A. What is Logic?

1. Logic is the study of the laws of thought - the general principles of reasoning. For example, from a sentence of the form: P or Q, and another of the form: not P, it follows that Q.
2. Logic is abstract in the sense that its principles do not depend on the content or topic of the sentences P, Q, etc. that make up the reasoning.
3. The ability to reason and draw conclusions according to the rules of logic can be computerized.
4. If one believes that human intelligence is explained by an ability to reason, and that reasoning can be explained by the content free principles of logic, then the idea that the brain contains a computer-like mechanism that follows the rules of logic will be an attractive one. This idea supports the classical approach to cognitive science.

 

1. However, some cognitive scientists believe that logic is a poor model of human reasoning. They think that logic may be a fine standard for good reasoning, but it is a bad picture of what human beings actually do when they reason. Here is evidence for this claim.
2. Logic is too slow. The computation time for solving the real world problems humans manage to solve on a daily basis is very long. A lot of research in AI has been directed to solving the problem of speed in the use of logical mechanisms, but simulation of practical real-time reasoning similar to what humans can do is still beyond us. This suggests that Mother Nature found logic too expensive in designing human intelligence. Human reasoning probably does not live up to the rules of logic. It is fast and scruffy; and a good thing too, since humans needed to respond to the environment quickly.
3. Logic does not tell us what conclusion to draw. The rules of logic only say whether a given conclusion follows from what one thinks is true. For a given set of beliefs there are infinitely many conclusions one can draw, most of them silly or pointless. This objection is related to the famous frame problem, the problem of determining what is relevant in a given situation. Logical mechanisms by themselves do not solve this basic problem.
4. There is evidence that people do not reason according to the rules of logic. Here is a famous experiment to illustrate the point.
5. In Wason 's selection task , the subject is given four cards, each one with a letter (A-Z) on one side, and a number (0-9) on the other. The subject is then asked to determine which of the following cards:

 A B 3 7
must be turned over to determine whether the following rule holds:
If there is a vowel on one side, then there must be a number less than 4 on the other.
The correct answer is that you must turn over the A and the 7, but people rarely select this answer, saying instead that you need to turn over only the A, or that you should turn over the A and the 4.
However, poor performance on this task is highly dependent on the content. (We tried it in class with a rule in a bar: If there is there is A (for alcohol) on one side then there must be a number over 20 on the other.) That suggests that people's reasoning ability is not content independent as the classicist suggests.
6. It has been suggested that where people do get the answer right, it is because they are able to understand the problem as a matter of detecting cheaters. Pinker suggests that the development of reasoning abilities for detecting cheaters might be easy to explain through natural selection, but that a more general ability to reason might not have developed.
7. Classicists have several responses to this evidence. Speed problems can be overcome in a massively parallel system like the brain. Poor reasoning of humans does not show that people fail to have logical machinery in their brains. There are just limitations on how they use it: bad memory, poor attention, etc.. When given the time to think, they often come to see their errors and correct them.
8. Another response given by some classicists goes like this. So what if humans don't use logic. They ought to. Cognitive science is the study of intelligence not stupidity.

1. Pinker points out that 5-month old infants already understand basic mathematical concepts such as 1+1=2 and 2-1=1.
2. Both children and animals have the ability to detect differences in number (up to about 5 or 6) without actually counting using number names. They can also appreciate concepts like more and less, and that the shortest route goes in a straight line.
3. Such basic mathematical abilities are common to all cultures, and are as close innate as anything comes.
4. On the other hand, the formal mathematics that we learn in school is far from something that is easily learned. Pinker complains that mathematical education in America is poor because it ignores the fundamental difficulties humans face in developing mathematical concepts on their own.

 

1. Although humans have some basic abilities in reasoning about probabilities, on the whole, people seem "blind" to many of the fundamentals.
2. One disturbing error that humans are prone to is to judge probability without properly considering the base rate . For example, given a disease which infects one thousandth of the population, and a test with a false positive rate of 5%, they badly overestimate the likelihood that a person with a positive test has the disease. (The correct answer is that only 2% of such people have the disease.) We worked through the reasoning in class. People are very poor at judging how important the low basis rate of the disease (the overall probability that people have it) is in determining your odds of having it.
3. Tversky and Kahneman are famous for showing incorrectness in human probability judgements. Most people get the following task wrong.

Linda went to college and was involved in leftist political organizations. Which is more likely?
(1) Linda is a bank teller.

(2) Linda is a feminist bank teller.


They answer that 2 is more likely, even though as a matter of logic, A and B must be less likely than A . Tversky and Kahneman suggest that the salience or memorability of an event, and reasoning according to stereotypes are two factors that help explain the bad performance.
 

4. Pinker points out that our poor ability in probability judgements is a reason that we make poor political choices. Atomic explosions are memorable, while coal dust explosions are old hat. So we worry about the dangers to the public of atomic power all out of proportion to the actual threat, and even though the overall risk for coal power exceeds that for atomic power.
5. Pinker also points out a mental trick that can help overcome our problems in judging probabilities. By describing the problems in term of frequencies rather than the probability of single events, human reasoning abilities are improved.

 

1. Pinker points out that much of language involves metaphor. For example, spatial metaphors are basic to how we think about ownership, change and many other more abstract ideas. (A gift goes from me to you. My troubles go from bad to worse. Activation values go from 0 to 1.)
2. Why do we talk and think in metaphors? Because the reasoning patterns we employ in one domain can be applied usefully in another domain. (After x goes from y to z, x is at z and no longer at y.)
3. Our abilities at so called abstract reasoning depend on this ability to apply metaphors to concrete and well understood cases.
4. Perhaps only a handful of basic conceptual schemes for (say) location, motion, agency, force, etc. might serve as a foundation for the majority of our thoughts.
5. The ability at metaphorical transfer from one domain to another could explain how an abstract language of thought developed. An original language designed to deal with (say) spatial motion, could have been modified to apply to a new area (say "movement" of ownership). The same machinery which originally was tied to spatial perception, could be detached from it and reused in a new way.
6. Pinker suggests that these metaphorical abilities are very basic, and found even in very young children, who create their own metaphors spontaneously.

 

1. There is a tendency to think of creativity and genius as special - a gift from God that has no scientific explanation. Geniuses appear to be radically different from ordinary people.
2. Pinker disagrees. Geniuses may be smart, but they are also hard workers, and that is their primary advantage. Generally, creative insight is gradually (and painfully) developed.