Transcript No Slide Title

Non-constructive inference
and conditionals
David Over
Psychology Department
Durham University
Thanks to:
• The organizers
• Nagoya University
• Japan Society for the Promotion of Science
Non-constructive reasoning
Modified example from Toplak &
Stanovich (2002):
Jack is looking at Ann, and Ann is
looking at George. Jack is a cheater, but
George is not. Is a cheater looking at a
non-cheater?
A) Yes
B) No
C) Cannot tell
The non-constructive aspect
• Jack is looking at Ann but Ann is looking
at George. Jack is a cheater but George is
not. Is a cheater looking at non-cheater?
• Ann is either a cheater or not. If she is,
then a cheater (Ann) is looking at a noncheater (George). If she is not, then a
cheater (Jack) is looking at a non-cheater
(Ann). Therefore, the answer is “Yes”.
A constructive approach
• Jack is looking at Ann but Ann is looking
at George. Jack is a cheater but George is
not. Is a cheater looking at non-cheater?
• We get hold of Ann and try to cooperate
with her in reciprocal altruism. She does
not cooperate. Our “cheater detection
mechanism” fires, and we conclude she is
a cheater. Therefore, the answer is “Yes”.
The distinction and dual
process theory
• Non-constructive inference is the purest
example of a type 2 analytic process. It
is an inference from “above”, using logic.
• Constructive inference is from “below”:
it is grounded in type 1 heuristic
processes, such as those of perception.
Jonathan Evans’s list of dual process theories
Perception & attention: Egeth & Yanis (1997), stimulus &
goal driven attention.
Skilled performance: Anderson (1983), procedural &
declarative knowledge.
Learning & memory: Reber (1993), implicit & explicit
learning.
Social cognition: Strack & Deustch (2004), impulsive &
reflective.
Reasoning & decision making: Evans (2006) and Evans &
Over (1996), heuristic & analytic; Barbey & Sloman (in
press), associative & rule based; Stanovich (1999) and
Kahneman & Frederick (2002), System 1 & System 2.
System 1 mental processes
(Stanovich, 2004)
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Associative
Holistic
Parallel
Automatic
Undemanding of cognitive capacity
Fast
Highly contextualized
“Old” in evolutionary terms
System 2 mental processes
(Stanovich, 2004)
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Rule based
Analytical
Serial
Controlled
Demanding of cognitive capacity
Slow
Decontextualized
“New” in evolutionary terms
Jonathan Evans’s characterization
Type 1 processes: Fast and automatic, with
high capacity and low effort.
Type 2 processes: Slow and controlled, with
limited capacity and high effort. These
processes make use of working memory.
Dual process theory is opposed to
the massive modularity hypothesis
• Many leading evolutionary psychologists
have argued for what has been called the
massive modularity hypothesis.
• Some leading evolutionary psychologists
do not accept this hypothesis, but the
following support it: Cosmides & Tooby,
Buss, Pinker, and Gigerenzer.
Massive modularity implies:
• There is no mental logic: no formal
system for performing valid inferences.
Massive modularity implies:
• There are no content independent
mechanisms for inference or learning.
Massive modularity implies:
• There are only content specific or domain
specific mechanisms – the modules - for
solving adaptive problems.
Massive modularity metaphor:
• The mind is a Swiss army knife - it has
many special blades for solving adaptive
problems but no general purpose blade.
Dual process theory implies:
• Type 1 processes result from content
specific mechanisms for perception,
memory, and heuristic inference - the
modules.
Dual process theory implies:
• Type 2 processes result from general
purpose mechanisms, including a means
of logical inference.
Dual process theory implies:
• That the mind has two systems, System 1
and System 2, or at least has two kinds of
processes, type 1 and type 2. The Swiss
army knife metaphor could also be used
for dual process theory.
My claim:
• The best example of a type 2 process is
non-constructive reasoning. A good
example of this reasoning is inferring a
disjunction, “p or q”, from “above”.
Validly inferring a disjunction
from “above”
• We may infer, “Ann is a cheater or not a
cheater”, from “above” using pure logic –
in this case we cannot say which disjunct
is true. We do not know which property
Ann has: being a cheater or not one.
Justifiably inferring a disjunction
from “above”
• We may infer, “Ann is a cheater or Jack
is a cheater”, from “above” using
probabilistic inference. Resources are
missing. Someone is taking more than
their share. Other general considerations
point to Ann or Jack, but we do not know
which is the cheater.
What use is non-constructive
reasoning?
• We may infer, “Ann is a cheater or Jack
is a cheater”, from “above”. This enables
us to infer, “If Ann is not the cheater then
Jack is.” That is a useful conditional to
infer. For when we later get evidence that
Ann is not the cheater, we may infer that
Jack is.
Constructive reasoning is not
useful in this way
• Suppose we infer, “Ann is a cheater or
Jack is a cheater”, from “below”, from
Ann is the cheater. We now cannot infer,
“If Ann is not the cheater then Jack is.” If
it does turn out that our information
from “below” was wrong, we do not have
any reason to suspect Jack.
Is non-constructive inference “old”?
• It is often said that type 2 processes are “new”
in evolutionary terms. Is this true of nonconstructive inference? Do any other animals
show signs of this kind of reasoning? The Stoic
logician Chrysippus claimed that a dog could
know that an animal went down one of three
roads and infer that, if it did not go down the
first two, then it went down the third. Is there
any scientific evidence of such inference?
The logical form of the inference
The form is that of inferring “if not-p then q”
from “p or q” or, equivalently inferring “if p
then q” from “not-p or q”. My claim is that
such inferences are justified only when the
disjunction is inferred non-constructively. But
in elementary logic, “if p then q” just means
“not-p or q” and so such inferences are always
justified, that is also so in the main
psychological theory of conditional reasoning.
The material conditional
In elementary extensional logic, “if p then
q” just means “not-p or q”, and so “if
not-p then q” means “p or q”. This kind
of conditional is the material conditional.
The mental model theory of JohnsonLaird & Byrne (2002) implies that the
ordinary conditional of natural language
is the material conditional.
What the mental model theory of
Johnson-Laird & Byrne implies
• Johnson-Laird & Byrne (2002, p. 650) hold
that the following inference is valid:
• In a hand of cards, there is an ace or a king or
both. So if there isn’t an ace in the hand, then
there is a king.
• Now if the above were valid, then the ordinary
conditional would be the material conditional,
which Johnson-Laird & Byrne deny in places,
but that is logically implied in what they write
down as their theory.
The form of the inference
referred to by Johnson-Laird &
Byrne (2002)
• Inferring “if not-p then q” from “p or q”.
• From “not-p or q”, we get “if not-not-p
then q” by the form, from which we infer
by double negation “if p then q”.
• So one could equally well study inferring
“if p then q” from “not-p or q”.
More logical points
For all conditionals we must have that
“if p then q”
logically implies
“not-p or q”
But only for the material conditional,
can the converse hold, as the material
conditional just means “not-p or q”.
If Johnson-Laird & Byrne
(2002) are right
• Then “if p then q” is logically equivalent
to the truth function material conditional,
“not-p or q”.
• And those of us who deny the equivalence
are wrong.
• But we deny the equivalence and so must
show that Johnson-Laird & Byrne (2002)
are wrong.
Our theory of ordinary conditionals in
natural language
• In our account, “if p then q” does not
mean “not-p or q”. According to us,
people evaluate “if p then q” by
supposing that p holds and judging q
under that supposition. This process is
called the Ramsey test in philosophical
logic.
The Ramsey test
• Ramsey (1931) suggested that people
could judge “if p then q” by “...adding
p hypothetically to their stock of
knowledge …” They would thus fix
'...their degrees of belief in q given
p…”, which is their conditional
subjective probability of q given p,
P(q/p).
What the Ramsey test implies
(Over, Hadjichristidis, Evans,
Handley, & Sloman, 2007)
• The probability of an indicative
conditional, P(if p then q), is the
conditional subjective probability,
P(q/p).
P(q/p) high implies high
P(not-p or q)
Suppose we find that P(q/p) is high.
Then we will find that P(not-p or q),
the material conditional, is high.
P(not-p or q) =
P(not-p) + P(q) - P(not-p & q) =
P(not-p) + P(q/p) - P(not-p)P(q/p)
P(not-p or q) high does not
imply high P(q/p)
Suppose we find that P(not-p) is high.
Thus P(not-p or q) is high, but recall:
P(not-p or q) =
P(not-p) + P(q/p) - P(not-p)P(q/p)
And that means that P(q/p) can be low
when P(not-p or q) is high.
Validity and strength
• Inferring “if not-p then q” from
“p or q”is not logically valid,
as P(p or q) can be higher than
P(q/not-p).
• However, the inference can be a
strong probabilistic inference in
non-constructive reasoning.
Constructive example
• We think that we see Ann going into
the library. We infer with high
confidence that Ann is in the library or
the computer lab. But we could not
infer from this that, if she is not in the
library, then she is in the computer
lab.
Constructive details
P(library & lab) = 0
P(library & not-lab) = .9
P(not-library & lab) = .01
P(not-library & not-lab) = .09
P(library or lab) = .91
P(lab/not-library) = .01/.1 = .1
Non-constructive example
• We infer from reading the module
guide that everyone in the class is
in the library or the lab. Ann is in
the class. So Ann is in the library
or the lab. And so, if Ann is not in
the library, then she is in the lab.
Non-constructive details 1
P(library & lab) = 0
P(library & not-lab) = .5
P(not-library & lab) = .5
P(not-library & not-lab) = 0
P(library or lab) = 1
P(lab/not-library) = .5/.5 = 1
Non-constructive details 2
P(library & lab) = 0
P(library & not-lab) = .45
P(not-library & lab) = .45
P(not-library & not-lab) = .1
P(library or lab) = .9
P(lab/not-library) = .45/.55 = .81
Non-constructive details 3
P(library & lab) = 0
P(library & not-lab) = .7
P(not-library & lab) = .2
P(not-library & not-lab) = .1
P(library or lab) = .9
P(lab/not-library) = .2/.3 = .66
Non-constructive details 4
P(library & lab) = 0
P(library & not-lab) = .8
P(not-library & lab) = .1
P(not-library & not-lab) = .1
P(library or lab) = .9
P(lab/not-library) = .1/.2 = .5
Ramsey test example
• Hypothetically suppose I buy a lottery ticket.
Under this supposition, I can use knowledge of
the lottery and probability to infer I will
probably lose my money. Using the Ramsey
test, I disbelieve, If I buy a lottery ticket, I will
win millions.
• Johnson-Laird & Byrne (2002) imply that I
should believe, If I buy a lottery ticket, I will
win millions, as I will not buy a lottery ticket,
and If I buy a lottery ticket, I will win millions
supposedly means I do not buy a lottery ticket
or I will win millions.
The Ramsey test and heuristics
• The Ramsey test can be compared to the
simulation heuristic (Kahneman & Tversky,
1982). Both are high level processes that have to
be implemented by more specific ones.
• The availability heuristic (Tversky &
Kahneman, 1972) could be used to judge P(p &
q) is probable than P(p & not-q), i.e. P(q/p) is
relatively high.
Over, Hadjichristidis, Evans, Handley,
& Sloman (2007) show:
• For an indicative conditional, If global
warming continues, London will be flooded.
• The subjective probability of such indicative
conditionals, P(if p then q), is the conditional
subjective probability of q given p, P(q/p).
Over, Hadjichristidis, Evans, Handley,
& Sloman (2007)
• People explicitly assess P(if p then q).
•
•
•
•
•
They also explicitly judge:
P(p & q).
P(p & not-q).
P(not-p & q).
P(not-p & not-q).
What we can get from the
probabilistic truth table
• P(p) = P(p & q) + P(p & not-q)
• P(q/p) = P(p & q)/[P(p & q) + P(p & not-q)]
• P(q/not-p) = P(not-p & q)/[P(not-p & q) +
P(not-p & not-q)]
The analysis
• We performed multiple regression
analyses on P(if p then q) using P(p)
and P(q/p) as predictors
• If “If p then q” is the material
conditional and so means “not-p or q”,
then P(p) should have a significant
negative loading.
The results for participants
• Analyses across individual participants.
Cells = beta weights
EXP1
(indicatives)
True
False
EXP2
(indicatives)
P(p)
.02
.02
.16*
P(q/p)
.42*
-.38*
.51*
The results for items
• Analyses of item means on item means.
Cells = beta weights
EXP1
(indicatives)
True
False
EXP2
(indicatives)
P(p)
.05
.00
.14*
P(q/p)
.90*
-.93*
.93*
Summary of results
• P(q/p) was by far the strongest predictor.
• There was no negative effect of P(p) as there
would have to be if “if p then q” means “not-p
or q”.
• There was a smaller negative effect of
P(q/not-p). This could suggest a relation to the
delta-p rule, which takes P(q/p) - P(q/not-p) to
measure the degree of covariation between p
and q. Does this mean that a counterfactual
states a causal relation between p and q?
The results on disjunction
(not published so far)
• For all 81 participants in these
experiments:
mean P(not-p or q) > mean P(if p then q)
• The same was true in the analyses by
items. For all 64 items:
mean P(not-p or q) > mean P(if p then q)
Explicit probability judgments about
disjunction
• Have more recently studied participant’s
explicit subjective probability judgments
about disjunctions and conditionals.
• Order bias and the suppositional disjunction,
end of ESRC grant report, Shira Elqayam
Jonathan Evans, David Over, & Eyvind Ohm.
Experiment: Probabilistic evaluation task 1
Variation on the probability of conditionals task (Evans et
al., 2003; Oberauer & Wilhelm, 2003).
A pack contains cards which are either blue or yellow
and have either a triangle or a circle printed on them. In
total there are:
10 blue triangles
40 blue circles
40 yellow triangles
10 yellow circles
How likely is the following claim to be true of a card
drawn at random from the pack?
The card is either blue or has a triangle printed on it
1-----2-----3-----4-----5-----6-----7
Very unlikely
Very likely
Experiment: Probabilistic evaluation task 2
Variation on the probability of conditionals task (Evans et
al., 2003; Oberauer & Wilhelm, 2003).
A pack contains cards which are either blue or yellow
and have either a triangle or a circle printed on them. In
total there are:
10 blue triangles
40 blue circles
40 yellow triangles
10 yellow circles
How likely is the following claim to be true of a card
drawn at random from the pack?
If the card is not blue then it has a triangle printed on it
1-----2-----3-----4-----5-----6-----7
Very unlikely
Very likely
Experiment: Probabilistic evaluation task
• Do participants rate disjunctions as more
probable than conditionals?
• Mean rate (1-7 scale) across 16 items of
each linguistic form
Experiment: Probabilistic evaluation task
Comparisons of mean probability estimates,
disjunctions vs. conditionals (standard deviations
in parentheses).
Disjunctions
p or q
5.2
not-p or q
4.3
p or else q
4.8
not-p or else q
4.3
(0.6)
(1.0)
(0.8)
(0.9)
Conditionals
If not-p then q
if p then q
If and only if not-p then q
if and only if p then q
3.7
3.5
3.5
3.5
Ratings for disjunctions significantly higher than ratings for the
extensionally equivalent conditionals
(0.4)
(0.6)
(0.5)
(0.6)
Evidence that the natural
language conditional is not the
material conditional
• People do not judge the probability of
“if p then q”, P(if p then q), to be the
probability of the material conditional,
P(not-p or q).
• People judge P(p or q) to be higher
than P(if not-p then q).
• People judge P(not-p or q) to be higher
than P(if p then q).
Experiment: Probabilistic evaluation task
• Is this a fallacy?
• Compare to conjunction fallacy
• Does not behave like one
– Ps were given transparent frequency
distributions
– Typically makes conjunction fallacy
disappear
– In our results the pattern persisted
Belief versus assertion
• In people’s beliefs, P(p or q) is often
greater than P(if not-p then q).
• People will often infer “if not-p then q”
when “p or q” is asserted.
• How is this possible? People acquire
extra, pragmatic information from
assertions.
A pragmatic inference from an
assertion
• Suppose you ask me where Linda is, and I
reply, “She is in her office or the Library.”
You will think you are justified in inferring,
“If Linda is not in her office, she is in the
Library.” You will assume I have a nonconstructive justification for what I assert.
Why pragmatic?
• Suppose I know that Linda is in her
office and nothing about why she is
there. From this, I can infer that she is
in her office or the Library. If I assert
only this disjunction, however, I will
violate Grice’s Maxim of Quantity, and
you will be misled.
Return to belief
• When can we infer “if p then q” from
“not-p or q” in our beliefs?
• When can we infer “if not-p then q”
from “p or q” in our beliefs?
Justifications of “p or q”
• “p or q” could be justified from
“below”, constructively. Then we
cannot believe “if not-p then q”.
• “p or q” could be justified from
“above”, non-constructively. Then
we can believe “if not-p then q”.
Conclusion
• Inferring “if not-p then q” from
“p or q” is sometimes justified and
sometimes not. It is justified when
P(if not-p then q) is close to P(p or
q), and that is so when we have a
non-constructive justification of
“p or q”, inferred using a type 2
process.