Inductive logic is a generalisation of Boolean deductive logic. Use of the sum and product rules when all probabilities are 0 or 1 is isomorphic to Boolean algebra. When probs are inbetween, the sum and product rules allow you to reason inductively. Of course you need a new concept which is absent from Boolean algebra, namely probability. I take p(A|B) to be a number representing how strongly the binary proposition A is true upon supposing that B is true, based on ontological relations known between their referents. This quantity is what you actually want in any real problem involving uncertainty.

As for philosophy of quantitative science, it has several components: invention of a theory/hypothesis, by a bright scientist; deduction of the testable consequences of that theory; inductive comparison of that theory with its rivals in the light of the data.

In view of the buzzwords in that sentence, I don’t like to say that the war in philosophy of science is between those who say it is hypothetico-deductive and those who say it is inductive. In practice I agree with what the latter camp say, but the words used in the debate are deeply and needlessly confusing.

]]>I’m still looking for an explanation of Goedel’s theorem at my own level, ie not the full gory details but some way beyond “intelligent layman” stuff. I suspect also that Jaynes (the outstanding advocate of Bayesianism) was a bit quick to dismiss the predicate calculus as merely the propositional calculus plus some fairly trivial notational sleight of hand. But he wasn’t often wrong, so I don’t know.

]]>I fully agree wih you, and I think if scientists would now and then ponder this, it would make the schism between them and religion smaller.

I am not so sure about what is meant by falsification. If I understand it correctly, one cannot accept a hypothesis on a 95% probabliity if 5% of the data proves it false (not uncertain).

In any case, we will never be able to move away from induction, as it is such a powerful tool, taking cognisance of the fact that what we want to know is unprovable in our frame of reference – Godel’s incompleteness theorom.

And that is what interested me in this argument.

I regard induction as pretty much failsafe, at least now it has foundations in a reasonably unique generalisation of Boolean logic. If you are always going to question axioms then you will ultimately be driven back to Cogito ergo sum, which is true but cannot be built upon in the absence of any other axiom. So there you will be stuck. I agree that faith is needed to get anywhere, whether faith in what a divinity says or faith in other propositional axioms.

Also I am not disagreeing that science based on induction is fallilble – because we might be a long way from coming up with a decent theory, and theory invention is up to humans. To make this point, suppose that the datapoints lie close to a straight line with gradient 2, and the only two theories to have been dreamed up have gradient 1.0 and 1.5. you can do Bayesian inductive comparison between them (and 1.5 will win), but you will still think that both are implausible.

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