Talk:Vacuous truth

3234:. Pointing out the deficiencies of this article has been constructive and there is nothing wrong with the content of the material you tried to add to fix those problems. The issue lies with the fact that only verifiable content is acceptable. Content that has appeared elsewhere and can be directly referenced via reliable secondary sources. Editors are not allowed to espouse their own views on a topic or create new and different ways to look at something. One hallmark that an editor might be doing so is essay-like writing, especially if there is no accompanying documentation. It takes a considerable amount of work by other editors to change that into something with an encyclopedic tone replete with sources–so much so that it is often better to scratch the whole thing and start over again. If you are familiar with the literature you can contribute to this project by referencing this material and indicating what is said there. That is the, albeit limited, role of a Knowledge editor. Contributions of this ilk are always welcome. -- 966:. 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 3141:

it amounts to "essay-like" content or "original research" etc. However, there is no other way to explain the reason why vacuous truth exists except by walking through the truth tables and illustrating that the other possible truth tables one might choose don't work, and doing so amounts to *direct evidence* rather than research. Direct logical proof of extremely simple facts like this should not require citation, especially in the math sections. It isn't original research when it is trivially obvious from the basic truth tables.

1766:'real world' of the example statements' content and our experiential interpretation, assumptions, cultural knowledge, etc. In a world of tiny pink elephants, or just in talking to an Eskimo ... we'd get into subjective experience vs. formalism. We need to be able to replace tokens like pink, elephant, and loaf-of-bread with things that have definite formal relationships and (possible and/or definite) characteristics, before we can talk about true/false (in the propositional logic sense), let alone vacuous, meaningful, etc. 1414:

language, by knowing we mean Boston, Massachusetts) in a new direction: Say you live in California. So: If you live in Boston, you live in New York. It's valid, because you cannot deal with rules outside of reality, so you cannot verify (perhaps related to Aristotle's future contingents). So when someone says, "Well that argument cannot be valid because if one lived in Boston, one would live in Massachusetts," the problem is that the premise has been switched from False to True, thus the argument becomes invalid T-: -->

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more so false because the first part is false. So not only is a vacuous truth false, it's ESPECIALLY false. "Were present King of France goes prancing around naked in pubic, nobody would even bothers to take a photograph" is ESPECIALLY false because it is false enough to say that were the king of France to prance around naked in public nobody would even bother to take a photograph, but it is even MORE ESPECIALLY false because there is no present king of France. Thus the article should be written thusly:

21: 171: 150: 1666:

since this means that "Elephant X being pink" would be both true and false (whereas the vacuous truth means Elephant X is both pink and not pink"). In general, illogic cannot exist, even from its own point of view, and the whole point of the vacuous truth is to show that a "what if scenario" doesn't exist, as counter-intuitive as it might seem to our bias, since we unconsciously presumed the Elephant to exist within the loaf of bread in the first place.

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it (it should be mentioned which and how perhaps?), but I thought of the following: since there are no elephants inside any loaves of bread, then clearly they can't be pink. So the statement would be false (F). But clearly they can't be anything else, so its inverse, "not pink" is also false (F). Under this condition, both A and ~A are false. This makes the logical argument "A or ~A" to be false. Using the equivalency laws, "A or ~A" = "~A-: -->

380: 2762:(In case you think this is semantics, consider the two-day case. The teacher will give the test Monday or Tuesday and yet surprise the students. The student could reason "if the test is not given today, then if tomorrow it is not given the condition that it is a surprise is broken". This contains an ESPECIALLY FALSE condition either in the first part (the test is not given today) or in the second (if tomorrow it is not given). 3137:

all members of the empty set have a certain property". The second paragraph says that vacuous refers to "a conditional statement with a false antecedent". These are not the same thing, although both are indeed sometimes referred to as "vacuous truth". This should be explained to the reader implicitly, rather than giving two conflicting definitions in a poorly phrased way like the article currently does.

1467:

then x + 2 is even" is unambiguously true. So we should define "if 3 is even then 3+2 is even" to be true because it is a special case of, and thus a logical consequence of, a true sentence. In your proposed counterexample, "it is false that (if 3 is even then 3=4)" is not a logical consequence of any unambiguously true statement. Perhaps a better writer than I can explain this in the article.

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nonsense/irrelevant, or true/false at the same time like a vacuous truth (basically asking "Can God destroy Himself?"). 2. The rules of "illogic" are "guided" by logic (and thus cannot exist). For example if we say that an object/power "A" both exists and doesn't exist (or whatever), then (A ∧ ~A) is true, and (A ∨ ~A) is false, as is the case with the vacuous truth. This means that ~A-: -->

874:, 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. — 1653:

force, electromagnetism (string theory related?)) is possible, as impossibility does not exist (not simply by definition(?)). I am by no means implying or necessitating that this omnipotent force(s) be God/gods, but I think this is related to an argument by Geoffrey Berg in his "The Six Ways of Atheism," where he tries to refute the logical possibility of the existence of omnipotence.

282: 255: 1544:!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. 3182:. Originally, I might have been willing to, but it seems to me that the community is too hostile to clarity for it to be worth me investing any more time to making these kinds of major contributions to Knowledge articles (for example, I wrote pretty much the entire article on Trapezoidal Distribution recently, which used to be a much less useful stub). 292: 2376:. I think we should go back to the version about conditionals with false antecedents and statements universally quantified over the empty set. If there are sources that describe all analytic truths as "vacuous", then we could mention that as a minority usage, but this is not the best article to discuss that concept--it should go at 761:, 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.) — 3163:) where I resurrected someone else's text from years earlier (with some minor edits) that explained very clearly why vacuous truth exists, at least for the case of classical truth dealing with simple true-false values. By not including a section that explains the why behind vacuous truth like this, you are 3156:

math section of Knowledge could use being re-written, but the editing culture is currently too hostile for any editors to do so. The community scares away all of the new editors who care about clarity, thus causing the articles to accumulate lots of cruft and superficial obscurantism that adds no value.

1533:

paradox, in that it is inherently meaningless to assign definite properties to things which cannot by definition exist. It is arguable that an argument based upon an ontologically meaningless statement is neither true, nor false, but simply a construct of words with the superficial appearance of sense.

3544:

It is definitely the case that in standard forms of mathematical logic the implication A→B is defined to be true when both A and B are false, regardless of whether A and B have any causal connection to each other. There are other forms of logic that attempt to get closer to the more common linguistic

3207:

Perhaps, however, there is some faint chance that me writing this here will reduce this toxic cultural trend at least somewhat. Maybe at least someone out there is sane. It's ludicrous that something so essential and fundamental to understanding a subject would be deleted. Knowledge really has become

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in many of the other subsets of Knowledge would never be tolerated in many of the other subsets of Knowledge, but apparently in math (where people are in love with obscurantism) including the fundamentals of any concept in a way that is easy for the general public to understand is viewed with nothing

3132:

The vacuous truth article has become poor quality. At one time, this article was even a featured article (way back in January 2004, admittedly) but the long chain of edits since then have stagnated the article, causing its quality to not keep pace with other Knowledge articles. There are at least two

2716:

Let's go through it: "Insofar as (if) 7 is even, then 9 is even" is ESPECIALLY FALSE, because the whole statement is baloney, but even more so the whole premise about seven being even is even more hogwash. Thus it is an even more especially false statement as regards odd x's. (A point to be careful

2680:

The correct resolution for the conundrums raised in this article is that vacuous truths are especially false: they are false because they would be false, were the first part true the second part would not follow or be false, or the statement as a whole would be false but they are even more especially

2549:

Again, I think the article should be primarily from the perspective of logic and philosophy. Mathematicians know about and use this notion but not, I would say, in a really deep or essential way, i.e., it's certainly "no big deal" to us and almost any sentence in a math paper which contains the term

2420:

The claim that all analytic truths are in some sense vacuous is one which has been held by many prominent philosophers (as has the negation of the claim, as usual in philosophy!). The fact that very few mathematicians would find this position appealing does not change this. It seems to me that this

2151:

As I recall, from hearing about it back in the 1980s, it was about anti-metric spaces. (Having looked at your link, it may be a different story entirely.) In such a space, the distance from A to C is always greater than or equal to AB plus BC. Lots of interesting results can be proved about them, all

1883:

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

1652:

F is false). From this, I could draw the (perhaps unwarranted) conclusion, that impossibility does not exist, and therefore there is an omnipotent force that exists by definition and natural (logical) law since anything for this force (or cumulative effect of forces - e.g. gravity, strong force, weak

1014:

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

3922:

In addition, a vacuous truth is often used colloquially with absurd statements, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle" to strongly claim that the dog was red), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the

3502:

b. The consequence, which could be true or false. B1: Bicycles roll on their wheels. which is factually correct. Or: B2. Bicycles walk on their feet. which is factually incorrect. Or B3. Triangles have three angles - which is correct by definition. B4. Triangles have 4 angles - which is incorrect by

2814:

The reason for not allowing zero in multiplication, is because the inflection point of addition & subtraction is NOT the same as the inflection point for multiplication & division , therefore the intersection at that point does not exist for all elements consisting of (number;operator) pairs

2810:

That same error in logic is derived from educational sessions where it is assumed that numbers stand on their own, without the presence of operators. That is not true, a description of a mathematical process contains two element subsets, the set of numbers onto which it is applicable, AND the subset

2754:

And early Monday morning, the first chance to talk before possibly being given the test, the students can furiously debate whether it is merely false or ESPECIALLY FALSE to say that if they do not get the test tomorrow (Tuesday), it is either false or ESPECIALLY false that if he does not give it the

2297:

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

1688:

An interesting addition I considered: the above proves the whole statement, "If an elephant is inside a loaf of bread, then it's pink," to be valid. But can we prove that the conclusion "it's pink" is both true and false as the article states? The article mentions that one of De Morgan's Laws proves

1091:

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

989:

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

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than the current versions of them. I see similar problems a lot on StackOverflow, where the moderators often "close as not constructive" some of the most useful threads (in fact, the majority of the most useful pages I've seen on StackOverflow were "closed as not constructive" by mods, as I seem to

3144:

This would be like the article on water not giving any explanation of the molecular chemistry of H20, or like a physics article giving no mention of the underlying principles. It is absurd. Explaining why vacuous truth exists is not original research. It is the bare minimum of what should be in the

3140:

2. Little to no explanation for why vacuous truth exists is given in the article, even though such information is readily available. Previous attempts by editors (dating back years) of explaining why vacuous truth has to apply have been deleted based on false/misguided notions by other editors that

2712:

Mathematicians need to be careful that when they make a general statement, such as "if x is a positive integer, then insofar as x is even, x + 2 is even", the last part can be ESPECIALLY FALSE. It is ESPECIALLY FALSE to say "insofar as x is even, x + 2 is even" when x is odd, for example when x is

2340:

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

1911:

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

1451:

However, I can make up another statement: "For all integers x, if x is even, then x=4". Clearly, this is false. But once again, the statement "if 3 is even, then 3=4" is vacuous, but I would argue that it is intuitively false, and therefore this isn't a good reason to define vacuous statements as

1447:

The "Intuition from Mathematical Arguments" section seems flawed. It says that the statement "For all integers x, if x is even, then x + 2 is even." is true, therefore the related vacuous statement "if 3 is even, then 3+2 is even" should be true too. Intuitively, I agree that the latter statement

3275:

I did have the option to correct the statement, but chose not to do so since I don't think that the example was a very good one for the level of readership of this article. Being familiar with topological concepts would be a prerequisite to understanding this example and if you have that knowledge

3136:

1. The page is inconsistent in its stance on what vacuous truth is, but doesn't make this explicit (potentially causing confusion). The top section of the article gives two different definitions of what vacuous truth is. The first paragraph says that vacuous truth is "a statement that asserts that

2988:

that has nothing to do with vacuous truths: the speaker is supposed to infer that, since Grandpa Joe isn't a Vermicious Knid, then "she" must not be a lady. Although the two statements have no logical connection, we're supposed to infer the antecedent must be false, which is distinct from it being

2533:

From a practical point of view, the large majority of the links to this article are from mathematics articles, a trend that I think we can expect to continue. So while it may well be that we need a bit more acknowledgment of the other usage, we certainly don't want the first paragraph to be mainly

1765:

Frankly, the whole example is a bit unwieldy (not surprisingly for elephants, painted or not, and possibly freakishly large loaves of bread. ;-) And all the talk about 'ontologically meaningless', implicit knowledge, 'objects which, etc. -- is getting down into the weeds of the projection into the

1532:

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 fallacy, or

1514:

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

1060:

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

721:

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

3155:

The community has a culture of deliberate obscurity and systematic hostility to anything that is easy to understand. It's almost as if you don't want the public to understand, just so that you can feel smarter than them by pretending to know more by being deliberately obscure. Huge swathes of the

2750:

likewise on Tuesday the students will wonder if their statement is false or ESPECIALLY false that, if the teacher does not give the exam today, it is either false or ESPECIALLY false that if he does not give it tomorrow (Wed), it is either false or ESPECIALLY false that if he does not give it the

1756:

at least one thing wrong with this — starting with the "implicit statement" example of Boston (...is a city in Mass. ... and USA, Earth, Milky Way, etc. if one want's to get ana...err, 'rigorous'.) -- NOT being carried forward to the elephant-in-the-loaf-of-bread thing. Again, being rigorous: one

1466:

Kjsharke: The main idea of that section is that it's clear what truth or falsity means for sentences of the form "whenever P(x) is true, Q(x) is true" even if it is a priori unclear for sentences of the form "if P is true, Q is true". In the example you cited, "for all integers x, if x is even

1364:

Speaking of distractions from the main (not Maine) point: yes! this example is tiresome and distracting. (Regrettably I just spent some time addressing it below on this Talk page — let me fix that here. ;-) RATHER than futzing around with taxonomy, referent, definition(s) of 'Boston', or even the

1351:

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

1162:

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

818:

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

588:

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

3498:

a. The antecedent A which could be true or false: A1: Bicycles have two wheels - which is always true. A2: Bicycles grow on trees - which is factually always false. A2b. Squares have exactly three angles - false by definition. A3: My bicycle is broken - can vary depending on who's bicycle we are

3254:

So why delete my submit and not add a source / adjust it? Aren't the empty set and its properties the most common vacuous truth example in mathematics? If find the example regarding the closed and open property very important for basic topology and maybe more important than the other two ones.

2911:

Isn't a vacuous truth also a tautology? I was looking for a term analogous to vacuous truth, in the sense that just as vacuous truth speaks of a implication whose antecedent is a contradiction (or always false), the term I'm looking for speaks of a proposition whose consequent is a tautology (or

2802:

A set is delimited by the empty element & infinite element in that set, those being the boundary conditions onto that set, and therefore no statement made about the elements of a set include the empty set & infinite set. Inside the set, the intersection of the elements with the empty set

2209:

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

1665:

A (we can't change that to true because of "illogic" since illogic works only with respect to A (I think)). This means that both "A" and "~A" are false, contradicting our starting point that (A ∧ ~A) is true, requiring both to be true. This I think contradicts the vacuous truth we've established

1373:

by it) — let's change the example. I'm going to use "Salt Lake City", and "Utah". I'm not aware of any other referents of either which could possibly be conflated by a reasonable person. (Frankly, the whole 'ooooh, there's a Boston in England' was a little ... let's be polite and say 'fussy' ...

3851:

It's important to also include the "universally quantified" parts of the lead. For instance, "all horse-sized ducks can talk" is vacuously true, because there are no horse-sized ducks. It can be converted into a vacuously-true implication (if x is a duck and is horse-sized, then x can talk) but

728:

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 Knowledge. I don't think that a lot of mathematicians appreciate the connections of foundations to philosophy, nor the

3186:

If this is how the community acts, where even the most basic most fundamentally important aspects of what an article on a piece of subject matter needs to cover is viewed with contempt, then editing here on Knowledge probably isn't a very good use of time, in most cases. Attempts at clarifying

2231:

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

1415:

F is false. That's because when one assumes a certain premise, one isn't interested in looking at it from the point of view of it being false, which serves no purpose. So there are two subtleties at work here: the definition of true/false in logic versus semantics and the subtle switch of the

1413:

I think much of the confusion can be avoided by realizing that in formal/symbolic logic the term "true" means "valid". So an argument with false premise(s) and conclusion(s) is valid but not sound (true premises and conclusions). Moreover, let's take the Boston example (without the fallacy of

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of which turn out to be trivial, since it can be shown quite easily that an anti-metric space can contain at most a single point. The first mention I found when Googling it just now doesn't claim that any journal published anything on the topic though. Well, if anyone feels like creating the

778:

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.

1492:. He argues that the reason why vacuously true statements seem difficult to swallow is that in non-mathematical English, the construction "if X then Y" typically implies causation. Therefore, when X is false, we tend to intuitively leap to thinking of the statement as a counterfactual. 2341:

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.

1662:

There are a few more implications of this, imo: 1. logic is an immutable relationship to whatever power (and its subordinate powers/objects) exists. An example of this would be, if we assume the existence of God, the question "Can God create a rock so heavy not even He can lift?" is

3880:

In the example given at the top of the page, the sentence "her cellphone is off" implies that "she has a cellphone", or mathematically speaking, "exists (her cellphone) such that is_off(her cellphone)". This is clearly false, and so should not be a good example of a vacuous truth

1891:

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:

537:

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

3276:

you are probably mathematically sophisticated enough to not need to look at an article on vacuous truth. Also, I believe that if you want to include a statement, you have the responsibility of providing a citation ... asking others to do your work for you is a bit gauche. --

527:

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

1085:

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 :

3204:

but hostility. How very counterproductive. As long as this culture continues to dominate on Knowledge, it will never reach its full potential, and more and more editors will continue abandoning the site (which is what the current trends are if you look at the charts).

3174:

of vacuous truth is ESSENTIAL to the article being useful, just as H20 chemistry (the underlying why) would be essential to the article on water. The exact wording and title of the explanation can vary, but it DEFINITELY should be in the article in at least some form.

2746:

likewise on Wednesday the students will wonder if their statement "if the teacher does not give the exam today, it is either false or ESPECIALLY false that if he does not give it tomorrow (Thur), he will give it Friday and yet surprise us" is false or ESPECIALLY false

1761:

big loaf of bread. (For that matter, one could paint an elephant pink, but now we're mixing antecedent and consequent w.r.t. that example.) So discussing 'objects which do not (or cannot) exist' is at least on-the-face-of-it irrelevant — but it goes deeper than that.

990:

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.

2806:

That mistake in logic is very similar to the assertion that division by zero, is the exception, not the rule, where in fact, multiplication is the exception, not the rule, the rule being that multiplication does NOT include zero nor infinite as elements of the set.

634:

This is a low-traffic article and does not attract a lot of attention on the talk page, so as far as I can tell it has never been archived. Often good to check how old a comment is before responding to it. The comments you are responding to are from August 2002.

2731:

realizing that the statement "if the teacher will give the exam on Friday, then we will know this fact come thursday night" is ESPECIALLY FALSE if the teacher is not giving the exam on Friday enables you to see the correct resolution to the hanging (non)-paradox.

2568:

My own opinion on what the term vacuous truth is that the statement itself is in some way meaningless, essentially the implication/quantifing over the empty set example given in the article, and this is very much the common usage of the term within mathematics.

3507:

B meaning that A being true implies that B is true, which itself can be correct or not. (A being true but not leading to B being true would mean the implication is false. But when A is false that does not imply anything about B, which could be true and could be

1561:~x is true (we can easily prove this. We state the obvious: 1. Elephant X is not inside a loaf of bread (~x). 2. "~x" is the same as "~x or y" (~x ∨ y). 3. (~x ∨ y) is the same as "y or ~x" (y ∨ ~x). 4. by the laws of equivalence (y ∨ ~x) is the same as ~y-: --> 2478:

This statement is not nonsense, and my agreement that it is true is not in the least grudging. I am not sure exactly how publication creeps into the discussion of truth, but mathematicians seek to publish (papers containing one or more) theorems, not truths.

1610:

x is false, because one can make a pink elephant by painting it, but one cannot fit an elephant in a loaf (by definition an impossibility). By definition, "x" is false (F). By definition, "y" is true (T) - we have presumed the existence of a pink elephant. T-:

1559:!x), but I think I found another way. I'm hoping I'm not mistaken. (!x means the negation of statement x which is the same as ~x). x is "Elephant x is inside a loaf of bread", and y is "Elephant x is pink." Therefore, the question is, is the statement x-: --> 3358:

of a statement (meaning Truth from a Vacuum, truth obtained from something missing) is implied by the fact that one cannot even begin to consider the hypothesis as being wrong, since there is no way that would imply or lead to the conclusion being wrong.

2618:

Not necessarily. In set theory, much of the definition and existence of the null set is pretty much determined by the vacuous truth that, for every member x of the null set, x is also a member of any other given set (which is of course a vacuous truth.)

2742:

The teacher can give the test on any day except Friday. on Thursday morning the students would wonder if their statement "If the teacher does not give the exam today, he will give it tomorrow and yet surprise us" is ESPECIALLY false, or merely false.

482:

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

2738:

It is ESPECIALLY false to say, then, in this case, the teacher must give it by thursday if the students have not had it by thursday, and so on, growing ever more especailly more especially false with each ESPECIALLY false especially bad deduction.

1650:

y means that "If something impossible occurs, then something impossible and possible occurs (or results?)." Since something that is impossible and possible cannot occur , then something impossible cannot occur (since "x" must be F, or else T-:

2239:. 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 3350:

I propose a more intuitive explanation using the terminology of the field but also shortly explaining each new term as we encounter it, which could, I hope, better convey the meaning of this term, to anyone studying philosophy, logic or math:

2700:" would also be considered "true" -- and not 'especially false', which is the correct resolution. Likewise mathematicians have not traditionally accepted as even more especially false the statement: "all cell phones in the room are turned on 2205:

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:

2684:

Confused mathematicians have called a "vacuous truth" any especially false statement that is not only false in its conclusion but especially false in that its premise is not even close to being an appropriate one to draw conclusions from.

2500:

as a technical term, not necessarily the same as "truth in some sense vacuous", so whether all analytic truths are in some sense vacuous doesn't really enter into the discussion much. What I don't really get is, what's wrong with the term

1308:

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

789:

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

2933:

I doubt a specific term for this exists, because such implications are not very interesting to logicians — if you already know that Q is true, why bother investigating if P implies it? (But if you insist, this trivially follows from the

2312:

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.

1347:

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.

2781:

So you say that for "if A then B" to be true, A must be true. Usually, when someone (including mathematicians) says that "if P then..." he does not know whether P is actually true, he just says what happens in the case that P is true

2758:

Naturally whether these statements are false or ESPECIALLY false has no bearing on whether the studetns are being given a test, and, resigned, they must simply await the teacher to come in and tell them whether they are being tested.

1028:

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

477:

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

2361:

I hadn't looked at this article in quite some time. I just looked at it today, and the concept now described (apparently it was changed back in April) now does not agree at all with the concept that I am used to hearing described as

2692:

in the room are turned off" is especially false if there are no cell phones in the room: it is false in that no cell phones are turned of, but it is even more especially false in that there are not even any cell phones in the room.

1743:

delete) the section). More generally, would you also need to put in '...U.S.A., Earth, Milky Way...' etc.? If not, why not? Where's the demarcation point on the spectrum of 'implicit knowledge'? Maybe that's a bit fussy/anal — but

2395:

I checked the Cambridge dictionary (my copy of the Oxford is in storage). Much to my own surprise and chagrin, the new material is verified by the entry there. I suppose there isn't much to do unless more sources can be found.

1769:

You could restate your point(s), though, e.g.: "if , then . Or, better: you could start out with "there are no pink elephants", and "an elephant cannot fit inside a loaf of bread" and get on with it. Similarly, we can say "Boston

2765:

This resolution is clearly the correct one, and you should change the whole article (or any other relevant mathematics) to reflect this. You may publish something in some glossified paper form first if you need to reference it.

2447:

There are however vacuous truths that even most mathematicians will outright dismiss as "nonsense" and would never publish in a mathematical journal (even if grudgingly admitting that they are true). An example would be the true

1799:

just fix it?" Good question. Answer: honestly (and I mean it - this is not a duck-out), it's not my field, and I'm not at all confident that my "fix" would be solid, either. To use the old quote: "one doesn't have to actually be

1100:" 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. -- 1621:

Let's prove this with our equivalences above. Knowing that "x" is false and "y" is true, the statement (~y ∨ x) is (F ∨ F) false . Another way: "x" is false, thus "~x" is true. "y" is true, therefore "~y" is false. Thus ~x-:

903:

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. —

543:

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.

1108:

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 :

846:

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

711:) 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 Knowledge when I learn it, which I intend to.) 3559:

OK thank you! So I think it should be conveyed clearly and explicitely, in the opening sentence, so that the layman who knows a bit of logic and a bit of math but is not an expert can immediately understand the

3430:- a conditional statement (a statement in which A implies B) discussing and considering what would be if something (A) would be different from the way it in fact is, or has been, and asking if it would imply B: 3010:

This is an insanely poor example. To use it as an example of a "Vacuous Truth", you have to completely ignore the meaning of the word "If", and how if/then operators actually function. The argument "If 2 :

1789:

both of these -- implicitly w.r.t. 'ontologically meaningless', and defining objects' possibility of existence, and then x's and y's...! So - again, why not just fix the statements? 'Cause you kind of did...

3436:"A subjunctive conditional's truth can be vacuosly trivial if its antecedent is impossible or non-counterfactually trivial, as I shall shortly explain, if its antecedent and consequent are both true."(ref) 1087:

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 : -->

722:

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? --

34: 3013:

3" is completely true, and undeniably logical. Because 2 is NOT greater than 5, so the argument stops there. There's no need to compare 2 to 3, because it's already failed the first conditional test.

1069:

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.

3153:

Stop being so actively hostile to clarity and comprehension. There is a reason why the math pages on Knowledge are regarded as the most incomprehensible subset of Knowledge by the general public.

1693: 70: 3091:"All my children are cats" is a vacuous truth, when spoken by someone without children. Similarly, "None of my children are cats" would also be a vacuous truth, when spoken by the same person. 89: 753:

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,

3792: 2550:

could easily be rewritten not to. On the other hand, there are (I gather) legitimate philosophical issues surrounding the term. It would be nice if the article said something about them.

916:

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.) —

237: 2837:

The set {1, 2, ⑀} has the elements 1, 2, ⑀. Which of those is the "empty element" and which the "infinite element"? Are you confusing "sets" with "elements", among other category errors?

2517:

prevalent that use is in contemporary philosophy -- this usage is for real: i.e., reputable and sourced. While it would absolutely be a good idea to be clear about these distinct uses of

1187:

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? -

1597:

Much simpler, faster way: 1. we know "~x" is true (Elephant X is not inside a loaf). 2. Thus (~x or y) is true (Either Elephant X is not inside a loaf or Elephant X is pink). 3. (x-: -->

3837:

If the above is correct, (non-vacuously true), and satisfactory, I'll find two refs to substantiate what I just wrote, one for the universal statement, one for the "regardless" clause.

1157: 2565:

So it seems this article should be completely re-done to separate the mathematical definition of vacuous truth from the philosophical debate about it, of which I am rather ignorant.

1541:

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 =: -->

1549:

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?

3199:. I know better than to invest in a community that can't even value the most basic and fundamental essentials of what a piece of content should have. Not including the underlying 927:

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

3170:

Just because someone from years ago labeled the content as "essay-like" doesn't mean you should continue blindly assuming that it is. Including an explanation for the underlying

1515:

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.

1649:

Implications (in philosophy) I see for this. Replace "x" with "impossible/illogical/untrue" and "y" with "possible/logical/true" (in which case "x" is the same as "~y"). x-: -->

3466:

B is vacuously true as long as A is false, regardless of whether A has any connection to B. For instance, "If this sentence is written in Japanese, then we should delete the

1195:

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"? ] 23:29, 8 Jul 2004 (UTC)

598:

need to get a better browser! Because it's so outdated, not only may it not recognize more current versions of HTML, but you'll be hideously susceptible to malware exploits.

3412:(a statement that can only be true or false) can be converted into a conditional (where A implies that B), it too, of course, can be determined to be true in the same way. 2020: 1323:

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

1785:

Or at least just state assumptions up front when talking about elephants and pinkness and bread loaves and qualia thereof. It's particularly interesting that you kind of

1374:

anyhow.) One other point: making the link to the particular city entry to Boston on WP should've taken care of it. If not, we could use VIAFs. And then if need be cite

757:

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

1991: 1949: 823:

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. —

313:

on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

3465:

I think you're missing the point. Your definitions of vacuous truth have too much about the consequent in them, but really it is purely about the antecedent. A-: -->

2803:

element & infinite set element pertain to the boundary limitations but never reach those boundary limitations, therefore those are limits towards, and never on.

2170:

Well, that's not quite vacuity, although it's degeneracy of some kind. There is another similar story about a dissertation concerning α-Hölder functions with α : -->

2696:

To confused mathematicians, who have historically called this a vacuous "truth", the statement would be true; however, tge "all cell phones in the room are turned

2040:

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

749:

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.

3167:

the value of this article for the users. It's like if the article on water had no content whatsoever on the chemistry of water (as I said earlier). It is absurd.

1416:

validity/truth value of the premises from false to true. This could perhaps clarify some things, though I'm sure I wrote it in the most confusing manner possible.

951:.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. 3832: 3812: 3752: 3732: 3712: 3692: 3672: 3652: 3632: 3612: 2755:

next day (Wednesday), it is either false or ESPECIALLY false to say that if he does not give it the next day (Thur), he will give it Friday and yet surprise us."

1262: 859:

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.) —

1355:

In the spirit of that memory, I changed Boston/Massachusetts/Seattle to Massachusetts/North America/Europe to lessen ambiguity. I hope it meets approval. --

746:

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.

2270:. However, boolean algebra is usually not well-regarded as a foundation for logic, at least not philosophically, which is what makes vacuous truths tricky. 3977: 3957: 350: 340: 3450:

Please consider and tell me what you think. And if you have a positive and simple example (which I cannot think of) in plain English I would be grateful.

739:

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 "

3967: 227: 3518:

But if A is false, and B is in fact true - or ...and B is in fact false, perhaps I can find a linguistic justification to call this implication a truth.

2281:

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.

2022:, or in words: (P implies Q) is equivalent with (Q or not P). This equivalence implies all vacuous thruths. Be aware, this implication does not hold in 454:

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? ;-) --

3178:

In addition to the need for adding back in some kind of explanation for why vacuous truth exits, this article in general seems like it may even need a

912:

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

3962: 1637:

y) is true whether "y" is true or false. This is simple enough, because we already know that "x" is false, therefore "y" can be T/F since both F-: -->

3972: 2513:. All I was saying was, so far as I know -- and (damn it, Jim) I'm a mathematician, not a philosopher, so I can't speak to why they do it or even 3440:

Eugene Mills, in American Philosophical Quarterly Vol. 33, No. 1 (Jan., 1996), pp. 105-117 (13 pages) Published By: University of Illinois Press (

1365:

entire example (look, 'Boston' is not the point, nor are context, implicit knowledge, culture, etc. — the point is that the referent of 'Boston'

675:, when → is material implication; but it may not be true for other forms of implication. (Material implication, of course, is the implication of 2084:

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.

3987: 2976: 1339:

problem, since besides the Boston in England (the original Boston) and the one in Massachusetts, there are, after all, various other Bostons.)

203: 3068:. As far as I understand Russel's concept, the statement "all my children are cats" is an idenfinite description which is to be interpreted as 2434:

At the same time, the treatment of mathematics and mathematicians in the relevant section rings false to me, especially the following passage:

1691:

A" (because "A or ~A" = "~A or A"). Those two statements must also be false. But the only way an if-then statement is false is if it's T-: -->

1511:

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.

2844: 2822: 315: 3484:

OK, perhaps so. I'm searching for the LINGUISTIC justification for the use of "truth" here, and perhaps that's what made me err. (If I did).

2505:, if that's what you mean? Is it just a way of saying "I want to talk about analytic truth but I don't want people to think I'm a Kantian"? 1110:

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). --

432:

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... --

2919: 2872: 2811:

of operators which are applicable. Defacto, the operators themselves, are proper elements of that set, due the union of both into the set.

2767: 2668: 1474: 1291: 616: 589:

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

1719:

A) "All unsigned Talk edits are suspect." (Or at least, a bit harder to parse who-said-what, esp. with a response that may or may not be

1452:

being true. Perhaps an example using empty sets would be better? Honestly, I am not convinced 'intuition' should enter into it at all.

3982: 3882: 2783: 2735:

if the teacher does not give the exam on Friday, all the deductions following thereafter are not only false, they are ESPECIALLY false.

2639: 3110:

No. It means that your item 1 is not part of the meaning of "all my children are cats". If Russell said it was, Russell was wrong. --

1832:

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.

3992: 3419:

because there is no mathematical processing involved or needed) when no statement (A) can lead to the conclusion of (B) being wrong.

3149:

behind vacuous truth, the article is massively less valuable to the public, to the point of bordering on being relatively worthless.

3046: 2157: 1864:

The "inner statement" is trivial to prove, therefore the Law of Contrapositives states that the first ultimate vacuous truth is true.

1697: 30: 3926:

Are these actually vacuous truths though? Neither of them is a conditional statement as-written. The first one seems to be closer to

2521:, and so far as I know one could appease the other usage by including a very short section saying essentially "Some people have used 1735:

page — don't delete it, at least not just for that. If the statement can be tweaked to serve the point, tweak it; if not, fix (or if

2990: 2708:

And so on. Please update the whole article to reflect the correct resolution. You may also publish this correct resolution first.

2236: 870:, but instead an arbitrary statement that happened to be false. Similarly, we needn't put in the symbol {} for (what I have called) 194: 155: 1873:

Boom, we have used a simple mathematical law to prove half of all the vacuous truths seconds after the other half. Any counters? --

3415:

In mathematics the term vacuous triviality is used in a similar manner for the truth of a term (B) derived by definition (called

3187:

articles to even a basic standard of usability are frequently destroyed. Many of the deleted edits in many articles are actually

2346:

Since it appears that you missed the point of that section of the article, I'm going to remove the "disputed" tag you added. --

305: 260: 20: 1243: 881:

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

2895:. Or you could say that (P → ⊤) ↔ (⊥ → ¬P), making it yet another example of a vacuous truth. Unless you refuse to believe in 1906:

Proof: If some falsehoods are not true, then no falsehood is true. Trivial. By contrapositive, this is #1. #2 is trivial. QED.

1202:

Anything that'd be informative and helpful in explaining the concept but would nevertheless look good on the main page ;-) -

2243:, but even the unitary product! It can be stated much more succinctly as "every positive integer is the product of a unique 1519:

Unlike your elephant statement, I agree completely with this and, unless someone says otherwise, this should be deleted. --

1121:

I removed the paragraph on the contrapositive. It tried to prove the truth of vacuous truths using the following argument

819:

formal language. I'm going to at least add an example for typed logic (which the "even primes greater than 2" example can

467:

Are the two concepts of "false implies anything" and "anything is true for the empty set" really that closely related? --

2572:

As for the philosophical/linguistic debate about the term this should be kept separate, whilst still in the same article.

2421:

article -- on a topic which belongs most to logic and philosophy -- is being overly skewed towards mathematics. This is

1077:

I hope nobody is hurt by my removing the whole "New York" and "crazy" discussion. I don't think it can convince anybody.

3900:

How was this reasoned in mathematical history? I've heard that Greek philosophers got it backwards, as vacuously false.

1967:, together with it's assumptions about implication, disjunction, reductio ad absurdum and negation, you can proove that 1529:

Another type of example, the elephant in the loaf type, calls into question whether any such statement is in fact true.

1436:"the fact that the result of multiplying no numbers at all is 1 -- which is useful in a variety of mathematical fields" 1401: 729:

somewhat arbitrary nature of choosing classical logic and set theory (only the axiom of choice is acknowledged widely).

130: 3145:

article, but isn't because misguided/rigid-minded editors keep deleting it. Without an explanation for the underlying

3022:

I argue that the nephew example is not an example of vacuous truth, because the set of nephews involved is not empty.

2720:

Statements "If the present king of france is bald..." are ESPECIALLY false (whatever the final part of the sentence).

1313:

The only problem is that there are a number of people in Britain who are managing, without any great effort, to be in

522:

AxelBoldt: What an improvement in the introduction! A few more changes like that and this article might make sense. =)

387: 265: 3488:

B is vacuously true even when B is actually false, (which is not what I originally understood from "a conditional...

2328:

Thus, if vacuously true statements are assumed to be false, it can be deduced (P → Q) & ~(P → Q). Contradiction.

1170:

Stuff like this "All elephants inside a loaf of bread are pink" and the prime number example seem as much vacuously

1836:

Any vacuous truth is an example of one of these, therefore proving both results in the proof of all vacuous truths.

1556:

You are a genius and you will have a place in the halls of Valhalla. I don't know how you got your argument (x-: -->

2325:

But P is false and Q is false, so ~(P → Q) is ~(false → false), i.e. false. ~~(false → false) must therefore hold.

3764: 3226:

Perhaps you would have had a better understanding of why you have run into these problems had you read the essays

33:. Please see the links under Article milestones below for its original nomination page (for older articles, check 2848: 1439:

The only utility I can think of is that it saves us the bother of writing "except zero" repeatedly in our proofs.

488:

Then there's the standard connection between logical implication and set theory: possible world semantics. Here,

2106:

Possibly related to Skolem's Paradox? Maybe also the Copenhagen interpretation of QM (i.e. Schrondinger's Cat)?

495:"the set of conceivable worlds in which A holds" is a subset of "the set of conceivable worlds in which B holds" 3857: 3755: 3550: 3515:

B (where #2 is read: When A is true A DOES lead to B being true) then we can say that all three parts are true.

3475: 3060:

First, I'm by far not an expert, and this is shaky ground. But I would like to put your attention to Russell's

2826: 2664: 2251: 1537:

This can be restated as "If an elephant x is inside a loaf of bread, then elephant x is pink." We know that it

974: 612: 2923: 2876: 2771: 1963:

is true, because the implication doesn't say anything about what Q should be. Ok, this is informal talk. With

1884:

is not "proving" that vacuous truths are true. They are true by definition. The question is, by definition of

1731:

relevant, or at least essential, here -- on a Talk page especially (ok, just go put in Massachusetts — on the

1478: 1380:

Which is why P → Q avoids this whole thing, and people should not get fussy about the human-convenient use of

866:

I made the empty set example follow the same form as the others. That is, we never put in a contradiction for

499:

3 is said to be the null set, and the null set is a subset of every set. Therefore, by the isomorphism, 3: -->

118: 3886: 2787: 2643: 1853:

Suppose that x is the set of all falsehoods. If not all members of x are true, then no members of x are true.

1774:

in the state of Massachusetts." (While we're at it: get rid of the conflation of 'in' vs. 'resident-of' : -->

1217:

I couldn't do it myself; not so handy with the images. But anybody who can, should. ] 23:44, 8 Jul 2004 (UTC)

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Article quality has become poor. Has inconsistencies and also needs explanation of why vacuous truth exists.

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meaning of implication, where a causal connection is implied, but that's not what this article is about. —

3085: 3065: 2939: 2577: 2403: 2211: 2176: 2128: 1497: 838:'s new additions clarify anything? If not, perhaps some further examples or restructuring are in order. -- 3437: 2723:

Finally, correcting this article enables you to resolve correctly the exam paradox (execution paradox).

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3 implies anything. I'm not sure exactly how this is related, but I can't help feel that it is. --Ryguasu

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Suppose that x is the set of all falsehoods. If any member of x is true, then all members of x are true.

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Is the statement in the first paragraph about even primes greater than 2 true in any sense? --rmhermen

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Q where Q is a tautology? I believe that if such term exists, it should be presented in this article.

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When we say a claim is true, we DO usually look at the consequent. But perhaps in this case we don't.

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Here's one that I claim to be a vacuous truth, since I have no children: "all my children are cats".

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Lots of articles on the basic features of logic, such as implication. Many of these already exist.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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For proofs, a vacuous truth of the kind you describe is nothing more or less than accepting that

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cannot be proven true and have no basis for truth. Or, say, "Many of the registered users on the

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leading to B cannot be false. In terms of formal logic: The truth of a conditional because the

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Quite possibly tgere is, for all we know. So, what's wrong with that? 10:42, 9 May 2020 (UTC)

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By the same arguments in this erroneous article, there is such a thing as a vacuous falsehood.

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I agree. I have changed the page to include your first example instead of the nephew example.

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obviously false on its own ("if Uluru is in France") and rendering the consequent irrelevant.

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party is in this town." Thirty minutes later I learned that I was in Scituate, Rhode Island.

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You don't have to go with MSIE; other browsers are available, especially Chrome and Firefox.

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The character formatting is not working on my browser (Netscape 4.7). What's the problem? --

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The most recent example of this (for this article) was when Wcherowi removed my recent edit (

2891:? P → ⊤ is equivalent to ¬P ∨ ⊤ which is itself a tautology. Unless you refuse to believe in 1640:~y) is true, since "x" is false. The statement is also valid, even though (y ∧ ~y) is false. 1088:

2 (not a multiple of two, an even number - I don't know how) it could not be divisible by 3.

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article" is a vacuously true implication, because the sentence is not written in Japanese. —

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Veracity of section "Arguments of the semantic "truth" of vacuously true logical statements"

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And if anyone can give a good example of such a proof it would be nice to see in there too.

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A pink unicorn, perhaps? With text in front that says if X is a unicorn, then X is pink?

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Bt. Right? Is this correct? If so I'll put it into a shorter more concise suggestion.

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B said nothing about the case of NOT A. So we are NOT CONTRADICTING the case of Af-: -->

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A post of mine was correctly deleted from here, and I moved the discussion about it to

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This statement is always wrong, because the first condition is wrong. At least, if the

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luxury in this case, or it's completely unclear what you've proven in the first place.

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Q, where Q is a tautology? Something like if I go to school today, then 2 + 2 = 4. --

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The set analogy is interesting. That certainly does put a connection between them. --

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because the antecedent cannot be satisfied") then where is the sense of "truth" here?

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But both of these appeals to inuition have more problems than just disbelieving that

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I would get rid of these entirely; they're fairly obscure arguments anyway (you did

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Also, all my extraterrestrial friends are beavers and love ice skating on the sun.

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This is quite possibly why mathematicians/logicians tend to mind their Ps and Qs.

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A lot of the stuff towards the end of the article would work well in a page on

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For example, the term vacuous triviality is used in a scientific article on

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Depends. Was it conclusively established that there were no spaces of type

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Q should be true. If P is false, there are no restrictions to what Q is. So

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page. I never had any particularly good reasons for believing it myself. --

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That link is to a post on my blog. My question above predates the post. —

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as elements, and the intersection at that point becomes the empty set .

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https://en.wikipedia.org/search/?title=Vacuous_truth&oldid=783859142

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is wrong; lying should be defined as knowingly making a false statement

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Now let's prove the first vacuous truth. By the Law of Contrapositives:

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of...') here thus massively expanding the domain-space of this entry.)

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But I think the context makes it clear that one particular city called

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Since it was just based on a misunderstanding of me, I'll remove it. —

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didn't understand if Ed Poor's comment was in jest. Ed Poor clarified:

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and are somewhat counterintuitive, making them worthy of an own name.

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Note that lying could be defined as knowingly making a false statement

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Every infinite subset of the set {1,2,3} has precisely seven elements.

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proof than that for a nonempty one, and this can be used to start an

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Is this true? Is it worth mentioning or discussing in this article?

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A vacuous truth is an example of either of the following statements:

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There exist at least one x with the property that x is a child; and

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Finally, to prove the vacuous truth, we need to prove that (x-: -->

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What do we call a true implication with a tautological consequence?

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just an article on vacuous truth in mathematics, nor should it be.

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King of England" to disagree with a previously made statement).

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is a contradiction (or more generally an explosive statement in

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On the other hand we can also look at it like this: if Af-: -->

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next day (Thur), he will give it Friday and yet surprise us."

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3 implies anything; the set of conceivable worlds where 3: -->

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Would it be worth including a separate section that explains

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to make a souffle' correctly to know that one's fallen...".)

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Comments from the earlier, especially incoherent versions of

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Anyway, this time round, if you want the article to improve

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B) The Boston example opens an entire can of worms probably

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doesn't need to be converted to be called vacuously true. —

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Then if any member of x is true, all members of x are true.

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If any falsehood is true, then all falsehoods are true.

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F; meaning ~A and A are both T and F. Is this correct?

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No wonder you people have trouble keeping new editors.

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A similar term for when the consequent is a tautology?

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What do we call true implications of the type P -: -->

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can be converted to a conditional statement, such as

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can never be true, it follows that this conditional,

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recent (April) expansion to analytic truth in general

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harder to justify than the truth of vacuous truths.

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Scrap this anti-logical, anti-mathematical nonsense

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If any falsehood is true, then all truths are true.

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as vacuously true. Are these really valid examples?

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Try 530:"x doesn't really say anything" implies "x is true" 361: 3826: 3806: 3786: 3746: 3726: 3706: 3686: 3666: 3646: 3626: 3606: 2864:Does anyone know of a term for a statement P-: --> 2727:http://en.wikipedia.org/Unexpected_hanging_paradox 2014: 1985: 1943: 1151: 973:The real way to deal with the statement is to use 420:The whole thing sounds pretty vacuous to me (see 3919:The introduction ends with the following lines: 1819:How to prove all vacuous truths at the same time 3521:(I'm thinking aloud here) When in fact Af-: --> 3346:A more intuitive explanation in simpler English 1867:The second statement is also trivial to prove. 450:article seemed so surrealistic that when I saw 446:I confess that I was just clowning around: the 3930:((A∨B) ∧ ~B = A), and the second represents a 3133:major problems with the article as it stands. 2970:, as in the line "If she's a lady, then I'm a 2798:empty element & infinite element in a set. 2156:article, I'll leave further research to them. 1823:The "ultimate vacuous truths" are as follows: 3787:{\displaystyle {\text{if }}A{\text{ then }}B} 3754:is in fact true, false or undetermined. If a 3096:Does this mean that LEM is not applied here? 2938:.) Whereas vacuous truths arise naturally in 1575:2. ~~y is the same as y, so (y ∨ ~x) is true. 1263:Knowledge:Featured article removal candidates 8: 3506:c. And then we have the implication. A-: --> 1061:green" vacuously true as simply mistaken. — 2509:I have no problem whatsoever with the term 2214:to prove the property for a class of sets. 1625:~y)=(~~x ∨ ~y)=(x ∨ ~y)=(~y ∨ x)=(~~y-: --> 1585:5. ~~x is the same as x, therefore (x-: --> 1560:y true? We know, as you said, that ~y-: --> 997:with a truth table, thinks the reader?). — 497:. This gives a justification for why 3: --> 116: 3208:a victim of its own culture and policies. 2838: 2654: 2087:(Whether it's true at all, I don't know.) 1490:Timothy Gowers' take on this kind of thing 1432:useful in a variety of mathematical fields 602: 358: 249: 144: 42: 15: 3819: 3799: 3776: 3768: 3766: 3739: 3719: 3699: 3679: 3659: 3639: 3619: 3599: 3495:There are three parts to the conditional: 3329:User talk:Pashute#From Talk:Vacuous truth 2227:Actually, the advantage you state really 1998: 1972: 1930: 1132: 594:Dude! If you're still using MSIE6.x, you 3523:B is showing its colors. We said A-: --> 1888:. This is a point the article addresses. 1335:was intended. (Besides, that's not the 1152:{\displaystyle \neg Q\Rightarrow \neg P} 695:) is vacuously true when we interpret ∀ 2977:Willy Wonka & the Chocolate Factory 251: 146: 2293:Vacuously true statements in real life 1694:2602:306:CD96:CC10:F010:96A4:5969:C5BE 319:about philosophy content on Knowledge. 3876:Implied falshood in example statement 2912:always true). Both are tautologies. 2298:situations such as giving testamony. 1609:Let's test this! We know that y-: --> 1581:4. (~x ∨ y) is the same as (~~x-: --> 517:Historically important (??) revisions 7: 3438:Interactionism and Overdetermination 1578:3. (y ∨ ~x) is the same as (~x ∨ y). 1378:... and down the rabbit hole we go. 1037:vacuously true, and tautological if 463:Clarifying the vacuous truth concept 303:This article is within the scope of 192:This article is within the scope of 3578:(meaning truth from a vacuum) is a 3197:you'll have to do the work yourself 1294:. Wondering, how I missed that! -- 135:It is of interest to the following 3978:Mid-importance Philosophy articles 3958:Knowledge former featured articles 3250:Vacuous truth example on empty set 2984:This is a humorous application of 2957: 2000: 1624:F statement, thus false. (~x-: --> 1143: 1134: 1056:true instead, and tautological if 977:to rephrase it as "For any person 14: 3968:Low-priority mathematics articles 3524:B. But A is NOT, so B is not too. 3487:If that is the case, that A-: --> 2966:Such statements can also be used 2237:fundamental theorem of arithmetic 212:Knowledge:WikiProject Mathematics 3963:Start-Class mathematics articles 2688:For example, the statement "all 1723:of one author, or two....) : --> 851:did months ago, but redirecting 559:Issues with mathematical symbols 325:Knowledge:WikiProject Philosophy 290: 280: 253: 215:Template:WikiProject Mathematics 179: 169: 148: 117: 19: 3973:Start-Class Philosophy articles 3915:Accuracy of colloquial examples 2496:(outdent) The thing is, I hear 1664:~A is false, as well as A-: --> 1639:T are valid. Moreover, (x-: --> 1571:1. Since we know that (~y-: --> 1021:You're right; I'll fix this. — 345:This article has been rated as 328:Template:WikiProject Philosophy 232:This article has been rated as 3514:If 1) A is true and 2) A-: --> 3385:vacuous truth is defined as a 1977: 1935: 1598:y) by the law of equivalence . 1582:y) by the laws of equivalence. 1140: 1: 3988:Mid-importance logic articles 3910:18:12, 30 November 2023 (UTC) 3891:13:37, 10 November 2022 (UTC) 3862:17:55, 15 December 2021 (UTC) 3847:16:52, 15 December 2021 (UTC) 3555:00:52, 15 December 2021 (UTC) 3540:00:31, 15 December 2021 (UTC) 3480:00:52, 14 December 2021 (UTC) 3460:00:03, 14 December 2021 (UTC) 3341:23:31, 13 December 2021 (UTC) 3004:For any integer x, if x : --> 1814:18:31, 18 November 2013 (UTC) 1748:getting kind of 'meta'. : --> 1406:18:59, 18 November 2013 (UTC) 1125:If P is false, then for any Q 206:and see a list of open tasks. 3528:Bt(!), Our assertion A-: --> 3318:18:13, 7 November 2017 (UTC) 2999:12:30, 9 November 2015 (UTC) 2853:00:27, 23 January 2021 (UTC) 2831:13:25, 8 February 2013 (UTC) 2609:17:41, 16 October 2008 (UTC) 2597:with the intention of deceit 2351:16:44, 26 October 2006 (UTC) 2335:03:37, 26 October 2006 (UTC) 2288:14:02, 6 February 2006 (UTC) 2275:09:32, 3 February 2006 (UTC) 2222:18:30, 29 January 2006 (UTC) 2181:21:17, 24 October 2021 (UTC) 2133:04:15, 3 November 2016 (UTC) 2116:23:06, 28 October 2016 (UTC) 2015:{\displaystyle \neg P\vee Q} 1502:21:25, 24 October 2021 (UTC) 1360:14:57, 22 October 2006 (UTC) 813:is greater than 2, then .... 3522:Bf our assertion of A-: --> 3304:and we already have a page 3018:The nephew example is wrong 2958:If she's an X, then I'm a Y 2629:22:54, 10 August 2009 (UTC) 2560:23:16, 10 August 2008 (UTC) 2544:22:18, 10 August 2008 (UTC) 2489:20:35, 10 August 2008 (UTC) 2412:00:40, 10 August 2008 (UTC) 2096:17:53, 3 January 2006 (UTC) 2071:17:29, 3 January 2006 (UTC) 2031:15:07, 3 January 2007 (UTC) 1921:17:49, 3 January 2006 (UTC) 1878:02:27, 15 August 2005 (UTC) 1702:07:55, 6 January 2016 (UTC) 1676:13:37, 1 January 2016 (UTC) 1524:20:01, 13 August 2005 (UTC) 1426:10:28, 1 January 2016 (UTC) 1286:It's called trivially true. 1015:seems almost misleading. -- 4009: 3983:Start-Class logic articles 2792:14:08, 21 March 2012 (UTC) 2776:22:57, 20 March 2012 (UTC) 2648:03:45, 14 April 2010 (UTC) 2582:14:53, 29 March 2011 (UTC) 1483:21:37, 28 April 2012 (UTC) 1282:09:28, Sep 30, 2004 (UTC) 1248:06:04, 23 April 2008 (UTC) 985:is in New York City, then 683:.) One could argue that ∀ 679:, what we normally use in 351:project's importance scale 71:Featured article candidate 3993:Logic task force articles 3944:00:23, 5 April 2024 (UTC) 3331:on my personal talk page. 2950:09:09, 7 April 2014 (UTC) 2928:21:42, 6 April 2014 (UTC) 2907:19:30, 4 April 2014 (UTC) 2881:12:41, 4 April 2014 (UTC) 2591:Not that it matters, but 2390:18:56, 21 June 2008 (UTC) 2368:logically necessary truth 2303:18:57, 21 July 2006 (UTC) 2166:00:42, 8 April 2017 (UTC) 2147:01:03, 8 April 2017 (UTC) 1961:'false implies anything' 1462:20:24, 22 July 2010 (UTC) 1298:07:04, Jan 6, 2005 (UTC) 1167:13:20, 24 Nov 2003 (UTC) 1081:00:25 Sep 30, 2002 (UTC) 1073:00:08 Sep 30, 2002 (UTC) 1065:10:10 Sep 29, 2002 (UTC) 1025:09:59 Sep 29, 2002 (UTC) 1001:10:28 Sep 28, 2002 (UTC) 939:, even if you still pick 920:10:28 Sep 28, 2002 (UTC) 908:08:35 Sep 28, 2002 (UTC) 895:09:59 Sep 29, 2002 (UTC) 878:10:28 Sep 28, 2002 (UTC) 863:08:33 Sep 28, 2002 (UTC) 765:11:13 Sep 22, 2002 (UTC) 718:08:39 Sep 20, 2002 (UTC) 386: 357: 344: 275: 231: 164: 143: 103: 45: 41: 37:) and why it was removed. 3594:. So for a conditional 3428:subjunctive conditionals 3286:03:51, 9 June 2017 (UTC) 3269:23:42, 8 June 2017 (UTC) 3244:18:00, 5 June 2017 (UTC) 3221:17:16, 5 June 2017 (UTC) 3055:10:34, 30 May 2017 (UTC) 3041:13:12, 19 May 2017 (UTC) 2534:about analytic truth. -- 2254:over an empty domain is 2252:universal quantification 1443:04:06, 6 Jun 2005 (UTC) 1343:02:09, 19 Mar 2005 (UTC) 1320:being in Massachusetts. 1302:Boston example is flawed 1269:21:40, 4 Sep 2004 (UTC) 1191:23:25, 8 Jul 2004 (UTC) 1178:19:05, 6 Feb 2004 (UTC) 1104:01:04 24 Jun 2003 (UTC) 975:universal quantification 970:true for the Buffaloan. 827:08:27 Sep 28, 2002 (UTC) 553:17:08 Aug 27, 2002 (PDT) 408:Talk:Vacuous truth humor 238:project's priority scale 3499:talking about and when. 3120:20:36, 8 May 2020 (UTC) 3106:19:22, 8 May 2020 (UTC) 2219:Confusing Manifestation 1690:~A" as well as "A-: --> 1327:in fact actually true? 1206:23:31, 8 Jul 2004 (UTC) 1096:Knowledge are hopeless 671:is false dates back to 645:18:32, 9 May 2020 (UTC) 621:09:59, 9 May 2020 (UTC) 532:seems rather confusing. 362:Associated task forces: 195:WikiProject Mathematics 106:Former featured article 90:Featured article review 31:former featured article 3828: 3808: 3788: 3748: 3728: 3708: 3688: 3668: 3648: 3628: 3608: 3370:that would imply that 3086:Law of excluded middle 3066:Theory of descriptions 2940:mathematical induction 2016: 1987: 1986:{\displaystyle P\to Q} 1945: 1944:{\displaystyle P\to Q} 1153: 383: 306:WikiProject Philosophy 125:This article is rated 35:the nomination archive 3928:disjunctive syllogism 3829: 3809: 3789: 3749: 3729: 3709: 3689: 3669: 3649: 3629: 3609: 2893:material conditionals 2017: 1988: 1946: 1846:If following is true: 1154: 732:What I visualise is: 382: 3818: 3798: 3765: 3738: 3718: 3698: 3678: 3658: 3638: 3618: 3598: 2232:doesn't matter much. 2024:intuitionistic logic 1997: 1971: 1929: 1488:You might also like 1257:FA removal candidate 1131: 755:Intuitionistic logic 657:material implication 549:Yup, "but" is fine. 218:mathematics articles 3410:universal statement 1993:is equivalent with 1965:deductive reasoning 770:Additional comments 759:Logical disjunction 331:Philosophy articles 3824: 3804: 3784: 3744: 3724: 3704: 3684: 3664: 3644: 3624: 3604: 3424:Philosophy of Mind 3259:comment added by ( 3165:massively reducing 3076:x is my child; and 2676:correct resolution 2614:Devoid of content? 2121:It is an old story 2012: 1983: 1941: 1306:From the article: 1149: 384: 316:general discussion 187:Mathematics portal 131:content assessment 46:Article milestones 3827:{\displaystyle B} 3807:{\displaystyle A} 3779: 3771: 3747:{\displaystyle B} 3727:{\displaystyle B} 3707:{\displaystyle A} 3687:{\displaystyle A} 3667:{\displaystyle B} 3647:{\displaystyle A} 3627:{\displaystyle B} 3607:{\displaystyle A} 3300:These are called 3180:complete re-write 2936:deduction theorem 2918:comment added by 2871:comment added by 2855: 2843:comment added by 2821:comment added by 2672: 2659:comment added by 2154:anti-metric space 1473:comment added by 1409: 1392:comment added by 1250: 1234:comment added by 899:Vacuous truth is 667:is true whenever 651:Future Directions 623: 607:comment added by 493:is isomorphic to 405: 404: 401: 400: 397: 396: 393: 392: 298:Philosophy portal 248: 247: 244: 243: 111: 110: 99: 98: 83:September 4, 2004 4000: 3934:((A∧B)∧~B = F). 3896:Historic context 3833: 3831: 3830: 3825: 3813: 3811: 3810: 3805: 3793: 3791: 3790: 3785: 3780: 3777: 3772: 3769: 3753: 3751: 3750: 3745: 3733: 3731: 3730: 3725: 3713: 3711: 3710: 3705: 3693: 3691: 3690: 3685: 3673: 3671: 3670: 3665: 3653: 3651: 3650: 3645: 3633: 3631: 3630: 3625: 3613: 3611: 3610: 3605: 3426:when discussing 3310:Bill Cherowitzo 3278:Bill Cherowitzo 3236:Bill Cherowitzo 2948: 2930: 2905: 2883: 2833: 2408: 2407: 2400: 2264:identity element 2021: 2019: 2018: 2013: 1992: 1990: 1989: 1984: 1950: 1948: 1947: 1942: 1752:C) Yes, , there 1485: 1408: 1386: 1357:House of Scandal 1229: 1158: 1156: 1155: 1150: 1045:); but I'd call 792:Peano arithmetic 633: 369: 359: 333: 332: 329: 326: 323: 300: 295: 294: 293: 284: 277: 276: 271: 268: 257: 250: 220: 219: 216: 213: 210: 189: 184: 183: 173: 166: 165: 160: 152: 145: 128: 122: 121: 113: 104:Current status: 85: 66: 43: 23: 16: 4008: 4007: 4003: 4002: 4001: 3999: 3998: 3997: 3948: 3947: 3917: 3898: 3878: 3816: 3815: 3796: 3795: 3763: 3762: 3736: 3735: 3716: 3715: 3696: 3695: 3676: 3675: 3656: 3655: 3636: 3635: 3616: 3615: 3596: 3595: 3348: 3325: 3298: 3252: 3130: 3020: 3012:5, then 2 : --> 3008: 2972:Vermicious Knid 2960: 2943: 2913: 2900: 2897:contrapositives 2866: 2862: 2845:125.239.154.159 2823:201.209.217.235 2816: 2800: 2678: 2636: 2616: 2589: 2405: 2404: 2398: 2359: 2310: 2295: 2199: 2038: 1995: 1994: 1969: 1968: 1927: 1926: 1821: 1623:~y is a T-: --> 1509: 1468: 1434: 1387: 1304: 1275: 1259: 1185: 1129: 1128: 1043:Brazilian logic 772: 677:classical logic 653: 627: 561: 519: 465: 422:talk:surrealism 410: 367: 330: 327: 324: 321: 320: 296: 291: 289: 269: 263: 217: 214: 211: 208: 207: 185: 178: 158: 129:on Knowledge's 126: 81: 64:January 9, 2004 62: 12: 11: 5: 4006: 4004: 3996: 3995: 3990: 3985: 3980: 3975: 3970: 3965: 3960: 3950: 3949: 3916: 3913: 3897: 3894: 3877: 3874: 3873: 3872: 3871: 3870: 3869: 3868: 3867: 3866: 3865: 3864: 3854:David Eppstein 3839:פשוט pashute ♫ 3835: 3823: 3803: 3783: 3775: 3743: 3723: 3703: 3683: 3663: 3643: 3623: 3603: 3564: 3561: 3547:David Eppstein 3532:פשוט pashute ♫ 3525: 3519: 3516: 3512: 3509: 3504: 3500: 3496: 3493: 3485: 3472:David Eppstein 3452:פשוט pashute ♫ 3448: 3447: 3446: 3445: 3347: 3344: 3333:פשוט pashute ♫ 3324: 3321: 3297: 3290: 3289: 3288: 3251: 3248: 3247: 3246: 3129: 3126: 3125: 3124: 3123: 3122: 3094: 3093: 3092: 3081: 3080: 3077: 3074: 3070: 3069: 3019: 3016: 3007: 3005:5 then x : --> 3002: 2982: 2981: 2962:Removed this: 2959: 2956: 2955: 2954: 2953: 2952: 2920:213.57.104.221 2909: 2873:46.121.232.249 2861: 2858: 2857: 2856: 2799: 2796: 2795: 2794: 2768:188.157.169.36 2711: 2707: 2677: 2674: 2661:TheGrandRascal 2651: 2635: 2632: 2615: 2612: 2588: 2585: 2563: 2562: 2531: 2530: 2527:analytic truth 2511:analytic truth 2503:analytic truth 2494: 2493: 2492: 2491: 2473: 2472: 2471: 2470: 2469: 2468: 2467: 2466: 2456: 2455: 2454: 2453: 2452: 2451: 2450: 2449: 2438: 2437: 2436: 2435: 2429: 2428: 2427: 2426: 2415: 2414: 2378:analytic truth 2373:analytic truth 2358: 2355: 2354: 2353: 2343: 2342: 2321: 2319: 2317: 2309: 2306: 2294: 2291: 2280: 2278: 2277: 2248: 2233: 2198: 2195: 2194: 2193: 2192: 2191: 2190: 2189: 2188: 2187: 2186: 2185: 2184: 2183: 2149: 2099: 2098: 2085: 2056:had property ¬ 2037: 2034: 2011: 2008: 2005: 2002: 1982: 1979: 1976: 1940: 1937: 1934: 1924: 1923: 1909: 1908: 1907: 1904: 1903: 1902: 1899: 1889: 1870: 1862: 1861: 1857: 1856: 1855: 1854: 1848: 1847: 1839: 1834: 1833: 1829: 1828: 1820: 1817: 1739:not possible, 1713: 1711: 1709: 1708: 1707: 1706: 1705: 1704: 1681: 1680: 1679: 1678: 1657: 1656: 1655: 1654: 1644: 1643: 1642: 1641: 1631: 1630: 1629: 1628: 1616: 1615: 1614: 1613: 1604: 1603: 1602: 1601: 1600: 1599: 1590: 1589: 1588: 1587: 1583: 1579: 1576: 1573: 1566: 1565: 1564: 1563: 1551: 1550: 1546: 1545: 1527: 1526: 1508: 1505: 1475:24.141.106.132 1446: 1433: 1430: 1429: 1428: 1345: 1344: 1303: 1300: 1292:67.171.229.101 1288: 1287: 1274: 1271: 1258: 1255: 1254: 1253: 1252: 1251: 1223: 1222: 1221: 1220: 1219: 1218: 1210: 1209: 1208: 1207: 1197: 1196: 1184: 1181: 1180: 1160: 1159: 1148: 1145: 1142: 1139: 1136: 1126: 1119: 1117: 1115: 1114: 1098:Wikipediholics 1083: 1076: 1068: 1012: 1011: 1003: 925: 923: 910: 897: 844: 843: 842: 831: 830: 829: 828: 816: 815: 814: 784: 783: 771: 768: 767: 751: 750: 747: 744: 737: 673:ancient Greece 652: 649: 648: 647: 630:TheGrandRascal 609:TheGrandRascal 592: 591: 590: 585: 584: 569: 568: 560: 557: 556: 555: 554: 546: 545: 540: 539: 534: 533: 524: 523: 518: 515: 514: 513: 512: 504: 502: 501: 485: 484: 473: 464: 461: 460: 459: 458: 437: 436: 429: 428: 409: 406: 403: 402: 399: 398: 395: 394: 391: 390: 385: 375: 374: 372: 370: 364: 363: 355: 354: 347:Mid-importance 343: 337: 336: 334: 302: 301: 285: 273: 272: 270:Mid‑importance 258: 246: 245: 242: 241: 230: 224: 223: 221: 204:the discussion 191: 190: 174: 162: 161: 153: 141: 140: 134: 123: 109: 108: 101: 100: 97: 96: 93: 86: 78: 77: 74: 67: 59: 58: 55: 52: 48: 47: 39: 38: 24: 13: 10: 9: 6: 4: 3: 2: 4005: 3994: 3991: 3989: 3986: 3984: 3981: 3979: 3976: 3974: 3971: 3969: 3966: 3964: 3961: 3959: 3956: 3955: 3953: 3946: 3945: 3941: 3937: 3933: 3932:contradiction 3929: 3924: 3920: 3914: 3912: 3911: 3907: 3903: 3895: 3893: 3892: 3888: 3884: 3883:176.12.196.63 3875: 3863: 3859: 3855: 3850: 3849: 3848: 3844: 3840: 3836: 3821: 3801: 3781: 3773: 3760: 3757: 3741: 3721: 3701: 3681: 3661: 3641: 3621: 3601: 3593: 3589: 3585: 3581: 3577: 3576:vacuous truth 3573: 3569: 3565: 3562: 3558: 3557: 3556: 3552: 3548: 3543: 3542: 3541: 3537: 3533: 3526: 3520: 3517: 3513: 3510: 3505: 3501: 3497: 3494: 3491: 3486: 3483: 3482: 3481: 3477: 3473: 3469: 3468:vacuous truth 3464: 3463: 3462: 3461: 3457: 3453: 3443: 3439: 3435: 3434: 3433: 3432: 3431: 3429: 3425: 3420: 3418: 3413: 3411: 3406: 3404: 3400: 3396: 3392: 3388: 3384: 3380: 3375: 3373: 3369: 3365: 3360: 3357: 3356:vacuous truth 3352: 3345: 3343: 3342: 3338: 3334: 3330: 3322: 3320: 3319: 3315: 3311: 3307: 3303: 3295: 3291: 3287: 3283: 3279: 3274: 3273: 3272: 3270: 3266: 3262: 3261:Tensorproduct 3258: 3249: 3245: 3241: 3237: 3233: 3229: 3225: 3224: 3223: 3222: 3218: 3214: 3209: 3205: 3202: 3198: 3193: 3190: 3185: 3181: 3176: 3173: 3168: 3166: 3162: 3157: 3154: 3150: 3148: 3142: 3138: 3134: 3127: 3121: 3117: 3113: 3109: 3108: 3107: 3103: 3099: 3095: 3090: 3089: 3087: 3083: 3082: 3078: 3075: 3072: 3071: 3067: 3063: 3059: 3058: 3057: 3056: 3052: 3048: 3043: 3042: 3038: 3034: 3029: 3026: 3023: 3017: 3015: 3003: 3001: 3000: 2996: 2992: 2987: 2986:modus tollens 2979: 2978: 2973: 2969: 2968:sarcastically 2965: 2964: 2963: 2951: 2947: 2941: 2937: 2932: 2931: 2929: 2925: 2921: 2917: 2910: 2908: 2904: 2898: 2894: 2890: 2886: 2885: 2884: 2882: 2878: 2874: 2870: 2859: 2854: 2850: 2846: 2842: 2836: 2835: 2834: 2832: 2828: 2824: 2820: 2812: 2808: 2804: 2797: 2793: 2789: 2785: 2784:84.229.25.134 2780: 2779: 2778: 2777: 2773: 2769: 2763: 2760: 2756: 2752: 2748: 2744: 2740: 2736: 2733: 2729: 2728: 2724: 2721: 2718: 2714: 2709: 2705: 2704:turned off". 2703: 2699: 2694: 2691: 2686: 2682: 2675: 2673: 2670: 2666: 2662: 2658: 2650: 2649: 2645: 2641: 2640:61.132.87.130 2633: 2631: 2630: 2626: 2622: 2613: 2611: 2610: 2606: 2602: 2598: 2594: 2586: 2584: 2583: 2579: 2575: 2570: 2566: 2561: 2557: 2553: 2548: 2547: 2546: 2545: 2541: 2537: 2528: 2525:also to mean 2524: 2523:vacuous truth 2520: 2519:vacuous truth 2516: 2512: 2508: 2507: 2506: 2504: 2499: 2498:vacuous truth 2490: 2486: 2482: 2477: 2476: 2475: 2474: 2464: 2463: 2462: 2461: 2460: 2459: 2458: 2457: 2446: 2445: 2444: 2443: 2442: 2441: 2440: 2439: 2433: 2432: 2431: 2430: 2424: 2419: 2418: 2417: 2416: 2413: 2409: 2401: 2394: 2393: 2392: 2391: 2387: 2383: 2379: 2375: 2374: 2369: 2365: 2364:vacuous truth 2356: 2352: 2349: 2345: 2344: 2339: 2338: 2337: 2336: 2333: 2329: 2326: 2323: 2314: 2307: 2305: 2304: 2301: 2300:163.192.21.43 2292: 2290: 2289: 2286: 2282: 2276: 2273: 2269: 2265: 2261: 2257: 2253: 2249: 2246: 2242: 2241:empty product 2238: 2234: 2230: 2226: 2225: 2224: 2223: 2220: 2215: 2213: 2207: 2204: 2196: 2182: 2178: 2174: 2169: 2168: 2167: 2163: 2159: 2155: 2150: 2148: 2144: 2140: 2136: 2135: 2134: 2130: 2126: 2122: 2119: 2118: 2117: 2113: 2109: 2105: 2104: 2103: 2102: 2101: 2100: 2097: 2094: 2090: 2086: 2083: 2079: 2075: 2074: 2073: 2072: 2069: 2064: 2061: 2059: 2055: 2051: 2048:had property 2047: 2043: 2035: 2033: 2032: 2029: 2025: 2009: 2006: 2003: 1980: 1974: 1966: 1962: 1958: 1954: 1938: 1932: 1922: 1919: 1915: 1910: 1905: 1900: 1897: 1896: 1894: 1893: 1890: 1887: 1882: 1881: 1880: 1879: 1876: 1871: 1868: 1865: 1859: 1858: 1852: 1851: 1850: 1849: 1845: 1844: 1843: 1840: 1837: 1831: 1830: 1826: 1825: 1824: 1818: 1816: 1815: 1811: 1807: 1803: 1798: 1793: 1792: 1788: 1784: 1780: 1778: 1773: 1767: 1763: 1760: 1757:could bake a 1755: 1750: 1747: 1742: 1738: 1734: 1730: 1725: 1722: 1721:in medias res 1717: 1716: 1712: 1703: 1699: 1695: 1687: 1686: 1685: 1684: 1683: 1682: 1677: 1673: 1669: 1661: 1660: 1659: 1658: 1648: 1647: 1646: 1645: 1638:F and F-: --> 1635: 1634: 1633: 1632: 1620: 1619: 1618: 1617: 1608: 1607: 1606: 1605: 1596: 1595: 1594: 1593: 1592: 1591: 1584: 1580: 1577: 1574: 1570: 1569: 1568: 1567: 1557:y) <-: --> 1555: 1554: 1553: 1552: 1548: 1547: 1542:y) <=: --> 1540: 1536: 1535: 1534: 1530: 1525: 1522: 1521:67.172.99.160 1518: 1517: 1516: 1512: 1506: 1504: 1503: 1499: 1495: 1491: 1486: 1484: 1480: 1476: 1472: 1464: 1463: 1459: 1455: 1449: 1444: 1442: 1441:24.64.166.191 1437: 1431: 1427: 1423: 1419: 1412: 1411: 1410: 1407: 1403: 1399: 1395: 1391: 1385: 1383: 1377: 1372: 1368: 1362: 1361: 1358: 1353: 1349: 1342: 1341:Michael Hardy 1338: 1334: 1330: 1329: 1328: 1326: 1321: 1319: 1316: 1311: 1310: 1301: 1299: 1297: 1293: 1285: 1284: 1283: 1281: 1272: 1270: 1268: 1264: 1256: 1249: 1245: 1241: 1237: 1233: 1227: 1226: 1225: 1224: 1216: 1215: 1214: 1213: 1212: 1211: 1205: 1201: 1200: 1199: 1198: 1194: 1193: 1192: 1190: 1182: 1179: 1177: 1173: 1168: 1166: 1146: 1137: 1127: 1124: 1123: 1122: 1118: 1113: 1107: 1106: 1105: 1103: 1099: 1095: 1089: 1082: 1080: 1074: 1072: 1066: 1064: 1059: 1055: 1052: 1048: 1044: 1040: 1036: 1032: 1026: 1024: 1019: 1018: 1009: 1005: 1004: 1002: 1000: 996: 991: 988: 984: 980: 976: 971: 969: 965: 961: 957: 952: 950: 946: 942: 938: 934: 930: 924: 921: 919: 915: 909: 907: 902: 896: 894: 889: 888: 884: 879: 877: 873: 869: 864: 862: 858: 854: 853:Vacuous truth 850: 841: 837: 833: 832: 826: 822: 817: 812: 809:is prime and 808: 804: 800: 796: 795: 793: 788: 787: 786: 785: 782: 777: 776: 775: 769: 766: 764: 760: 756: 748: 745: 742: 738: 735: 734: 733: 730: 726: 725: 719: 717: 712: 710: 706: 702: 698: 694: 690: 686: 682: 678: 674: 670: 666: 662: 658: 650: 646: 642: 638: 631: 626: 625: 624: 622: 618: 614: 610: 606: 599: 597: 587: 586: 583: 579: 575: 571: 570: 567: 563: 562: 558: 552: 548: 547: 542: 541: 536: 535: 531: 526: 525: 521: 520: 516: 511: 507: 506: 505: 496: 492: 487: 486: 481: 476: 475: 474: 471: 470: 462: 457: 453: 452:vacuous truth 449: 445: 444: 443: 441: 435: 431: 430: 427: 423: 419: 418: 417: 415: 414:Vacuous truth 407: 389: 381: 377: 376: 373: 371: 366: 365: 360: 356: 352: 348: 342: 339: 338: 335: 318: 317: 312: 308: 307: 299: 288: 286: 283: 279: 278: 274: 267: 262: 259: 256: 252: 239: 235: 229: 226: 225: 222: 205: 201: 197: 196: 188: 182: 177: 175: 172: 168: 167: 163: 157: 154: 151: 147: 142: 138: 132: 124: 120: 115: 114: 107: 102: 94: 92: 91: 87: 84: 80: 79: 75: 73: 72: 68: 65: 61: 60: 56: 53: 50: 49: 44: 40: 36: 32: 28: 27:Vacuous truth 25: 22: 18: 17: 3925: 3921: 3918: 3899: 3879: 3586:because the 3575: 3490:that is true 3489: 3449: 3421: 3414: 3407: 3376: 3374:is false. 3371: 3367: 3363: 3361: 3355: 3353: 3349: 3326: 3301: 3299: 3255:— Preceding 3253: 3210: 3206: 3200: 3196: 3194: 3188: 3183: 3179: 3177: 3171: 3169: 3164: 3158: 3152: 3151: 3146: 3143: 3139: 3135: 3131: 3047:50.35.82.244 3044: 3030: 3027: 3024: 3021: 3009: 2983: 2975: 2961: 2914:— Preceding 2867:— Preceding 2863: 2839:— Preceding 2817:— Preceding 2813: 2809: 2805: 2801: 2764: 2761: 2757: 2753: 2749: 2745: 2741: 2737: 2734: 2730: 2725: 2722: 2719: 2715: 2710: 2706: 2701: 2697: 2695: 2687: 2683: 2679: 2655:— Preceding 2652: 2637: 2617: 2596: 2592: 2590: 2571: 2567: 2564: 2532: 2526: 2522: 2518: 2514: 2510: 2502: 2497: 2495: 2422: 2399:siℓℓy rabbit 2371: 2367: 2363: 2360: 2332:Pcu123456789 2330: 2327: 2324: 2315: 2311: 2296: 2283: 2279: 2272:81.58.51.131 2259: 2255: 2235:Compare the 2228: 2216: 2208: 2202: 2200: 2173:Double sharp 2158:79.73.144.18 2139:Mark Dominus 2125:Double sharp 2081: 2077: 2065: 2062: 2057: 2053: 2049: 2045: 2041: 2039: 1960: 1956: 1952: 1925: 1885: 1872: 1869: 1866: 1863: 1841: 1838: 1835: 1822: 1801: 1796: 1794: 1790: 1786: 1782: 1781: 1776: 1771: 1768: 1764: 1758: 1753: 1751: 1745: 1740: 1736: 1732: 1728: 1726: 1720: 1718: 1715: 1714: 1710: 1538: 1531: 1528: 1513: 1510: 1494:Double sharp 1487: 1469:— Preceding 1465: 1450: 1445: 1438: 1435: 1388:— Preceding 1381: 1379: 1375: 1370: 1366: 1363: 1354: 1350: 1346: 1336: 1332: 1324: 1322: 1317: 1312: 1307: 1305: 1289: 1276: 1260: 1204:David Gerard 1189:David Gerard 1186: 1171: 1169: 1161: 1120: 1116: 1090: 1084: 1075: 1067: 1057: 1053: 1050: 1046: 1038: 1034: 1030: 1027: 1020: 1013: 1008:tautological 994: 992: 986: 982: 978: 972: 967: 963: 959: 955: 953: 948: 944: 940: 936: 932: 928: 926: 922: 913: 911: 900: 898: 890: 880: 871: 867: 865: 848: 845: 820: 810: 806: 805:is even and 802: 798: 773: 752: 731: 727: 720: 713: 708: 704: 700: 696: 692: 688: 684: 668: 664: 660: 654: 603:— Preceding 600: 595: 593: 529: 503: 494: 489: 479: 472: 466: 438: 411: 346: 314: 304: 234:Low-priority 233: 193: 159:Low‑priority 137:WikiProjects 105: 88: 69: 26: 3674:) in which 3580:conditional 3568:mathematics 3503:definition. 3387:conditional 3383:mathematics 3323:erased post 3294:false truth 3292:Make page: 3213:MagneticInk 3079:x is a cat. 3062:On Denoting 2991:31.169.57.1 2690:cell phones 2268:conjunction 2258:, that is, 2247:of primes". 2197:Usefulness? 1955:P is true, 1951:means that 1626:x)=(y-: --> 1612:F is false. 1586:y) is true. 1230:—Preceding 681:mathematics 596:desperately 566:John Knouse 209:Mathematics 200:mathematics 156:Mathematics 127:Start-class 3952:Categories 3590:cannot be 3588:antecedent 3563:How about: 3401:cannot be 3399:antecedent 3395:hypothesis 3393:of of any 3377:In formal 3362:Formally: 2621:MarcelB612 2574:KingStuart 1543:(!y =: --> 834:Do any of 448:surrealism 322:Philosophy 311:philosophy 261:Philosophy 3759:statement 3756:universal 3654:implies 3592:satisfied 3403:satisfied 3192:recall). 3112:Trovatore 2889:tautology 2536:Trovatore 2448:statement 2382:Trovatore 2322:~(P → Q) 2316:Premises: 2212:induction 2108:Cornelius 1668:Cornelius 1558:(!y-: --> 1448:is true. 1418:Cornelius 1236:Mike40033 1165:AxelBoldt 1112:Camembert 1094:Pig Latin 1079:AxelBoldt 1071:AxelBoldt 1054:trivially 968:vacuously 883:empty set 857:Empty set 836:AxelBoldt 637:Trovatore 551:AxelBoldt 544:--Ryguasu 3761:such as 3582:that is 3560:concept. 3257:unsigned 3098:Heiko242 3064:and the 2916:unsigned 2869:unsigned 2841:unsigned 2819:unsigned 2669:contribs 2657:unsigned 2245:multiset 2036:Folklore 1875:Ihope127 1791:here.... 1507:Examples 1471:unsigned 1454:Kjsharke 1402:contribs 1390:unsigned 1382:examples 1290:Thanks, 1244:contribs 1232:unsigned 1102:Geoffrey 797:For all 617:contribs 605:unsigned 538:article. 490:A -: --> 76:Promoted 3902:DAVilla 3634:(read: 3444:)(/ref) 3417:trivial 3391:premise 2601:Albmont 2552:Plclark 2481:Plclark 2348:Dominus 2285:Rintrah 2262:is the 2068:Dominus 2028:Leoremy 1318:without 1183:Picture 1017:Ryguasu 887:Ryguasu 840:Ryguasu 724:Ryguasu 659:. That 582:Tarquin 578:Mozilla 510:Tarquin 480:nothing 469:Tarquin 456:Ed Poor 440:Ryguasu 434:Tarquin 426:Ed Poor 349:on the 236:on the 95:Demoted 54:Process 3936:Anerdw 3508:false. 3228:WP:HOW 3189:better 2887:Well, 1806:A Doon 1759:really 1746:that's 1737:that's 1733:actual 1394:A Doon 1333:Boston 1315:Boston 1296:Sundar 1280:Sundar 1176:168... 133:scale. 57:Result 3778:then 3572:logic 3442:JSTOR 3408:If a 3379:logic 3232:WP:5P 3033:Gzorg 2717:on.) 2587:Lying 1777:(sic) 1371:meant 1309:true. 1172:false 981:, if 914:prior 801:, if 687:∈{}, 574:Opera 483:them. 424:). -- 388:Logic 266:Logic 29:is a 3940:talk 3906:talk 3887:talk 3858:talk 3843:talk 3584:true 3574:, a 3570:and 3551:talk 3536:talk 3476:talk 3456:talk 3381:and 3354:The 3337:talk 3314:talk 3308:. -- 3302:lies 3282:talk 3265:talk 3240:talk 3230:and 3217:talk 3116:talk 3102:talk 3051:talk 3037:talk 2995:talk 2946:Keφr 2924:talk 2903:Keφr 2877:talk 2849:talk 2827:talk 2788:talk 2772:talk 2665:talk 2644:talk 2625:talk 2605:talk 2578:talk 2556:talk 2540:talk 2485:talk 2406:talk 2386:talk 2380:. -- 2260:true 2256:true 2177:talk 2162:talk 2143:talk 2129:talk 2112:talk 2093:Talk 2060:. 1957:then 1918:Talk 1886:what 1810:talk 1802:able 1749:;-) 1741:then 1724:;-) 1698:talk 1672:talk 1562:~x). 1498:talk 1479:talk 1458:talk 1422:talk 1398:talk 1376:that 1337:only 1265:. -- 1261:See 1240:talk 1063:Toby 1023:Toby 999:Toby 995:what 918:Toby 906:Toby 893:Toby 876:Toby 861:Toby 825:Toby 821:also 763:Toby 716:Toby 641:talk 613:talk 51:Date 3770:if 3566:In 3405:. 3306:Lie 3201:why 3172:why 3147:why 3011:--> 2713:7. 2702:and 2515:how 2423:not 2370:or 2266:of 2203:why 2171:1. 2089:JRM 2066:-- 1914:JRM 1797:you 1729:not 1651:--> 1627:x). 1622:--> 1611:--> 1267:mav 1109:--> 1086:--> 901:not 855:to 781:LDC 741:and 580:-- 576:or 341:Mid 228:Low 3954:: 3942:) 3908:) 3889:) 3860:) 3845:) 3553:) 3538:) 3478:) 3458:) 3339:) 3316:) 3284:) 3271:) 3267:) 3242:) 3219:) 3211:-- 3118:) 3104:) 3053:) 3039:) 3031:-- 3006:3. 2997:) 2944:— 2926:) 2901:— 2899:. 2879:) 2851:) 2829:) 2790:) 2782:-- 2774:) 2698:on 2671:) 2667:• 2646:) 2627:) 2607:) 2599:. 2580:) 2558:) 2542:) 2487:) 2410:) 2388:) 2320:~Q 2318:~P 2229:is 2179:) 2164:) 2145:) 2131:) 2123:! 2114:) 2091:· 2007:∨ 2001:¬ 1978:→ 1953:if 1936:→ 1916:· 1812:) 1787:do 1772:is 1754:is 1700:) 1674:) 1539:is 1500:) 1481:) 1460:) 1424:) 1404:) 1400:• 1367:is 1325:is 1246:) 1242:• 1144:¬ 1141:⇒ 1135:¬ 1049:→ 1033:→ 962:= 958:→ 947:= 943:→ 935:= 931:→ 779:-- 714:— 703:, 663:→ 643:) 635:-- 619:) 615:• 416:: 368:/ 264:: 3938:( 3904:( 3885:( 3856:( 3841:( 3822:B 3814:→ 3802:A 3782:B 3774:A 3742:B 3722:B 3714:→ 3702:A 3682:A 3662:B 3642:A 3622:B 3614:→ 3602:A 3549:( 3534:( 3474:( 3454:( 3372:B 3368:A 3364:B 3335:( 3312:( 3280:( 3263:( 3238:( 3215:( 3114:( 3100:( 3049:( 3035:( 2993:( 2980:. 2922:( 2875:( 2847:( 2825:( 2786:( 2770:( 2663:( 2642:( 2623:( 2603:( 2576:( 2554:( 2538:( 2483:( 2402:( 2384:( 2175:( 2160:( 2141:( 2127:( 2110:( 2082:X 2078:X 2058:A 2054:X 2050:A 2046:X 2042:X 2010:Q 2004:P 1981:Q 1975:P 1939:Q 1933:P 1808:( 1696:( 1670:( 1496:( 1477:( 1456:( 1420:( 1396:( 1384:! 1238:( 1147:P 1138:Q 1058:T 1051:T 1047:p 1039:F 1035:p 1031:F 987:x 983:x 979:x 964:T 960:p 956:F 949:F 945:F 941:F 937:T 933:T 929:F 872:A 868:P 849:I 811:x 807:x 803:x 799:x 709:x 707:( 705:P 701:S 699:∈ 697:x 693:x 691:( 689:P 685:x 669:P 665:Q 661:P 639:( 632:: 628:@ 611:( 491:B 478:" 353:. 240:. 139::

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