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Whereas some of our (propositional) knowledge is based directly on perception, say, or indirectly on what we have been told, much of the knowledge we have is justified by means of reasoning or (logical) argument, based on other knowledge we have.

Knowledge, according to our definition above, is justified true belief, and one way of justifying knowledge is to derive it by valid arguments from knowledge we already have. Logic is the branch of philosophy concerned with analysing the patterns of reasoning by which a conclusion is properly drawn from a set of premises.

If we are not willing to use logical reasoning, our beliefs or values might well be inconsistent, and then we would have no knowledge or proper judgement at all. If someone, for instance, claims to have certain general beliefs and to hold certain values, but their particular actions are logically inconsistent with these, then we would be right – on these grounds alone – not to accept the assertions they make or trust them, (unless they are willing to recognise the inconsistency and change accordingly.)

The need for correct reasoning and valid arguments is fundamental to all thinking, and the origins of logic date back as far as the beginning of philosophy, to the ancient Greek philosophers, like Socrates (?470 - 399,) Plato (427 - 347) and Aristotle (384 - 322,) and medieval Scholasticism.

After considering the nature of rationality and logical arguments, we shall look at two formal ways of deciding the validity of certain kinds of arguments, and at some of the ways in which arguments can fail to be valid; and then at other ways of reasoning than pure logic.

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1. Rationality

As a first approach, what I shall mean by rationality (from Lat. ratio, reason) is saying or doing things for good reasons. Our reasons for saying or doing things are "good," if we have taken into account the available facts and have thought through things carefully; or perhaps if we would be able to explain them to someone else, especially someone who at first might not agree with us. The opposite of being rational is being irrational.

(This definition, and other ones like it, are of course my personal ones, but they are trying to capture and make explicit what people generally mean by the terms; most of what follows could be the same if one chose a somewhat different definition.)

Here is an example from real life, of the relation between logic and rationality; it is also an example of why in real life we should try to be rational.

It is the case, at least in English-speaking Western countries, that

• If a student is called Ruth, then that student is a girl;

from this it follows that

• If a student is a boy (i.e. not a girl,) he will not be called Ruth;

but clearly it does not follow that

• If a student is a girl, she will be called Ruth.

This is easy enough. Now similarly, (and this comes from real life,) suppose a teacher holds the view – possibly mistakenly, but that is a separate issue – that

• If a student is lazy, they will take Maths Studies (rather than Standard Level,) – (i)

then he should also agree that

• If a student does not take Maths Studies, they are not lazy; – (ii)

but the teacher does not need to also believe that

• If a student takes Maths Studies, they are lazy. – (iii)

It would be a mistake to complain that the teacher said that (iii) when in fact he had expressed the view in (i): since (iii) does not logically follow from (i), one is acting irrationally by complaining and may be getting upset quite needlessly. (We shall look below at the formal argument underlying this example.)

As in this example, the problem with irrationality is, very simply, that it can make things go wrong, and often does so on a much larger scale than in this case – among the consequences of people's irrationality are prejudice, violence and wars.

1. descriptive propositions, or statements of fact,
2. normative propositions, or value judgements,
3. metaphysical propositions, expressing one's most basic world view.

Like morality, rationality is normative. One ought to be moral and one ought to be rational. One is wicked if not moral and foolish if not rational.

When we say that it is morally right to relieve famine victims, we are expressing our approval of famine relief, and we are at least suggesting that people ought to come to the aid of those in danger of starvation. Morality is thus both expressive and normative.

So is rationality. When we say that it is rational for individuals to have medical insurance, we are expressing approval of doing so and suggesting that people ought to make sure they are insured. Similarly, to characterize a choice as irrational (or immoral) is to condemn it, and not simply to describe it.

Hausman, McPherson, Economic Analysis and Moral Philosophy, 1996.

1. the validity of both is independent of what they are about, so they are just formal rules, empty of meaning;
2. they are both purely descriptive, without moral or other normative significance in themselves;
3. but still, we ought to apply them, if we want to make good moral (or other) judgements.

Now that we have gained some understanding, I hope, of what we mean by the terms, there are two main problems with rationality and logic. The first is that they have a bad reputation; but this bad reputation, it seems to me, is the result of misunderstanding or muddled thinking.

1. There are a lot of people who are suspicious of rationality and logic because they view them as opposed to emotions: a rational person is often assumed to be lacking in feelings, to be cold. This seems to me a false opposition.

   While it is true that the main reason that people hold irrational beliefs or act irrationally is that they get carried away by their emotions, or because their emotions are inappropriate to the occasion, it does not follow that a rational person does not have feelings. What makes someone a rational person is not that they don't have feelings, but that they will not be governed by them. Thus, it is not irrational to have a craving for something that the doctor has told me is bad for me, but – assuming that I want to remain alive and healthy – indulging in that thing to excess is so. And it is not irrational to be in love with someone, but such things as i. needing to 'own' them to be sure of their love in return, or ii. constantly and without good reason suspecting one's lover of being unfaithful, are.

   (It may be that all the people who are suspicious of rationality for this kind of reason have in common a certain view of themselves and others: they view themselves as just the victims of their emotions, and out of control of their own lives, rather than as persons making choices for themselves based on their desires and their circumstances.)

2. Another common objection to rationality may arise from its use in economics: economic theory, at least traditionally, has assumed – or has had to assume – that human beings are self-interested. and that they generally act rationally. Thus, if I have the choice between paying a higher and a lower price for the same good, my self-interest and rationality will lead me to pay the lower price.

   However, these are two separate assumptions, and it does not follow that being rational means being selfish, or that to be unselfish requires that one act irrationally: in fact, it is quite possible to act quite rationally without being self-serving – there is nothing irrational about voting for higher petrol taxes, even when one always tries to buy the cheapest patrol for one's own car.

   Exercise 1.5.:
   Recent economic models have increasingly taken into account the fact that people do not always (i) act rationally and (ii) out of self-interest, and that they may not be (iii) in full possession of the facts.

   For each of these assumptions, try to think of examples in which 'classical' economic theory does not apply because the assumption is not satisfied.

3. There are of course many cases where we feel that using arithmetic is not appropriate, such as when two friends have been out for a meal, and one carefully adds up for the other how much more he has spent. Similarly there are situations where we feel that rationality and logical argument are inappropriate: when a relative has died it is not much consolation to be told that they would have died some time anyhow.

   However, we must be careful not to dismiss rationality and logic in general because of these cases. When I owe someone money it is not right for me to complain when they add up how much it is I owe them. And when I am about to lose an argument, or to be shown that I have been wrong, it is not right for me to complain that the other person is being "too logical." (Similarly, some people, IB teachers no less than students, are liable to say, when they don't like a conclusion, that something is "just ThoK" ...)

   Some people we have encountered ... still persist in acts that may be tinged with illogicality. ... But they do all these things with full knowledge that they are being self-indulgent or irrational. When irrational acts are committed knowingly they become a sort of delicious luxury.

   It would be a dreary world if we all had to be rational, right-thinking, non-neurotic people all the time, even though we may hope we are making general gains in that direction.

   At times it is pleasanter or easier to be non-logical, But I prefer being non-logical by my own free will and impulse rather than find myself manipulated into such acts.

   Vance Packard, The Hidden Persuaders, 1957.

   On this kind of view it may be that our lives would be poorer if we thought about everything rationally; but on the other hand, as argued above, much evil comes from people not thinking and acting rationally. So I would like to put forward the following rough moral rule: there may be many situations in which there is no need to think logically or to be all rational, or when it is even inappropriate; but whenever our opinions or actions might affect others in a negative way, we must allow the logic of our reasoning to be challenged, and the rationality of our actions to be critically examined: we ought to be rational.
   Exercise 1.6.:
   When someone does not want to, or cannot, give a reason for what they have done, they often say (or shout) that they "just felt like it." This lack of rationality is acceptable when it is an artist who is asked about a painting or a composition, but not when it is a murderer who is asked why he has killed.

   Try to think of other cases where the requirement to think rationally (i) is justified, and (ii) may not be justified.

And then there is anchoring: people are often influenced by outside suggestion, ... even when they know that the suggestion is not being made by someone who is better informed. In one experiment, volunteers were asked a series of questions whose answers were given in percentages – such as what percentage of African countries is in the UN? A wheel with numbers from 1 to 100 was spun in front of them; they were then asked to say whether their answer was higher or lower than the number on the wheel, and then to give their answer. These answers were strongly influenced by the randomly selected, irrelevant number on the wheel. The average guess when the wheel showed 10 was 25%; when it showed 65 it was 45%.

Experiments show that most people apparently also suffer from status quo bias: they are willing to take bigger gambles to maintain the status quo than they would be to acquire it in the first place. In one common experiment, mugs are allocated randomly to some people in a group. Those who have them are asked to name a price to sell their mug; those without one are asked to name a price at which they will buy. Usually, the average sales price is considerably higher than the average offer price. [Closer to home, a student who has for days been undecided which of two Service activities to join will often be very upset when they are told that there is no space in the one that they did finally choose, perhaps even after tossing a coin, and that they have to join the other one.] ...

Most people, say psychologists, are also vulnerable to hindsight bias: once something happens, they overestimate the extent to which they could have predicted it. Closely related to this is memory bias: when something happens people often persuade themselves that they actually predicted it.

Finally, who can deny that people often become emotional, cutting off their noses to spite their faces. One of the psychologists' favourite experiments is the "ultimatum game" in which one player, the proposer, is given a sum of money, say $ 10, and offers some portion of it to the other player, the responder. The responder can either accept the offer, in which case he gets the sum offered and the proposer gets the rest, or reject the offer in which case both players get nothing. ... [See the Exercise below.]

And the article quoted above goes on to say that many of the cases of irrationality described can be understood if we assume that instead of always trying to maximise their gain, people often act from "loss-aversion."

Several real-world examples of how this theory can explain human decisions are reported in a forthcoming paper, ... Many New York taxi drivers ... decide when to finish work each day by setting themselves a daily income target, and on reaching it they stop. This means that they typically work fewer hours on a busy day than on a slow day. Rational labour-market theory predicts that they will do the opposite, working longer on the busy day when their effective hourly wage-rate is higher, and less on the slow day when their wage-rate is lower. Prospect theory can explain this irrational behaviour: ... [See the Exercise below.]

There may even be a reason that we are not consistently rational, namely that it would just be too emotionally demanding if we were.

A peculiar feature of beliefs about politics and religion, he [Bryan Caplan] says, is that the costs to an individual of error are "virtually non-existent, setting the private cost of irrationality at zero; it is therefore in these areas that irrational views are most apparent." ...

... Observing that "those who are most likely to make unbiased cognitive assessments are the clinically depressed," he [Jon Elster] argues that the "emotional price to pay for cognitive rationality may be too high."

The Economist, December 18th 1999.

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2. Logical Arguments

According to Bertrand Russell (British philosopher, pacifist and mathematician, 1872 -1970,) a proposition is what different sentences, even in different languages, with the same meaning have in common. We 'express' propositions in particular sentences.

Propositions are such that they are capable of being true or false. When we talk about 'knowledge that', the thing that is known, that is believed to be true, is a proposition.

(This is the main meaning of the word "proposition", which is quite different from the meanings of some of its cognates, such as "to propose" and "proposal", which mean something like "to put forward" or "suggest".)

Note that this use of the word "argument" must be distinguished from other meanings it can have, e.g. when we say: "A heated argument followed," or: "They had good arguments on their side."

Note that whereas propositions are either true or false, arguments are valid or not valid: they cannot be true or false.

It might be thought that this distinction between truth and validity is theoretical, or 'purely academic'; however, in every-day reasoning too we need to be concerned not only with the truth of a conclusion, but with how it has been arrived at: if we arrive at a true conclusion but for the wrong reasons, then those same wrong reasons may in future lead us to false conclusions.

But there are ethical reasons why human cloning should not at present go ahead even in single cases. ...

This Catch-22 may well stymie human cloning forever. So why bother to dissent from the howl of protest that attended the advent of Dolly [a lamb, the first mammal to be have been successfully cloned]? Because it is an error to reach the right decision [that human cloning would be morally wrong or should be controlled] for the wrong reason. The vague feeling that cloning is an unnecessary offence against the natural scheme may very well solidify into a backlash against many of the other efforts of biologists. That would be a pity. ...

The fact that new technologies feel scary or strange [which is presumably why there were "howls of protest" concerning the cloning of Dolly] should not be enough to rule them out. The careful application of biotechnology to plants and animals is already bringing benefits: ...

The Economist, March 1st 1997.

There are different branches of logic which deal with different forms of logical arguments.

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3. Syllogisms

Syllogistic logic distinguishes four types of quantified propositions. Each type has a traditional representation by a lower-case letter, while capital letters are here used to represent concepts or classes:

i.	positive (– "a" and "i", from Lat. affirmo, I assert):
	All mammals are animals.	= M a A
	Some mammals are human beings.	= M i H
ii.	negative (– "e" and "o" from Lat. nego, I deny):
	No stones are animals.	= S e A
	Some mammals are not human beings.	= M o H

The following must be the most famous example of a syllogistic argument:

All human beings are mortal.   Greeks are human beings.	All A are B.   All C are A.	= A a B = C a A
Hence Greeks are mortal.	All C are B.	= C a B (barbara)

No horse is human. Some mammals are horses.	No A is B. Some C are A.	= A e B = C i A
So some mammals are not human.	Some C are not B.	= C o B (ferio)

In the examples so far, the types of quantified statements – denoted by "a", "i", "e" and "o" – have differed, but the patterns of the concepts A, B and C have been the same in all of them.

There are 256 such arrangements of three propositions, but only 24 of them are valid arguments or 'modi': there are six of each of four different patterns of the concepts A, B and C, or 'figures'. – Here are two more syllogisms, of a different figure from the examples above, one valid and the other not:

Some students like art. All students do maths.	Some A are B. All A are C.	= A i B = A a C
Some who do maths like art.	Some C are B.	= C i B (disamis)
Some students like art. Some students do physics.	Some A are B. Some A are C.	= A i B = A i C
Some who do physics like art.	Some C are B.	= C i B

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4. Propositional Logic: Truth Tables

A syllogism, as we have seen, is an argument of a standard format, with the two premises and the conclusion being all of the same form, such as: "some A are B," for instance.

In propositional logic, on the other hand, the basic constituents are simple propositions which are combined into more complex propositions by means of logical words, like "not" and "and". For example, in the proposition

"If a student is from Hong Kong and works too little, then (s)he is from Hong Kong."

p = "a student is from Hong Kong," and
q = "(s)he works too little,"

"If p and q, then p."

Thus the logical form of a complex proposition consists of

1. propositional variables:
   p, q, ..., and
2. logical connectives:
   ¬ ("not": negation),   & ("and": conjunction),   v ("or": disjunction),   => ("if ... then": implication).
   (Note that in logic, as in mathematics, "or" always has an inclusive meaning: so "p v q" means "p or q, or both.")

p & q => p .

"Even if a student is from Hong Kong but works too little, (s)he is still from Hong Kong."

The idea of variables and connectives should be familiar from mathematics: we can speak of arithmetical forms consisting of

1. numerical variables:
   a, b, ..., and
2. arithmetical connectives:
   = (equality), + (addition), – (subtraction), ...

a + b = b + a ,

p & q => p

Fortunately there is a simple procedure for deciding whether a particular logical form expresses a valid argument or not: we can set up a 'truth table' for each logical connective, based on what we mean by it, and from the truth tables for the connectives we can construct the truth tables of forms of argument.

Truth tables for the connectives:

p	¬ p
T F	F T

p	q	p & q	p v q	p => q
T T F F	T F T F	T F F F	T T T F	T F T T

For each combination of truth-values, T(rue) or F(alse), of each of the simple propositional variables, p and q, the truth-value of the complex proposition is given.

For instance, "p & q" is only T when both p and q are T, otherwise it is F – that in fact is the 'meaning' of the connective "and".

Examples:

p	¬ p	¬ ¬ p
T F	F T	T F

If two logical expressions, such as "p" and "¬ ¬ p" in this example, always have the same truth value, they can be said to have the same meaning: whenever one is true then so is other.


p	q	p & q	p & q => p
T T F F	T F T F	T F F F	T T T T

A logical formula which is always true, such as "p & q => p" in this example, is called a tautology. So an argument is valid if its truth table shows that it is a tautology.

A logical formula that is always false is called a contradiction.

Similar truth tables can be completed to show that the following formulae are tautologies and therefore are the forms of valid arguments:

• modus ponens:
  ((p => q) & p ) => q ,
• modus tollens:
  ((p => q) & ¬ q) => ¬ p .

5. Fallacies

The word "fallacy" actually comes from Lat. fallere, to deceive.

What follows is a small 'menagerie', or zoo, of fallacies, but there are many more kinds; (most of the classification and many of the examples are from Madsen Pirie, The Book of the Fallacy, 1985.) Many types of fallacies were already described by ancient and medieval thinkers, which is why still often have Latin names.

• Logical or Formal Fallacies:
  1. denying the antecedent:
     (( p => q ) & ¬ p ) => ¬ q .
     
     • "If he is slow, he will lose. Since he isn't slow, he won't lose."
  2. affirming the consequent:
     (( p => q ) & q ) => p .
     
     • The prosecution in a murder case: "If he intended poison, he would have bought some. He did in fact buy some weed killer ..."
     • "If someone is a communist, then he is an atheist. But Jones is an atheist. Therefore he must be a communist."
  3. conversion of a universal positive proposition, as in:
     "All A are B. Hence all B are A."
     • "Everyone in that company knows some Japanese, so you better learn some quickly to make sure they hire you."
  4. undistributed middle:
     when the 'middle term' in a syllogistic argument, i.e. the one which appears in both premises but not in the conclusion, is 'undistributed,' so that there is no class to all the members of which it applies in both premises, as in:

     "All A are B; some C are B.	or:	"All A are B; all C are B.	or:	"All A are B; some C are not A.
     So some C are A."		So all A are C."		So some C are not B."

     • "The worst oppressors of the working class are landlords. Jones is a landlord. So Jones is one of the worst oppressors of the working class." (– try: "... wear shoes" instead.)
  5. illicit process:
     when the conclusion of a syllogistic argument is about a whole class, whereas the premise in which it appears is only about some of the class, as in:

     "All A are B; no C is A.	or:	"All A are B; all A are C.
     So no C is B."		So all B are C."

     • "All cyclists are economical people, and no farmers are cyclists, so no farmers are economical people."
• Informal Linguistic Fallacies:
  6. equivocation:
     using words or phrases ambiguously.
     
     • "Happiness is the end of life. The end of life is death. So happiness is death."
• Informal Fallacies of Relevance:
  7. special pleading:
     using double standards, that some cases are to be judged differently from others of the same kind.
     
     • A newspaper man defending himself: "While it is not normally right to invade on someone's privacy, it is alright for us, as journalists, to do so, because we serve a public need."
  8. ad hominem (= "to the person"):
     attacking or appealing to the arguer instead of attacking or supporting an argument.
     
     • An MP in parliament: "I would remind the House that when my questioner was in office, unemployment doubled; and he has the temerity to ask me about the future of the mining industry."
       
     • From a political speech: "You, as members of the working class, will agree that ..."
  9. post hoc ergo propter hoc (= "after this, therefore because of it"):
     • "Ever since you arrived the weather has been great. You must have brought it with you."
     • "In Sweden they have been trying, since 1955, to understand and cure criminals; and look what's happened: suicides, moral degeneracy and drunks everywhere. Do we want that here?"
  10. circular reasoning, begging the question, or petitio principii:
      assuming the conclusion in the premise already.
      • "Justice requires higher wages, because it is right that people should earn more."
      • "We know about God from the Bible; and we know we can trust the Bible because it is the inspired word of God."
  11. category mistakes,
      in which the conclusion is of a different propositional type from the premise(s).
      • "Everyone of us has a mother. That is why everyone should honour their mother."
      • "You are students, therefore you should do as I say."

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6. Necessary and Sufficient Conditions

Some of the premises of arguments are simple statements of fact, such as: "John is a Catholic." But other premises express more complicated conditional relationships between concepts, such as: "If someone is a Catholic, they will object to abortions," or: "All Catholics object to abortions." (Note that from these two premises we can deduce that: "John objects to abortions.")

For the purpose of pure logic it is of no interest what a conditional proposition is based on – there need not even be any connection between the antecedent and the consequent. For instance, it is clear from the truth table for logical implication that the following is true: "if 1 + 1 = 2, then the law of gravitation holds."

However, the interesting conditional propositions are of course ones that express 'real' relations between events or states. In so far as these relations are causal relations between events, they do of course depend on the present state of our knowledge of science. Thus there was a time when people believed that: "if clouds collide, then there will be thunder and lightening" – something we don't hold true any more. But many conditional statements do not express causal relations, such as: "if you say this, then you are a liar."

One way of talking about these conditional propositions is in terms of necessary and sufficient conditions. If A and B are two states or events, we say that:

1. A is a necessary condition for B, if B cannot happen without A preceding, and that
2. A is a sufficient condition for B, if A cannot happen without B following.

1. a necessary condition for B, or
2. a sufficient condition, or
3. both, or
4. neither.

"A bachelor is an unmarried man" is an analytic truth,

"John is a bachelor," no matter if true or false, is synthetic.

"If it is brillig or mimsy, and it is not mimsy, then it is brillig."

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7. Alternative Ways of Reasoning

1. Deductive logic, which we use to derive (or 'deduce') conclusions from premises, though often useful for gaining new knowledge, is in a certain sense empty: the truth of the conclusion must always already be contained in the truth of the premises. If all human beings die, and Socrates is a human being, then Socrates will die.

   'Induction', on the other hand, is a way of reasoning in which the conclusion goes beyond what can be logically deduced from the premises. If someone questioned our belief that the sun will rise tomorrow, for instance, we would presumably point to the fact that it has risen on all previous days. However, it does not logically follow from this evidence that it will rise again tomorrow. (We might try to justify our belief by deducing it from scientific laws – but then the same problem recurs, because it is only by induction that we can justify our belief that the laws of science will continue to be valid, at least until tomorrow morning.)

   Inductive reasoning is characteristic of science, because there we make general statements – such as: "All human beings die." – on the basis of what can always be only a finite number of particular observations, such as: "This person has died, and that person has died, ..." Induction requires 'inventing', in a reasonable way, a conclusion on the basis of necessarily limited evidence; and this is one of the ways in which science, contrary to what people often think, needs creativity for its progress.

   To put it briefly, deduction argues from the general to the particular, whereas induction argues from the particular to the general. Whereas valid deductive reasoning from true premises guarantees the truth of the conclusion, induction can go wrong.

   Exercise 7.1.:
   1. Induction is notoriously difficult to justify. Which fallacy is committed in the following attempt: "Experience shows that induction is a valid form of argument in many situations."
   2. Try to think of cases in which the results of inductive reasoning were generally accepted but later turned out to be false.
2. Edward de Bono (British writer, 1933- ) has argued that top-down logical thinking is not a good method for problem solving; over the years he has put forward his alternative approach in a series of popular books and taught it in lucrative management-training courses.

   De Bono introduced the distinction between deductive 'vertical' thinking and creative 'lateral' (sideways) thinking.

   Warning! Lateral Thinking will change the way you think.
   

   There is nothing more exciting ......than thinking of a new idea.
   There is nothing more rewarding ......than seeing a new idea work.
   There is nothing more useful .....than a new idea that helps you meet a goal.

   Lateral Thinking is:

   • seeking to solve problems by apparently illogical means,
   • a process and willingness to look at things in a different way,
   • a relatively new type of thinking that complements analytical and critical thinking,
   • not part of our mainstream education – yet,
   • a fast, effective tool used to help individuals, companies and teams solve tough problems and create new ideas, new products, new processes and new services,
   • a term that is used interchangeably with creativity. ...

   Our minds are trained to find typical and predictable solutions to problems. Sometimes we do not look beyond the obvious alternatives. Sometimes we do not look for alternatives at all. ...

   from http://www.edwdebono.com/debono/worklt.htm, 1996.
   [I have omitted the fancy formatting of this publicity page.]

   Differences between lateral and vertical thinking:

   • Vertical thinking is selective, lateral thinking is generative.
   • Vertical thinking is analytical, lateral thinking is provocative.
   • Vertical thinking is sequential, lateral thinking can make jumps.
   • With vertical thinking one has to be correct at every step, with lateral thinking one does not have to be.
   • With vertical thinking categories, classifications and labels are fixed, with lateral thinking they are not.
   • Vertical thinking follows the most likely paths, lateral thinking explores the least likely.
   • Vertical thinking is a finite process, lateral thinking is a probabilistic one.

   from http://www.admin.upm.edu.my/~mzbd/lateral.html, 1996.

   Edward de Bono has written extensively about the process of lateral thinking – the generation of novel solutions to problems. The point of lateral thinking is that many problems require a different perspective to solve successfully. De Bono identifies four critical factors associated with lateral thinking:

   1. recognize dominant ideas that polarize the perception of a problem;
   2. searching for differ ways of looking at things;
   3. relaxation of rigid control of thinking; and
   4. use of chance to encourage other ideas.

   This last factor has to do with the fact that lateral thinking involves low-probability ideas which are unlikely to occur in the normal course of events. DeBono's work is highly relevant to the concept of creativity.

   from http://www.oltc.edu.au/cp/04i.html, 1996.

   Exercise 7.2.:
   The following problems – which may have been intended as examples, or for practice – are typical of the kind that lateral thinking tries to deal with: it is the clear outcome of some situation that is given, and we need to work backwards, to find how it came about.

   Try to answer the following questions. The 'right' answer is easily recognised because once the whole story has been found, its logical sequence feels quite obvious.

   1. A man living on the 14th floor of a building takes the lift to the ground floor every morning when he goes to work, but when he returns home in the evening he can only take the lift to the 12th floor and then walks up the remaining two flights – except on rainy days. Why?
   2. A beautiful (and clever) princess has fallen into the hands of an evil ruler, but for the sake of his reputation he gives her a chance in public to gain her freedom: he picks up two stones, one black and one white he claims, from the gravel path on which they stand and puts them in a bag. If she picks the white stone she will be free. She is quite sure that he has put two black stones in the bag, but she doesn't dare to say so. What should she do?
   3. A man is lying dead in the middle of a field. There is nothing in the field apart from an unopened package. He caused his own death. What happened?

   The following situations are more complex and require one person to know the full story, who will then answer the other people's yes/no-questions until they have figured it out. What is interesting are the difficulties we often have to even think of questions to ask.

   4. A man cannot sleep at night, so he turns off the light, but turns on the radio. Half an hour later he gets up and shoots himself. Why?
   5. A man is driving in a car listening to the radio. He stops at the side of the road and shoots himself. Why?
   Implicit in the concept of lateral thinking is a criticism of rule-bound, top-down logical reasoning: while formal logic enables us to check the validity of our arguments, it is not often a good way to find solutions to practical problems.
3. Formal logic is also very abstract, requiring that one disregard the particular aspects of a situation; this abstraction is artificial and not always appropriate when reasoning in a real-life setting. Consider the following excerpt from a conversation with a member of the Liberian Kpelle tribe, "an articulate group, enjoying argument and debate."

   Experimenter:	Flumo and Yakpalo always drink cane juice [rum] together. Flumo is drinking cane juice. Is Yakpalo drinking cane juice?
   Subject:	Flumo and Yakpalo drink cane juice together, but the time Flumo was drinking the first one Yakpalo was not there on that day.
   Experimenter:	But I told you that Flumo and Yakpalo always drink cane juice together. One day Flumo was drinking cane juice. Was Yakpalo drinking cane juice?
   Subject:	The day Flumo was drinking the cane juice Yakpalo was not there that day.
   Experimenter:	What is the reason?
   Subject:	The reason is that Yakpalo went to his farm on that day and Flumo remained in town that day.

   The example is not atypical; Cole's subjects often say things like "Yakpalo isn't here at the moment; why don't you go and ask him about the matter?" The psychologist Ulric Neisser, who excerpted this dialogue, notes that these answers are by no means stupid. They are just not answers to the experimenter's question.

   A ground rule when you solve a problem at school is to base your reasoning on the premises mentioned in a question, ignoring everything else you know. This attitude is important in modern schooling. ...

   But outside of school of course, it never makes sense to ignore what you know. A Kpelle could be forgiven for asking, Look, do you want to know whether Yakpalo is drinking cane juice, or don't you?

   Stephen Pinker, How the Mind Works, 1997.

   Logical thinking does not come to us naturally, but we seem to be able to deal with certain situations more easily than with others.
   Exercise 7.2.:
   1. Four cards are lying on the table, each has a letter on one side and a number on the other:

      D

      F

      3

      7

      Which cards do you have to turn over to test the following rule: "If a card has a D on one side, it has a 3 on the other side?"
   2. As a bouncer in a bar, to test the rule "If a person is drinking beer, they must be 18 or over," what do you check:
      1. the age of someone drinking beer,
      2. the age of someone drinking coke,
      3. what a 25-year old is drinking, or
      4. what a 16-year old is drinking?
   3. To test the rule "If a person eats hot chili peppers, then they drink cold beer," who do you have to check:
      1. someone eating hot chili peppers,
      2. someone eating spaghetti,
      3. someone drinking cold beer, or
      4. someone drinking wine?

   These three logical problems have the same structure. Which of them is easiest to figure out? Try to think of a reason why. (Also from Stephen Pinker, How the Mind Works, 1997.)

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'Green' Readings :

THINKING
[before 12c; from Old E. thencan, to think.]

Using your intelligence to the full. Some psychotherapies, like co-counselling, emphasise the importance of clear thinking in order to make the distinction between behaviour which is the result of intelligent thought and behaviour which is the result of distress. Clear thinking is the only reliable guide to appropriate behaviour, and it is important to trust it: 'Other people's thinking can be good information for you, but it can't replace your thinking. Your thinking is good. Depend on your thinking. This is your only guide to what you would like to do, what your best judgement is; not anybody's shoulds or society's rules or anything of the sort. You'll make some mistakes, of course, if you trust your own thinking; but if you make the mistake while trusting your own thinking, you'll be alert to the fact that the results aren't working and you'll correct it quickly. ... Trust your own thinking, and it will work out fine' (Harvey Jackins, 1983.)

REASON
[13c; from Lat. ratio, reckoning]

The innate human capacity for intelligence, leading to appropriate action. Patriarchy has gone to even greater lengths than usual to subvert the idea of reason, redefining it to mean 'a cold and unemotional approach to decision-making,' an approach that women (being 'emotional') are deemed incapable of undertaking. The righteous anger of women at being excluded is then conveniently interpreted as justification for their further exclusion (on the grounds of 'emotional insecurity.') Many people, including a number of feminist women, have turned against reason and logic with a will, proclaiming that what is needed is more sensitivity and understanding. This is to ignore the fact that what has been labelled 'logical' and 'reasonable' by those who maintain power over other people is nothing of the sort. Crime, killing, aggressive defence policies, economic competition and pollution are the antithesis of rational behaviour; these antisocial and exploitative behaviours are the result of thoughtless and uncontrolled emotion, justified in the name of 'reason.' In a world where emotion has led us to the brink of disaster, it is time to reinstate reason as the creative spark for radical change. 'Though we would never be so foolhardy as to assume that reason alone is sufficient to build a caring, civilized society, the politics of ecology is none the less profoundly rational' (Jonathan Porritt, 1984.) 'The essential point is that holistic, or ecological, thinking is not a retreat from reason; it is an enlargement of it to more comprehensive and hence more efficient means of analysis' (Carlene Spretnak, 1986.)

PATRIARCHY
[1561; from Gk. pater + archein, father-rule.]

(Also: 'male supremacy.') The institutionalised and internalised system by which men maintain their privileges at the expense of women; the social structure which fosters sexist oppression ... The apparent universality of patriarchy provides a convenient 'proof' for (mostly male) commentators that it is therefore the natural state of things, but green-thinkers believe strongly that 'biology is not destiny' ...

COUNTER-INTUITIVE BEHAVIOUR
[1968]

Behaviour which is the result of acting blindly to cure a symptom, without bothering to look for and deal with the underlying problem: thus we see traffic jams and think: 'More roads!' – or see increasing cancer rates and think: 'More hospitals!'

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Lecture :

Rationality and Logic

nothing from Britannica or Encarta,
but have talked with friends, read around.
Hence, first, my own cautious analysis of what we mean by the term,
not normative but descriptive, (as is much of philosophy):
no attempt to say what rationality should mean,
but trying to state precisely what we do mean by it;
you may not agree with my examples,
but try to follow the broad argument.
Then: how is rationality related to logic?

("normative" = telling us how we ought to be behave,
e.g. moral judgments are not statistical statements):
I ought not to act foolishly, just as I ought not act wickedly,
so "irrational" is a term of criticism.

someone can act irrationally,
if they allow their actions to be determined by feelings alone;
someone can be said to be 'too rational' (or 'too logical'),
if they have no (strong) emotions, apparently.
Explanations in economics assume everyone to be rational,
in the sense of maximising their own profit,
but economics doesn't deal well with altruism, environmental concerns:
e.g. is it irrational to choose 5% interest rather than 6%
1. other things being equal,
2. if the higher-interest account takes 10 hours to set up,
3. if the lower-interest account involves no exploitation?
Applied to
beliefs,
and arguments, which express/ defend beliefs;
behaviour,
and policies/ plans, which prepare behaviour;
persons, e.g.
mother, woken up by her son at 8 o'clock, as she had requested:
"What do you wake me up for, at this time?"
son, trying to explain: "But you asked me to ..."
mother, scolding: "You could have done it more gently!"
but: not to feelings, desires (– it seems to me):
the desire to eat one's 3 lb of chocolate now is not irrational,
but actually doing so is:
one will be sick, not have it later, etc.
Requires judgment:
an apparently irrational action may turn out to be rational when we investigate further and understand it better:
e.g. taking an umbrella on a sunny day: maybe for self-defence, return it to a friend, etc.

reasonable, reasoning, reasons, giving reasons;
acting rationally means being able to give reasons, to explain oneself to others,
hence part of society, relationships.
Everyone is mostly rational
– just as everyone mostly tells the truth, (otherwise language could not work):
to get through life in practical terms requires reasoning and logic;
according to the ancient Greeks man is "the rational animal".
A typical 'explanation' of irrational behaviour: "I just felt like it"
– usually said with resentment, irritation;
or 'rationalise' one's behaviour:
covering up one's irrationality with specious reasons
– often people believe them themselves:
change to Maths Studies "to have more time for other subjects",
parents physically abusing a child "to teach him a lesson",
[other examples from school ...]
Extreme cases of irrationality:
insanity, mental illness, e.g.
hand-washing compulsion, "to get rid of germs;"
anorexia, "to lose weight to be more attractive."

(– this has some personal significance for me ...):
rationality can mean acting selfishly, for one's own benefit only,
cf. in economics, above:
however, one can both choose the cheapest petrol and vote for higher petrol taxes;
rationality v. emotions is a false contradiction,
one can think rationally about one's emotions, and then act,
instead of allowing one's emotions to control one's actions,
which often means disregarding others, the consequences, etc.;
one can rationally decide not to be rational about some relationship;
ultimately perhaps a defence of irrational people:
after doing s.th. bad: "you don't understand, you are too logical".

a student who has been good at maths in the past receives a low grade on a test in the first term.
Possible irrational responses
(we are remarkably patient with people reacting irrationally ...):
beliefs:
"Not my fault: the test was too difficult."
"(I am good at maths but) the teacher hates me."
"I am no good at maths."
ordering of preferences:
want to always do homework regularly, but then always be distracted;
behaviour:
being short-tempered with others, hitting the wall;
study less maths, rather than more, miss lessons;
go on sick list for the next test.
Possible rational responses:
beliefs:
"I am less good than I thought."
"This is harder than I had realised."
ordering of preferences:
"Not working too much is more important than doing better."
or vice versa;
behaviour:
opt out and change to Maths Studies,
doing homework more regularly.

beliefs: must be based on evidence,
and be consistent;
consistent ordering of preferences
(– this is an important assumption in economics;)
appropriateness of means to ends.

not to decide what is true and what is false,
but check validity of reasoning
(– except in two cases, I do not believe the arguments in the examples in the rest of the lecture are valid or rational):
deduce beliefs from evidence:
• A common American view: "That so many blacks face the death penalty shows that blacks are more likely to commit violent crimes."
• A typical conservative view: "Foreigners don't have the right to live here, because British people have the right to live here, and foreigners are not British."
• "S.A. is the country with the highest crime rate, and Jo'burg is the city in S.A. with the highest crime rate, so Jo'burg is the city with the highest crime rate in the world."
check consistency of beliefs:
• KA: "When we change time, there will be one extra hour's sleep that night, and from then on the sun will set an hour later each day."
  (– here, as often, irrationality may have arisen as a result of wishful thinking ...)
  the more important/dangerous examples are less easy to see;
choose means appropriate to ends:
• Israeli policy: "In the long run, we promote our safety by building more settlements in occupied territory."
• According to Mr. Mandela and others, in order to develop, "African countries cannot afford to lower their trade barriers. They must wait until their companies, infrastructure and people are 'ready' " (The Economist, April 4th 1998.)
• A drug squad detective: "We must keep marijuana banned, because practically all users of harder drugs started by smoking it" (– they also all drank milk ...)
– like that of maths in physics and economics,
not interesting in itself but necessary:
the criteria for rationality call upon logic
[or: logic is the formalisation of criteria for rationality?]

the formal means for testing the validity of arguments:
a logical argument:
deduces (arrives at) some proposition, the 'conclusion', from other propositions the 'premises';
(cf. above examples,)
a valid argument:
must guarantee that if the premises are true then the conclusion will be true,
• ( (p => q) & ¬ q) => ¬ p)   (= modus tollens) :
  "If he liked me he would have helped me, and he has not helped me, so he doesn't like me."
a fallacy:
an apparently valid argument that is not,
• ( (p => q) & ¬ p) => ¬ q)   (= 'denying the antecedent') :
  "If he likes me he will help me, and he does not like me, so he won't help me."
(cf. also above examples.)
Ancient Greeks: syllogisms,
• "All students at the lecture have a hand-out;
  not all students at the College have a hand-out.
  So not all students at the College are at the lecture."
  valid;
• "All students here are academically able people;
  some academically able people are very selfish.
  So some students are very selfish."
  not valid: even though the conclusion is true it does not follow –
  try "are dwarfs" instead of "are very selfish;"
• the "foreigner" -example above is not a valid syllogism either.
Modern philosophy: symbolic logic,
(cf. modus tollens and denying the antecedent, above,)
truth tables even in Maths Studies;
now much developed, many branches.
But this is not the place ...

needs more careful investigation:
e.g. knowing that I may often not feel like going to a concert, but that when I do go, I always enjoy it, I may buy tickets for a whole series, to 'make myself' go regularly;
on the economists' doctrine, 'sunk costs' should be ignored:
since I have paid already, when I don't feel like going on an evening, isn't it irrational to go?
how then is committing oneself, and following through, rational?

despite some people's suspicion/ dislike of it,
if we were not generally rational we would be in (more of) a mess,
and logic is the method to test for rationality.

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Teaching Notes :

One could start by asking for examples of reasoning in everyday life.

1. Rationality :

   Exercise 1.2.:

   My personal answers look like this:
   actions – yes	beliefs, views – yes
   conclusions – no; right or wrong	desires – yes
   events – no	feelings – no: we just have them
   hopes, fears – yes	persons – yes
   places – no	periods in history – possibly
   policies – yes	relationships – no
   stones, trees – no	subjects (academic) – possibly

   Exercise 1.3.:
   10. consistency: if someone believes that p, they cannot also believe that not p;
   11. conjunctive disposition: if someone believes that p and they believe that q, then they must also believe that p and q;
       
   12. minimal inferential capacity: if someone believes that p, and q is an obvious consequence of p, then they must also believe that q;
   13. stability of preferences;
   14. transitivity: if someone prefers a to b, and b to c, then they must prefer a to c;
   15. explanatory relevance of reasons.

   Exercise 1.4.:

   1. being obvious (– by following logically from generally accepted, supposedly 'obvious' premises;)
   2. contrast with feelings (– so that logical arguments rather than emotional conclusions hold sway;)
   3. being a matter of logic (– i.e. of the general rules by which we decide if an argument is valid or not;)
   4. having a certain skill (– being good at giving/ following abstract arguments;)
   5. basing oneself on reasoning (– accepting conclusions based on logical arguments.)

   Exercise 1.7.:

   1. "In experiments, very low offers ... are often rejected, even though it is rational for the responder to accept any offer (even 1 cent!) which the proposer makes. And yet responders ... seem to get more satisfaction from taking revenge on the proposer than in maximising their own financial gain. Mr Spock would be appalled if a Vulcan made this mistake."
   2. "failing to achieve the daily income target feels like incurring a loss, so drivers put in longer hours to avoid it, and beating the target feels like a win, so once they have done that, there is less incentive to keep working."
   3. "Expected utility theory assumes that people look at individual decisions in the context of the big picture. But psychologists have found that, in fact, they tend to compartmentalise, often on superficial grounds. They then make choices abut things in one particular mental compartment without taking account of the implications for things in other compartments.

      There is also a huge amount of evidence that people are persistently, and irrationally, over-confident. Asked to answer a factual question, then asked to give the probability that their answer was correct, people typically overestimate this probability. ...

      Another delightful human habit is magical thinking: attribute to one's own actions something that had nothing to do with them, and thus assuming that one has a greater influence over events than is actually the case. For instance, an investor who luckily buys a share that goes on to beat the market may become convinced that he is a skilful investor rather than a merely fortunate one. He may also fall prey to quasi-magical thinking – behaving as if he believes his thoughts can influence events, even though he knows that they can't."

      The Economist, December 18th 1999.

2. Logical Arguments :

   Exercise 2.1.: For example:
   • questions: to find out information,
   • commands/orders: to make someone do something,
   • promises: to commit ourselves,
   • declarations: of war, that two people are now man and wife, etc.,
   • naming things or people,
   • threats, etc.
   Often the phrase: "I hereby ..." indicates that the utterance performs an action. (Cf. J.L. Austin, How to Do Things with Words, 1962.)

   Exercise 2.2.:

   1. "Two languages are spoken by no-one in this place."
      "The target wasn't hit by many arrows."
      Explanation: the order of 'operator'-words like "not" and "many" is different in the active and the passive sentences.
   2. Hobsbawm has the "not" in the second place.
   3. Hobsbawm has the "not".

   Exercise 2.3.:

   Only in c. and e. does a conclusion follow from (a) premise(s): look for words like "therefore", "consequently", "because", ...

   Note that in h. it is not asserted that Max being in the library follows from my seeing him there: it is just that my knowledge that he is depends on my having seen him there.

   Exercise 2.4.:
   The following should, typically, show how we justify some knowledge indirectly: using logical arguments to derive it from other knowledge we have.
   1. If a number is divisible by 10, the last digit is zero. But the last digit of 4096 is not zero. Hence ...
   2. All human beings die eventually, and the teacher of the class is a human being, so ...
   3. It is a condition for gaining an IB diploma that one write an Extended Essay. Since I want to gain an IB diploma, ...
   Exercise 2.5.:
   None of the arguments is valid: for each we can find concepts which make the premises true but the conclusion false. (Note that a single counter-example is enough to discredit the form of argument.)
   1. "Every mathematician can add 2 and 3, but ..."
   2. "Some Americans are black, and some Americans are white. Therefore ..."
   3. "If someone is a doctor, they ..."

3. Syllogisms :

   The starting point for the ancient Greeks may have been their hierarchical organization of concepts, which is familiar from the modern biological classification into genus, species, sub-species etc. The elements of the lowest classes are particular individuals: "Socrates is a Greek."

   A little exercise might be to draw a tree structure representing the hierarchical organisation of the following concepts: animals – 'barbarians' – birds – dogs – fish – Greeks – horses – humans – living things – mammals – non-living things – plants – stones.

   Exercise 3.1.:

   1. No A are B;
      all C are A.

      No C are B.

   2. All A are B;
      some C are A.

      Some C are B.

   
   Exercise 3.2.:
   All A are B;
   no C is A.

   No C is B.

   In a. the premises and the conclusion are true, in b. the premises are true but the conclusion false: hence not a valid argument (– fallacy of 'illicit process'.)
   Exercise 3.3.:
   1. valid: ferio
   2. not valid – try: "All those in favour of the motion drink milk."
   3. valid: barbara
   4. valid: darii
   5. not valid – try: "All islands are on earth."
   6. valid: celarent
   Exercise 3.4.:
   1. "If it is not a good orchestra, it cannot be a good concert."
   2. "All used-car salesmen are liars."
   3. "Only Brits have a right to live here."
   4. "All Soviet politicians were Marxist communists."
   5. "Only islands are surrounded by the sea."

4. Propositional Logic :

   Exercise 4.1.:
   1. r => r
   2. r v ¬r
   3. ( p => ¬q ) & ( ¬p => q )
   4. ( s & ¬m ) v ( ¬s & e )
      – because of the s and ¬s, the "or" is already exclusive.
   The truth table definition of p => q is not quite obvious:
   One way of arguing for it is that we are willing to accept a logical argument except in the case when the premise p is true and the conclusion q turns out false.

   Another argument comes from counterfactuals: e.g. "If fried birds could fly with knife and fork in their back, ready to be eaten, then anything could happen." – i.e. if the premise p is false the implication is true, no matter whether the conclusion q is true or false.

   Exercise 4.2.:
   a.  

   p	q	either p or q
   T T F F	T F T F	F T T F

   either p or q
   = ( p v q ) & ¬ ( p & q )
   = ( p v q ) & ( ¬ p v ¬ q )
   
   c.  

   p	q	m v l
   T T F F	T F T F	T T T F

   p	q	¬ m	¬ m => l
   T T F F	T F T F	F F T T	T T T F

   So the expressions do have the same meaning.
   
   d.  

   l	s	l => s	s => l	(l => s) => (s => l)
   T T F F	T F T F	T F T T	T T F T	T T F T

   So this is not a valid argument.
   
    

   l	s	¬ l	¬ s	l => s	¬ s => ¬ l	(l => s) => (¬ s => ¬ l)
   T T F F	T F T F	F F T T	F T F T	T F T T	T F T T	T T T T

   So this is a valid argument.
   
   e.  

   c	a	c => a	(c => a) & a	((c => a) & a) => c
   T T F F	T F T F	T F T T	T F T F	T T F T

   So it is not a valid argument.
   

   Other valid forms of propositional arguments:

   1. Proof by contradiction:
      this is quite common in mathematics: to prove that p is true, suppose that it is not and derive a consequence q which is always false, i.e.
      
      (( ¬p => q ) & ¬q ) => p ,
      – which can be derived from modus tollens by replacing "p" by "¬p".
   2. Disjunctive argument:
      
      (( p v q ) & ¬p ) => q ,
      
      e.g. a teacher at the end of a lesson: "This lesson is now over, or 2 + 2 = 5 ; so ..."
   In the case of simple arguments their validity may be obvious, but in the case of complex propositions a truth table is the only way to check it. For instance, the argument:
   If Jones did not meet Smith last night, then either Smith was the murderer or Jones is lying. If Smith was not the murderer, then Jones did not meet Smith last night and the murder took place after midnight. If the murder took place after midnight, then either Smith was the murderer or Jones is lying. Hence, Smith was the murderer.
   is of the form
   [ ( ¬p => ( q V r )) & ( ¬q => ( ¬p & s )) & ( s => ( q V r )) ] => q
   – which has a 16-line truth table! (The symbol "V" here denotes the exclusive "either ... or", as in the truth table above.)

   Exercise 4.3.:

   1. not valid (– denial of the antecedent)
   2. (( w => ¬m ) & m ) => ¬w ,
      which can be obtained from modus tollens
      (( w => n ) & ¬n ) => ¬w
      by replacing n by ¬m : so it is valid.
   3. not valid (– affirmation of the consequent)

   There are other branches of logic, apart from syllogistic and propositional logic, each one formalising an aspect of our ability to reason.

   1. Predicate logic:

      A predicate is a concept applied to one or more objects (or 'arguments'.) For instance, if F(x, y) is the predicate "x is the father of y" and G(x, y) is "x is the grandfather of y", then

      F(b, a) & F(c, b) => G(c, a) for all a, b, c.
      There are two 'quantifiers' in predicate logic, ∀ and ∃:
      (∀ x) P(x) = "for all x, P(x)"
      (∃ x) P(x) = "for some x, P(x)"
      Thus we can write the propostion that every number has a negative as
      if x and y are integers, (∀ x)(∃ y)( x + y = 0 ) ,
      and the proposition that every number has a unique negative as
      (∀ x)(∃ y)( x + y = 0 & (∀ z)( x + z = 0 => z = y ) )
   2. Modal logic:

      In everyday language things may not only be true or false, they can be possible or impossible, or may be necessary. This idea is formalised in modal logic. If p is a proposition then

      Nec p  = "it is necessary that p"
      Poss p = "it is possible that p"
      Because these modal operators are not just the invention of logicians but modelled on our everyday language, we can immediately recognize certain formal properties they must have:
      ¬( Nec p ) => Poss( ¬p ) ,
      but not
      Poss p & Poss q => Poss( p & q ) .
      (To see the latter, let p be getting Heads and q getting Tails when tossing a coin.)

5. Fallacies :

   Exercise 5.1.:

   a. x.	b. iv.	c. ii.	d. viii.	e. ix.	f. vi.
   g. i.	h. valid	i. xi.	j.  vii.	k. iii.	l. v.

   Some other common fallacies:
   12. ad ignorantiam:

       supposing that if a proposition has not been proved false it must be true, or if it has not been proved true it must be false:

       • "I am right to believe in ghosts, because you cannot prove that ghosts don't exist."
       (This one seems to me to be particularly common amongst Americans, for some reason.)
   13. tu quoque (= "and you too"):

       justifying an unsound argument by claiming the unsoundness of another.

   14. that when it is difficult or even impossible to draw a definite line between two things, there is no real difference between them:
       • "We cannot know what is right and wrong anyhow - all the discussions about capital punishment, euthanasia, abortion and so on never reach any conclusion."
       (– however, as Dr. Johnson observed: "Though no man can draw a line between darkness and light, still, night and day are tolerably distinguishable.")
   15. fallacy of many questions:
       • "When did you last beat your wife?"

6. Necessary and Sufficient Conditions :

   Examples:

   presence of light – seeing with one's eyes:	necessary,
   a fire burning – heat being given off:	sufficient,
   passing the driving test – being able to get a full licence:	both,
   passing the driving test – being able to drive well:	neither.

   Exercise 6.1.:

   a.	necessary	b.	both	c.	neither	d.	necessary
   e.	neither	f.	sufficient	g.	sufficient	h.	necessary
   i.	sufficient	j.	neither	k.	necessary	l.	sufficient
   m.	necessary	n.	sufficient	o.	neither

7. Other Ways of Reasoning :

   Exercise 7.2.:
   1. The man is a midget and cannot reach all the buttons in the elevator, but he can use his umbrella on rainy days.
   2. She picks a stone but immediately drops it on the path, and says: "We can see what colour it was by looking at the colour of the remaining stone."
   3. The man is a skydiver, and the unopened package is his parachute which failed to open.
   4. The man is a lighthouse keeper, and the light he turns off is that of the lighthouse; what he hears on the radio is the news of a shipping disaster.
   5. The man is a DJ who has just driven home briefly to murder his wife; he is listening to his own programme on the radio, and what he hears is the record skipping, so his alibi is blown.
   Exercise 7.3.:
   1. Turn over the D and the 7. Checking the 7 is an instance of testing an hypthesis by trying to falsify it, as in science.
   2. Check the beer-drinker and the 16-year old.
   3. Check the person eating hot chili peppers and the wine-drinker.

      b. is generally found to be easiest. Pinker suggests that this is because we have a 'cheat-detecting module' which has evolved and functions quite separately from standard logic.

   Reading :

   J.L. Austin, How to do Things with Words, 1962. OUP.

   A. Diemer, I. Frenzel (ed.) Fischer Lexikon - Philosophie, 1967. Fischer. pp. 130-156.

   Encyclopaedia Britannica, XI. "Logic, Applied", 28b - 29f; "Logic, Formal", 39a - 42b, 50f - 51e; "Logic, History of", 56c-59b.

   A. Foster, G. Shute, Propositional Logic - A Student Introduction, 1976. Aston Educational Enquiry Monograph.

   A.M. Frye, A.W. Levi, Rational Belief - An Introduction to Logic, 1941. Harcourt, Brace and Co.

   R.W. Jepson, Clear Thinking - An Elementary Course of Preparation for Citizenship, 1936. Longman.

   M. Pirie, The Book of the Fallacy, 1985. Routledge & Kegan Paul.

   L.S. Stebbing, A Modern Elementary Logic, 1952. Methuen.

   D.N. Walton, Informal Logic, 1989. Cambridge University Press.