Individuals versus aggregrates

“Winwood Reade is good upon the subject,” said Holmes. “He remarks that, while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician.”

The Sign of Four by Sir Arthur Conan Doyle (hat-tip MP)

Dysrationalia — if you disagree with me, off with your head

Some quotations from a critique of (an early version of) Stanovich’s theory by Sternberg (1994). Firstly, theory:

“Why do we need a theory? Because we’ve had too many fly-by-night constructs in the abilities business, and we don’t need more of them. We do need serious new constructs — and the way to present them is via a theory […] — and construct validation of that theory.”

What is an irrational belief?

“In the real world, few problems truly lend themselves to the kind of deductive (rational) reasoning we learn in logic classes. The vast majority of problems are inductive, so that arguments can be stronger or weaker, but not logically valid or invalid. I am afraid that Stanovich has fallen into a trap that of labeling people as “dysrational” who have beliefs that he does not accept. And therein lies frightening potential for misuse. And if you disagree with me, off with your head. Here, it’s a joke. Historically, it’s not.”

Reference

Robert J. Sternberg (1994). What If the Construct of Dysrationalia Were an Example of Itself? Educational Researcher, 23, pp. 22-23+27

Mao Tse-tung on the psychology of problem solving and the empirical method

“You can’t solve a problem? Well, get down and investigate the present facts and its past history! When you have investigated the problem thoroughly, you will know how to solve it. Conclusions invariably come after investigation, and not before. Only a blockhead cudgels his brains on his own, or together with a group, to “find a solution” or “evolve an idea” without making any investigation. It must be stressed that this cannot possibly lead to any effective solution or any good idea.”

Oppose Book Worship (May 1930)

🙂

The rational agent

“… the rational agent is not simply the one who follows the normative canons of logic and probability theory, and neither is she the one who follows adapted heuristics for action choice or ‘somatic markers’. Rather the rational agent is the critically self-aware agent; the one who is aware why she acts, and who modifies her own behaviour according to her self-knowledge. As Karl Popper (1990) wrote, ‘A rationalist is simply someone for whom it is more important to learn than to be proved right’…”

Lambie, J. A. (2008). On the irrationality of emotion and the rationality of awareness. Consciousness and Cognition, 17, 946-971

The Wisdom of Many in One Mind

Averaging many people’s estimates of, e.g., when a famous event occurred tend to be better than asking any one arbitrary person.   Herzog and Hertwig (2009) investigated whether the average of two estimates from one person tended to be better than their first estimate, using the years of 40 historical events, e.g., when electricity was invented.

There were three conditions:

  1. Repeated sampling: just giving an estimate twice.
  2. So-called “dialectical” sampling (they cite Hegel here), where participants were told: “First, assume that your first estimate is off the mark. Second, think about a few reasons why that could be. Which assumptions and considerations could have been wrong? Third, what do these new considerations imply? Was the first estimate rather too high or too low? Fourth, based on this new perspective, make a second, alternative estimate.”
  3. Pairing each participant’s guess with a random other participant.

Results are below:

dialectical

The instruction to consider you were wrong increases accuracy beyond that with simple repeated measurement.  Best of all is averaging with another person.

Reference

Herzog, S. M. & Hertwig, R. (2009). The Wisdom of Many in One Mind: Improving Individual Judgments With Dialectical Bootstrapping. Psychological Science, 20, 231-237

Prover9 and Mace4

Just found two fantastic programs and a GUI for exploring first-order classical models and also automated proof, Prover9 and Mace4.  There are many other theorem provers and model checkers out there.  This one is special as it comes as a self-contained and easy to use package for Windows and Macs.

There are many impressive examples built in which you can play with.  To start easy, I gave it a little syllogism:

all B are A
no B are C

with existential presupposition, which is expressed simply:

exists x a(x).
exists x b(x).
exists x c(x).
all x (b(x) -> a(x)).
all x (b(x) -> -c(x)).

and asked it to find a model. Out popped a model with two individuals, named 0 and 1:

a(0).
- a(1).

b(0).
- b(1).

- c(0).
c(1).

So individual 0 is an A, a B, but not a C. Individual 1 is not an A, nor a B, but is a C.

Then I requested a counterexample to the conclusion no C are A:

a(0).
a(1).

b(0).
- b(1).

- c(0).
c(1).

The premises are true in this model, but the conclusion is false.

Finally, does the conclusion some A are not C follow from the premises?

2 (exists x b(x)) [assumption].
4 (all x (b(x) -> a(x))) [assumption].
5 (all x (b(x) -> -c(x))) [assumption].
6 (exists x (a(x) & -c(x))) [goal].
7 -a(x) | c(x). [deny(6)].
9 -b(x) | a(x). [clausify(4)].
10 -b(x) | -c(x). [clausify(5)].
11 b(c2). [clausify(2)].
12 c(x) | -b(x). [resolve(7,a,9,b)].
13 -c(c2). [resolve(10,a,11,a)].
16 c(c2). [resolve(12,b,11,a)].
17 $F. [resolve(16,a,13,a)].

Indeed it does. Unfortunately the proofs aren’t very pretty as everything is rewritten in normal forms.  One thing I want to play with is how non-classical logics may be embedded in this system.

Competence vs. performance

It’s all Chomsky’s fault (Chomsky 1965, p. 4):

“We thus make a fundamental distinction between competence (the speaker-hearer’s knowledge of his language) and performance (the actual use of language in concrete situations). […] A record of natural speech will show numerous false starts, deviations from rules, changes of plan in mid-course, and so on. The problem for the linguist, as well as for the child learning the language, is to determine from the data of performance the underlying system of rules that have been mastered by the speaker-hearer and that he puts to use in actual performance.”

So the idea is that people are trying to do C but only manage to do P, because of various constraints. We (children, adults, theorists) see (imperfect) P, and want to infer C. We go to school and go through various rigmaroles to better approximate C. The same distinction is applied in reasoning. Various options: people are irrational (with respect to C); maybe C = P, if we look hard enough to see it. Or bright people have P = C. Or bright people want P = C.

What fascinates me in reasoning is the role played by small groups of experts who produce particular systems of reasoning—logical calculi, probabilistic machinery—along with proofs that they have properties which they argue are reasonable properties to have. Then others come along to use the systems. Hey, this looks like a good logic to know; maybe it’ll help make my arguments better if I use it. Maybe this probability calculus will make it easier to diagnose illness in my patients. And so forth. Then somebody else comes along and decides whether or not we’re consistent with a competence theory’s judgements, or whether we’re interpreting things a different way; whether another competence theory (application thereof) might be more appropriate for a given situation or a different psychological model of the situation.

Easy to get tied up in knots.

References

Chomsky, N. (1965). Aspects of the Theory of Syntax. MIT Press.

“Conscious” reasoning

Read this:

For us, however, a key difference is that only conscious reasoning can make use of working memory to hold intermediate conclusions, and accordingly reason in a recursive way (Johnson-Laird, 2006, p. 69): primitive recursion, by definition, calls for a memory of the results of intermediate computations (Hopcroft & Ulmann, 1979). [… example task omitted …] The non-recursive processes of intuition cannot make this inference, but when we deliberate about it consciously, we grasp its validity (Cherubini & Johnson-Laird, 2004). Conscious reasoning therefore has a greater computational power than unconscious reasoning, and so it can on occasion overrule our intuitions.

There’s no evidence that whatever bits of memory intuition uses cannot do recursion.  Hunting through semantic memory structures can be viewed as a recursive process and the process is not (at least always) accessible to consciousness.  Aside from this, you can impose recursion on just about any process you care to analyse, and you can often remove recursion from a process description depending on what primitives are available.  Questioning whether a process “is” or “isn’t” recursive isn’t a healthy activity.  Also the jump from “recursive” to “primitive recursive”, as if they were one and the same, is deeply confusing.  See the Stanford Encyclopaedia of Philosophy for details of other flavours of recursion.

Bucciarelli, M.; Khemlani, S. & Johnson-Laird, P. N. (2008). The psychology of moral reasoning. Judgment and Decision Making, 3, 121-139

Sexual arousal and decision making

Interesting study by Ariely and Loewenstein (2006) [The Heat of the Moment: The Effect of Sexual Arousal on Sexual Decision Making. J. Behav. Dec. Making, 19, 87–98]. From the abstract:

Despite the social importance of decisions taken in the “heat of the moment,” very little research has examined the effect of sexual arousal on judgment and decision making. Here we examine the effect of sexual arousal, induced by self-stimulation, on judgments and hypothetical decisions made by male college students. Students were assigned to be in either a state of sexual arousal or a neutral state and were asked to: (1) indicate how appealing they find a wide range of sexual stimuli and activities, (2) report their willingness to engage in morally questionable behavior in order to obtain sexual gratification, and (3) describe their willingness to engage in unsafe sex when sexually aroused. The results show that sexual arousal had a strong impact on all three areas of judgment and decision making, demonstrating the importance of situational forces on preferences, as well as subjects’ inability to predict these influences on their own behavior.

[Hat tip: Deborah Frisch on the Society for Judgment and Decision Making mailing list]

A good way to think about truth

“My intention was not to deal with the problem of truth, but with the problem of truth-teller or truth-telling as an activity. By this I mean that, for me, it was not a question of analyzing the internal or external criteria that would enable the Greeks and Romans, or anyone else, to recognize whether a statement or proposition is true or not. At issue for me was rather the attempt to consider truth-telling as a specific activity, or as a role.”

Discourse & Truth, Concluding remarks by Foucault. (Spotted here.)