[lit-ideas] Re: Back to Popper (and further back to Hume)
- From: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sat, 9 Dec 2006 13:18:35 +0000 (GMT)
Eric Yost asked about weather forecasting and polls re induction. (Robert
Paul?s post requires a separate reply though some of what is said here is, I
think, relevant).
Some comments:-
An underlying issue is whether these forecasts and polls vindicate the
existence of induction? Can they be accounted for without recourse to
induction? Is the concept of ?probability?, as used in certain contexts,
irredeemably ?inductive??
I am suggesting these and other phenomena do not vindicate the existence of
induction, can be accounted for without recourse to inductive reasoning, and
that the concept of ?probability?, in all contexts where it is used validly,
is irredeemably non-inductive.
What is not being suggested:-
It is not suggested that we cannot deploy the concept of ?probability? in any
inductive sense ? just that we cannot do so validly (cf. we can deploy the
concept of unicorns, or leprechauns, in a sense in which such things are said
to actually exist ? but not validly).
It is not suggested that we can conclusively disprove the existence of
induction ? to do so we would need some incontrovertible premises from which
we could deduce that induction does not exist; but, while we can show that
inductive reasoning lacks deductive validity, we cannot deduce from this that
there is no such thing as valid inductive reasoning. In this sense, induction
is, from the standpoint of deductive logic, a logical possibility ? but of
course so is the existence of the unicorn and the leprechaun: the sense in
which induction is a logical possibility in no way shows there is such a
thing as induction as a valid process (as opposed to a kind of myth or
optical illusion).
We need to clearly distinguish ?logical probability? from probability in some
other sense (including the sense of ?inductive probability?). Say, we have
two unbiased dice where there is a 1/6 chance of throwing any number, A, in
any particular throw: the odds of throwing A in two consecutive throws is the
?probability of A? x ?probability of A? i.e. 1/6 x 1/6. The odds of this are
1/36. These odds (or this probability) can be deduced. The odds reflect
?logical probability?. It can be deduced without recourse to any inductive
reasoning i.e. ?logical probability? is deductively valid.
In the above example it does not matter which of the two dice we throw:
because they are both unbiased, the probability is the same no matter which
die we throw.
But let?s say we know only that the first die is unbiased. Consider now the
question: if the probability of throwing A is 1/6 with a first die, what is
the chance of throwing A with a second die? This is no longer a matter of
?logical probability? because the second die may be biased and we cannot
deduce whether it is (or not) on the basis of the first die.
Turn now to ?statistics?, since these are central to the validity of ?polls?
and ?weather forecasts?.
A central problem in statistics is finding a ?representative sample?. If we
can assume that our sample is properly representative, then we may deduce, as
a matter of logical probability, that if the probability of A is ¼ for our
sample, it is ¼ more generally. This is like saying that, if we know our
first unbiased die is the same (in respect of the probability of throwing A)
as some wider class of dice, then we can deduce that for that wider class of
dice the probability of throwing A is 1/6.
The view I am about to briefly defend is that the decision as to whether a
sample is ?representative? (or not) is best understood as a matter of
?critical evaluation/falsifiability? and not of ?inductive probability?.
From a ?critical/falsificationist? perspective, we might begin by asking:
?representative sample? of what? What ?wider class? are we seeking to test
using our sample? Say we want to poll the likelihood of the voting in the
next Presidential election. Clearly the ?class? we want to test is that of
?American voters?. So we would not easily regard a survey of how Chinese or
Canadians would vote in the American Presidential Election, if they could, as
a survey of a ?representative sample? ? it might be they would vote as
American voters would, but such a sample is obviously open to too many
critical objections (for example, Chinese people might have little
understanding of the issues that affect American voters, or they might have a
better understanding than most Americans do but their votes would differ
accordingly).
What about a sample of American voters: will any sample do? No, because we
know from prior research that the spread of votes alters from area to area,
according to sex, class, according to degree of political commitment [?the
party diehard? is different from the ?floating voter?] and the likelihood of
people bothering to vote etc. This prior research is not a bedrock for
inductive probabilising. Rather it falsifies many assumptions, that we might
otherwise naively make, as to whether a particular sample is likely to be
?representative?: and so it sets a basis for a critical debate as to how we
should select our sample in order to best guard against the dangers that it
is unrepresentative.
This is why even random polling may be seen as, generally, more prone to
error than critically-selective polling, provided the selection is made after
the closest possible critical evaluation of all the available information ?
information that would falsify naïve assumptions that a certain random sample
is representative. (It also helps explains why random-polling is open to
attack by politicians when its results are against them). Since we cannot
ensure the sample is representative by inductive means, the best we can do is
critically guard against the dangers that it is unrepresentative.
This critical debate may of course be changed by new research and,
historically, this is how ?statistical research? advances. Say new research
shows that, other things being equal, the main determinant of how ?swing
voters? decide is on how upbeat the candidate is perceived to be ? or some
other aspect of personality or presentation. This might mean polls as to
voting intention at the next election would be largely worthless until the
candidates were known, and even then rendered worthless if some scandal came
to light - or something else that significantly changed the perception of a
candidate?s character or personality. We might even develop enough prior
research to indicate that it is the ?swing voter? than determines the
election and that, even if they are unrepresentative as a proportion of the
general voters, sampling should focus on ?swing voters? because they are
nevertheless disproportionately ?representative? as a sample predictive of
the election result.
The above comments are offered not because they state anything new or very
surprising but to show that we can account for the predictive usefulness of
polling without recourse to any concept of ?inductive probability?. In fact,
we may see that ?inductive probability? cannot do the job of the accounting
for the predictive usefulness of polling because such probability is either
(a) naïve or question-begging as to whether a sample is representative or (b)
based on a critically examined view as to whether the sample is
representative. But the problem with (b) is that it is not really inductive
probability at all because the ?critical examination? is not based on
induction but on the falsification of mistaken assumptions or theories as to
what would best make a ?sample? representative for a prediction. This is
obvious when we accept that, even if our sample is chosen according to the
best critical debate, it may turn out to be ?unrepresentative? for some
hitherto undiscovered reason i.e. whatever decision we make here is fallible
and open to be overthrown ? overthrown not by induction but by falsification.
To approach this another way: we might all agree that a bigger sample is
generally more ?representative? in some sense than a smaller one (though the
example of the ?swing voter? shows one reason why this assumption may be
naïve ? a sample of 1000 ?swing voters? may be more predictive of the
election result than a sample of 10,000 diehard voters, and thus more
?representative? as critical evidence testing the hypothesis ?Democrats will
win the election?). An inductive probabilist might suggest that this is
because the sample is ?supporting evidence? for this or that conclusion, and
that therefore the bigger the sample the bigger the ?supporting evidence?:
i.e the more ?positive instances? of Democrat voters, the more supporting
evidence there is for the view that it is probable most will vote Democrat.
On the critical rationalist view, however, the bigger sample is generally
better because it is more of a test than a smaller sample: that is, the
indications we get when our sample is 1,000 (when 750 said they would vote
Democrat) may be overturned/falsified when we poll another 10,000 (and find
only 4,500 say they would vote Democrat).
One of the beliefs that underpins belief in induction is that a ?positive
instance? is (somehow, to be spelt out by inductivists) ?supporting
evidence?. This belief seems plausible enough, especially to those given a
basic education where often when they are asked to support a view they are
effectively being asked to give an example. But it is naïve a belief.
Popper speculates [in ?Schilpp?] that this naïve and uncritical belief chimes
with the inductive syllogism:-
1. Socrates is a man and a mortal
2. Plato is a man and a mortal
3. Aristotle is a man and a mortal
4. Therefore, all men are mortal.
But, logically, steps 1-3 equally are positive instances of the rule ?All
mortals are men?. Why do we not see them as ?supporting evidence? for such a
conclusion? Because we can easily find counterexamples.
The only sense in which a ?positive instance? is ?supporting evidence? is not
inductive but consists in the fact it is the absence of a counter-example.
This is the logical point that lies at the heart of Popper?s attack on
induction, including the idea of ?inductive probability?. (See Quine on
?Negative Methodology? and Popper?s reply, both in ?Schilpp?).
Of course, if anyone takes on board this logical point and still insists on
calling their views ?inductive?, or as based on ?inductive probability?, they
may do so ? but they are using these terms in a way quite contrary to their
traditional meaning and in a way that is not likely to clarify but is likely
to mislead.
Donal
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