Talk:Vacuous truth
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[edit] Talk:Vacuous truth humor
Comments from the earlier, especially incoherent versions of Vacuous truth:
- The whole thing sounds pretty vacuous to me (see talk:surrealism). --Ed Poor
- No, this is mathematics. It's real. There's just parts of it that only make sense after the imbibing of certain quantities of alcohol... -- Tarquin
Ryguasu didn't understand if Ed Poor's comment was in jest. Ed Poor clarified:
- I confess that I was just clowning around: the surrealism article seemed so surrealistic that when I saw vacuous truth right beside it on Recent Changes, I couldn't resist. Frankly, I don't understand either article, maybe washing my brain with alcohol would help? ;-) --Ed Poor
[edit] Clarifying the vacuous truth concept
Are the two concepts of "false implies anything" and "anything is true for the empty set" really that closely related? -- Tarquin
- As for the relatedness of those concepts, Tarquin, I can say at least this much: they both have the quality that you could legitimately "legislate in several manners". That is, there is certainly a reasonable point of view from which "nothing is true for the empty set", rather than everything. This seems remarkably parallel to the point of view from which "false implies nothing". I guess I need to think some more about whether "vacuous truth" is the best name for what is in common between them.
- Then there's the standard connection between logical implication and set theory: possible world semantics. Here, A -> B is isomorphic to "the set of conceivable worlds in which A holds" is a subset of "the set of conceivable worlds in which B holds". This gives a justification for why 3>3 implies anything; the set of conceivable worlds where 3>3 is said to be the null set, and the null set is a subset of every set. Therefore, by the isomorphism, 3>3 implies anything. I'm not sure exactly how this is related, but I can't help feel that it is. --Ryguasu
- The set analogy is interesting. That certainly does put a connection between them. -- Tarquin
[edit] Historically important (??) revisions
- AxelBoldt: What an improvement in the introduction! A few more changes like that and this article might make sense. =)
- Do you object to my rewording the first sentence as "Informally, a statement is vacuously true if it is true but it doesn't really say anything"? In particular I'm wondering if replacing your "because" with my "but" is acceptable. I did this because I think the implication "x doesn't really say anything" implies "x is true" seems rather confusing.
- My sentence leaves the reader with the question "well, why should those example sentences be said to be true in the first place?". I think this is okay, because I think I'd rather leave the reader with the question, and then delve into the answer in the more formal part of the article.
- Your sentence seems to try to answer that same question right off that bat. I think this would be fine if you could deal with all the subtleties of the answer in that one sentence, but I don't think that is possible. --Ryguasu
- Yup, "but" is fine. AxelBoldt 17:08 Aug 27, 2002 (PDT)
[edit] Issues with mathematical symbols
- The character formatting is not working on my browser (Netscape 4.7). What's the problem? --John Knouse
- You can't make netscape 4.7 work with the right fonts? (What's the problem anyway? Are math symbols ignored altogether? Do they come out as little squares?) Speaking of fonts, anyone know what fonts to install to make IE 6 work right with the math symbols? As shipped, most of the math symbols are displayed as little squares. As IE is probably the most popular browser, it would be good to get the math symbols working on it. --Ryguasu
[edit] Future Directions
A lot of the stuff towards the end of the article would work well in a page on material implication. That P → Q is true whenever P is false dates back to ancient Greece, when → is material implication; but it may not be true for other forms of implication. (Material implication, of course, is the implication of classical logic, what we normally use in mathematics.) One could argue that ∀ x∈{}, P(x) is vacuously true when we interpret ∀ x∈S, P(x) using material implication, but not using other forms of implication. (Unfortunately, I don't know anything about this stuff, but I'll have a lot to write all over Wikipedia when I learn it, which I intend to.)
— Toby 08:39 Sep 20, 2002 (UTC)
This is not a bad idea. I was thinking that it might be fruitful to move most of this page to a page about comparing our intuitive concepts of logical connectives and their formalized "equivalants". This isn't exactly what you're suggesting, but I imagine such an effort, combined with a historical perspective, would make everything here much richer. Do you have any particularly interesting historical references? Did the Greeks talk much about what (I suppose) you'd call non-material implication? --Ryguasu
I don't know the answers to these questions yet. I actually hope to learn a lot of this over the course of the next school year, and if so I certainly intend to write it up for Wikipedia. I don't think that a lot of mathematicians appreciate the connections of foundations to philosophy, nor the somewhat arbitrary nature of choosing classical logic and set theory (only the axiom of choice is acknowledged widely).
What I visualise is:
- Lots of articles on the basic features of logic, such as implication. Many of these already exist.
- Lots of links from articles on mathematics (and other subjects that use logic precisely) to logic articles, such as linking the first technical usage of "and". I include these links whenever I edit a math article, which I do often (that being my field).
- Articles on nonclassical approaches to logic. I'm not sure that any of these exist yet, and this is what I can't write yet but hope to be able to write soon.
- Bits towards the end of each logic article on how that article's feature fits into nonclassical logic, and the implications that this has for how we reason with it.
Your article on comparing intuition with formalism isn't in this vision directly, but it's certainly not antithetical to it. And an article of type 3 would include much of this; for example, Intuitionistic logic would explain why intuitionists and constructivists reject the law of the excluded middle. What I would really find neat, however, is a brief bit on this at the end of Logical disjunction, showing how intuitionists and constructivists differ from Aristotle on what "or" means, which goes a long way towards explaining why they reject that law. (Of course, this would not infect the beginning of the article, which should spend its time explaining that logicians' "or"s are inclusive and things like that.) — Toby 11:13 Sep 22, 2002 (UTC)
[edit] Additional comments
Is the statement in the first paragraph about even primes greater than 2 true in any sense? --rmhermen
- Yes, of course; the English phrase "even primes greater than 2" translates into the mathematical expression "The intersection of the sets (even integers), (prime numbers), and (integers greater than 2)"; that is, the empty set. The statement is then a vacuously true one of the third form. --LDC
-
- Ha! I knew that this list of "every form" of vacuously true statements was incomplete. The problem with LDC's reduction of this example to the 3rd form is that the statement can be made in a logical system that does not have sets (such as Peano arithmetic). Of course, we can reduce it the second form by similar tricks:
- For all x, if x is even and x is prime and x is greater than 2, then ....
- but the same is true for the 3rd form after all, yet we list it. Of course, the 3rd form is in formal language, while "even primes greater than 2" is in informal language; LDC and I have simply described two different ways to formalise it. But my point is that there are yet more possibilities for formal language. I'm going to at least add an example for typed logic (which the "even primes greater than 2" example can also be reduced to if one wishes), but the important thing is that we shouldn't insist that these are the only possibilities. The imaginations of logicians are endless. — Toby 08:27 Sep 28, 2002 (UTC)
- Ha! I knew that this list of "every form" of vacuously true statements was incomplete. The problem with LDC's reduction of this example to the 3rd form is that the statement can be made in a logical system that does not have sets (such as Peano arithmetic). Of course, we can reduce it the second form by similar tricks:
- Do any of AxelBoldt's new additions clarify anything? If not, perhaps some further examples or restructuring are in order. --Ryguasu
Who thinks that the empty set case is in some sense the most basic? Most logics have no notion of set in the first place, yet any classical logic has vacuous truth! (Ironically, this may be based on something that I did months ago, but redirecting Vacuous truth to Empty set on the grounds that the empty set version of vacuous truth was discussed there. That was when I was unskilled with redirects; I never should have done that.) — Toby 08:33 Sep 28, 2002 (UTC)
I made the empty set example follow the same form as the others. That is, we never put in a contradiction for P, but instead an arbitrary statement that happened to be false. Similarly, we needn't put in the symbol {} for (what I have called) A, since we can use an arbitrary set that happens to be empty. That change, I just noticed, makes the only reason given for the primacy of the empty set example vanish; I'll remove that reason so as not to make the alleged supporters of that position look silly ^_^. But I'd still like to know if any actually exist, or if this was simply a misinterpretation of my poor redirecting skills. — Toby 10:28 Sep 28, 2002 (UTC)
I admit it - I put in the idea that some view the "empty set version" as more canonical in order to appease the authors of the empty set page. I never had any particularly good reasons for believing it myself. --Ryguasu
Since it was just based on a misunderstanding of me, I'll remove it. — Toby 09:59 Sep 29, 2002 (UTC)
Vacuous truth is not limited to two-valued logics; only some of the arguments in favour of it (the ones that say "Well, it can hardly be false!") are so limited. In particular, intuitionistic logic has the same concept of vacuous truth. — Toby 08:35 Sep 28, 2002 (UTC)
I changed "two big questions" to "one big question", since there didn't seem to be any attempt to answer the 2nd. Indeed, the 2nd is largely discussed prior to the listing of the questions, while this part of the article is identified by the header as being about the 1st. (I suppose that this was the result of a rearrangement of the article.) — Toby 10:28 Sep 28, 2002 (UTC)
The New York argument and its cousin seem quite tenuous to me. For the case of the commutative law (called "symmetric" in the article), this problem disappears if you assume that F → T = T, even if you still pick F → F = F.The other example didn't even make sense as written, since both statement simply said that I'm sane (given that 3 indeed equals 3). I put a new example in there.
But both of these appeals to inuition have more problems than just disbelieving that F → p = T. For example, the New York argument works on the assumption that people want to reject "If I'm in New York State, then I'm in New York City". Yet that is precisely what we want the vast majority of humanity (residents of Buffalo being a principal exception) to accept! I myself am in California, so this statement is as true its converse. To avoid taking advantage of false assumptions (thereby reinforcing them), we need to focus on a specific person, such as one of the aforementiond residents of Buffalo. But when we do that, now people will start to doubt that "If I'm in New York City, then I'm in New York State" is really true, since it's only vacuously true for the Buffaloan.
The real way to deal with the statement is to use universal quantification to rephrase it as "For any person x, if x is in New York City, then x is in New York State". (Arguably, this is what the original phrasing about "being in New York city" was supposed to mean.) But now we're no longer dealing with any of the forms of vacuous truth discussed in the article. (Alternatively, we could keep my "I" and quantify over possible worlds, or speak of future probabilities, but that doesn't fit the templates any better.) My new mathematical example succumbs even more obviously to a missing quantifier, and I suspect that any mathematical example would do so, since it's hard for mathematics to avoid being precise.
I would get rid of these entirely; they're fairly obscure arguments anyway (you did what with a truth table, thinks the reader?). — Toby 10:28 Sep 28, 2002 (UTC)
- "Vacuously true" is also sometimes also used as a synonym for tautological. This article, however, attempts a more technical analysis of a more limited concept of vacuous truth.
In what sense is the concept of vacuously true discussed in this article "more limited" than the concept of tautological? Is the idea that all vacuously true statements are tautological? If so, could someone please flesh out the concept of tautological? If not, saying the one is a subset of the other seems almost misleading. --Ryguasu
You're right; I'll fix this. — Toby 09:59 Sep 29, 2002 (UTC)
While we're on the subject, who uses "vacuously true" as a synonym for "tautological". (The external reference mentions the possibility of such use, but even the person that mentions it there disparages it.) For example, I would call F → p vacuously true, and tautological if F is a contradiction (or more generally an explosive statement in Brazilian logic); but I'd call p → T trivially true instead, and tautological if T is a tautology. Thus, the notion of vacuous truth is quite independent of the notion of tautology; as you were quite right to point out above, neither is stronger or weaker than the other. And I would regard somebody that said that "If the sky is blue, then if the grass is green, then the grass is green" vacuously true as simply mistaken. — Toby 10:10 Sep 29, 2002 (UTC)
I vote for the removal of the usage of tautological==vacuous. There may be some people who use the word in this sense, but they are probably confused. No need to reinforce or justify this mistake. AxelBoldt 00:08 Sep 30, 2002 (UTC)
I hope nobody is hurt by my removing the whole "New York" and "crazy" discussion. I don't think it can convince anybody. AxelBoldt 00:25 Sep 30, 2002 (UTC)
I don't get the first two examples - in fact, I intuitively say the statements are false. "All elephants inside loaves of bread are pink" - why? If you said "...have ears", that would be true because all elephants - loaf-inhabiting or not - have ears. "All even primes >2 are multiples of 3" cannot be true because 3 itself does not satisfy the even requirement and any greater multiple of 3 does not satisfy the prime requirement. If I theoretically had an even prime >2 (not a multiple of two, an even number - I don't know how) it could not be divisible by 3.
I could say, however, "All non-exponential functions that are their own integrals are also their own second derivatives" because that can be proven for any function f(x) - exponential or not. Here it is of no consequence that only e^x, an exponential function, is its own integral. The examples given cannot be proven true and have no basis for truth. Or, say, "Many of the registered users on the Pig Latin Wikipedia are hopeless Wikipediholics" can be assumed true because, despite there being no Pig Latin 'pedia, many users of other language Wikipedias (myself included) are hopeless Wikipediholics. But the two statements given are only absurd, as far as I can tell. --Geoffrey 01:04 24 Jun 2003 (UTC)
- That's the point, really. You can't have such a prime such as the one descibred in the article - that's what makes the truth vacuous. Show me an elephant in a loaf of bread that isn't pink. Show me an even prime >2 not divisible by 3. You can't, because such things don't exist. The statements are thus true, but vacuously true (as the article says). --Camembert
I removed the paragraph on the contrapositive. It tried to prove the truth of vacuous truths using the following argument
- If P is false, then for any Q
is also true, apparently using the fact that an implication is automatically true if its conclusion is true. Some vacuous truths have a true conclusion however, so we use what we want to prove in the proof. Furthermore, the equivalence of every implication with its contrapositive is arguably much harder to justify than the truth of vacuous truths. AxelBoldt 13:20, 24 Nov 2003 (UTC)
Stuff like this "All elephants inside a loaf of bread are pink" and the prime number example seem as much vacuously false as vacuously true. Are these really valid examples?168... 19:05, 6 Feb 2004 (UTC)
[edit] Picture
This is a Featured Article that doesn't have a picture. This would stop it appearing as a Main Page feature, for example. Is there a useful picture possible? A Venn diagram or similar? - David Gerard 23:25, 8 Jul 2004 (UTC)
- Hate to nix it, but a Venn diagram would communicate precisely the wrong thing. How about a larger form of that "P with a double arrow going to Q"? [[User:Meelar|Meelar (talk)]] 23:29, 8 Jul 2004 (UTC)
-
- Anything that'd be informative and helpful in explaining the concept but would nevertheless look good on the main page ;-) - David Gerard 23:31, 8 Jul 2004 (UTC)
[edit] FA removal candidate
See Wikipedia:Featured article removal candidates. --mav 21:40, 4 Sep 2004 (UTC)
[edit] What do we call a true implication with a tautological consequence?
What do we call true implications of the type P -> Q, where Q is a tautology? Something like if I go to school today, then 2 + 2 = 4. -- Sundar 09:28, Sep 30, 2004 (UTC)
- It's called trivially true.
Thanks, 67.171.229.101. Wondering, how I missed that! -- Sundar 07:04, Jan 6, 2005 (UTC)
[edit] Boston example is flawed
From the article: Consider the implication "if I am in Boston, then I am in Massachusetts." [...] There is something inherently reasonable about this claim, even if one is not currently in Boston. [...] Thus at least one vacuously true statement seems to actually be true.
The only problem is that there are a number of people in Britain who are managing, without any great effort, to be in Boston without being in Massachusetts.
Might I therefore suggest that the example of an "actually true" vacuous statement be changed, and the first step, in selecting a better one, should be to check whether it is in fact actually true?
- But I think the context makes it clear that one particular city called Boston was intended. (Besides, that's not the only problem, since besides the Boston in England (the original Boston) and the one in Massachusetts, there are, after all, various other Bostons.) Michael Hardy 02:09, 19 Mar 2005 (UTC)
Yes, the context denotes that the Boston mentioned is Boston, Massachusetts. But since we are tackling weighty logical arguments here, I think the knowledge of other locations named "Boston" is a distraction that weakens the example.
True Story: I was once travelling from a location in Rhode Island to a party in the town of Scituate, Massachusetts. After getting directions from a friend, driving, and confirming that I had arrived in Scituate, I made the logical assumption that "the party is in Scituate, I am in Scituate, the party is in this town." Thirty minutes later I learned that I was in Scituate, Rhode Island.
In the spirit of that memory, I changed Boston/Massachusetts/Seattle to Massachusetts/North America/Europe to lessen ambiguity. I hope it meets approval. -- House of Scandal 14:57, 22 October 2006 (UTC)
[edit] useful in a variety of mathematical fields
"the fact that the result of multiplying no numbers at all is 1 -- which is useful in a variety of mathematical fields"
The only utility I can think of is that it saves us the bother of writing "except zero" repeatedly in our proofs.24.64.166.191 04:06, 6 Jun 2005 (UTC)
[edit] Examples
By the definition given ("a logical statement is vacuously true if it is true but doesn't say anything"), some of the examples appear flawed.
The Boston example: "If I were in Boston, then I would be in Massachusetts" is an implicit statement of "Boston is a city in Massachusetts." or "Residents of Boston are a sub-set of residents of Massachusetts." Simply because this fact is reasonably well known does not suggest in any way that the statement tells one nothing. If that were so, then any true statement of any sort would be vacuously true if one happened to already be familiar with the content of the message.
- Unlike your elephant statement, I agree completely with this and, unless someone says otherwise, this should be deleted. --67.172.99.160 20:01, 13 August 2005 (UTC)
Another type of example, the elephant in the loaf type, calls into question whether any such statement is in fact true.
The elephant example (and similarly constructed examples): "All elephants inside a loaf of bread are pink." Examples of this type rely upon the assumption that arbitrary characteristics may be meaningfully assigned to objects which do not (or cannot exist). This appears to be a type of falacy, or paradox, in that it is inherently meaningless to assign definite properties to things which cannot by definition exist. It is arguable that an arguement based upon an ontologically meaningless statement is neither true, nor false, but simply a construct of words with the superficial appearance of sense.
- This can be restated as "If an elephant x is inside a loaf of bread, then elephant x is pink." We know that it is true that "If an elephant x is not pink, then elephant x is not inside a loaf of bread." We can assert that this new statement is perfectly valid. Also we can assert that "(x => y) <=> (!y => !x)", or, in other words, if x implies y, then not-y implies not-x. Our new elephant statement (let's call it y) along with the negation thing (we'll call it z) proves the old elephant statement, therefore to deny the old elephant statement is to deny either these logical postulates or the fact that all non-pink elephants are outside all loaves of bread.
- Put simply: the statement "All non-pink elephants are outside all loaves of bread" can be restated as "All elephants inside a loaf of bread are pink." Is something wrong with that?
[edit] How to prove all vacuous truths at the same time
The "ultimate vacuous truths" are as follows:
- Suppose that x is the set of all falsehoods. If any member of x is true, then all members of x are true.
- Suppose that x is the set of all falsehoods and y is the set of all truths. If any member of x is true, then all members of y are true.
Any vacuous truth is an example of one of these, therefore proving both results in the proof of all vacuous truths.
Now let's prove the first vacuous truth. By the Law of Contrapositives:
- If following is true:
-
- Suppose that x is the set of all falsehoods. If not all members of x are true, then no members of x are true.
- Then if any member of x is true, all members of x are true.
The "inner statement" is trivial to prove, therefore the Law of Contrapositives states that the first ultimate vacuous truth is true.
The second statement is also trivial to prove.
Boom, we have used a simple mathematical law to prove half of all the vacuous truths seconds after the other half. Any counters? --Ihope127 02:27, 15 August 2005 (UTC)
- Yes, this is hardly complete. The vacuous truth "every infinite subset of the set {1,2,3} has seven elements", as mentioned in the article, doesn't seem to be an instance of either of the molds for vacuous truth you give. It's unclear "vacuous truth" even admits a single obvious definition. The trick is not "proving" that vacuous truths are true. They are true by definition. The question is, by definition of what. This is a point the article addresses.
- Your proof is in fact distinctly unsatisfying. We can easily restate it without those sets, by simply rendering "x is a member of the set of all falsehoods" by "x is a falsehood" and "x is a member of the set of all truths" by "x is a truth". Then your "proof" goes as follows:
- A vacuous truth is an example of either of the following statements:
- If any falsehood is true, then all falsehoods are true.
- If any falsehood is true, then all truths are true.
- Proof: If some falsehoods are not true, then no falsehood is true. Trivial. By contrapositive, this is #1. #2 is trivial. QED.
- A vacuous truth is an example of either of the following statements:
- The distinct and obvious problem I have with this is that you leave out all the interesting bits. What is an "example", in this case? What happens at "trivial"? What axioms are you appealing to? This is not much better than stating "vacuous truths are true". Absolute and rigorous precision is not a luxury in this case, or it's completely unclear what you've proven in the first place. JRM · Talk 17:49, 3 January 2006 (UTC)
[edit] Folklore
There's a piece of mathematical folklore that concerns a topology journal that published a series of papers from various authors about properties of spaces of type X. One of the papers proved that all spaces of type X had property A. A subsequent paper proved that all spaces of type X had property ¬A.
Is this true? Is it worth mentioning or discussing in this article?
-- Dominus 17:29, 3 January 2006 (UTC)
- Depends. Was it conclusively established that there were no spaces of type X, either in another proof or by verifying that both proofs were correct and type X therefore had to be empty lest a contradiction occurred? If so, then both papers technically proved vacuous truths, and I'd say that's relevant.
- (Whether it's true at all, I don't know.) JRM · Talk 17:53, 3 January 2006 (UTC)
[edit] Usefulness?
Would it be worth including a separate section that explains why the vacuous truth is actually a useful concept and not just something to make the truth tables work out? (I realise some of it is mentioned throughout the article, but an explicit explanation might be good too.) Something along the lines of:
While it may seem counterproductive to bother about proving cases that don't actually exist, the use of a vacuous truth is helpful in proofs that seek to prove a property for a large range of cases, including the vacuous one. For example, proving a property of the empty set may be a much simpler proof than that for a nonempty one, and this can be used to start an induction to prove the property for a class of sets.
And if anyone can give a good example of such a proof it would be nice to see in there too. Confusing Manifestation 18:30, 29 January 2006 (UTC)
- Actually, the advantage you state really is just something to make the truth tables work out, though the importance of this should not be underestimated. You can, after all, always start an induction argument "one step later", so to speak. Of course this will be much less convenient than a proof over the empty set, but mathematically it doesn't matter much.
- Compare the fundamental theorem of arithmetic. Our definition is "Every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. Furthermore this factorization is unique except for the order." This definition not only implicitly denies the empty product, but even the unitary product! It can be stated much more succinctly as "every positive integer is the product of a unique multiset of primes".
- For proofs, a vacuous truth of the kind you describe is nothing more or less than accepting that universal quantification over an empty domain is true, that is, true is the identity element of conjunction. However, boolean algebra is usually not well-regarded as a foundation for logic, at least not philosophically, which is what makes vacuous truths tricky. 81.58.51.131 09:32, 3 February 2006 (UTC)
The Rota quote inserted in the introduction is misplaced. It should be in some other part of the article. And the language of it should be changed too.
Rintrah 14:02, 6 February 2006 (UTC)
[edit] Vacuously true statements in real life
I added a sentence about how vacuously true statements can be used to mislead people. I feel that should be expanded on, and include quotes from famous movies (I would have included one but I couldn't think of any!). The reason is that these statements are very important when speaking, especially in situations such as giving testamony. 163.192.21.43 18:57, 21 July 2006 (UTC)
[edit] Veracity of section "Arguments of the semantic "truth" of vacuously true logical statements"
Parts of this section, including the overall conclusion, state that a vacuously true statement (e.g. P → Q assuming that P and Q are false) is not necessarily true. However, "P → Q" in standard logic is by definition true if P is false.
Premises:
~P
~Q
~(P → Q)
But P is false and Q is false, so ~(P → Q) is ~(false → false), i.e. false. ~~(false → false) must therefore hold.
Thus, if vacuously true statements are assumed to be false, it can be deduced (P → Q) & ~(P → Q). Contradiction.
Pcu123456789 03:37, 26 October 2006 (UTC)
- The distinction here is between statements that are logically true and those that are semantically true. If logic is correct, then logical truth and semantic truth will be identical. But the whole point of this section is to claim that logical truth may not be a good model for semantic truth, and to present counterexamples of places where logcially true statements may not be judged to be semantically true. Your argument that "P → Q in standard logic is by definition true if P is false" is therefore begging the question; nobody disputed that this was true in the logical sense.
- Since it appears that you missed the point of that section of the article, I'm going to remove the "disputed" tag you added. -- Dominus 16:44, 26 October 2006 (UTC)
