Why the hell is induction considered a logically sound method?

[yeti]

Quote:

Quote:Inductive reasoning is not considered a logically sound method by the majority of people.

Hope you caught the irony there.

What irony? :biggrin:

Quote:Anyway, inductive reasoning is held to a different standard of 'soundness' than deductive reasoning. Admittedly it is a lower standard and therefore not as strong. However you made a deductive argument in your Og example with inductive support.

The inductive form would be:

The bankers I study are Talmudic Jews.
Therefore, a random banker will probably also be a Talmudic Jew.
(of course there would be various concerns such as sample size, representation etc...)

The example I gave is exactly like the ones in the wikipedia article:

Quote:All observed crows are black.
Therefore:
All crows are black.

This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain.

I always hang pictures on nails.
Therefore:
All pictures hang from nails.

Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalisation that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak.

I have read articles that support your model. Wikipedia is probably not the best source for info, but their article is how I define deductive vs. inductive reasoning from the point of view of logic, which is rigid, vs. research methodologies. In other words, we're talking about 2 different things. I just assumed from the article heading that we are dealing with logic.

Quote:Social sciences would be impossible without using induction.

It depends on whether one is treating assumptions as conclusions.*

* Bear in mind that I'm using the wikipedia article as my definition of inductive reasoning. There is another article here that defines induction vs. deduction. That definition more closely matches what you're saying.

[shZ]

Quote:~ is equal (for predicates)
=> is implication

I don't know how much it will help you, since it's still same thing, just differently put down. We really are in circles now:)

Ahhhh! I did all that editing to my post for nothing :tongue:

Quote:

Quote:Hm...

P(n)...is valid when the formula we want to prove when we insert n is valid

P(1) ~ 1

We assume P(n) and start proving P(n+1) so P(n)=>P(n+1) once we have it.
This implication works when both are valid.
So:
P(1) => P(n+1)
P(1) ~ 1
1 => P(n+1)
P(n) => P(n+1) for all P(n)~1
from all this => P(1 + 1) ~ 1
if P(1+1)~1 => P(1+1+1) ~ 1
etc

I'm assuming by tilda (~) you mean "approximately equal" ... ? and is "=>" less than or equal too or different notation for "-->" or a conditional?

EDIT:

Alright, so far this is what we have (up till the point before your post that is):

Peano's Axiom
There is a relational system <Nat, 0, s>, which satisfies the following axioms:[list=1]
[*]There is a special element of Nat denoted by 0.<>
[*]s is the successor function from Nat to Nat<>
[*]s is one-one<>
[*]~ (∀ k ∈ Nat : s(k) == 0)<>
[*]The Induction Axiom
Let S be any subset of Nat,
IF: (i ) (0 ∈ S) and (ii) (∀ k ∈ Nat) ∧ (∃ k ∈ S) ⇒ (s(k) ∈ S)
THEN: S == Nat<>
[st]These three "assumptions"[list=1]
[*]Nat is well-ordered.<>
[*]∀ k, n ∈ Nat: (k == 0) ∨ (k == n +1)<>
[*]∀ n ∈ Nat: (n << n +1)<>
[st]Now the proof goes:[list]
[*]We have a proposition P(n) defined on Nat where n is any natural number.<>
[*]∃ m ∈ Nat: m < n<>
[*]S ⊆ Nat: (k ∈ Nat ∧ k < m)<>
[*]P(base case) is shown to be true.<>
[*]P(k) is shown to be true. (or that P is true for S)<>
[*]P(m) is shown to be true.<>
[st]Now the last step left is to show that P(n) is true via 'Transfinite induction.' That's the step I want to understand the proof for, and below is a standard symbol table I put together to make it easier to read... which took a while I'll admit (the formatting etc):huh:

Notation:[list]
[*]∃ <-- existential quantifier<>
[*]∀ <-- universal quantifier<>
[*]~ <-- negation<>
[*]∧ <-- conjunction<>
[*]∨ <-- disjunction<>
[*]⇒ <-- implication<>
[*]⇔ <-- material equivalence<>
[*]== <-- (mathematicaly) equality<>
[*]!= <-- not equal<>
[*] <, > <-- less than, greater than<>
[*] ≤, ≥ <-- less than or equal, greater than or equal<>
[*]∈ <-- is a member<>
[*]⊆ <-- is a subset<>
[*]ø <-- empty set<>
[*]Nat <-- the set of natural numbers<>
[st]

:scream:



:smile:

EDIT: I'm also going to take a quick break as I can't think anymore:dead:

[Melchor]

Quote:

Quote:Anyway, inductive reasoning is held to a different standard of 'soundness' than deductive reasoning. Admittedly it is a lower standard and therefore not as strong. However you made a deductive argument in your Og example with inductive support.

The inductive form would be:

The bankers I study are Talmudic Jews.
Therefore, a random banker will probably also be a Talmudic Jew.
(of course there would be various concerns such as sample size, representation etc...)

The example I gave is exactly like the ones in the wikipedia article:

Quote:All observed crows are black.
Therefore:
All crows are black.

This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain.

I always hang pictures on nails.
Therefore:
All pictures hang from nails.

Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalisation that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak.

I have read articles that support your model. Wikipedia is probably not the best source for info, but their article is how I define deductive vs. inductive reasoning from the point of view of logic, which is rigid, vs. research methodologies. In other words, we're talking about 2 different things. I just assumed from the article heading that we are dealing with logic.

Well I think we just got crossed in the nomenclature. There is inductive logic and deductive logic. Specifically, an inductive argument (inductive fallacy) is actually a non-valid fallacious deductive argument, which is separate from inductive logic which is what I was talking about. Anyhow, I'm going to stop clogging up the math thread with my philosophy spam. :biggrin:

[shZ]

Quote:Well I think we just got crossed in the nomenclature. There is inductive logic and deductive logic. Specifically, an inductive argument (inductive fallacy) is actually a non-valid fallacious deductive argument, which is separate from inductive logic which is what I was talking about. Anyhow, I'm going to stop clogging up the math thread with my philosophy spam. :biggrin:

No, this is good spam that got me obsessed with mathematical induction in the first place and revisit it... please continue with the interesting spam :tongue: :biggrin: