Let me start by saying I'm an amateur insofar as I took one propositional logic course in college and have self-taught everything else so I may be missing something obvious. However, I was reading an article by Niiniluoto (1972) and (with respect to induction) he makes the following statement:(adsbygoogle = window.adsbygoogle || []).push({});

(2) P(h[itex]\supset[/itex]b|e) = P([itex]\neg[/itex]h[itex]\vee[/itex]b|e)

Can someone confirm the rationale for this? I assume this is the stastical calculus version of Hempel's paradox. Specifically, the deductive version of this argument (and presumably the assumption used by Nicod and Niiniluoto) arises from the idea that:

(a[itex]\rightarrow[/itex]b) [itex]\rightarrow[/itex] ([itex]\neg[/itex]b[itex]\rightarrow[/itex][itex]\neg[/itex]a)

The idea being... since the second is deduced from the first, evidence for the second supports the first. However, it's trivial to create an inductive state where this is not true. Assuming this is a table of observations (column labels above, row labels to the right):

| a | [itex]\neg[/itex]a |

| 8 | _0 | _b__

| 2 | _1 | [itex]\neg[/itex]b__

In this situation

a[itex]\rightarrow[/itex]b (80%)

is clearly probable. However,

[itex]\neg[/itex]b[itex]\rightarrow[/itex][itex]\neg[/itex]a (33%)

does not follow. Intuitively, it seems to me that (in an inductive context at least) the observation of a shoe, no matter its color, can never disconfirm the hypothesis. As a consequence, it ought not confirm the hypothesis. "All" should only apply to objects/observations that can disconfirm the theory.

What got me to the above explanation was my initial intuition that we care about the proportion of b[itex]\wedge[/itex]e that falls into h[itex]\wedge[/itex]e rather than including, for some reason, all of [itex]\neg[/itex]h including the region that is both [itex]\neg[/itex]h and [itex]\neg[/itex]b. For example, I believe what I'm saying is properly stated as:

P(h[itex]\supset[/itex]b|e) = P(h|b[itex]\wedge[/itex]e)

I have trouble imagining that there isn't some kind of response within Nicod's paradox for what I've described... but I can't for the life of me figure it out (or even how to look for it).

P.S. appologize in advance for any mistakes in Latex formatting (or propositional logic)

Ilkka Niiniluoto (1972) Inductive Systematization: Definition and a Critical Survey. Synthese, Vol. 25, No. 1/2, Theoretical Concepts and Their Operationalization (Nov. -Dec., 1972), pp. 25-81

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# Probability of being a Superset

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