Dear Friends,
Like many of you, I have done some work with chains & webs of
inferences. However, there are a number of questions that have bothered
me from time to time and I would like to get your reading of whether my
concerns are warranted or coherent.
*******
One can begin by thinking of a series of inferences in the following
way:
r ----> s -----> t and so on
where (let us assume) r, s & t are "factual propositions" or
"propositions about facts" or "propositions about events in space and
time."
One may then think it necessary to enrich this picture by adding
"generalizations" or "ancillary hypotheses" or "warrants" some such
thing to help explain, or to help think about, why or how "r ---> s"
and why or how "s --> t" and so on. Thus one may picture inference in
the following way:
r -------------> s -----------> t and so on
|__________| |________|
H-1 H-2
where H-1 and H-2 represent generalizations, ancillary hypotheses,
warrants, or some such thing.
One may add more to this picture by supposing that evidence -- ancillary
evidence -- may support such generalizations, ancillary hypotheses,
warrants, or what have you, and that a series of inference may thus look
like something like this:
primary evidence -------------->| r ---------------->| s
-------------->| t and so on
|________________| |__________| |____________|
H-1
H-2 H-3
^
^ ^
|
| |
|
| |
ancillary evidence #1 ancillary e-2
ancillary e-3
Now much of this sort of picture looks intuitively correct to some
(many) observers. For example, there is a longstanding tradition in
legal scholarship on evidence that sees inference in roughly the above
fashion. However, there are several questions or objections one might
advance against the above sort of picture of inference.
1. One might object on "formal" grounds to the mixing of two (types of)
graphs: (i) the (horizontal) chain of inferences and (ii) the (vertical)
supporting or ancillary chain. I am not much troubled by this. Should I
be? Is the above picture -- the last diagram above -- an example of a
graph embedded in a graph? Is that an incoherent or nonsensical notion?
If so, why?
2. In the hands of some legal scholars and also in the hands of some
scholars of the non-legal sort , the above sort of picture of inference
seem always to end up with supporting generalizations or ancillary
hypotheses or warrants that look very much like relative frequency
statements (e.g., "people usually tell the truth when they are asked by
a grand jury what they had for breakfast"). This troubles me greatly.
Should it? I am troubled, first, because most of the examples of
ancillary hypotheses that are given sound positively silly (such as the
one in the parenthetical above). That's a clue, isn't it? It seems to me
that (i) ancillary hypotheses or warrants must have a more complex
structure than "the proportion or frequency of p in r = x." Am I right?
I suspect that the warrant for an inference must be some sort of complex
hypothesis -- one that usually includes hypotheses of a causal sort --
and that such a complex warrant cannot and should not be reduced to
relative frequency statements. If I am right, the following sort of
picture is a more adequate depiction of the sorts of things that "go on"
in ancillary reasoning, a more adequate hint of the way that warrants
for inferences actually work (and ought to work?):
primary evidence -------------->| r ---------------->| s
-------------->| t and so on
|________________| |__________| |____________|
Warrant 1 =
thoughts such as =
f = ma
"the developed superego
is distantly related to the id
but is prone to great vanity
in moments of stress" &
"dark windshield glass can
often distort images"
The immediately-preceding thing is admittedly not a pretty picture of
inferential reasoning. But isn't it nevertheless at least suggestive --
more realistically suggestive -- of how the human mind sometimes or
often "passes" from, e.g., proposition (fact) r to proposition (fact)
s, and isn't the sort of reasoning (so roughly depicted in my inelegant
picture) often (sometimes?) quite effective and rational?
3. The questions in par. 2 above bring me to an alternative to the sort
of picture of inference that I have sketched above: Pearl-style causal
reasoning. Isn't it the case that Pearl-style causal reasoning --
despite being avowedly _"causal"_ -- isn't it the case that Pearl-style
causal reasoning fails to "capture" the complexity of the sort of
(jumbled) causal reasoning that is involved in my last example? Pearl's
diagrams look "causal" to me _only_ in the sense that they posit that
all events have an antecedent cause or causes. In every other respect,
his diagrams make the links between events or propositions look like
relative frequencies rather than like the sort of complex relationships
that one finds in even simple "real" causal hypotheses such as F = MA.
(Some comments made by Christian Borgelt prompted these present thoughts
of mine about Pearl-style causal reasoning -- though Borgelt cannot be
and must not be blamed for the egregious errors in my reasoning here!)
I would be grateful for any enlightenment that any of you might offer on
any of the above questions and issues. But please don't be too brutal
with your responses. Keep in mind that I am a simple law teacher, not
(like many of you) an eminent logician, mathematician, statistician, or
philosopher.
Sincerely yours,
Peter Tillers
*********************************************************
Tillers' Dynamic Evidence Site
--with "frames":
http://www.tiac.net/users/tillers/index.html
--sans "frames":
http://www.tiac.net/users/tillers/home.html
**********************************************************
Peter Tillers, Professor of Law
Benjamin N. Cardozo School of Law, Yeshiva University
55 Fifth Avenue, New York, New York 10003, U.S.A.
(212) 790-0334; FAX (212) 790-0205
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Dear Friends,
Like many of you, I have done some work with chains & webs of inferences. However, there are a number of questions that have bothered me from time to time and I would like to get your reading of whether my concerns are warranted or coherent.
One can begin by thinking of a series of inferences in the following way:
r ----> s -----> t and so on
where (let us assume) r, s & t are "factual propositions" or "propositions about facts" or "propositions about events in space and time."
One may then think it necessary to enrich this picture by adding "generalizations" or "ancillary hypotheses" or "warrants" some such thing to help explain, or to help think about, why or how "r ---> s" and why or how "s --> t" and so on. Thus one may picture inference in the following way:
r -------------> s -----------> t and so on
|__________| |________|
H-1
H-2
where H-1 and H-2 represent generalizations, ancillary hypotheses, warrants, or some such thing.
One may add more to this picture by supposing that evidence -- ancillary evidence -- may support such generalizations, ancillary hypotheses, warrants, or what have you, and that a series of inference may thus look like something like this:
primary evidence -------------->| r ---------------->| s -------------->| t and so on
|________________| |__________| |____________|
H-1
H-2
H-3
ancillary evidence #1 ancillary e-2 ancillary e-3^ ^ ^
| | |
| | |
Now much of this sort of picture looks intuitively correct to some (many) observers. For example, there is a longstanding tradition in legal scholarship on evidence that sees inference in roughly the above fashion. However, there are several questions or objections one might advance against the above sort of picture of inference.
1. One might object on "formal" grounds to the mixing of two (types of) graphs: (i) the (horizontal) chain of inferences and (ii) the (vertical) supporting or ancillary chain. I am not much troubled by this. Should I be? Is the above picture -- the last diagram above -- an example of a graph embedded in a graph? Is that an incoherent or nonsensical notion? If so, why?
2. In the hands of some legal scholars and also in the hands of some
scholars of the non-legal sort , the above sort of picture of inference
seem always to end up with supporting generalizations or ancillary hypotheses
or warrants that look very much like relative frequency statements (e.g.,
"people usually tell the truth when they are asked by a grand jury what
they had for breakfast"). This troubles me greatly. Should it? I am troubled,
first, because most of the examples of ancillary hypotheses that are given
sound positively silly (such as the one in the parenthetical above). That's
a clue, isn't it? It seems to me that (i) ancillary hypotheses or warrants
must have a more complex structure than "the proportion or frequency of
p in r = x." Am I right? I suspect that the warrant for an inference must
be some sort of complex hypothesis -- one that usually includes hypotheses
of a causal sort -- and that such a complex warrant cannot and should not
be reduced to relative frequency statements. If I am right, the following
sort of picture is a more adequate depiction of the sorts of things that
"go on" in ancillary reasoning, a more adequate hint of the way that warrants
for inferences actually work (and ought to work?):
primary evidence -------------->| r ---------------->| s -------------->| t and so on
|________________| |__________| |____________|
Warrant 1 =
thoughts such as =
f = ma
"the developed superego
is distantly related to the id
but is prone to great vanity
in moments of stress" &
"dark windshield glass can
often distort images"
The immediately-preceding thing is admittedly not a pretty picture of inferential reasoning. But isn't it nevertheless at least suggestive -- more realistically suggestive -- of how the human mind sometimes or often "passes" from, e.g., proposition (fact) r to proposition (fact) s, and isn't the sort of reasoning (so roughly depicted in my inelegant picture) often (sometimes?) quite effective and rational?
3. The questions in par. 2 above bring me to an alternative to the sort of picture of inference that I have sketched above: Pearl-style causal reasoning. Isn't it the case that Pearl-style causal reasoning -- despite being avowedly _"causal"_ -- isn't it the case that Pearl-style causal reasoning fails to "capture" the complexity of the sort of (jumbled) causal reasoning that is involved in my last example? Pearl's diagrams look "causal" to me _only_ in the sense that they posit that all events have an antecedent cause or causes. In every other respect, his diagrams make the links between events or propositions look like relative frequencies rather than like the sort of complex relationships that one finds in even simple "real" causal hypotheses such as F = MA. (Some comments made by Christian Borgelt prompted these present thoughts of mine about Pearl-style causal reasoning -- though Borgelt cannot be and must not be blamed for the egregious errors in my reasoning here!)
I would be grateful for any enlightenment that any of you might offer on any of the above questions and issues. But please don't be too brutal with your responses. Keep in mind that I am a simple law teacher, not (like many of you) an eminent logician, mathematician, statistician, or philosopher.
Sincerely yours,
Peter Tillers*********************************************************
Tillers' Dynamic Evidence Site**********************************************************--with "frames": http://www.tiac.net/users/tillers/index.html
--sans "frames": http://www.tiac.net/users/tillers/home.html