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And?
And you've missed the point is several different ways.
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How does this negate the fact that by being a square, it is necessarily not a circle?
This is true only within a defined ruleset-- a defined ruleset which, by the way, is itself self-contradictory. There is no guarantee that it applies to the real world. Sorry. Your incredulity is irrelevant.
The second way you missed it, and more directly the intent of my statements, is that the functional component of a proof-- any proof-- is the proving of something about the relationships between your premises-- proving a true statement about that/those relationships. Any proof, if it is correct, is a true statement-- true not false, positive, not negative. Constructing a proof, by default means constructing a true statement. One can construct a hundred false arguments-- the conclusion does not follow from the premises-- and it simply does not matter. The arguments are irrelevant. They don't matter. The only ones that do matter are the true ones.
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You have to keep your implications going in the correct direction.
And you have to think more carefully about the statements being made. If I say 'square' you know what I mean. You can draw it. If I say 'not a circle' it conveys virtually no information. It eliminates one of an infinite set of shapes. They are quite different statements. If you can draw 'not a circle' in the same way that you can draw 'square' -- with a compass and strait edge-- then lets see it. Until then, you cannot claim they are the same type of statement. The fact is that the only things you can produce are things-- circles, squares, etc. And you could draw shapes to infinity and beyond and not run out of shapes. But you wish to draw upon the limited set of shapes, generalize to all shapes-- that is what geometry is-- and conclude that nothing violating your sample set of shapes can exist. This is absurd. It is generalization from specific to universal. You'll find that that is fallacious. All we can do is accept the information we have, draw conclusions, and run with it, all the while knowing the conclusions must be tentative and the next turn may change everything.
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Who said anything about the subatomic level?
If you want an adequate definition, you need to include all components in the definition.
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Incorrect. If the statement uses the existential operator, then it is an existential statement. That is by definition.
You are not using a relevant definition and are stubbornly refusing to use, or to even attempt to understand, the relevant definition.
Skipping the Santa bits as you've misread the article and once again missed the reason I posted the article in the first place, which was to demonstrate a negative existential statement of the type at issue.
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There is no largest prime number. It does not exist.
What you have proven is that if there is a largest prime our system of mathematics is contradictory. Ignoring that our system of mathematics may be contradictory-- and in fact is-- lets look at how your proof works. You assume a largest prime and prove a contradiction, and set this up so that it applies to any prime you fill into the blank. What you've actually got is an infinite series of assumptions and an infinite series of proofs that there is a larger prime than the assumed one, or for any prime there must be a larger one. You can translate this into English as a negative, but that isn't how the proof works. The proof is an infinite series of positives.
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So? One can show that an infinite number of things are of a certain set by showing that they are not members of the complement.
So... do this in the real world. ''
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As Holmes put it, when you have eliminated the impossible, whatever remains, however improbable, must be the truth.
Holmes was not really much of a logician.
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Why not?
Because talking about things for which there is no evidence is just making things up. Why is this so hard to understand? A idea remains a phantasm, a thought experiment, until there is some positive proof of its existence. You can bitch and moan about this or that being impossible, but educated people throughout history have done the same; and been wrong about it. Any statement about what is imposible, or possible even, is conditional upon our understanding of the universe. If a relevant portion of that understanding is flawed, then the impossible suddenly becomes possible. Only when there is evidence for a thing does it step out of that limbo and into actuality.
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I can tell you everything about what a "square-circle" is: It has all the properties of a square and a circle.
Can you? How? How do you know the properties of a square? Initially, someone measured one, then another, and generalized. Same with circles, same with triangles. And we end up with something we call geometry. It is the generalization from specific cases to the universal and as such will always be questionable. Hate to break it to you.
Without a square-circle to measure, you can't know with certainty, its properties. You can't know if you've named all of them, or if you've got all the properties correct. You can't check your assumptions, in other words. What you've got is thought experiment. What you've got is "if the assumptions of plane geometry are true, then square circles can't exist." Notice how the whole structure is conditional, and that there is no guarantee that it applies to the real world?
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But such an object still doesn't exist.
hmmm... if space were to be contracted to a point as it is a a singularity, would not the dimensions of a square and a circle match? The radius of a circle under such circumstances would be zero, as would any measurement you made of a square. Thus, they would appear to be the same.
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Information only exists for what actually is defined.
Why are you equating definition with existence?
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No, what is meant by "you cannot prove a negative" is acceding to the logical error of ad hoc argumentation.
It is getting damned frustrating to attemp a debate with you when you choose to remain ignorant of the meaning of the phrase which started the debate. You can't win by arguing against the wrong definition.
Additionally, you really ought to brush up on what is meant by 'ad hoc.' There is some question as to whether such hypothesis are technically fallacious.
Strictly, 'ad hoc' means 'for this case only.' No one is making that claim. The claim being made is we don't know what is outside our experience. Since our experience is limited, we have to conclude that our claims knowledge are tentative. There may be things we don't know. To do otherwise is to commit the most-definitely-a-fallacy of generalizing from the specific to the universal. This ought to be common sense, though such seems to be none too common.
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You mean Descartes was wrong?
Frequently.
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A difference that makes no difference really is a difference?
Could be. Of two things which appear to us to be the same, or make no difference, one or the other may well be true and the other false. Whether we are ever able to sort this out is another thing. This is a rule of thumb, not hard logic.
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Not quite. It isn't proving a negative.
Here you go agian, refusing to understand what is being stated.
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There are cases where lack of evidence for S is relevant to the truth or falsity of S.
What would those cases be? The author doesn't say, and I can't think of any. Nor can I find a similar statement elsewhere, as here:
Page not found - Intrepid Software
Surely you must know what these two cases are, as you know that your case falls into one of them. Or perhaps this is just what it looks like-- avoidance.
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Or that I wasn't proving the non-existence of something?
... that you are proving only within a defined rule set. And that you are translating positive mathematics into negative English and calling it proof of a negative.
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But if the thing's existence requires that we have evidence and we don't, then we necessarily conclude that it doesn't exist.
More like 'assume.' Unless you have absolute and infallible knowledge you are arguing from ignorance.
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Again, it is nothing more than a special case of the False Dilemma. If the dilemma is not false, then absence of evidence is evidence of absence.
How is it not a false dilemma? For every real thingie-- say, cat-- there are three possibilities.
1) Evidence we have that supports the proposition. One case of 'cat' and its over.
2) Evidence we have which does not support 'cat.' Finding a dog does not prove or disprove 'cat.' Finding that dog is not 'cat' does not prove or disprove 'cat' thought you seem to think it ought to.
3) Then there is evidence we don't have which might go either way. Until we have the evidence, we won't know which way it goes. Until we have absolute and infallible knowledge, there will always be this option.
What you want to do is deny the third possibility. You can't do so without making the idiotic claim that we have absolute and infallible knowledge.
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They allow possibilities that have yet to be elucidated.
No kidding?
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But if you can elucidate all possibilites....
You really aren't getting this are you? You can't elucidate all possibilities-- ever, not in the real world.
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How can you possibly say "~p" without proving a negative? You just negated p!
Boy, do you love the trivial!
The 'item is not P', or 'we do not have P', is not the same as 'P is not.'
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It is tautological, but it is not pointless. It is not trivial.
Sorry, but it is. A thing is a thing is a thing is a thing. There is no logic involved. It is saying the same thing with extra words. A tautology is true no matter what. Its truth value is built in-- assumed a priori. It can be a fallacy, but isn't always. Inductive systems are based on such assumptions. Doesn't make them true in the real world. They are just assumed to be.
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For many processes, we can't because the objects which we are examining aren't sufficiently defined.
There are no processes sufficiently defined. Assuming certain limitations is necessary but you don't get absolute proofs, only conditional ones, ones conditional upon your assumptions.
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That is because that's precisely what happens. By proving what something is, you necessarily prove what it isn't.
Then try doing it the other way around. There is a rat on my desk who keeps walking over my keyboard. Prove what he isn't, to prove what he is. He isn't a cat. He isn't a bird. He isn't a she. He isn't a martian. He isn't.... You can never complete the task. That is a large part of the point. That should tell you there is something quite different about the two statements-- "He is a rat" and "He is not a rat." You can prove the first without bothering with the second. But trying to prove the second without bothering with the first leads you into an impossible task.
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How is that insufficient?
Mathematics is a made up system. It is based upon assumption and is internally inconsistent. How exactly can it be SUFFICIENT?
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Have you checked the definition of "strawman" in those logic sites of yours? That's where you take an argument that isn't the one your interlocutor is trying to make, and show that it isn't valid.
Indeed. You've been constructing and arguing against such straw men since this discussion began. And now appear to be getting a bit testy that we are't taking you scarecrows seriously.
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By your logic, if you were to say, "It is physically impossible for a heavier-than-air object to fly," and I were to show you an airplane and explain the concept of Bernoulli's Principle, for you to come back and say, "Yeah, but try and do that without using any physics," you would understand why I'd blink at you. How can I show something physically without using physics?
You've got the analogy backwards. You've given a positive proof-- that of some principles of aerodynamics. You've given proof that such objects can fly. The difficulty would be if I tried to prove it impossible for heavier that air objects to fly. A few hundred years ago I might have been able to do so, based on all the best human knowledge; but we all know that that proof would have been flawed. To prove that something can happen, you just have to find a case demonstrating it. You don't have to understand all the reasons why. You can stumble upon an example and figure out what is going on later. To prove something can't happen, you have to assume that you know all the relevant information. It is this assumption I am asking you to drop.
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I am arguing the validity of deduction and how one can logically deduce negative propositions.
Then you are not arguing the fundamental point.
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Strawman.
Was the phrase that started this exchange, "You can't prove a negative"?
You are funny guy. You stubbornly refuse to have the statement explained to you. You argue a different point. And complain that we don't argue this different point. Did you mean this to as funny as it is? Or should I be laughing at you, rather than with you?
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For you to switch to non-Euclidean geometry in the middle of a sentence is a logical error.
As illustration of the limitations of axiomatic systems it is quite appropriate. You assume one system and insist that it describes everything everywhere. But there are multiple such systems and none of them, necessarily, apply to the universe.
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Why?
Because it is quite easy to prove that something does exist without relying on a system of axioms. One just have to find an example of the object. If the object doesn't fit into some pre-existing logical construct, then tough. The object exists. There it is. It can contradict everything we think we know. Doesn't matter. There it is. It does not work the other way around.
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I wholeheartedly agree that if there is something wrong with the axioms, then the deductions we make from them cannot be trusted.
But I am not arguing that the axioms are faulty.
Then there is not much debate, and you have admitted to arguing a straw man.
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Sure I can. If apples are inconsistent, then apples don't exist.
BS. There is absolutely no reason real things have to be consistent. And at very fundamental levels things appear to be radically at odds with what we'd call consistency.
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But it's my argument. Therefore we use my definitions.
You're arguing a strawman.
You... YOU... jumped into a discussion that was already in progress. You used definitions not like those in use ( though not directly stated ). You refused to accept definitions appropriate to the discussion, and instead insist that we use yours. Amazing ... !!! It is you who is arguing a straw man.
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Incorrect. Even at singularities, logical things happen.
You have got to be joking?
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Do not confuse our personal knowledge of something with its ability to exist or not exist.
Actually, this is pretty obviously your error.
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Question: Do you know about the mathematical concept of "Platonism"?
Question: Do you know that Platonism isn't a mathematical concept?
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A Platonist would then say that there is an answer to the question, we just don't know what it is.
This is just about the most pitiful thumbnail version of Platonism I have ever seen.
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The size of the Reals does exist, we just have no tools to let us know what it is.
So... there is a largest Real number then?
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Thus, inside the event horizon of a black hole, at atomic scales, etc., things happen according to their internal logic...we just don't know what it is.
And you know this how? Because you say so? Because you think so? I don't know if there is an internal logic or not, but that logic, if it exists, is damn sure not the logic we know and love.
I'm not sure what to call this... equivocation? Arguing from ignorance? Just plain 'it is cause I say it is'?
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Mathematics would still exist, even if there were no people around to think about it.
So a system of definitions has existence outside the minds of the people who made made up the definitions? You're joking.
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I didn't say we did.
Sorry, but you've been implying it all along. Every time you insist that something is mathematically or logically impossible and thus it IS IN FACT impossible, you are implying that math and/or logic is in fact an accurate representation of the world.
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You seem to be arguing that because I cannot disprove the existence of any possible description of god you might be able to come up with, that means I am incapable of disproving the existence of any specific description of god.
This is the root of the most common problem with attempting to prove a negative and probably the impetus for most chat-room utterances of the phrase. You do end up being asked to disprove an infinite series of 'nots.' And there is no way to settle on a definition if there is nothing to define, if there are no dimensions that can be measured-- whatever. This is the colloquial meaning of the phrase, if you will.
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I don't hve to "observe and record everything that is, was, and will be across all of space and any other spaces there might be" in order to define a square.
But to make some claim of actual existence or non-existence, you do. Otherwise what you have is a generaliation from the specific to the universal. I can't be more plain than that. This is induction's dirty little secret. If you want to make the conclusions conditional upon some set of, possibly, arbitrary definitions that is another thing altogether. And I don't think this is the case. Your statements about mathmatics make my think that you consider math to be far from arbitrary.
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I have my definition.
Why must you confuse definition with reality?
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If my definition leads to a contradiction, then I necessarily conclude that it does not exist.
Your whole point is that you manipulate definitions? You rest your case upon the manipulation of definitions? Defining doesn't make a thing, just describes it-- in a very limited way, usually.
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Before you said you were talking about existentials. Now you say you're not.
I made a statement about existentials. And defined the usage at great length. I believe it was a logician named Adler(?) who first useged the phrase as I have. You've thus far ignored this and continue to harp on another usage. This is textbook equivocation on your part, though I doubt it is intentional. You must pay attention to meaning and not get stuck on the language. Browse through the dictionaries at Onelook. There are more ways than yours to use a word.
Definitions of existential - OneLook Dictionary Search
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Yes.
Numbers exist? Show me one.
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And that means there is no largest prime number because of what, precisely?
'cause there is always a larger one. What is your point?
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Definitions exist, too.
??????????
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And even if the definition of ghosts is that they can be captured on film, that we don't have any pictures of them is not sufficient to say that they don't exist.
Definitions don't make the thing. We make definitions to fit the thing, not the thing to fit the definition. You appear to be equating our ideas of things with the things themselves.
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For some things, yes.
You actually believe we have infallible knowledge of something?
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I'm a mathematican by training. Do we really need to go through credentialing before we hit the argument from authority?
I know you are a mathematician. It shows. You reek of it.
Credentialing? Seems you are the first one to get to that. I haven't said anything about what formal training I have had or haven't had, not to you anyway. So... bullseye on that argument from authority, bud.
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Incorrect.
Deductive logic is basically subtraction, hence the name. Inductive logic is an inference from specific cases to the universal. Math is essentially the same. We count five rocks, then split the pile and count again. hmmm... two in one pile, three in another. And so on. Eventually we-- our ancestors, really-- recognized a pattern and formalized that pattern into a basic arithmetic. This is a generalization from the specific to the universal and it is a violation of one of the rules of deductive logic.
Now, take
i. This is a number which when multiplied by itself equals negative 1. hmmm..... but a positive times a positive is a positive. And a negative times a negative is a positive. So
i quite blatantly implies a violation of the rules of multiplication. How is that for consistency?
# added by edit
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By the way, "inconsistent" is equivalent to "incomplete." It all depends upon how you look at it.
Ya sure about this?
And for any question A, for any affirmation A, you can ask if it's possible to settle the matter by either proving A or the opposite of A, not A. ... That's called completeness.
...
Another interesting question is if you can prove an assertion (A) and you can also prove the contrary assertion (A). That's called inconsistency, and if that happens it's very bad!
Page Not Found
Inconsistent does not look synonymous with incomplete. You are saying that there is a transformation that allows you to equate 'can't answer one way or the other' with 'can get two answers that contradict one another'?
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[This message has been edited by John, 05-15-2003]
[This message has been edited by John, 05-15-2003]
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