -- and you seem pretty intelligent. You say "the whole story is being neglected here". Perhaps you can fill in the details? Or, the article could stay much as it is, but make more clear reference to the other kind of inverse, in another article. Another thought: there are wikipedians who are mathematicians, and computer scientists, and perhaps logicians? Can we get a couple of them to work this out once and for all? --kier07 23:24, 20 July 2006 (UTC)

• I am an educated philosopher, or at least that is my training and teaching was once my calling. Logic is a major branch of philosophy and general dictionary definitions are often ambiguous and even vague for terms used in the technical sense of a specific discipline. I have often found dictionaries in err for such technical usages.

• I wrote the articles on contraposition, conversion, transposition, and obversion, but I write no more at Wiki, just minor edits and a vandal watchdog.. Ironically, I found myself spending way too much time on talk pages as a consequence (nonetheless I have enjoyed OUR conversation). I had not run across full and partial inversion, but it would make sense and perhaps would be explained by set theory and the fact that a proposition and its inverse have the same symbolization.

• Thanks for the compliment but I don't feel competent enough to write on inversion and explain the apparent inconsistencies I find. But the article should explain alot more than it does. I would have to understand how to get around what appears to me as the fallcy of affirming the consequence in conversion. What you refer to as "partial inverse" would probably be the traditional conception, and "full inversion" as what the traditional formal logician may refer to as the "obverted inverse", which incorporates obversion in the definition much like set theory. If you know "wikipedians who are mathematicians, and computer scientists, and perhaps logicians?", invite ALL three. Amerindianarts 01:42, 21 July 2006 (UTC)

---

As a relative layman (or putting myself in the position of one, anyway), the second part of the article makes not one jot of sense at all in its current form. An encyclopedia should enlighten those for whom the subject is new, not just those who already know this stuff GRAHAMUK 12:28, 30 Jul 2003 (UTC)

Converse or inverse

This article should be merged with Converse (logic)? Gene.arboit 02:15, 11 September 2005 (UTC)

I suggest that the discussion be carried out at Talk:Converse (logic). -- Dominus 03:50, 12 September 2005 (UTC)

Contrapositive

The current article seems to indicate that inverse is the same as contrapositive, but I am not sure that is true. If it is true, then it should merge with contrapositive (and possibly contraposition), rather than converse. If it isn't the same as contrapostive, then is it the same as converse? 134.250.72.141

No. You are confusing necessary and sufficient conditions, and antecendence and consequence. Obtaining an inverse requires the process of conversion. Amerindianarts 18:06, 8 November 2005 (UTC)

Also, the inverse is not logically equivalent. Conversion, obversion, and contraposition are methods yielding equivalent propositions. Amerindianarts 18:09, 8 November 2005 (UTC)

That is, the converse/contrapositive/obverse of P=>Q is ~Q=>~P, whereas the inverse is ~P=>~Q.

Be careful! normal P->Q, inverse Q->P, contrapositive ~Q->~P, converse ~P->~Q. 68.144.80.168 (talk) 13:51, 23 June 2008 (UTC)

tags

After researching the use of inversion in logic I have found the following problems with this article:

• Inversion is a process and the inverse is the product of that process. This article should be moved to Inversion (logic) and the current article redirected to there. The current direction scheme is ass backwards.
• Wiki references to "inversion" and "inverse" are all directed towards this article basically from inverse (mathematics). I do not know the definition of inverse in mathematics, but that is what the current article appears to be about because it is not about inversion as a concept of traditional, philosophic logic (and its non-reference in modern logic).
• Once again, there may be a use of "inverse" in mathematics I am not aware of, but in modern logic it is not referred to (see Copi) and that is what the current article implies (e.g truth tables are not applicable). For traditional logic, the current definition of "inverse" in this article is wrong (See L Susan Stebbing and the glossary of terms in the McMillan Encyclopedia of Philosophy). The standard definition that has been handed down since the term was invented by John Maynard Keynes, and referred to by logicians in the standard defintion which can be considered public domain makes absolutely no reference to changes in the predicate. The true inverse has a contradictory subject, and either the same predicate or the contradictory of the original predicate. The inverse stated in the article is an obverted inverse. Meaning, either "All not S are P" or "All not S are not P" can be the inverse. Thus the recent addition of "(or, equivalently, the converse of the contrapositive)" is a contradiction. This is because neither inverse is logically equivalent to the original and either one can be the inverse.
• The question of the validity of "inversion" is examined by Keynes and Stebbing, where there may be a possible valid inverse in "All S are P" and "Some not S are not P", but it is questionable. Once again, it is not examined by Copi in his modern Symbolic Logic,
• Since logicians reference to the process of inversion as an immediate inference is limited to traditional logic, truth tables have a questionable application. Thus, material implication or the hypothetical is not a correct format for examining inversion, and you won't find it referred to in Copi at all.

I have made no changes, hoping for source and citation of the current article. Amerindianarts 12:37, 25 June 2006 (UTC)

This article says that the inverse of the statement P ⇒ Q is the statement ~P ⇒ ~Q, but the inverse is the statement Q ⇒ P, which is equivalent. They are equivalent because the converse (~P ⇒ ~Q) is the contrapositive of the inverse.

As the table indicates, when a statement is false the inverse is true and when the inverse is false the statement is true. When both the statement and the inverse are true, P ⇔ Q.

Inverse and converse are terms used in symbolic logic, and when they are used in traditional logic, the meaning is different. User:Awis 3:40, 7 July 2006 (UTC)

• In traditional logic the inverse of "All S are P" is either "All non-S are P" or "All non-S are not P" The definition of inversion is "In traditional logic, a type of immediate inference in which from a given proposition another proposition is inferred whose subject is the contradictory of the subject of the original proposition" Hence, because the predicate can be either positive or negative, the truth tables do not apply. I did my grad degree in philosophy, and I never found "inverse" or "inversion" referred to in symbolic logic, anywhere. So you need to cite a source. The only reference to "inverse" in symbolic logic is the "law of Inverse variation", which is not the same thing as inversion, but follows a similar form in terms of extension and intension of propositions.
• Q ⇒ P is not the equivalent or the inverse of P ⇒ Q. This statement commits the fallacy of affirming the consequence. To state that they are equivalent is true only if you can show a bi-conditional relation between P and Q. I would like to know where you get your info, because I don't think it is correct. In terms of material implication the inverse of P ⇒ Q is either ~P ⇒ Q or ~P ⇒ ~Q, but this is not used in any rules of inference I am familiar with. Even if what you say can be sourced, the current article is limited in scope given that it is an article on logic, and not mathematics, which is merely a species of logic. Amerindianarts 04:35, 8 July 2006 (UTC)

• Q ⇒ P is not the inverse. It is known as the converse, and it is equivalent to ~P ⇒ ~Q. I never said the inverse was equivalent to the original statement, I said it was equivalent to the converse. Sorry if I was unclear.
• There are two ways to form the inverse of a statement "All S are P" but what is the inverse of the statement "Some S are P" in traditional logic?
• Does traditional logic only deal with universal propositions as the main premise, or is an existential proposition allowed?
• If a statement is a universal propostion and so is the inverse, it's

possible that both of them are false. A single object can't refute a statement and refute the inverse of a statement at the same time, unless it's made of two parts.

• For the statement "all ravens are black" the inverse would be either "all non-ravens are black" or "all non-ravens are non-black".

A white raven would refute the statement but wouldn't affect the truth of the inverse. Awis 01:55, 11 July 2006 (UTC)

• • Q ⇒ P is equivalent to ~P ⇒ ~Q by the rule of transposition (or the law of contraposition by logicians such as A.N. Prior), but the converse of P ⇒ Q must be a particular statement, e.g. The converse of "All S are P" is "Some P is S". Conversion of a universal affirmative is a particular statement. So, I don't know what you mean when you say Q ⇒ P is the conversion of P ⇒ Q. From reading this statement "This article says that the inverse of the statement P ⇒ Q is the statement ~P ⇒ ~Q, but the inverse is the statement Q ⇒ P, which is equivalent. They are equivalent because the converse (~P ⇒ ~Q) is the contrapositive of the inverse." and then your subsequent comments I am not sure what you are trying to say. In set theory and class algebra some of the tradition immediate inferences of categorical propositions (which presume existence) are factored into notation,e.g. the class "a and not b"=0 means the symbolization for All S are P (or ~P ⇒ ~Q) and its inverse are the same. Can you explain how this might effect truth table results according to the traditional definition of inversion? What has been factored in to this symbol is the process of obversion, but it also implies that P ⇒ Q and Q ⇒ P can be equivalent and the fallacy of affirming the consequence is a chimera.
  • Yes, traditional logic does deal with particulars, but the problem of existentiality was dealt with by Boole and led to the the development of modern logic and quantification, and Aristotle did deal with particulars as a main premise, but the concept of inversion was of minimal importance.
  • "If a statement is a universal propostion and so is the inverse, it's possible that both of them are false. A single object can't refute a statement and refute the inverse of a statement at the same time, unless it's made of two parts." Did I state something to the contrary?? I am not following your argument.
  • "all ravens are black" the inverse would be either "all non-ravens are black", both of these can be true or false at the same time. This may or may not be true for "all ravens are black" the inverse would be "all non-ravens are non-black". I do not recall stating anything to the contrary to "A white raven would refute the statement but wouldn't affect the truth of the inverse."

In conclusion I'm not sure where you are going and if you are defending the article, or what. I don't agree with you on conversion. P ⇒ Q has a valid converse, but it is not Q ⇒ P. You can convert or transpose P ⇒ Q to Q ⇒ P and call it the converse, but they are not converse in the sense that ~P ⇒ ~Q is the inverse of P ⇒ Q.Amerindianarts 03:58, 11 July 2006 (UTC)

I'm not defending the original article. I'll rewrite it in a day or so, now that I know what changes to make. Awis 11:12, 11 July 2006 (UTC)

Good. One final note to clarify my position in regard to the current article content.

• In class algebra "a and not b=0" appears to be a definition of what is in traditional logic the "obverted inverse", but it is not referred to in class algebra as the definition of "inversion". The process of obversion of categorical statements is factored in, but categorical propositions assume existence, that is, a universal statement cannot refer to an empty class, otherwise it is not a proposition. But I think the equivocation of "a and not b=0" in notation of a proposition and its obverted inverse is only apparent, and may be worked out in further premises in a proof. I am not that familiar with set theory and its notation.
• The current article assumes the same definition, and I cannot find anything in the definitions and rules of inference for quantification logic to substantiate this. Nor can I find anything in the definitions and rules of inference for quantification logic to substantiate that it uses the traditional definition of inversion. Nothing. Inversion isn't a term or definition explicitly stated in the rules. Thus, unless it can be sourced through citation, even if the fundamentals are correct, the use of the term must be considered original research or work, which is not allowed at Wiki.
• This is why I find the current article confusing and contradictory. The page title is inclusive of "Inversion (logic)", thus it needs to be expanded to begin with the traditional definition and work from there, explaining the ins and outs of further definition. Amerindianarts 17:48, 11 July 2006 (UTC)

article rewrite

I would like to comment on the term inverse. I have noticed one user who claims to be a professor of philosophy comment at this site about a definition of inverse applying to a form of early logic. At my school, however, if there was ever such a term, it is now a dead term. Instead, the common definition refers to the rather simple property: P=>Q has inverse ~P=>~Q. It is well discussed in logic, where we learn such principles such as if both a conditional and it's inverse hold, then the equivalency P==Q holds. (the inverse also holds)

Through the research I undertook on wikipedia, I discovered that conditionals are currently being called material conditionals (bizarre). I reccomend see also: material conditional, converse, contrapositive. I have never heard of those other bizarre terms (obverse, etc) so I am led to the conclusion that they have little significance for anyone except PHDs (and therefore should be delegated to the bottom of the article).

Added note: converse (Q=>P) article seems to be screwed up in similar way 68.144.80.168 (talk) 14:37, 23 June 2008 (UTC)

I have one HUGE problem with the inverse/converse/contrapositive articles as they stand: they aren't comprehensible to the vast majority of people. Whenever possible statements should be put in plain enough English for a typical high school student utilizing Wikipedia to benefit. When propositions are stated in symbols and nothing more, the result is an article people will look at and avoid.Dismalscholar (talk) 18:26, 4 March 2010 (UTC)