3. No-Go Result
for deductively definable, asymptotically stable inductive logics

Or... Why they all fail.

The results so far would remain interesting and unproblematic if each algebra of propositions could persist as a distant island, isolated from all others. The basic problem with all deductively definable logics is that this is not so. Any deductive structure is expandable. We can always find ways to replace a proposition by its proper disjunctive parts. For any deductively definable logic of induction, these expansions can always be done in such a way as to defeat the stabilization of the inductive logic. Any inductive judgment beyond the most banal can be targeted for reversal by a suitably chosen expansion. This problem makes non-trivial, deductively definable inductive logics impossible.

Taking disjunctive parts serious

In our world, finite algebras of propositions arise only because we neglect to note that each proposition in turn can represent a disjunction--that is an "OR' ing" together--of many more propositions. We do need to take these parts seriously, for they can provide essential information for an inductive logic.

Here's a simple example that illustrates the potential importance of the disjunctive parts. Imagine that we consider the proposition:

The sun will rise on Friday.

We ask what inductive support it gets from the evidence in propositions like:

The sun rose on Thursday.
The sun rose on Wednesday.
The sun rose on Tuesday.
The sun rose on Monday.

If we can only describe the information available to us at the coarser level of description of these proposition, the direction most logics of induction will take us is clear enough. We will have good support for the proposition that the sun will rise on Friday. Now let us consider that each of these propositions has disjunctive parts. For example:

The sun rose on Thursday.
= The sun rose on Thursday for 1 minute.
OR The sun rose on Thursday for 2 minutes.
OR The sun rose on Thursday for 3 minutes.
OR ...

If we are able to use this finer level of description, it is now possible for us to represent more complicated inductive relations. Our evidence might turn out to be:

The sun rose on Thursday for 10 minutes.
The sun rose on Wednesday for 20 minutes.
The sun rose on Tuesday for 30 minutes.
The sun rose on Monday for 40 minutes.

Now we might well expect an inductive logic NOT to support a Friday sunrise. Indeed this is just the sort of thing that might happen on a Monday, Tuesday, Wednesday and Thursday in the Arctic circle just prior to the failure of the sun to rise on the Friday.

Disjunctive refinements

We introduce the disjunctive parts of propositions here by means of the notion of a disjunctive refinement. Here's the notion illustrated by means of the "tile" example. We start with ten tiles as before

1 2 3 4 5 6 7 8 9 10

Now imagine that the option of tile 1 is really a summary for the possibility of tiles of many different sizes labeled 1.x. That is, the outcome can be written as

1 = 1.1 or 1.2 or 1.3 or 1.4

That means that our original set of options of ten tiles is really a set of thirteen options

1.1 1.2 1.3 1.4 2 3 4 5 6 7 8 9 10

Presenting the tiles this way is a little misleading. It looks as if the first four are somehow different from the other nine tiles. As far as deductive relations are concerned, there is no difference between the thirteen tiles. As we saw before, the deductive relations remain unchanged no matter how we switch the labels around. As far as deductive relations are concerned, our thirteen tiles might just as well be

1 2 3 4 5 6 7 8 9 10 11 12 13

The transformation just displayed from ten to thirteen tiles is an example of a "disjunctive refinement." Many disjunctive refinements are possible. Instead of expanding tile 1, we could have expanded any of the other tiles or even all of them.

Malicious refinements

The possibility of these different disjunctive refinements generates considerable problems for the inductive strengths [A|B], if the refinements are carried out maliciously. To see the problems, consider the two equal strengths

[ 1|1 or 2] = f₁₀(1, 0, 1) and [ 2| 1 or 2] = f₁₀(1, 0, 1)

where the function f₁₀ is used since we are assessing the strengths in the 10 tile set.

We get very different results if we form the same strengths in the 13 tile set:

[ 1 | 1 or 2] = [ 1.1 or 1.2 or 1.3 or 1.4 | 1.1 or 1.2 or 1.3 or 1.4 or 2 ] = f₁₃(4, 0, 1)

whereas

[ 2 | 1 or 2] = [ 2 | 1.1 or 1.2 or 1.3 or 1.4 or 2 ] = f₁₃(1, 0, 4)

The effect of the disjunctive refinement has been to move the inductive strengths in favor of 1 at the expense of 2. Clearly this process could be continued to favor 1 still further. Instead of refining 1 to four component options, we could have refined it to one hundred and four. Then we would have recovered

[1|1 or 2] = f₁₁₃(104, 0, 1) and [2|1 or 2] = f₁₁₃(1, 0, 104)

Does this mean that we should have favored 1 over 2 all along? That is obviously not right. In our original ten tile set, the two tiles were equivalent. Instead of refining 1, we could have applied the same refinements to 2 and ended up deciding that 2 is inductively favored over 1.

This is the no-go result. Deductively definable inductive logics do not stabilize as we enlarge the set of options by disjunctive refinement. By choosing our refinements carefully, we can end up favoring any option over any other. More precisely, the no-go result tells us that deductive definability is incompatible with

Asymptotic Stability under Disjunctive Refinement.
An inductive logic should be stable either immediately or eventually under repeated disjunctive refinement.

This is a Bad Thing. We do expect our logics to be stable under inductive refinement. To begin with, we might expect our assessments to be adjusted when we start disjunctive refinement. On the evidence of a poll, do we expect political party A or political party B to win? We might re-assess when we allow that these two options are really many more. We find that party A is fielding 100 candidates, but party B only has 2. Eventually the continuing refinement adds nothing of inductive relevance. We might split the proposition

Candidate 23 of party A wins.

into

(Candidate 23 wins AND one bat nests in the belfry.)
OR (Candidate 23 wins AND two bats nest in the belfry.)
OR (Candidate 23 wins AND three bats nest in the belfry.)
OR...etc

We would not expect that allowing for these extra options would affect our inductive assessment of which party wins.

However a deductively definable logic is simply unable to resist accommodating new refinements no matter how unconnected they are inductively to the question at hand. Even the most absurd of hair splitting is taken seriously by the logic and treated with all the honors of distinctions of direct inductive moment. The result is a capricious assignment of inductive strengths that can never settle on any stable values.

Loose Ends

To conclude discussion of the no-go result, we need to tie off a few loose ends.

First, what we have above is not a precise proof. It relies heavily on our intuition that quantities like f₁₁₃(104, 0, 1) are very different from f₁₁₃(1, 0, 104) and that the first represents a much greater strength than the second. These are natural presumptions but they are not necessarily forced on us. A more careful proof of the no-go result is possible in which no such presumption is needed. It works by assuming that there is some inequality somewhere allowed by the function f_N. It is then a matter of mechanical construction to find different ways of refining the options so that the inequality favors one option or another in ways that are stable.

In the end, we see that the only deductively definable inductive logic that escapes is trivial. It is one that allows no inequalities among inductive strengths. It just assigns the same strength to everything.

Second, one might respond to the problems generated by successive disjunctive refinements in this way. Why not, one may wonder, just do all the refinements first. Only then, when we have the fully refined set before us, should we proceed to form the inductive strengths, which will, of course, be unique.

The difficulty with this enticing proposal is that there is end to the disjunctive refinements. There is no "biggest" set of propositions. After every refinement has been effected, we can alway carry out one more. It is no help to image the refinements completed infinitely, so that we have infinitely many propositions (which would lie outside the present analysis). These propositions can still be refined more.