non-constructive inference and conditionals david over psychology department durham university

Non-constructive inferenceand conditionals

David Over

Psychology DepartmentDurham 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 non-cheater (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)

• Associative• Holistic• Parallel• Automatic• Undemanding of cognitive capacity• Fast• Highly contextualized• “Old” in evolutionary terms

System 2 mental processes (Stanovich, 2004)

• 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.


• There are no content independent mechanisms for inference or learning.


• 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.


• Type 2 processes result from general purpose mechanisms, including a means of logical inference.


• 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 non-constructive 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 Johnson-Laird & 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(not-library & not-lab) = 0

P(library or lab) = 1

P(lab/not-library) = .5/.5 = 1

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)

EXP2

(indicatives)

True False

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)

EXP2

(indicatives)

True False

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


Comparisons of mean probability estimates, disjunctions vs. conditionals (standard deviations in parentheses).

Ratings for disjunctions significantly higher than ratings for the extensionally equivalent conditionals

Disjunctions Conditionals p or q 5.2 (0.6) If not-p then q 3.7 (0.4)

not-p or q 4.3 (1.0) if p then q 3.5 (0.6)

p or else q 4.8 (0.8) If and only if not-p then q 3.5 (0.5)

not-p or else q 4.3 (0.9) if and only if p then q 3.5 (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).


• 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 non-constructive 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.

non-constructive inference and conditionals david over psychology department durham university

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