Cervelli, Persone e Società ***ABSTRACTS
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Brains, Persons, and Society *** ABSTRACTS Cervelli, Persone e Società ***ABSTRACTS |
Vincenzo
Crupi
University of Trento
Michel
Gonzalez
On the logic and psychology
of evidential support
Epistemologists
and
philosophers of science have often attempted to define a formal
relationship
which
adequately expresses the impact of a piece of evidence on the
credibility of
a
hypothesis. The
present talk will focus on the Bayesian approach to evidential support.
A new
formal
treatment of the
notion of degree of confirmation will be proposed which
overcomes
some
limitations of the
currently available approaches.
A
plurality of
non-equivalent Bayesian measures of confirmation or evidential support
have
been
devised [1,2]. One
way to handle such plurality is to resort to the long standing and
traditional
view of
inductive logic as an “extension” of classical deductive logic [3]. We
will
consider
two basic
adequacy requirements for such an estension.
Consider
a set of
sentences , closed under negation and
conjunction, on which a (regular)
probability
function p
is defined. Let a function v from
into {–1,
0, 1} be defined on the
basis
of classical
deductive logic, so that v(e,h) = 1 iff e |=
h,
v(e,h) = –1 iff e |= not-h, and
v(e,h)
= 0
otherwise. A first adequacy requirement for a (Bayesian) confirmation
measure c
will
be:
(i)
if v(e1,h1)
> v(e2,h2), then c(e1,h1)
> c(e2,h2)
A
second proposed
adequacy requirement will extend the treatment of a set of symmetries
and
asymmetries,
which have
been recently discussed as constraints on the selection of a
normatively
adequate
Bayesian confirmation measure [4].
Let
a symmetry be
a function from into such that (e,h)
is obtained from (e,h) by
negating
either e or
h (or both) and/or by inverting them. On the whole, there are
seven such
symmetries
(which will
be displayed in details during the talk). Then let s (for
“sign”) be a
function
from into {–1, 0, 1}, defined as
follows: s(e,h)
= 1 if p(h|e) > p(h); s(e,h)
= 0 if
p(h|e)
= p(h);
s(e,h) = –1 if p(h|e) < p(h).
A confirmation measure c is said to mirror a
symmetry
in case of confirmation [disconfirmation] iff, for
any e,h such that s(e,h)
= 1
[–1], s(e,h)
• c(e,h)
= s((e,h)) • c((e,h)). (Any measure c trivially
mirrors any symmetry in
case
of neutrality,
i.e., when p(h|e) = p(h).) Our second
adequacy
requirement will be:
(ii)
c mirrors in case of confirmation
[disconfirmation]
iff
v mirrors in case of confirmation
[disconfirmation]
In
the talk, it will be
argued that the normative plausibility of both (logically independent)
principles
(i) and (ii)
is supported by simple intuitive examples. Then, two major formal
results
will
be presented: first,
none of the currently available Bayesian confirmation measures
satisfy
both
(i) and (ii);
second, the following new measure does satisfy both requirements [5]:
Z(e,h)
= [p(h|e)
– p(h)]/p(not-h) if p(h|e) p(h)
[p(h|e)
– p(h)]/p(h)
otherwise
It
will also be observed
that measure Z enjoys several desirable properties involved in
standard
Bayesian
treatments of
traditional epistemological puzzles, such as the “ravens paradox” [6],
the
“grue paradox” [7]
and the paradox of “irrelevant conjunction” [8].
Finally,
the discussion
will be extended from the context of normative (epistemological)
analysis
to the
descriptive (psychological) issue of capturing naïve individuals’
judgments of
inductive
strength.
Empirical results will be reported showing that, as compared to its
major
competitors,
measure Z
is a reliably better predictor of reported confirmation judgments
in an
experimental
setting
involving the evidential impact of extractions from an urn on beliefs
about
the
urn’s composition
[9]. It will then be proposed that the virtues of measure Z might
not be
confined
to the
normative level of epistemological reflection, but extend to the
descriptive
dimension
of the
psychology of confirmation.
[1]
Festa, R., “Bayesian
confirmation”, in M. Galavotti & A. Pagnini (eds.), Experience,
Reality,
and Scientific Explanation,
[2]
Fitelson, B., “The
plurality of Bayesian measures of confirmation and the problem of
measure
sensitivity”, Philosophy
of Science, 66 (1999), S362–S378
[3]
Carnap, R., Logical
Foundations of Probability,
[4]
Eells, E., &
Fitelson, B., “Symmetries and asymmetries in evidential support”, Philosophical
Studies, 107 (2002), 129–
142
[5]
Crupi, V., Tentori,
K., & Gonzalez, M., “On Bayesian theories of evidential support:
normative
and descriptive
considerations”,
manuscript submitted
[6]
Horwich, P., Probability
and Evidence, CUP,
[7]
Sober, E., “No
model, no inference: a Bayesian primer on the grue problem”, in D.
Stalker
(ed.), Grue! The New Riddle
of Induction, Open Court, Chicago,
1994, 225-240
[8]
Science,
forthcoming
[9]
Tentori, K., Crupi,
V., Bonini, N. & Osherson, D., “Comparison of confirmation
measures”, Cognition,
forthcoming