John responds to me:
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If you claim that X is a circle and I show that it is a square, then I have proven a negative: X is not a circle.
Actually, what you have prove is that it IS a square. You said this yourself-- "I show that it is a square."
And? How does this negate the fact that by being a square, it is necessarily not a circle? Are you saying that it is possible to be both a square and a circle?
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And then the slight of hand... "X is not a circle." This is a qualitatively different statement.
Yes and no. Indeed, showing that something is not a circle does not mean it is a square. However, showing that something is a square does mean it is not a circle.
You have to keep your implications going in the correct direction.
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To illustrate, the phrase "X is a square" tells you precisely what X looks like when drawn on a flat sheet of paper.
And thus, it is not a circle.
Or are you saying that it is possible for a planar object to be both a square and a circle?
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The phrase "X is not a circle" eliminates one of an infinite set of possibilities.
You've reversed the implications.
A -> B does not imply B -> A.
X being a square implies X is not a circle. X is not a circle does not imply X is a square.
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In other words, you prove a positive-- you prove something about the object in question, then make the assertion that within some some axiomatic system inconsistent properties cannot also be true.
And how is that not proving a negative? When you draw a boundary, you are also defining the outside as well as the inside. It's a common technique in drawing when you're having trouble with an object: Don't draw the object but instead draw the space around it. By defining everything that is not the object, you'll be left with the object.
In mathematics, a similar thing happens. It is often difficult to describe the elements of a set but much easier to describe the elements that are not in the set. By accounting for all of the non-elements, you are left with the elements and having proven what you were trying to prove in the first place.
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You cannot prove the negative other than by proving something about the object.
So? How is that not proving a negative? Once again, you're saying that I must prove something mathematically without being allowed to use mathematics to do so.
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Are you saying you don't have absolute knowledge about some things like what your car keys look like?
At the subatomic level, we have very little idea of what they "look like."
Who said anything about the subatomic level?
You don't know what your keys look like?
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I know. But it is still an existential room. There exists no element X in the set. That is an existential statement.
It is a conditional. It is conditional upon the items in the set.
Incorrect. If the statement uses the existential operator, then it is an existential statement. That is by definition.
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I provided a link to the following earlier. I assumed you read it. Guess I was wrong.
Incorrect. Didn't you read my response?
Oy...they equivocated. They shifted from the object to the name of the object. That is, they shifted from "Santa Claus does not exist" to "The name 'Santa Claus' does not exist" and then acted as if the two statements were equivalent.
In short, the article made a fundamental error. It confused an object for its name. An object and the name of the object are separate objects. To say that the statement "Santa Claus does not exist" cannot be true because the physical words "Santa Claus" exist is to completely misunderstand the point behind the statement, "Santa Claus does not exist."
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Has everything to do with it. All of your efforts go into proving what is present. None go into proving what is not.
I have.
There is no largest prime number. It does not exist.
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Why is that? 1) There are a finite number of 'is presents' while there are an infinite number of 'is not presents.'
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. Take, for example, the definition of an infinite set: "A set that is not finite is infinite." From one of my Analysis texts (
Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, 1976):
2.4 Definition For any positive integer
n, let
Jn be the set whose elements are the integers 1, 2, ...,
n; let
J be the set consisting of all positive integers. For any set
A, we say:
(
a)
A is
finite if
A ~
Jn for some
n (the empty set is also considered to be finite).
(
b)
A is
infinite if
A is not finite.
(
c)
A is
countable if
A ~
J.
(
d)
A is
uncountable if
A is neither finite nor countable.
(
e)
A is
at most countable if
A is finite or countable.
You will notice that both "infinite" and "uncountable" are defined in terms of not being something else. It is much easier to describe a finite set than it is to define an infinite set. Ergo, infinite sets are defined as that which is not finite.
One aspect of sets is that the complement of an open set is closed. From the same source:
2.23 Theorem A set E is open if and only if its complement is closed.
Proof First, suppose
Ec is closed. Choose
x element of
E. Then
x not element of
Ec, and
x is not a limit point of
Ec. Hence there exists a neighborhood
N of
x such that
Ec intersection
N is empty, that is,
N subset
E. Thus
x is an interior point of
E, and
E is open.
Next, suppose
E is open. Let
x be a limit point of
Ec. Then every neighborhood of
x contains a point of
Ec, so that
x is not an interior point of
E. Since
E is open, this means that
x element
Ec. It follows that
Ec is closed.
Corollary A set F is closed if and only if its complement is open.
From this, we can then talk about the "closure" of a set in a metric space. One aspect of a closure of a set is that it is closed. The way we prove that is by taking a point
p that is in the metric space but not an element of the closure,
E-bar. This means that
p is neither a point in
E nor a limit point of
E. This means that
p has a neighborhood which does not intersect
E. This means that the complement of
E-bar is open (since neighborhoods are open) and therefore,
E-bar is closed.
In other words, you often prove things by showing that only certain possibilities necessarily exist and that the specific scenario which we have necessitates that all possibilities but one are impossible. Therefore, you are left necessarily concluding that the specific scenario is the one that is left behind.
As Holmes put it, when you have eliminated the impossible, whatever remains, however improbable, must be the truth.
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2) Proving directly what is not present would require having information about what is not present, and such information does not exist.
Why not? I can tell you everything about what a "square-circle" is: It has all the properties of a square and a circle.
But such an object still doesn't exist.
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Information only exists for what actually is present.
Incorrect. Information only exists for what actually is defined. If you leave something undefined, then you lose information.
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Within a limited set-- say, a collection of marbles-- where you have knowledge of what lies outside the set you can determine if something is not in the set of marbles. But in an absolute sense, as in the case of the universe, we have no knowledge of what isn't in the set-- only of what is. This is what is meant by "you cannot prove a negative."
No, what is meant by "you cannot prove a negative" is acceding to the logical error of ad hoc argumentation. It surrenders to the vague definition rather than meeting it head on to directly state that the definition is vague and therefore not only can we have a problem proving negatives about it, but we will also have problems proving positives about it.
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If it is not detectable no matter what, then it is the same as if it doesn't exist.
This is a textbook case of the fallacy of 'proving a negative.'
You mean Descartes was wrong? A difference that makes no difference really is a difference?
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Appeal to Ignorance (Proving a Negative): an argument that asserts a claim is true because no one can prove it is wrong; this shifts the burden of proof to the audience or opponent rather than the claimant.
http://chuma.cas.usf.edu/~pinsky/logicguide.htm
Not quite. It isn't proving a negative. It is, however, a misuse of the "not" operator. Argumentum ad ignorantiam goes both ways and is just as invalid when applied to a positive:
You cannot prove that X > Y, therefore X <= Y.
The problem is the inability to prove something does not give us information about it. Argumentum ad ignorantiam is a special instance of the error of False Dilemma.
However, if the dilemma is not false, then there is no error of False Dilemma. And with no error of false dilemma, there is no argumentum ad ignorantiam.
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Argumentum ad Ignorantiam [appeal to ignorance]
To recognize the fallacy of appeal to ignorance, look for a conclusion based upon an absence of proof or evidence. Be aware of the two types of cases in which lack of evidence for S is relevant to the truth or falsity of S. But also, be aware that the onus of proof is on the claimant and that no one can prove a negative.
Page Not Found | Embry-Riddle Aeronautical University - Faculty
The final statement simply isn't true and is directly contradicted by the statement immediately preceding: There are cases where lack of evidence for S is relevant to the truth or falsity of S.
There is no largest prime.
Or are you saying that my proof of it is unjustified? Or that I wasn't proving the non-existence of something?
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If it doesn't exist, wasn't that what we were trying to demonstrate?
You haven't demonstrated that it doesn't exist, only that you have no evidence for its existence. Review the fallacies cited above.
You mean I didn't prove the non-existence of a largest prime?
I have reviewed the fallacies you've quoted. There is an error. You can prove a negative.
Again, if X is a square, then it is not a circle is a logical statement. It's reversal is not, however.
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But doesn't that mean we've just succeeded in showing that it doesn't exist?
No.
You mean there is a largest prime? Or that I cannot prove that there isn't a largest prime? Strange...I thought I could prove there isn't a largest prime.
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It proves that we have no evidence for the thing's existence.
But if the thing's existence requires that we have evidence and we don't, then we necessarily conclude that it doesn't exist.
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.
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Two thousand years ago, no one on Earth had evidence for sub-atomic particles. Does this mean that sub-atomic particles did not exist? Thus, the reason it is a fallacy to conclude that something does not exist because you have no evidence for its existence.
Agreed.
But the problem is not proving a negative. It's that the definitions are too vague. They allow possibilities that have yet to be elucidated.
But if you can elucidate all possibilites, then it is sufficient to prove that something is of one of them by showing that it is none of the others.
F'rinstance, all integers are either odd or even. If I can show that it is not odd, then I have necessarily shown that it is even. There is nothing else for it to be. The dilemma here is not false...there really are only two possibilities.
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Isn't it a truism that A v ~A?
Yeah, sort-of. Via the rule of inference called addition ( in the symbolic system set out by Irwin Copi ) if you have a proposition A, you can add anything to the right of it seperated by 'or' -- k, l, x, whatever. Then, if you can infer the negation of one side, you can infer the other.
That's what I've been saying. If you have all possibilities staked out (which requires well-defined objects behaving in well-defined ways), then by showing that something is not of all but one, then you necessarily have shown that it is that last one.
It may be difficult to show that something is of a particular set. It may be easier to show that it is not of its complement. Therefore, if we do show that it is not of the complement, we have shown that it is of the set.
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p v q
~p
Therefore, q
Now, look what happens when you use A v ~A.
A v ~A
A
Therefore.... nothing. This isn't a valid operation.
No...you have A, therefore A. Similarly, A, therefore ~(~A).
And notice in your first statement, you said "~p."
How can you possibly say "~p" without proving a negative? You just negated p!
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A v ~A
~A
Therefore, ~A. This one is pointless. We already had ~A. It is tautalogical.
It is tautological, but it is not pointless. It is not trivial.
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What you are looking for is an inference from lack of evidence, and such cannot be done.
Unless evidence is necessarily required. If evidence must necessarily exist and it doesn't, then we necessarily conclude absence.
The problem is not the ability to prove a negative in general. It is the ability to do it in specific. That is, the error of argumentum ad ignorantiam is not "If evidence must necessarily exist and it doesn't," it is "If evidence must necessarily exist and it doesn't that we know of." The question immediately then becomes, "How do you know you've taken everything into account"? For many processes, we can't because the objects which we are examining aren't sufficiently defined.
But if they are, we can show precisely why the evidence that must necessarily exist doesn't.
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Why the restriction?
Because I can't get you to understand that proving what IS does not prove what ISn't.
That is because that's precisely what happens. By proving what something is, you necessarily prove what it isn't.
Something that is a square is necessarily not a circle.
What you are complaining about, and it is a legitimate complaint, is that proving what something is not does not necessarily prove what it is. That only happens when you can account for all possibilities for what it might not be. That can often be quite difficult if not impossible.
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Proving what is, only proves what is; or proves something about what is.
You mean if I show that something is a square, there is a possibility that it might be a circle, too?
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That you have a circle, may imply within our system of mathematics that this circle is not a square. It does not prove that squares do not exist or even that circles and squares cannot exist, in reality, in the same object;
Actually, it does. Are you seriously saying that there can be a planar object that has the properties of both a circle and a square?
Some properties of a circle:
The longest chord of a circle is its diameter.
All diameters pass through the center.
If a chord passes through the center, it is a diameter.
Two non-identical diameters intersect at the center.
All diameters are the same length.
Let's see what happens when we look at a square, now:
The longest chord of a square are its diagonals.
The diagonals of a square cross at a certain point, which we'll call the center as defined above.
It is possible to draw a chord through the center that is not the diagonal.
Therefore, since we know that all chords through the center of a circle are of the same length, this must mean that the line through the center of the square must be the same length as the diagonal.
However, it isn't.
Therefore, there can be no planar object that has the properties of both a circle and a square. By having one, it necessarily cannot have the other.
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but proves only that our mathematics forbid it.
How is that insufficient?
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A property of the circle-- a positive thing-- is that it doesn't have the properties of a square. You haven't proven the non-existence of squares.
I wasn't trying to.
Instead, I was trying to prove the non-existence of "circle-squares."
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.
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Of course. You're trying to challenge me to do something mathematically without letting me use mathematics to do it.
Something like that.
Then you can understand why I refuse to concede to such a request.
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?
So how can I show you something mathematically without using any mathematics?
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I am trying to get you to prove a negative without reliance on an axiomatic system.
Why?
I am arguing the validity of deduction and how one can logically deduce negative propositions.
You're arguing whether or not the axioms from which we make our deductions can be accepted.
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. I am arguing that deductive statements are logically valid, even when those deductions are negative.
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Prove an apple. Point to it. Pick it up. Now prove that apples do not exist. You can't point to not-apple.
Sure I can. If apples are inconsistent, then apples don't exist.
I agree that for this specific example, it's going to be very hard to do so. But, that depends upon the definition of "apple." If the definition allows it to be anywhere in the universe, then I'm going to have an extremely tough time, conceivably impossible. But if the definition requires it to be sitting right here on my desk, then the task will be much easier.
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In fact, there is no information about not-apple.
Incorrect. A banana, for example, is something that is not an apple. It is not the only thing that is not an apple, but that's something.
Again, you've got your directions reversed. If it's a square, then that necessarily means it is not a circle. That is what I'm arguing.
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Why can't you rely on an axiomatic system? Because no such system can be proven.
Of course not. That's why they're called "axioms." Those are the statements that are true without being proven and form the basis by which all are statements are derived.
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More formally, an existential statement is one that uses the existential operator: "There exists." This is in contrast to the universal operator: "For all."
Different use of the term
But it's my argument. Therefore we use my definitions.
You're arguing a strawman.
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But the relevant formulation would be that there is no case where X exists. Period.
Strawman. That is not the argument I am making.
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Can you prove that no right triangles have angles adding to more than 180 degrees? Yep. Can you prove that no right triangles have angles adding to 180 degrees? Yup. You can prove that too. You just have to switch geometries.
Since when were we switching geometries?
You've just argued yet another strawman. That things are true for non-Euclidean geometry does not mean they are true for Euclidean. For you to switch to non-Euclidean geometry in the middle of a sentence is a logical error.
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You should consider what happens at a singularity, or inside the event horizon of a black hole, or at sub-atomic scales. Illogical things, as logic is defined, happen all the time. Nature glibly ignores your convictions.
Incorrect. Even at singularities, logical things happen.
Do not confuse our personal knowledge of something with its ability to exist or not exist.
You just committed argumentum ad ignorantiam.
Question: Do you know about the mathematical concept of "Platonism"?
Here's an example of a question which would define a Platonist from a non-Platonist:
Does the Continuum Hypothesis have an answer?
The Continuum Hypothesis has to do with the size of the Real Numbers. We know they're bigger than aleph-null, but we don't know if they're aleph-one. We do know that assuming they're aleph-one does not lead to a contradiction. But, we also know that assuming they're not aleph-one does not lead to a contradiction.
A Platonist would then say that there is an answer to the question, we just don't know what it is. The size of the Reals does exist, we just have no tools to let us know what it is.
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.
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We've invented math over the past few thousand years.
No, we discovered math. Mathematics would still exist, even if there were no people around to think about it.
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There is no guarantee that what we have invented is an accurate representation of the world.
I didn't say we did.
However, you were saying that there is something wrong with a deductive system. But that simply isn't true. Deductive systems are necessarily logical.
If you want to argue the validity of the axioms by which the deductive statements are made, that's another question entirely.
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There is no largest prime number.
Which is shorthand for "For every number that is prime, there is a larger number that is also prime." Simply rephrasing it doesn't change the meaning.
But you prove that by demonstrating a negative: Any prime that you think is the largest one really isn't.
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Oy...they equivocated. They shifted from the object to the name of the object. That is, they shifted from "Santa Claus does not exist" to "The name 'Santa Claus' does not exist" and then acted as if the two statements were equivalent.
I think you misread the article,
Strange...just above you were saying I hadn't read the article at all.
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I think I get to be the arbiter of what negative I was trying to prove. If you are concerned about a different negative statement, then don't hold that against me. I wasn't dealing with that one.
Ummm... you jumped in, rather arrogantly, to explain to crash that he was wrong to claim that you cannot prove a negative. So, no, you don't get to be arbiter.
Yes, I do. Crashfrog was stating a universal. The way to disprove a universal (there's that "proving a negative" thing again) is to prove an existential. That is, if you say, "for all," then I can disprove that by showing, "there exists for which it isn't true."
Therefore, if crashfrog has a specific negative in mind, then it is quite possible that I won't be able to prove it.
Instead, however, he claimed that all negatives were impossible to prove.
Therefore, I simply need to show one negative that can be proven in order to show that his universal statement is false.
And in the process, I end up proving
two negatives: Not only do I end up proving my negative, but I also prove the negative that "You can't prove a negative" is false.
If you are arguing a different negative, then you are not arguing the POINT. Remember the first thing I said to you? "That isn't what he is talking about."
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... that you are stubbornly arguing the wrong point-- wrong in reference to the meaning of the phrase that started this exchange.
Strawman.
Was the phrase that started this exchange, "You can't prove a negative"?
Therefore, wouldn't it be sufficient to demonstrate a negative you can prove in order to show that statement false?
Is it not true that there is no largest prime?
Ergo, I just proved a negative.
Ergo, "You can't prove a negative" is false.
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You are confused.
Strange...I've been saying the same thing to you.
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This is equivalent to "X is not Y."
Isn't that sufficient to show that you can prove a negative?
What I have been saying all along is that proving a negative requires well-defined objects behaving in well-defined manners.
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Have you ever heard anyone say "Prove that God is not a piano, a redhead, a martian?" Nope.
That just proves my point. The reason why you don't hear that is because there is no well-defined description of "god."
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.
I, however, am arguing the other direction: Given a specific description of god, it may be possible to show that god as defined by that description does not exist.
If you then change your definition of "god," then my proof of non-existence may no longer work, but we wouldn't expect to work since, after all, you changed your definition.
You seem to want me to prove a universal without a definition. I am insisting that you give me a definition before I try to do anything. I may or may not be able to do so, but until you give me that definition, I can't do anything.
People say "Prove that God does not exist." This should illustrate to you the inherent differences of the claims you are making and what is intended by the phrase "you cannot prove a negative." It is about existence and non-existence, not about qualities.
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What you want to do is prove Y, or define Y. That means observing and recording everything that is, was and will be across all of space and any other spaces there might be.
No, it means making a definition. 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.
Or are you saying that there is no definition of a square?
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Once you have done that, you still cannot infer that X doesn't exist,
Sure I can. I have my definition. If my definition leads to a contradiction, then I necessarily conclude that it does not exist.
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or that other things do not exist.
Whose talking about other things? I thought we were talking about the one thing that we were specifically trying to disprove. You're absolutely right that my ability to disprove the existence of square-circles says nothing about my ability to prove or disprove the existence of triangles.
But then again, we weren't talking about triangles.
Why are you setting up a strawman?
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Inferring based on lack of evidence is a fallacy.
Only if you are engaging in False Dilemma.
If you aren't, then lack of evidence isn't a fallacy.
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You're not talking about the existential operator...you're talking about the universal.
No, I believe I am talking about something quite different from either.
Now I admit I am confused.
Before you said you were talking about existentials. Now you say you're not.
So which is it? Are you talking about existentials or not? Are you talking about the existential operator or not? The negation of the universal is the existential and vice versa.
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Sure it can. There does not exist a largest prime number, remember?
Just for the record, do any numbers exist?
Yes.
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Saying a number exists is not like saying a rock exists.
Perhaps...perhaps not. It is, however, like saying love exists.
Does love exist?
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Even your proof of 'no largest prime' works by proving what DOES exist within the rules of the system. Remember?
So? How is that not proving a negative? If something is either X or Y and it is X, then it is necessarily not Y.
You're arguing a strawman again.
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The relevant claim is that NO CASE EXISTS.
Strawman.
Stop arguing your point and start arguing mine.
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There is no largest prime number.
There is always a larger prime.
And that means there is no largest prime number because of what, precisely?
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Thus, by definition, there can be no planar object that has the properties of both a square and a circle.
That's right, BY DEFINITION.
And that means we haven't proven a negative because of what, precisely?
Definitions exist, too.
You seem to be arguing that because axioms may be questionable, the ability to make valid deductive arguments based upon those axioms is impossible.
That simply is true. Indeed, if the axioms are questionable, then the results of the deductive process may not be accurate, but the process, itself, is still valid. The problem is not the deductive process, it is the axioms that aren't.
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This may be the case on a plane as we define it,
Who cares about the plane as we don't define it? We're not talking about that. If you want to talk about that, then we'll shift gears and deal with that other definition. But until you come up with that other definition, we can't say anything at all, positive or negative.
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but does such actually exist anywhere?
Who cares? We're not dealing with those other possibilities. We're only concerned with the definition before us.
It does not matter that triangles have more than 180 degrees on a sphere when we're dealing with a plane. Because on a plane, triangles have exactly 180 degrees, they always will, and there is no way they can have anything other than 180 degrees.
The question of whether or not we are in a plane is a completely different question from how triangles behave in a plane. Too, the existence of triangles that are consistent with a planar triangle does not mean we are in a plane (unless we also know that "planar triangles" can only happen in a plane...which isn't true, they can happen in non-planes.)
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You simply need to have a solid definition of the purple people eaters
How does one, in the real world, get a solid definition of something that does not exist?
Carefully.
A "square-circle" is a planar object that has the properties of both a square and a circle.
The "largest prime" is the prime number for which there is no larger.
Neither one exists, but both have solid definitions.
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But that is the case only if we have no definitive method of verification.
With things that do not exist you can't really know what must happen if they did exist.
Incorrect.
The largest prime number doesn't exist, but we know exactly what must happen if it did. That's how we go about proving that it doesn't: Assume it does.
The existence of a "largest prime number" means that there are a finite number of prime numbers. That means we can list them (the Sieve of Eratosthenes is sufficient to get them all.) That means we can construct a number that is the product of all the primes...plus 1. That means that either this new number is prime or that there is a prime between the "largest prime" and this new number.
Things that don't exist can easily have definitions. That's the only way we can know they don't exist: Something about their definitions makes them disallowable.
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For example, lets test for ghosts. If ghosts exist we ought to be able to film them?
I don't know. Can we? You'll need to define that. If they can, then yes. But if they can't always be filmed, then no. The term "ghost," in and of itself, is not sufficient.
Therefore, since we have an ill-defined definition, we are going to have a hard time saying anything, positive or negative, about ghosts.
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Well... why? We don't know anything about ghosts. We have no evidence.
That we have no evidence of ghosts doesn't mean we don't have a definition of ghosts.
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. If we are going to be using pictures as a test, then we need to be able to set up a scenario whereby a picture of a ghost must necessarily result.
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Logical error: Ad hoc.
When making absolute claims one must include every case imaginable. This is not fallacious. And certainly isn't this fallacy.
It is when you are making an ad hoc addition to the defintion. When making absolute claims, one must include every case imaginable
within the context of the definition.
If we're going to talk about the angular sum of planar triangles, it doesn't matter what spherical triangles are like because we're not talking about spherical geometry. We can say with absolute certainty that all triangles have 180 degrees in the plane.
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Logical error: Incredulity.
Do you assert that we have infalible knowledge?
For some things, yes. For other things, no.
We should not hold the instances where we don't have infallible knowledge against those instances where we do.
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ummm... you might brush up on your informal logic, because you've mis-applied this one as well.
Hah!
I'm a mathematican by training. Do we really need to go through credentialing before we hit the argument from authority?
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And you just proved a negative. Newton's mechanics are not true.
Newton's mechanics are a description, not a thing.
Descriptions are things, too.
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Strike this one up as "still just not getting it."
Indeed, you don't.
Now that we've gotten the catty remarks out of the way, can we return to the issue at hand?
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Incorrect. As crashfrog said, an inductive conclusion might be wrong. A deductive conclusion cannot be.
You apparently missed Gdel.
Incorrect.
Wrote about him in 10th grade.
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Ever hear of the Russell paradox?
Still have the paper I wrote on it somewhere.
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Math and logic are riddled with contradiction and based upon assumption. Why exactly can these not be wrong?
Incorrect.
What Godel showed is that for an axiomatic system complex enough to model simple arithmetic, there will be statements within the system that can not be proven within the system.
Not all axiomatic systems are complex enough to model simple arithmetic and indeed, for some of them, there are no undecideable propositions.
By the way, "inconsistent" is equivalent to "incomplete." It all depends upon how you look at it.
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No, not "without absolute knowledge." Without a sufficient definition.
A sufficient definition would require absolute knowledge,
But only so far as required to create the definition. I don't need to know anything about spherical geometry in order to make definitions in planar.
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We 'solve' this by introducing assumptions-- things like 'within the confines of plane geometry' or 'visible things inside this room.' We limit our scope. But it is assumption.
So?
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You started with the definition that "Santa Claus" was an object that lived at the North Pole and delivered presents.
What you prove in this paragraph is that either the definition is wrong -- ie. it does refer to a real thing but that are mistaken about what that thing is-- or that Santa doesn't exist.
But the thing is its definition. You are perfectly free to say that this other definition is what you really meant by "Santa Claus," that's fine. It's ad hoc and thus does not alter the fact that we showed the non-existence of the original object as described by the original definition.
Your new definition may require a new proof and we will have to examine it to be certain.
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In the absence of Santa's existence, there is no way to sort out the definition.
Why not? Why is it impossible to define Santa? I thought we had one: A "jolly old elf," male, white, beard, overweight, lives at the North Pole at the surface, not invisible, delivers presents to all of the good boys and girls (seemingly only the Christian ones) on Christmas Eve.
Well, we've been to the North Pole. There's nothing there. In fact, the ice moves at the northern polar ice cap so quickly that it is impractical to put a "pole" there like we have at the South Pole. If Santa built a workshop at the North Pole, it would no longer be there within hours. I recall a program when Hugh Downes was at the North Pole talking about what is happening in each section of the world as he walked around the North Pole and he mentioned that by the time the program is over, he'd be nowhere near the pole. Another program showed a baseball game that took place at the North Pole with the pitcher's mound at the pole...and again, the comment that by the time the game was over, they'd be nowhere near the pole.
Ergo, we've shown that Santa Claus doesn't exist. His definition does not correspond to reality.
Now, if you want to come along and change the definition...workshop is at the bottom of the ocean, not the surface, he can make himself invisible, whatever...then you're dealing with something else entirely. The fact that you're calling this new thing "Santa Claus" does not make it the same thing as the old thing.
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The only way to sort out the definition is to have a Santa to investigate, which of course, would mean you were investigating and proving a positive.
Incorrect. I only need the definition. I do not need to have a Santa to investigate.
We don't have any electrons to directly investigate...they've always been detected by inference...and yet we have a definition of them.
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Rrhain
WWJD? JWRTFM!
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