| A Bayesian Approach to the Argument from Ignorance |
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| 作者：佚名 文章来源：本站原创 点击数： 更新时间：2006-8-28 |
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A Bayesian Approach to the Argument from Ignorance
Abstract In this paper, we re-examine a classic informal
reasoning fallacy, the so-called argumentam ad ignorantiam.
We argue that the structure of some versions of this
argument parallels examples of inductive reasoning that are
widely viewed as unproblematic. Viewed probabilistically,
these versions of the argument from ignorance constitute a
legitimate form of reasoning; the textbook examples are
inductive arguments that are not unsound but simply weak,
due to the nature of the premises and conclusions involved.
In an experiment, we demonstrated some of the variables
affecting the strength of the argument, and conclude with
some general considerations towards an empirical theory of
argument strength.
Argumentation, in one form or other, pervades most
aspects of our everyday lives. People argue about
whether someone should be convicted of a crime,
whether taking drugs is right or wrong, whether you
should allow a 15-year-old to stay out to midnight, and
so on. Most of these arguments are informal (i.e., they
cannot simply be encoded in standard logic). This is
for several reasons. First, these arguments are often
dialogical, that is, they are conducted between two or
more participants, possessing differing goals and background
knowledge (Perelman & Olbrechts-Tyteca,
1969; Walton, 1989). Second, the relationships that are
important are defined at a higher level than the microstructures
of reasoning investigated in formal logic
(Freeman, 1991). Third, by the standards of formal
logic these arguments are often invalid or fallacious.
Although informal argumentation has been studied
intensively in social (for a review see, Voss & Van
Dyke, 2001), developmental (for a review see, Felton
& Kuhn, 2001), and other areas of psychology (e.g.,
Rips, 1998, 2002), to our knowledge, with a few
exceptions (Neuman & Weizman, 2003; Rips, 2002),
the informal fallacies that philosophers, logicians, and
rhetoricians have identified over the last two millennia
have not been investigated experimentally. Moreover,
we know of no attempts to extend psychological theories
of reasoning to try and account for these patterns
of informal argument. We argue that a probabilistic
approach to human reasoning (Chater & Oaksford,
2001; Oaksford & Chater, 1998, 2001) may be extended
to many of these informal arguments. The goal of this
paper is to provide a Bayesian analysis of at least
some versions of the argument from ignorance
(Walton, 1992). This is “the mistake that is committed
whenever it is argued that a proposition is true simply
on the basis that it has not been proved false, or that it
is false because it has not been proved true.” For
example:
Ghosts exist because no one has proved
that they do not.
(1)
This argument seems unacceptable. However, we
argue that this is not because the argument structure it
embodies is fallacious as has traditionally been
assumed from a logician’s perspective. Rather, the
argument is structurally perfectly acceptable, but weak
due to its specific contents. We present a Bayesian
analysis and report an experiment showing that the
acceptability of the argument from ignorance is affected
by the factors that a Bayesian analysis would predict.
Generally, logic and the acceptability of informal
arguments dissociate. There are arguments that are logically
invalid but that are regarded as informally
acceptable, and there are arguments that are logically
valid but that are regarded as informally unacceptable.
For example, an argument of the form, the key was
turned because the car started and if you turn the key
the car starts, is an instance of the logical fallacy of
affirming the consequent. However, it also an instance
of inference to the best explanation (Harman, 1965),
that is, the best explanation of why the car started is
that the key was turned. Furthermore, the inference
from the key was turned to the key was turned can be
described as the claim that if the key was turned, then
the key was turned. This conditional is necessarily true
because it is truth functionally equivalent to the logical
law of the excluded middle, that is, the key was turned
or the key was not turned, which is necessarily true.
Mike Oaksford and Ulrike Hahn
School of Psychology, Cardiff University
Canadian Journal of Experimental Psychology, 2004, 58:2, 75-85
76 Oaksford and Hahn
However, the inference from God has all the virtues to
God is benevolent, considered as an informal argument
would be condemned as circular. That is, it assumes
what it is supposed to establish, even though it can be
viewed as logically valid (at least given the additional
premise “benevolence is a virtue”). So this argument
succeeds as a logical argument but fails as an informal
one (see e.g., Walton, 1989, 1996). These examples
make it clear that standard logic provides little guidance
regarding the acceptability of a pattern of informal
reasoning. This conclusion is further borne out by
recent studies showing that the ability to identify informal
reasoning fallacies is not correlated with deductive
reasoning performance (Neuman, in press; Ricco,
2003).
Furthermore, as many authors have observed, a pattern
of informal reasoning that is unacceptable in one
context may be acceptable in another. This can often
be shown by placing the argument in an appropriate
dialogical context (Walton, 1992, 1996). For example, if
a novice Christian was ignorant of God’s properties
and asked his vicar if God was benevolent, then the
reply, “Yes, God has all the virtues,” seems acceptable.
Certainly it seems no less acceptable than the novice
Classicist asking whether Hercules managed to clean
the Augean stables, to be told by his lecturer that,
“Yes, Hercules succeeded in all his labours.” We argue
that acceptability is a matter of degree and that a
Bayesian approach may provide a useful metric of
acceptability for informal arguments. This approach is
related to Kuhn’s (1993) attempts to relate scientific
and informal reasoning. The most general formal
model of scientific reasoning currently available is provided
by Bayesian probability theory (Earman, 1992;
Howson & Urbach, 1989).
In some psychological work, it is assumed that the
fallacies are instances of bad argumentation and the
focus is on the factors that allow people to avoid them
(e.g., Neuman & Weizman, 2003). By contrast, following
some recent philosophical work in this area (e.g.,
Copi & Burgess-Jackson, 1996; Eemeren &
Grootendorst, 1992; Walton, 1996), we argue that
whether an argument ad ignorantiam is fallacious
depends on the context in which it occurs. Moreover,
we attempt to show that at least some versions can be
viewed as an instance of – fundamentally sound –
inductive inference.
The Argument From Ignorance
Walton (1996) identifies the following form for the
argument from ignorance:
If A were true (false), it would be known
(proved, presumed) to be true (false).
(2)
A is not known (proved, presumed) to be true (false).
Therefore, A is (presumably) false (true).
In general, of course, lack of knowledge, evidence
or proof, is not sufficient to establish that a proposition
is false. Indeed, if it were, then all kinds of absurd
conclusions could be licensed by such arguments. For
example, the fact that we have no evidence that flying
pigs do not exist outside our solar system, does not
imply that we should conclude that they do (we thank
an anonymous reviewer for this example). Similarly,
the fact that we have no evidence that flying pigs do
exist outside our solar system, does not imply that we
should conclude that they do not. Both these arguments
are instances of the argument from ignorance,
and both seem to be strictly fallacies (although our
prior beliefs seem to suggest that the latter argument is
more acceptable).
Walton (1996) identifies three basic types of the
argument from ignorance where fallacies may arise:
shifting the burden of proof, epistemic closure, and
negative evidence.
Shifting the Burden of Proof
The classic example of shifting the burden of proof
comes from the anticommunist trials overseen by
Senator Joseph McCarthy in the 1950s. The proposition
in question is that the accused is a communist sympathizer.
In one case, the only evidence offered to support
this conclusion was the statement that “…there is
nothing in the files to disprove his Communist connections”
(Kahane, 1992, p. 64). This argument attempts to
place the burden of proof onto the accused person to
establish that he is not a Communist sympathizer.
Indeed, it is an attempt to reverse the normal burden
of proof in law that someone is innocent until proved
guilty, which itself licenses one of the few arguments
from ignorance that some philosophers regard as valid
(e.g., Copi & Cohen, 1990). That is, if the prosecution
cannot prove that a defendant is guilty, then he/she is
innocent. In the McCarthy example, it is clear that the
argument is open to question. The conditional premise
in this argument is, “if A were not a Communist sympathizer,
there would be something in the files to
prove it.” However, there is no reason at all to believe
that this should be the case.
Epistemic Closure
The second type of argument from ignorance is
knowledge-based and relies on the concept of epistemic
closure (De Cornulier, 1988; Walton, 1992) or
what is known as the closed world assumption in
Artificial Intelligence (AI) (Reiter, 1980, 1985). The
negation-as-failure procedure (Clark, 1978) is a clear
A BAYESIAN APPROACH TO THE ARGUMENT FROM IGNORANCE 77
example, where one argues that a proposition is false
and therefore that its negtion is true, because it cannot
be proved from the contents of a data base, . As a
result, the meaning of ¬A, is that A cannot be proved
from the other statements in , that is, negation is intuitionistic
(McCarty, 1983), not classical. This pattern of
reasoning assumes that all knowledge relevant to this
question is in the data base. However, the simple addition
of A to would override this assumption and
consequently this style of reasoning relativizes truth to
truth in a knowledge state. It is clearly also nonmonotonic
or defeasible (Oaksford & Chater, 1991)
(i.e., conclusions can be overridden by new information).
Negation-as-failure is a practical necessity in knowledge-
based systems but there are also more mundane
examples. Walton (1992) provides the example of a
railway timetable. Suppose the point at issue is
whether the 13:00 train from London, Kings Cross to
Newcastle stops at Hatfield. If the timetable is consulted
and it is found that Hatfield is not mentioned as
one of the stops, then it can be inferred that the train
does not stop there. That is, it is assumed that the
timetable is epistemically closed such that if there were
further stops they would have been included. The reason
why such arguments may fail is again related to
the conditional premise in the argument from ignorance.
In the real world, the closed world assumption
is rarely justified so it is not reasonable to assume that
if A were true this would be known.
Negative Evidence
The final type of the argument from ignorance that
Walton (1996) identifies is based on negative evidence
and so the conclusion of interest is a hypothesis under
test. If this hypothesis is true, then the experiments
that are conducted to test it would reveal positive
results (i.e., the predictions that can be deduced from
the hypothesis would actually be observed). However,
if they are not observed, then the hypothesis is false. A
mundane example of this style of reasoning is testing
new drugs for safety. The argument from ignorance
here is that a drug is safe if no toxic effects have been
observed in tests on laboratory animals (Copi &
Cohen, 1990). The critical point about such arguments
is that the tests are well conducted and performed in
sufficient number that if the drug were truly toxic the
tests would have revealed it. As with the other arguments
from ignorance, if this conditional premise cannot
be established then fallacious conclusions may
result.
There is some disagreement in the literature as to
whether these last two types are genuine arguments
from ignorance. Copi and Cohen (1990), for example,
argue that these arguments do in fact rely on knowledge
(i.e., for epistemic closure it is known that something
is not known and for negative evidence it is
known that there are failed tests of a hypothesis).
Hence, strictly they are not arguments from, at least
total, ignorance. However, we follow Walton (1996) in
grouping all these types of argument under the same
heading as they do have the same underlying form.
Walton (1992) points out that the conclusion of any
argument from ignorance is open to refutation (i.e., the
conclusion can really only be accepted tentatively). He
suggests dealing with this uncertainty by regarding the
argument from ignorance as a case of presumptive or
nonmonotonic reasoning. That is, conclusions based
on the results of the argument from ignorance are presumed
true until proven otherwise. In the next section,
we discuss some problems for this approach that we
think our probabilistic approach can resolve.
Coping With Uncertainty
Defeasible reasoning, where conclusions can be
defeated, create many problems in designing knowledge-
based systems in Artificial Intelligence (for extensive
discussion in relation to the psychology of human
reasoning, see Oaksford & Chater, 1991, 1993, 1995,
1998). Problems for this style of reasoning can be
readily illustrated using another form of presumptive
reasoning using conditional rules (Oaksford & Chater,
1991, 1993).
For example, if I know that Jane is a runner I may
infer that she is fit. This is a case of presumptive reasoning
because she may have a heart condition that
you do not know about. In an AI knowledge based
system, this style of presumptive reasoning is dealt
with in a similar way to negation-as-failure (Reiter,
1985). Jane is fit can be inferred from the fact that she
is a runner, as long as it cannot be proved from the
contents of your knowledge base that Jane is unfit.
While runners tend to be fit, academics tend not to be.
Thus, by the same pattern of reasoning, if you found
out that Fred is an academic, and you could not prove
from the contents of your knowledge base that Fred is
fit, you could conclude that Fred is unfit. What happens
when you find out that Amelia is an academic
runner? The problem here is that, depending on which
general rule is applied first, you can conclude, presumptively,
either that Amelia is fit or that Amelia is
unfit. Overall, therefore, all that can be concluded is
that Amelia is fit or she is unfit. But of course that was
known before any presumptive inferences were drawn
(i.e., this is an uninformative tautology). However,
intuitively, most people would probably infer that
Amelia was fit.
That presumptive, nonmonotonic reasoning systems
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