Proof Induction is Impossible

[JansenDowel]

Let O1, O2,...., On be observations 1 through n
Let Ci be Conclusion i

Suppose (O1, O2, .... , On |- Ci ) where Ci is a conclusion derived from some set of observations.
By the laws of logic, we can now use Ci as an observation

We know that observations cannot be inconsistent with each other. Therefore,
(O1, O2, ... , On, (not Ci) |- Cj )

Cj is consistent with (not Ci). But Cj is also consistent with Ci because Cj is consistent with (O1, O2, ... , On).
Thus Cj is consistent Ci and (not Ci), which is a contradiction.

Therefore, logical induction is impossible.

What say you?

 

[Cognisant]

Assume I'm an idiot and explain this "logic" in terms of apples and oranges.

 

[Ex-User (14663)]

What is “|-”, what is Cj, what is “some set of observations”, is the index i in the range 1,..,n?

Hard to judge a proof that uses a bunch of undefined symbols

 

[rlnb]

In inductive reasoning the conclusion is likely to be true given the observations. So I don't think it can be treated the same as an observation (which are true)

For example if you observe 1000 black crows (O1.....O1000) you might conclude that all crows are black(C1) but if you observe a white crow (O1001) you must revise your conclusion.

source : wiki_Inductive_reasoning

 

[Black Rose]

Can this also prove there is no way to disprove we are in a simulation?
No observation can prove inductively that this is base reality.

 

[rlnb]

We cannot make any extraordinary hypothesis and say, since there is no evidence to the contrary, we should accept the hypothesis.

So, Yes. There is no evidence to disprove that we are in a simulation. But that doesn't mean, we are in a simulation. There are much simpler explanations for our observed reality.

 

[Black Rose]

If all induction is at base perception. No perception can be proof of any conclusion. All empiricisms fall apart. We do not know if any part of the world is real or not. All we have is our perceptions which are inductions by default.

Kant's Rationalism comes in here I think?

 

[rlnb]

Subjective model of reality : me and my experience/perception.
Objective model of reality : there are things and rules that govern the interaction of these things.

In the first model, Yes, no perception can prove any conclusion definitively. But it is also a fact that there are patterns to my experiences. And it is possible to come up with an objective explanation that could explain these patterns. Which lead one to the second Model of reality.

My take is, there is both an objective reality and a subjective one. And there is a interplay of the two. The nature of this not entirely clear though.

 

[JansenDowel]

What is “|-”, what is Cj, what is “some set of observations”, is the index i in the range 1,..,n?

Hard to judge a proof that uses a bunch of undefined symbols

"|-" is Mathematical Logic notation. It means that whatever is on the right of "|-" is entailed by whatever is on the left. Whatever is on the right is logically derived by whatever is on the left.

Cj is Conclusion J.

"some set of observations" is just a random set of observations made by some person performing an inductive inference. Yes, index i is in range 1,...,n

 

[JansenDowel]

In inductive reasoning the conclusion is likely to be true given the observations. So I don't think it can be treated the same as an observation (which are true)

For example if you observe 1000 black crows (O1.....O1000) you might conclude that all crows are black(C1) but if you observe a white crow (O1001) you must revise your conclusion.

Yes you are right. And that's the problem. In logic, the conclusion is just another premise. But in induction, the conclusion is not just another observation. This is problematic since then you have to explain how one kind of information can derived from a completely different kind of information. Induction is not explained by logic, and no amount of "examples" will explain it.

 

[rlnb]

In inductive reasoning the conclusion is likely to be true given the observations. So I don't think it can be treated the same as an observation (which are true)

For example if you observe 1000 black crows (O1.....O1000) you might conclude that all crows are black(C1) but if you observe a white crow (O1001) you must revise your conclusion.

Yes you are right. And that's the problem. In logic, the conclusion is just another premise. But in induction, the conclusion is not just another observation. This is problematic since then you have to explain how one kind of information can derived from a completely different kind of information. Induction is not explained by logic, and no amount of "examples" will explain it.

Ok. So taking your logic forward with my crow example, O1...On, (not C1 ) would be : I observe n black crows and 'not all crows are black' (not C1) , you can maybe conclude that (C2) 'all crows that are visible to me are black'.
Yes. C2 is consistent with (O1...On) and C2 is consistent with C1 and C2 is consistent with (not C1) . But there is no contradiction here.

The proper mathematical framework for this is Hypothesis testing, which distinguishes between a hypothesis and an observation.

 

[JansenDowel]

In inductive reasoning the conclusion is likely to be true given the observations. So I don't think it can be treated the same as an observation (which are true)

For example if you observe 1000 black crows (O1.....O1000) you might conclude that all crows are black(C1) but if you observe a white crow (O1001) you must revise your conclusion.

Yes you are right. And that's the problem. In logic, the conclusion is just another premise. But in induction, the conclusion is not just another observation. This is problematic since then you have to explain how one kind of information can derived from a completely different kind of information. Induction is not explained by logic, and no amount of "examples" will explain it.

Ok. So taking your logic forward with my crow example, O1...On, (not C1 ) would be : I observe n black crows and 'not all crows are black' (not C1) , you can maybe conclude that (C2) 'all crows that are visible to me are black'.
Yes. C2 is consistent with (O1...On) and C2 is consistent with C1 and C2 is consistent with (not C1) . But there is no contradiction here.

The proper mathematical framework for this is Hypothesis testing, which distinguishes between a hypothesis and an observation.

Ok now you've drawn out the argument with a real example. In your example, its abundantly clear that C2 is consistent with C1. Its completely obvious that "all crows that are visible to me are black" is consistent with "all crows a black". The problem is,

If you logically derived C2 from (not C1), then C2 MUST be inconsistent with C1.

Which means that C2 cannot have been derived from (not C1). But if C2 can't be derived from (not C1), where did C2 come from? The answer, as Karl Popper proposed, is creative conjecture and criticism. Inductive inferences are really just creative leaps of imagination.

 

[rlnb]

Which means that C2 cannot have been derived from (not C1). But if C2 can't be derived from (not C1), where did C2 come from? The answer, as Karl Popper proposed, is creative conjecture and criticism. Inductive inferences are really just creative leaps of imagination.

Agreed. But we wouldn't have science without these 'creative leaps of imagination'

 

[JansenDowel]

Which means that C2 cannot have been derived from (not C1). But if C2 can't be derived from (not C1), where did C2 come from? The answer, as Karl Popper proposed, is creative conjecture and criticism. Inductive inferences are really just creative leaps of imagination.

Agreed. But we wouldn't have science without these 'creative leaps of imagination'

Yes, I agree! Creative leaps of imagination is the life force of science. You can't have science without it.

 

[JansenDowel]

Assume I'm an idiot and explain this "logic" in terms of apples and oranges.

I cannot explain unless you are familiar with formal logic. Its not possible to explain the concept in just one message.

 

[Ex-User (14663)]

the proof still looks like gibberish to me. When a conclusion is "consistent" with a set of observations, is it consistent with the conjunction of that set or the disjunction, or something else? I assume it's the first, but then the claim that "Cj is also consistent with Ci because Cj is consistent with (O1, O2, ... , On)." is a nonsensical argument because you defined Cj as being derived from that set and (not Ci).

 

[Cognisant]

Assume I'm an idiot and explain this "logic" in terms of apples and oranges.

I cannot explain unless you are familiar with formal logic. Its not possible to explain the concept in just one message.

Really because the impression I get is that you're trying to logically explain to us with "formal logic" why logic isn't logical but of course if logic wasn't logical it wouldn't be logic then would it?

See that's the thing about logic, it's logical and self evident, so it can't be that hard to explain, surely.

 

[QuickTwist]

-1/12

Sorry, the problem has an answer.

Also, I could say (not Ci) doesn't exist in reality, but only in mathematics, which is not real life because it's an over simplification of a symbolic reality in materialism.

 

[JansenDowel]

Assume I'm an idiot and explain this "logic" in terms of apples and oranges.

I cannot explain unless you are familiar with formal logic. Its not possible to explain the concept in just one message.

Really because the impression I get is that you're trying to logically explain to us with "formal logic" why logic isn't logical but of course if logic wasn't logical it wouldn't be logic then would it?

See that's the thing about logic, it's logical and self evident, so it can't be that hard to explain, surely.

No, I was trying to show why induction isn't possible. Not show why logic is illogical. Im not trying to talk down to you, its just very difficult to give a lecture on logical entailment since I am not a professor.

 

[JansenDowel]

-1/12

Sorry, the problem has an answer.

Also, I could say (not Ci) doesn't exist in reality, but only in mathematics, which is not real life because it's an over simplification of a symbolic reality in materialism.

Yes. This is another reason to doubt induction.

Anyway, what problem has an answer?

 

[Polaris]

The statement "Proof induction is impossible" seems like a contradiction in itself...one could say "Proof induction seems impossible".

Then again, perhaps you are just conflating mathematical proof induction with inductive reasoning (which, by definition is inherently uncertain), which I guess is what Serac and QuickTwist were getting at.

 

[JansenDowel]

The statement "Proof induction is impossible" seems like a contradiction in itself...one could say "Proof induction seems impossible".

Then again, perhaps you are just conflating mathematical proof induction with inductive reasoning (which, by definition is inherently uncertain), which I guess is what Serac and QuickTwist were getting at.

Well mathematical induction is completely different, so Im not conflating the two. Also, I don't see why "proof induction is impossible" is a contradiction? Where is the contradiction?

Further, induction may be inherently uncertain, but every characterization of induction in philosophy is 'logical'. Induction is always characterized as 'extrapolating inferences from observations', which is a logical process. So the fact that induction is 'inherently uncertain' means very little.

 

[mr_darker]

Let O1, O2,...., On be observations 1 through n
Let Ci be Conclusion i

Suppose (O1, O2, .... , On |- Ci ) where Ci is a conclusion derived from some set of observations.
By the laws of logic, we can now use Ci as an observation

We know that observations cannot be inconsistent with each other. Therefore,
(O1, O2, ... , On, (not Ci) |- Cj )

Cj is consistent with (not Ci). But Cj is also consistent with Ci because Cj is consistent with (O1, O2, ... , On).
Thus Cj is consistent Ci and (not Ci), which is a contradiction.

Therefore, logical induction is impossible.

What say you?

Ima try an translate to english, but define some stuff first:
--Instead of observations, we'll just use primary colors RED, GREEN, and BLUE.
--Each color is in a state of on or off, fully present or not present at all.
----The '!' operator will indicate a lack of a color. Red = Present, !Red = Not-Present
--Conclusions will just be the resulting colors from each color observed.

SO, with that said, following along with what you say....

RED + GREEN + BLUE = WHITE (Checks out)
BLACK = !WHITE (Checks out)
BUT
BLACK = WHITE (Doesn't check out, but that's your point)
BECAUSE
RED + GREEN + BLUE = BLACK (Doesn't check out, explained below)

What you're doing is inverting the conclusion, without inverting the definition too, which is impossible because they are the same thing.

Example where part of the definiton is inverted instead:
White = Red + Green + Blue.
Say we put a filter on the light and block out all red light.
This changes the definition to be "!Red + Green + Blue".
In your example, you'd have this still defined as white, but its not white, it's
Brown = !Red + Green + Blue

 

[ZenRaiden]

Therefore, logical induction is impossible.

What do you mean by all this?

 

[JansenDowel]

What do you mean by all this?

That it is impossible to do induction.

 

[DoIMustHaveAnUsername?]

Not sure what's your point. We already know induction is not like deduction. The negation of the inductive conclusion is often logically consistent with the observations.

The proof also seems trivial because you already assume that the negation of the induced conclusion Ci is consistent with other observations. So all those symbols doesn't really say much of anything that just plainly saying 'induction doesn't work'.


"But Cj is also consistent with Ci because Cj is consistent with (O1, O2, ... , On)."

Furthermore, this is wrong.

"There is no apple in the bag" is consistent with "there is an orange in the bag".
"There is one apple in the bag" is also consistent with "there is an orange in the bag".
But "There is no apple in the bag"
and "There is one apple in the bag" are not consistent with each other.

Just because Cj and Ci both are consistent with O1,O2...On doesn't mean Cj and Ci are consistent with each other.

 

[Ex-User (14663)]

@DoIMustHaveAnUsername? nice, thanks for clarifying. I kept wondering what sort of definition of “consistent” one would be using to make that particular sentence work.