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.
It may be precisely because rationality is such a fundamental concept, in different areas, that it has in the past tended not to be closely examined: rationality is the main feature that has distinguished human beings from other animals; and irrationality is the main reason for which people have been dispatched to mental hospitals. It has been only recently that philosophers and economists have started to examine what we mean by it: so even the Encyclopaedia Britannica is not of much help, although the concept does get mentioned in various, mostly very technical contexts.
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.)
The concept of rationality is a case in which we have to think about a concept more carefully than just taking its dictionary definition, which is likely to be something like: "1. the state or quality of being rational or logical. 2. the possession or utilization of reason or logic. 3. a reasonable or logical opinion. 4. Economics. ..." (Collins English Dictionary, 1991.)
Discuss in which ways this definition, though appropriate for a dictionary, is unhelpful or even misleading as the starting point of a serious investigation of the concept.
About what kinds of 'things' can we say that they are rational or irrational, or more or less rational? Consider examples of things like:
periods in history
(Rational and irrational numbers in mathematics are something quite different.)
Discuss which of the following are irrational, and say why you think so.
feeling like killing one's lover for being unfaithful;
killing one's lover for being unfaithful;
feeling like killing one's lover for burning the dinner;
imagining that one's lover burnt the dinner on purpose;
dreaming that one's lover has burnt the dinner;
'knocking on wood' so that one's lover will not burn the dinner;
getting angry with the person who brings the news that one's lover has burnt the dinner ('shooting the messenger';)
buying a lottery ticket to be able to employ a cook;
being so elated when the dinner isn't burnt that one cannot eat it.
The cases below are more 'technical': explain carefully in which ways person A is being irrational in the following situations, and hence try to formulate some minimal conditions for someone's (i) beliefs, (ii) preferences and (iii) actions to be rational.
A : " I am just too stupid for this subject. I never understand any of it." B : "But you did very well in the last test ..." A : "Well, that was easy, and I am not thick, am I?"
A : "Believe me, God loves all his faithful and will not let any ill befall them." B : "But you know that good Christians got killed too when that bridge collapsed." A : "..."
A (a bright 12-year old boy, in mental hospital at the time) : "I am a cat." B (a nurse) : "All cats have tails, and you don't have a tail." A : "Yes. ... I am a cat."
A : "What did you wake me up for?" B : "You asked me to, yesterday." A : "That was before I went to sleep!"
A : "I don't think this book is worth the $ 20 I just paid for it." B : "Can I have it? I'll give you $ 24 for it." A : "No way!"
A : "I have so much work! I have no time for anything else." B : "But haven't you just been playing games on your computer?" A : "That's been so I can do more work."
The dictionary definition above mentioned "logic" as an ingredient of rationality; (one of the meanings of "logical" in the same dictionary is "capable of or characterized by clear or valid reasoning.") What I shall mean by logic (from Gk. logikos, concerning speech or reasoning) is the formal system which we can use to decide whether some opinion or belief is consistent with or follows from other beliefs and opinions we hold; logic therefore concerns how propositions – which is how we express beliefs and opinions – are related by arguments, regardless of the content of those propositions. (For now you can take a proposition to be just a statement.)
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.
Describe the meaning with which the word "logical" is used in each of the sentences below, and try to say how it is related to the idea of 'a logical argument'?
It is logical that the earth is round.
Men are more logical than women.
If he is right, then they must be wrong. That's only logical.
She is a very logical person, so she works well with computers and is good at maths.
When it comes to my faith in God I cannot be very logical.
To further clarify what we mean by rationality and logic, and how they are related, it may help to go back to a distinction introduced in an earlier chapter, between
descriptive propositions, or statements of fact,
normative propositions, or value judgements,
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.
Logic, on the other hand, is purely descriptive: it describes the relation between propositions: whether some opinion or belief is consistent with, or indeed follows from, other beliefs and opinions someone may hold. But we need to apply logic in order to be able to be rational. It may help to draw a comparison, between logic and arithmetic:
the validity of both is independent of what they are about, so they are just formal rules, empty of meaning;
they are both purely descriptive, without moral or other normative significance in themselves;
but still, we ought to apply them, if we want to make good moral (or other) judgements.
Thus, to the fact that two apples and two oranges make four pieces of fruit it is not relevant that I like apples better than oranges. From the fact that one thing is twice as expensive as another it does not follow that to buy it would be a waste of money – but to be able to judge whether it would be a waste of money to buy it, I do need to be able to calculate that it is twice as expensive.
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.
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.)
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.
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.
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.
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.
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.
The other main problem with rationality and logic is that we are not very good at it. Here are some examples from an article "Rethinking Thinking," The Economist, December 18th 1999:
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.]
Many examples of irrationality can be explained, or explained away, when we allow a wider range of motives or causes. Thus Steven Pinker, in How the Mind Works, 1997, uses an argument from evolution: it is often advantageous (in game-theoretic terms) for it to be known that one's strategies may not be purely rational, so that one's threats and promises are 'guaranteed' by unfeigned anger and sincere friendship, and known not to be affected by what is in one's rational self-interest.
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.]
In the "ultimatum game" described in the article, what do you think responders will often do when they are made low offers, less than 20% of the sum, say?
How do you think the theory proposed explains the New York taxi drivers' irrational behaviour?
Other examples of irrationality in the article are that people tend to "compartmentalise," that they are "over-confident," and have a habit of "magical thinking." Try to think of examples of these.
What is meant by "wishful thinking," and in which way is it irrational?
Our ability to think logically is often greater than our willingness to act on the logical conclusions. What is meant by "weakness of will," and how can it lead to irrational behaviour?
In many areas of our lives, like the traffic or financial dealings or in relationships, people will of course in general act rationally: if they too often didn't, they would die soon, or go bankrupt, or have no friends. But in other areas we may think we can afford to be less rational.
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.
To see how difficult or easy you find logical reasoning, you can take a small logic quiz. To start it, press the button on the right.
Logic then is a tool that helps us to reason things out and act rationally, much like arithmetic is a tool that helps us to keep count of things. And just as the building blocks of arithmetic are numbers, so the building blocks of logic are propositions.
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".)
When we utter a sentence which expresses a proposition we make an assertion. Give examples of sentences which do not express propositions. What then do we do with such sentences?
We use logic and reasoning in every-day life, but the logical workings of natural languages are often far from clear.
Form the passive of the following two sentences, and try to explain why the meaning has changed:
"No-one in this place speaks two languages."
"Many arrows didn't hit the target."
Carefully read the following passage, and decide in which one of the two places you think the word "not" is required (– the author has it in the wrong place, it seems to me):
In fact ... we should be hard put to it to find a sizeable European state in which (not?) at least four out of every five inhabitants were (not?) country men. And even in England itself, the urban population only just outnumbered the rural population for the first time in 1851.
Eric Hobsbawm, The Age of Revolution, 1975. p. 11.
Carefully read the following sentence, and decide whether the last "not" is required:
None of this means that colonies were not acquired because some group of investors did (not?) expect to make a killing, or in defence of investments already made.
Eric Hobsbawm, The Age of Empire, 1987. p. 66.
Propositions are related by reasoning, or (logical) arguments: in a (logical) argument, propositions stand in a special relationship to one another, such that some 'follow from' others, (or are claimed to) – the 'conclusion' of such an argument is 'deduced' from the 'premises'.
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."
Which of the following are (logical) arguments ?
Close the window, because I am cold.
Two is not a safe number. There must be at least three.
Rainy days and Sundays always get me down; so it must be raining, because it isn't Sunday and I am down.
After the rain, the sky quickly cleared and it became a fine day.
Since there are only two of us, we must wait for another person.
"I am right." – "No, you're not." – "Yes, I am." – "No, you aren't."
Although he isn't here, we shall start the meeting.
Max is in the library, because I saw him there.
How would you go about justifying your knowledge of the following propositions? On what premises are they based, and what logical arguments are you using?
4096 is not divisible by 10.
The teacher of this class will die.
I must write an Extended Essay.
Since knowledge that p requires that the proposition p be true, we require that a valid logical argument must guarantee that whenever the premises are true then the conclusion is true; so an argument is not valid if one could find any case in which a false conclusion could, by that argument, be deduced from true premises.
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.
The premises and conclusions of the following arguments are all true, but the arguments by which the conclusions are arrived at are not all valid. Which ones are not, and why aren't they?
Every mathematician can solve this equation: sin z = 2 , but students doing SL Maths here are not mathematicians, so they cannot solve it.
Some Americans are black, and some Americans are criminals. Therefore some blacks are criminals.
If someone has a good heart, they will help others. And Mother Theresa definitely did help others, so she must have had a good heart.
So whether an argument is valid depends not on the truth of the conclusion, but on its form: not on whether the conclusion is true or not, but on whether the truth of the premises makes the conclusion true.
There are different branches of logic which deal with different forms of logical arguments.
A syllogism (C14: via Lat. from Gk. sullogismos, reckoning together) is a deducitve argument relating two premises and a conclusion, all of which are quantified propositions, i.e. propositions joining concepts by using words such as "some" and "all". Syllogistic logic goes back to ancient Greece and was then further developed by the 'schoolmen' in the Middle Ages.
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:
positive (– "a" and "i", from Lat. affirmo, I assert):
All mammals are animals.
= M a A
Some mammals are human beings.
= M i H
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)
The traditional name of this syllogism is "barbara", based on the sequence of vowels in the traditional representation. – Here is another example of a syllogism:
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)
Following the pattern of the two examples, write down the form of the following classical syllogisms, and give an example of each form of argument:
It is one of the attractions of syllogistic logic that the forms of argument can be nicely classified and arranged. Every syllogism relates three concepts, here denoted by capital letters; the subject of the conclusion is called the minor term and its predicate the major term; the middle term occurs in both premises but not in the conclusion.
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
Remember that a valid form of argument must guarantee that if the premises are true then so is the conclusion. It follows that a syllogism is valid only if there can be no concepts A, B and C such that the two premises are true but the conclusion false.
Write down the general form of the following argument, and check whether it is valid by substituting the different concepts, as suggested below:
All Brits have the right to live here.
No foreigner is British.
Foreigners don't have the right to live here.
A = human, B = mammal, C = stone.
A = human, B = mammal, C = horse.
To check the validity of syllogistic arguments, some people find it helpful to think of them in terms of the Venn-diagrams used in set theory – which the ancient Greeks were of course not yet able to do.
Which of the following are valid arguments? If you decide about an argument that it is not valid, explain why not. (Remember that an argument may not be valid, even if the conclusion is true, and that an argument may be valid even if the conclusion is not true.)
No man (i.e. human) is an island.
Some animals are human.
Hence some animals are not islands.
All those in favour of the motion will vote for it.
No member of the opposition is in favour of the motion.
Therefore no member of the opposition will vote for the motion.
Things that are only imagined cannot travel through space.
All UFOs are only imagined.
Hence UFOs cannot travel through space.
Those that can be fooled all the time are dangerous to democracy.
Some of the people can be fooled all the time.
Consequently some of the people are dangerous to democracy.
All islands are surrounded by the sea.
No man is an island.
So no man is surrounded by the sea.
Nothing that is free is worth having.
All the best things in life are free.
None of the best things in life are worth having.
An incomplete syllogism, in which one of the premissis is not explicitly stated but is taken for granted, is called an "enthymeme", (from Gk. en + thumos, in the mind.) Note that often an assumed premise is precisely the weak point of an argument.
What are the missing premises that would make the following arguments valid?
It won't be a good concert, because it is not a good orchestra.
Since you are a used car salesman, you must be a liar.
No foreigner is British, hence no foreigner has a right to live here.
All Marxist communists are atheists, and so all Soviet politicians were atheists.
No man is an island. So no man is surrounded by the sea.
A formal system of logic enables us to see whether an argument is valid, i.e. whether the truth of the premises guarantees the truth of the conclusion, regardless of the content of the propositions.
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."
there are two simple propositions,
p = "a student is from Hong Kong," and
q = "(s)he works too little,"
and the logical form of the argument is
"If p and q, then p."
– which is of course valid.
Thus the logical form of a complex proposition consists of
p, q, ..., and
¬ ("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.")
Using these symbols for the logical connectives, the logical form of the above proposition can be written as
p & q => p .
Of course, a sentence in a natural language can convey much that is not represented in its logical form. For instance, the following sentence has the same logical form:
"Even if a student is from Hong Kong but works too little, (s)he is still from Hong Kong."
But the additional information that is conveyed, about the speaker's feelings for instance, do not affect whether the proposition expressed by the sentence is true or false.
The idea of variables and connectives should be familiar from mathematics: we can speak of arithmetical forms consisting of
Using these symbols, the arithmetical form of "2 + 5 = 5 + 2" can be written as
a + b = b + a ,
and just as this is an arithmetical identity, (expressing that addition is commutative,) because it is true for all values of the variables a and b, so
p & q => p
is a logically valid argument, no matter what the propositions p and q are.
Using the logical connectives, and choosing propositional variables as appropriate, write down the logical forms of the following sentences.
If it rains, it rains.
Either it will rain, or it will not.
If Mr Jones is happy, Mrs Jones is unhappy, and if Mr Jones is unhappy, then Mrs Jones is happy.
Either Sam will come to the party and Max will not, or Sam will not come to the party, and Max will enjoy himself.
In serious work, the system of propositional logic is studied as an axiomatic system: all the theorems are derived, or proved, from a very small number of axioms, which are usually considered intuitively obvious.
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 & 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".
¬ ¬ p
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
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:
((p => q) & p ) => q ,
((p => q) & ¬ q) => ¬ p .
For example, the argument
"If someone is a communist, then he is an atheist. Jones is a communist. Therefore he is an atheist."
is a case of modus ponens, and hence a valid argument, no matter whether the premises are true or false.
Write down the truth table definition of the exclusive "or" ("either ... or.") Can you think of a way of representing exclusive "or" in terms of the connectives introduced here?
Use truth tables to show that "p v ¬ p" is a tautology, and that "p & ¬ p" is a contradiction.
(The first of these is called "the Law of the Excluded Middle," or in Latin: tertium non datur: there is no third possibility.)
Write down the logical forms of the following two expressions and then complete truth tables for them:
"Your money or your life."
"If not your money, then your life."
Do they have the same meaning?
Write down the logical forms of the following two expressions and then complete truth tables for them:
"If a student is lazy, they will do Maths Studies. Therefore, if a student does Maths Studies, they are lazy."
"If a student is lazy, they will do Maths Studies. Therefore, if a student does not do Maths Studies, they are not lazy."
Which one is a tautology, and therefore represents a valid argument?
Write down the logical form of the following argument, and then complete a truth table for it:
"If someone is a communist, then he is an atheist. But Jones is an atheist. Therefore he must be a communist."
Is it a tautology? Is the argument valid?
Invent some further arguments which are instances of modus ponens and modus tollens.
Which of the following show valid forms of arguments?
John doesn't hurry, because if he wanted to come, he would hurry, and he doesn't want to come.
If it is a club I want to join they won't have me as a member. But they will have me as a member. So it isn't a club I want to join.
If we cannot find it, it isn't here. But it isn't here, and so we cannot find it.
(Hint: Try it with "we can find it/ it is here" – if the premises are true, is the conclusion true?)
Fallacies are errors in reasoning that render an argument logically invalid; the 'dangerous' fallacies are those that can be mistaken for valid arguments, (especially of course by someone who is uncritical because (s)he agrees with the conclusion.)
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:
denying the antecedent:
(( p => q ) & ¬ p ) => ¬ q .
"If he is slow, he will lose. Since he isn't slow, he won't lose."
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."
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."
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.
"All A are B; all C are B.
"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.)
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.
"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:
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:
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."
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 ..."
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?"
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."
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."
For each of the following arguments state which, if any, fallacy is committed. In some cases more than one answer is possible; try to find one example of each type of fallacy.
We should not accept scholarships from RTZ, because it would be wrong for us to take money from that kind of company.
All communists are in favour of cooperative organizations, and some Labour Party members are in favour of cooperative organizations, so some members of the Labour Party are communists.
(Overheard on an American tourist bus:) "If it is Sunday we should be in Pisa. But over there is the leaning tower, hence today must be Sunday."
"Very smart, Maria, very smart." – "Oh no! Anita, no. You should know better. You were in love – or so you said. You should know better" (from the musical West Side Story.)
When we have come back from London Trip, every year a lot of students have ended up in sick bay; so if we want fewer people to get ill, we should cancel the Trip.
A leg of mutton is better than nothing. But nothing is better than heaven. Thus a leg of mutton is better than heaven (G. C. Lichtenberg, 1742 -1799.)
"If your Project Tutor came with you, he would make sure you do the project well." – "So because he is not coming, we won't do it well?"
No atheist believes in God. But no true communist isn't an atheist. Hence no true communist believes in God.
We all of us sometimes experience a feeling of awe and gratefulness when confronted with the beauty and magnificence of nature, and so there is a God to whom we are grateful.
Of course theft is a bad thing, because if it happened all the time society wouldn't function at all. But banks are so rich – just look at the buildings they have! They probably don't even notice a couple of thousand pounds gone.
(Quoted from a lecture by a member of the drugs squad:) "I have never met a user of hard drugs who had not started on soft drugs. So it is dangerous to experiment with soft drugs."
All inconsiderate smokers drop cigarette butts. But no students here are inconsiderate smokers, so no students here drop cigarette butts.
Invent an argument exemplifying one of the above types of fallacies, for the other students to try and identify.
Before the next lesson, try to find another type of fallacy, different from the ones above, and construct an example of it.
Logical arguments, as we have said, relate propositions as premises and conclusions, so that if the premises are true, then the conclusion is guaranteed to be true as well. This way we can gain new knowledge by deducing it logically from other knowledge we already have.
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:
A is a necessary condition for B, if B cannot happen without A preceding,
A is a sufficient condition for B, if A cannot happen without B following.
Given any two states or events, A and B, A can be
a necessary condition for B, or
a sufficient condition, or
State in each case whether the first event or state is a necessary or a sufficient condition for the second, or both, or neither.
presence of oxygen – combustion,
being bounded by three straight lines – being a triangle,
eating poisonous tablets – illness,
presence of moisture – growth of a plant,
attending Atlantic College – gaining an IB diploma,
rain falling on a path – path getting wet,
a stone hitting a fragile window hard – the window breaking,
turning on a light switch – the light goes on,
turning off a light switch – the light goes off,
taking an aspirin – being cured of appendicitis,
petrol being in the car's tank – the car actually starting,
a casualty being conscious – his or her heart beating,
a casualty's heart beating – the casualty breathing,
a casualty breathing – his or her heart beating,
a casualty being in pain – the casualty being seriously injured.
We can return here to the distinction made earlier between analytic and synthetic propositions: whereas analytic propositions are true (or false) by virtue of the meanings of the words alone, without reference to facts about the world, synthetic propositions depend for their truth value on facts about the world:
"A bachelor is an unmarried man" is an analytic truth,
"John is a bachelor," no matter if true or false, is synthetic.
Logical truths (and falsehoods) are a form of analytic proposition, whose truth depends only on the meanings of certain words, the logical connectives that are used. Thus, the proposition
"If it is brillig or mimsy, and it is not mimsy, then it is brillig."
is true, and the argument valid, regardless of what the words and "brillig" and "mimsy" mean, just because of the meanings of the words "if ... then", "or", "and" and "not".
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.
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."
Try to think of cases in which the results of inductive reasoning were generally accepted but later turned out to be false.
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:
recognize dominant ideas that polarize the perception of a problem;
searching for differ ways of looking at things;
relaxation of rigid control of thinking; and
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.
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.
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?
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?
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.
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?
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.
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."
Flumo and Yakpalo always drink cane juice [rum] together. Flumo is drinking cane juice. Is Yakpalo drinking cane juice?
Flumo and Yakpalo drink cane juice together, but the time Flumo was drinking the first one Yakpalo was not there on that day.
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?
The day Flumo was drinking the cane juice Yakpalo was not there that day.
What is the reason?
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?
[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.)
[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.)
[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' ...
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!'
A difficult topic:
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?
Rationality is a normative concept, like morality,
("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%
other things being equal,
if the higher-interest account takes 10 hours to set up,
if the lower-interest account involves no exploitation?
and arguments, which express/ defend beliefs;
and policies/ plans, which prepare behaviour;
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.
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.
Rationality and reason:
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."
The 'bad reputation' of rationality
(– 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 ...):
"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;
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:
"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;
opt out and change to Maths Studies,
doing homework more regularly.
Hence some criteria for rationality:
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.
The role of logic:
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."
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."
"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 ...
The concept of rationality is not at all simple,
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.
Keep emphasizing throughout that we need valid arguments to be able to justify claims of knowledge based on, deduced from other, perhaps more basic things we know. A conclusion may be false either because the premises it is derived from are, or because they the reasoning is bad. Even if our conclusion happens to be true, we cannot claim to know it if the argument for it is not valid.
One could start by asking for examples of reasoning in everyday life.
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
consistency: if someone believes that p, they cannot also believe that not p;
conjunctive disposition: if someone believes that p and they believe that q, then they must also believe that p and q;
minimal inferential capacity: if someone believes that p, and q is an obvious consequence of p, then they must also believe that q;
stability of preferences;
transitivity: if someone prefers a to b, and b to c, then they must prefer a to c;
explanatory relevance of reasons.
being obvious (– by following logically from generally accepted, supposedly 'obvious' premises;)
contrast with feelings (– so that logical arguments rather than emotional conclusions hold sway;)
being a matter of logic (– i.e. of the general rules by which we decide if an argument is valid or not;)
having a certain skill (– being good at giving/ following abstract arguments;)
basing oneself on reasoning (– accepting conclusions based on logical arguments.)
"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."
"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."
"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.
Logical Arguments :
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,
Often the phrase: "I hereby ..." indicates that the utterance performs an action. (Cf. J.L. Austin, How to Do Things with Words, 1962.)
"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.
Hobsbawm has the "not" in the second place.
Hobsbawm has the "not".
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.
The following should, typically, show how we justify some knowledge indirectly: using logical arguments to derive it from other knowledge we have.
If a number is divisible by 10, the last digit is zero. But the last digit of 4096 is not zero. Hence ...
All human beings die eventually, and the teacher of the class is a human being, so ...
It is a condition for gaining an IB diploma that one write an Extended Essay. Since I want to gain an IB diploma, ...
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.)
"Every mathematician can add 2 and 3, but ..."
"Some Americans are black, and some Americans are white. Therefore ..."
"If someone is a doctor, they ..."
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.
No A are B; all C are A.
No C are B.
All A are B; some C are A.
Some C are B.
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'.)
not valid – try: "All those in favour of the motion drink milk."
not valid – try: "All islands are on earth."
"If it is not a good orchestra, it cannot be a good concert."
"All used-car salesmen are liars."
"Only Brits have a right to live here."
"All Soviet politicians were Marxist communists."
"Only islands are surrounded by the sea."
Propositional Logic :
r => r
r v ¬r
( p => ¬q ) & ( ¬p => q )
( 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.
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 )
m v l
T T F F
T F T F
T T T F
¬ 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.
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
¬ 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.
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:
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".
(( 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.)
not valid (– denial of the antecedent)
(( w => ¬m ) & m ) => ¬w ,
which can be obtained from modus tollens
(( w => n ) & ¬n ) => ¬w
by replacing n by ¬m : so it is valid.
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.
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 ) )
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 ) ,
Poss p & Poss q => Poss( p & q ) .
(To see the latter, let p be getting Heads and q getting Tails when tossing a coin.)
Some other common fallacies:
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.)
tu quoque (= "and you too"):
justifying an unsound argument by claiming the unsoundness of another.
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.")
fallacy of many questions:
"When did you last beat your wife?"
Necessary and Sufficient Conditions :
presence of light – seeing with one's eyes:
a fire burning – heat being given off:
passing the driving test – being able to get a full licence:
passing the driving test – being able to drive well:
Other Ways of Reasoning :
The man is a midget and cannot reach all the buttons in the elevator, but he can use his umbrella on rainy days.
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."
The man is a skydiver, and the unopened package is his parachute which failed to open.
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.
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.
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.
Check the beer-drinker and the 16-year old.
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.
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.