philosophy and inductive logic

[Tumblehome]

But this requires

at age X, Y is permissable

at age X-a Y is not permissable.

and yet, a can be a measure as small as we like, let's say 10^-100th of a second.....

in a can we be said to cross a permissable/non permissable boundary?

This is where I'm probably in over my philosophical head. I can't accept the notion that a can be considered negligible and discounted. No matter how small it is, it is still part of X. At least, that's the way it works in real life. If we've concocted a hypothetical boundary beyond which a can be discounted, then it just isn't for me.

And I still stand by my argument that if a can be discounted, and thus all previous a's back to the beginning, then X cannot exist because it is defined by all the a's, no matter how small a is.

the trouble is with a notion of "imprecise quantity" that that one has to abandon descriptors such as "heap" and "bald" altogether - that is accept that they apply to nothing. There is no number of leaves on the ground that would constitue a "heap" - for if there were then we would have a precise quantity with which to ground our induction.

That's something that's nagged me since I first read your OP. Terms like "heap" and "bald" (and "responsible enough to drink alcohol") are human inventions that were never meant to be precisely measured, or can't be. So why bother trying?

The fact is that "heap" and "bald" do exist in our human consciousness, whether they're measurable or not. Again, if we've come up with a method of determining that they can't exist, it isn't for me.

As I said, I'm a beginner at this line of reasoning. If I'm missing the point of it, which I suspect I am, I'll bow out. But it's been a good introduction to it.

i'm not sure if the sorites problem is different - t may be continuous, but it's broken up into discrete quantities.

Maybe that's the problem. Is t really broken up into discrete quantities, or are we doing it hypothetically to fit the model of the argument?

 

[Tumblehome]

3)Give up on philosophical 'paradoxes' and head for the pub.

I'm right behind you. By the way, how do we determine how much beer is too much, if X is the number of beers consumed in t and--Oops, never mind.

 

[Jekyll]

I'm right behind you. By the way, how do we determine how much beer is too much, if X is the number of beers consumed in t and--Oops, never mind.

Well if one small sip can't possibly make me any drunker, then I appear to have finished my glass.

So who's round is it now?

 

[chriswl]

The problem isn't fundamentally about the sumation of infinietesimals [as in Zeno] but about what point the T/F boundary is crossed within such a sumation. if calculus can provide such an answer, i'd like to see it

Then I don't understand the OP.

I thought you were saying:

1) Assume there is a quantity x (maturity, common sense, intelligence, whatever) that increases smoothly and monotonically over time as one ages.

2) We agree there is some threshold T which x must be at least equal to before one is allowed to do Y.

3) On average, people reach x = T at age M, so we use this as a rough-and-ready proxy for judging whether someone ahould be able to do Y.

4) But, at age M - t, if t is small enough, the difference in x, x' is effectively zero. So one should be able to do Y at age M - t.

And from here on the argument is like Zeno's paradox, isn't it? We can take a point any number of time intervals t, before M and x will supposedly be unchanged, because x' is effectively zero and the sum of any number of things of size zero is still zero.

Except that "effectively zero" is not the same thing as zero, so this doesn't work. If that's not what you're saying then I don't understand your statement that "...however from these two premises, induction leads to a permissibility to do Y at any age."

 

[andyandy]

And from here on the argument is like Zeno's paradox, isn't it? We can take a point any number of time intervals t, before M and x will supposedly be unchanged, because x' is effectively zero and the sum of any number of things of size zero is still zero.

Except that "effectively zero" is not the same thing as zero, so this doesn't work. If that's not what you're saying then I don't understand your statement that "...however from these two premises, induction leads to a permissibility to do Y at any age."

It's not the same because zeno is concerned about the sum of infinitesimals which incorporates a defined interval -ie the inital distance between the runner and the tortoise (set at d). To apply calculus to this problem, one has to arrive at a value for d - in this case the distance between T and F. If there is no distance between T and F, as one would expect from classical bivalent logic for a discrete model, then d is zero, and you have a sum to infinity as d tends to a number it's already at


based on the acceptance of the premises

P₁ = true
p_n implies p_n+1
therefore p_n is true for all n.


If you accept the two premises, then induction leads to permissibility to do Y at any age. If you reject the premises, then which premise do you reject?
Do you make the argument that since "effectively zero" is not the same as zero, that one has to reject premise (2)? That an individual is sufficiently changed after arbitrary time a to mean that at age X, Y is permissable, but that at X-a it is not permissable?

 

[drkitten]

P₁ = true
p_n implies p_n+1
therefore p_n is true for all n.


If you accept the two premises, then induction leads to permissibility to do Y at any age. If you reject the premises, then which premise do you reject?

The second one, of course.

In particular, let P_t stand for the proposition that "I am alive at time t," setting t=0 to be "now." Your induction would suggest that I am immortal, which is patently untrue.

A better formulation of the second axiom would be that
P_n implies P_n+delta with probability 1-epsilon.

(in other words, my probability of dying in the next infinitesimal interval is infinitesimal, which is true.

... and at that point, you should see how it can be reformulated as a calculus-style limit.

 

[andyandy]

The second one, of course.

In particular, let P_t stand for the proposition that "I am alive at time t," setting t=0 to be "now." Your induction would suggest that I am immortal, which is patently untrue.

A better formulation of the second axiom would be that
P_n implies P_n+delta with probability 1-epsilon.

(in other words, my probability of dying in the next infinitesimal interval is infinitesimal, which is true.

... and at that point, you should see how it can be reformulated as a calculus-style limit.

well the problem with the whole paradox is that if you do accept both premises, and the inductive logic, then you do end up with rather large contradictions - because you can set up the problem from the other end, ie at age X, Y is not permissable, and by induction you've simultaneously proved that Y is not permissable at any age and permissable at any age

I do like the introduction of 1-epsilon - and thus a probabilistic framework - it fits nicely for age, and quite nicely for permissability. In both these cases, you have probabilistic elements in calculations.....

extending the case beyond the OP to sorites problems, such as "heaps," probability would seem less satisfactory, but still an option....

 

[UserGoogol]

The trouble is with a notion of "imprecise quantity" that that one has to abandon descriptors such as "heap" and "bald" altogether - that is accept that they apply to nothing. There is no number of leaves on the ground that would constitue a "heap" - for if there were then we would have a precise quantity with which to ground our induction.

Not really. A concept can be meaningful but vague. The heapiness of a collection of leaves is a quantity that varies relatively smoothly. As you remove leaves, you get "a pile" to "sort of a pile" to "barely a pile" to "vaguely pile-like" to "not a pile."

Similarly for permissibility. Of course, there does something inherently boolean in the definition of permissibility, (either you're allowed to do something or you're not) and in that sense, the word is somewhat contradictory and confused. But I see no reason why permissibility cannot be made somewhat rigorously fuzzy.

 

[Tumblehome]

Well if one small sip can't possibly make me any drunker, then I appear to have finished my glass.

So who's round is it now?

If one small sip can't make you any drunker, then you aren't drunk at all, no matter how many sips you've had. Best to drink in large gulps.

I believe Xeno is picking up the tab--if he ever gets here.

 

[drkitten]

well the problem with the whole paradox is that if you do accept both premises, and the inductive logic, then you do end up with rather large contradictions.

Well, yes. If you accept a set of contradictory premises, you should end up with large contradictions. That's part of how you know your premises are contradictory.

The solution, of course, is don't accept stupid premises.

 

[drkitten]

Similarly for permissibility. Of course, there does something inherently boolean in the definition of permissibility, (either you're allowed to do something or you're not) and in that sense, the word is somewhat contradictory and confused. But I see no reason why permissibility cannot be made somewhat rigorously fuzzy.

Actually, permissibility can be handled much more easily without fuzziness. Permissibility is a prescriptive, not a descriptive, attribute. You -- or the law -- can easily draw a thin bright line and say "everything on this side of the line is permitted, nothing on that side is."

You can get away with this because the law is not expected to correspond directly to reality. You can drink at age X, but not at age X-epsilon, because the law says so. In this case, permissibility is genuinely a discontinuous attribute of the world, and so the idea that small differences can be neglected is directly refuted.

 

[Bri]

We have the following 2 premises;

1) at age X, it is permissible to do Y.

2) in an arbitrarily short period of time, an individual is sufficiently similar to his/her previous self for any changes to his/her self to be negligible

It seems to me that you're using the term "negligible" in exactly the same way that "heap" and "bald" are used -- they are imprecise terms, and therefore probably aren't adequate for inductive reasoning without an arbitrarily precise definition. The question "what is the maximum number of hairs that can be present in order to describe the head as bald?" becomes "how small must a change be in order to be considered negligible?"

So the real problem with premise (2) is that you haven't precisely defined the term "negligible." And, of course, as soon as you do define it you will still have a problem with premise (2) in that your definition will undoubtedly be inadequate. You are using it to indicate that the change is both "small" (which is true in this case) and "irrelevant" (which is decidedly not true in this case). No nonzero amount of change can be considered irrelevant, since premise (1) is based on an accumulation of all changes between age 0 and age X. An arbitrarily small time slice is expected to yield a proportionally small change, but premise (2) is based on a faulty assumption that small changes are irrelevant (and of course, they're not).

-Bri