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I'm definitely open to critique here. I see ampliative reasoning (basically, induction or abduction) as different than explicative reasoning (deduction) in that ampliative reasoning adds additional information not contained in the premises. By some people's standards, this is never valid, and I suppose supporters of ampliative reasoning must show how this trick is possible.

The understanding that I've kind of developed is that ampliative reasoning adds information by decreasing likelihood. For instance, Pr(Socrates is a human mortal)=99% ⊢ Pr(All humans are mortal)=1%. What would otherwise be an audacious generalization is accounted for by decreasing the likelihood of the assertion.

My problem is that I don't even really know if this works out. The numbers in the Socrates example are completely made up, but my sense is that there are a set of probabilities that would make this inference valid. Peirce once wrote that with ampliative reasoning you trade in some security for uberty, and maybe I'm making too much of this relationship.

So my sense is that we're trying to quantify the information in both sentences, and for the inference to be valid, you can't add information to the inference, just like in deduction. But this means, in some sense, by saying that a proposition is merely probable, we say that the proposition contains less information.

For instance, "Socrates is a human mortal" contains a lot less information than "All human beings are mortal", because the latter tells us something about every human. But is there a way of quantifying the amount of information in these sentences?

If there is a way of quantifying the information in these two sentences, such that for instance the second sentence contains a thousand times the quantity of information as the first, then for the inference to be valid, would we just make the probability of the second sentence one-thousandth times the probability of the first?

What do you think?

  • This sounds a lot like Jaynes' The Logic of Science (that one a doomed approach IMHO) – Drux May 2 '15 at 20:52
  • @Drux can you say why it is doomed? I guess right now it makes so much sense. – Kevin Holmes May 2 '15 at 22:28
  • It's a long story. E.g. Mises' Human Action (since it was mentioned in another recent question) is relevant. – Drux May 3 '15 at 5:37
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Yes, these measures exist, and they work. But they are generally harder than we imagine.

A lynchpin of many variants of artificial intelligence (like car diagnostic equipment, for example) is the application of Bayesian statistics. You need to know not only the probabilities of your two truths, but the degree to which those are statistically related. Then you can use Bayes Formula: The chance of A given B is the chance of A and B occurring together divided by the chance of A occurring alone.

This is closely related to the notion of the risk of falsification in Popper's theory of science. If I keep making audacious predictions, which have a low odds of being true at random, and they keep coming up true, I can chip away at the likelihood they are unrelated by continually applying Bayes analysis to it.

Further, using the Law of Large Numbers, I can be assured that some estimates of probability are fairly safe to trust. If I have enough data, from a wide enough variety of sources, I can expect the distribution of their average parameters to be normally distributed.

I can use those two principles together to find those things which have a rather small likelihood of being false. This is the procedure used to keep folks honest in social science research. You do a lot of statistics to find out to what degree certain observations predict various consequences.

But first of all, it is far too easy to forget these are statistics, and that systematic errors or prejudices cannot be overcome by randomness. There may simply not be 'a wide enough variety of sources' because people won't let that variety exist. You cannot objectively study criminal behavior in a society that does not allow criminals freedom to exhibit their behavior!

Another problem is that only some kinds of things are easy to come by in the requisite numbers. So we have a lot of information on, for instance sex differences in the student-aged population. A lot of these are small, or inconsequential, but are very well proven. But we have also decided politically that this is not really important in the long run, because we want to look past trends and treat people less statistically in certain ways.

For any observation that is hard to make, induction cannot rely upon so demanding a standard.

For instance, medical knowledge on individual predictive numbers cannot use raw statistics to validate their predictions when they are new. They have to assume certain measures have certain distributions. (For example the definition of diabetic insulin resistance uses the notion that human blood sugar levels are normally distributed, which is both clearly disproven by the data, and literally logically impossible, because the normal distribution includes negative values. But it was the only easily checked definition available to start out, and now it is established.) They also have to assume certain causal factors can be safely ignored. (If being in a hospital makes cancer worse, we will not find out for a while, since we have no other safe place to study cancer patients.) Other domains have even bigger obstacles to actually getting good measures of probability.

So we rely all the time on inductive or abductive logic that is not really mathematically sound, simply because doing otherwise is intractable. Absolute adherence to Bayesian convergence is a great theory, but it does not actually get applied as often as we think it does, and it does in fact never establish truth beyond a constantly decreasing probability, which can easily be inadvertently manipulated by lack of objectivity.

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