Logic - Study Notes
LOGIC by Graham Priest [OUP, 2000]
Graham Priest is Professor of Philosophy at the University of Queensland, and he is the author of various books and many articles on logic and philosophy. Logic has wide applications in various fields including science, mathematics, history, language, law, computer science, not just philosophy.
So – what is logic about? In everyday life ‘being logical’ means much the same as being sensible, organized, rational, systematic, practical and so on, but logic as an academic field of study has a more precise and circumscribed meaning. It is primarily concerned with sentences conveying information and what follows from a sentence or group of sentences and what does not. For example if the sentence ‘Rosie is a cat’ is true and if the sentence ‘all cats are mammals’ is also true, it follows that Rosie is a mammal.
This simple example illustrates a key feature of a logically valid argument – the information in the conclusion (that Rosie is a mammal) is already implicit in the premises (that Rosie is a cat and that all cats are mammals). One point in passing, in everyday life the word ‘argument’ is often used as a synonym for ‘dispute’, ‘quarrel’ or ‘disagreement’ etc., but in logic and philosophy its meaning is more circumscribed. It simply means a set of statements which lead to a conclusion. Another point in passing, going back to Rosie the cat, it is not the logician’s business to determine whether cats are mammals or some other type of animal. That is someone else’s business which the logician takes as a datum.
Anyway - I shall start this résumé of Priest’s book not at Chapter 1, but at Chapter 14, for this is where he gives a brief history of logic. It seems to me that this might be a more appropriate starting point for a talk such as this one. Priest relates how the study of logic began in ancient Greece, mainly in Athens but also at Megara. Aristotle (384-322 BC) was the most notable logician. He developed a theory of deductive reasoning which followed a set pattern called a syllogism.
Here is an example of a valid syllogism:-
All A’s are B’s; all B’s are C’s; therefore, all A’s are C’s.
This argument-form is always valid, whatever A, B, or C stand for. For example ‘all dogs are mammals’; ‘all mammals are warm-blooded creatures’; therefore ‘all dogs are warm-blooded creatures’. Another example, ‘Jones is a man; all men are mortal; therefore, Jones is mortal’.
‘Validity’ has a very specialized meaning in logic. The validity of an argument depends upon its structure, not its content or subject-matter which is a separate question. The full elaboration of syllogistic logic is quite complex and intricate, and obviously we cannot go into all its ins and outs in a short talk such as this one. So as an Appendix to these notes I attach a few examples of valid and invalid syllogisms.
Validity in logic must be distinguished from ‘factually true’. For example the following argument is logically valid although it incorporates false premises:- ‘All men are reptiles, all reptiles have red hair, therefore all men have red hair’. The conclusion follows from the premises, and the logician is solely concerned with validity. The truth or falsity of the premises is somebody else’s business, such as the anthropologist or zoologist in this example. Likewise in mathematics, a formula such as ‘2a + 3a = 5a’ is always true, but the mathematician is not concerned with whether you’re counting flowers, planets, dollars, fairies or anything else.
Returning to the history of logic, the second growth period in Western logic took place in mediaeval times, important logicians being Duns Scotus (1266-1308) and William of Ockham (1285-1349). Priest also mentions the German philosopher Leibniz (1646-1716), but the third major growth period began in the late 19th and early 20th centuries with the work of German mathematician Gottlob Frege (1848-1925) and British philosopher Bertrand Russell (1872-1970). Their work inaugurated modern logic as distinct from traditional logic. It was characterized by an entirely new mathematical notation devised by the Italian logician Giuseppe Peano, and it incorporated new developments in set theory. Further work continued in the 20th century and up to the present day.
Before reverting to Chapter 1, a general comment about this book is that it frequently uses a specialized logical symbolism. It is not possible for me to include this in a general talk such as this one; quite apart from anything else, we’d need a blackboard and chalk and we’d be considerably longer than our usual couple of hours. I shall therefore have to discuss and hopefully explain things in general terms, without the symbolism.
So – Chapter 1 is ‘Validity: What Follows from What?’.
We have just covered some of this ground already, of course. Priest distinguishes between two types of validity. The first, deductive validity, we have already encountered. If Rosie is a cat and all cats are mammals, Rosie must be a mammal. The word ‘must’ is important. If the premises are true and deduction rules are correctly followed, the conclusion cannot fail to be true.
The second, inductive validity, is not quite so straightforward. For example Jones has nicotine stained fingers, so we infer that he is a smoker. Most people would agree, but it is not certain – it is not logically ‘watertight’. He could conceivably have stained his fingers yellow to make people think he was a smoker (for some reason). Another example, all observed ravens have been black and no ravens have been observed that are any other colour. We conclude that the next raven we observe will also be black, and that all ravens are black.
But these conclusions can never be guaranteed, because they go beyond what has been observed. We have never observed all ravens, and never will. On the other hand we know that inductive reasoning is usually reliable, both in everyday life and in science, and we use inductive reasoning all the time. The usual procedure is to draw conclusions that fit all the observable facts and to accept them in the absence of evidence to the contrary. In complex cases, of course, this is far from straightforward.
Chapter 2 is intriguingly entitled ‘Truth Functions – or Not?’.
The basic idea is very simple but its elaboration turns out to be rather complex and possibly baffling to a newcomer to logic. Priest introduces the reader to ‘truth tables’, a tabular way of breaking down complex statements into separate parts which individually turn out to be quite simple, although I do think his account is somewhat convoluted..
Sentences can be true, or false (some can be indeterminate but we’ll come back to that in a moment). Using ‘T’ for true and ‘F’ for false, a sentence can be given a ‘truth-value’ of either T or F (some logicians use 1 or 0). So, consider two simple sentences such as ‘Jack is in the kitchen’ (sentence A) and ‘Jill is in the garden’ (sentence B). Each of these can have the truth-value T or F respectively. They can be conjoined by the word ‘and’ to make the composite sentence ‘Jack is in the kitchen and Jill is in the garden’, sentence C. Is sentence C true or false?
If Sentence A is true and Sentence B is false, Sentence C is false.
If A is false and B is true, C is false.
If A is false and B is false, C is false.
If A is true and B is true, C is true.
We can put this into a tabular format called a truth-table as follows:-
A B C
T F F
F T F
F F F
T T T
Using column C which represents truth-values for the composite sentence ‘Jack is in the kitchen and Jill is in the garden’, we can see that sentence C only has the truth-value T when A and B both have the value T. So the truth-value of C is a function of A and B.
The truth-table we have just shown is for conjunction which is for the truth-values of ‘A and B’ as a composite statement. A truth-table can be given for negation which means that if ‘A’ is true, ‘not-A’ is false, and if ‘A’ is false, ‘not-A’ is true. Priest also provides a truth-table for disjunction, ‘A or B’, which is true if either A or B or both are true but false if both A and B are false. These truth-tables are set out at the Appendix. I think he could also have provided a truth-table for exclusive disjunction where A or B but not both can be true.
Of course, some sentences can be neither true nor false but indeterminate or where their truth-value is unknown. This suggests a triple-valued logic which logicians also study. On the other hand it is important to avoid a couple of misunderstandings. One is to say that truth is a relative concept from one person to another, meaning that if Jack says ‘x is true’ this only means that ‘x is true for Jack’ and if Jill says ‘x is not true’ this only means ‘x is not true for Jill’. On this basis, there is supposedly no contradiction between what they say – but plainly, there is. If Jill says ‘it’s not true that pigs can fly’ for example, she is saying something objective and determinate about pigs, she is not just expressing ‘a personal view’. Of course it is true that opinion is relative, but the study of opinion falls within the domain of psychology, not logic. Another fallacy is to say that ‘truth is always in the middle’. If this were true, then the truth must lie at B which is midway between A and C – but then it must also lie at the midpoints between A and B, B and C and so on ad infinitum so that truth is nowhere to be found. Also, if the sentence ‘truth is relative’ is true then that sentence is relative itself so it amounts to a contradiction in terms.
Chapter 3 is ‘Names and Quantifiers: Is Nothing Something?’
Traditional grammar, says Priest, tells us that the simplest whole sentences are composed of a subject and a predicate. For example, ‘Marcus saw the elephant’, ‘the cat sat on the mat’ and so on. The subject-word can be a name such as Marcus or a pronoun such as ‘I’ or ‘you’; it can also be an indefinite pronoun as in ‘someone left the light on’, ‘nobody lives here’ and so on. Words such as ‘someone’, ‘everyone’, ‘nobody’, ‘everything’ etc. are called quantifiers by logicians and although they can be the grammatical subjects of sentences, just like names or pronouns, their logical behaviour is different and can sometimes be misleading.
Obviously when we say ‘nobody lives here’ we do not mean there is a mysterious person or entity called ‘nobody’ who lives here. The sentence ‘nobody lives here’ is simply a grammatical shorthand or idiom for saying ‘it is not the case that somebody lives here’. That’s all it means, just as the word ‘nothing’ does not denote a thing. But a less obvious solecism arises with the notion of causality. We know from ample experience that events have causes, so we say, ‘everything has a cause’. But what is the cause of everything, we might ask? The mediaeval theologians had an answer – the cause of everything is God.
However, as Priest points out, this involves ‘an enormous logical fallacy’, as he puts it, based on the ambiguity of the sentence ‘everything has a cause’. It can mean, perfectly innocuously, that everything that happens has some cause or other – but it does not follow from this that there is one cause of everything. Likewise, the sentence ‘everyone has a mother’ is true, but this does not mean there is someone who is the mother of everyone..... There may have been a ‘big bang’ which caused everything, but this is a scientific hypothesis based upon research, not upon the platitude that ‘everything has a cause’.
Chapter 4 is ‘Descriptions and Existence: Did the Greeks Worship Zeus?’.
This chapter summarizes what is known as Bertrand Russell’s ‘theory of definite descriptions’ as set out in On Denoting published in 1905 in the philosophical journal Mind and reiterated in his two volume Principia Mathematica published in 1912. Priest uses Russell’s logical symbolism in his account but for the purposes of this talk, I shall use ordinary language.
If we say ‘the present king of France is bald’, is this true or false? If we say ‘false’, is it correct to say ‘the present king of France is not bald’? Clearly not, for there is no king of France; so we have two contradictory sentences, ‘the present king of France is bald’ and ‘the present king of France is not bald’, which are both false. But it is logically impossible for two contradictory sentences both to be true or both to be false (see Appendix c). Russell solved this problem by making the crucial point that although expressions such as ‘the present king of France’ have meaning they do not denote entities. He called such expressions logical fictions. Russell was concerned about these questions because some 19th century philosophers said that if a sentence such as ‘unicorns have four legs’ is true as a matter of definition then unicorns must in some sense ‘exist’. Russell’s theory of descriptions put an end to that sort of nonsense.1 So although the ancient Greeks thought they were worshipping an entity called ‘Zeus’, the sad reality is that they were worshipping a logical fiction.
Chapter 5 is ‘Self-Reference: What is this Chapter About?’.
It is possible for a sentence to refer to itself, e.g. ‘This sentence contains five words’. We note in passing that this particular sentence is true. Self-reference also occurs in regulations which can refer to ‘these regulations’, and this is likewise unproblematic.
Problems do occur if I say for example ‘the sentence I am saying at this moment is false’. If the sentence is true then it’s false and if it’s false then it’s true. There seems to be nothing that can settle the matter one way or the other. As Priest says, it would seem to be neither true nor false, adding that it is a paradox which has been known since ancient times – often called the liar paradox.
Other paradoxes can arise, for example in set theory. A set is a collection of objects, such as the set of all the books in my house, the set of all the people on this planet, the set of all the planets in the solar system, and so on. A set can be a member of another set, for example the set of cats is a subset of the set of mammals. Obviously the set of cats is not itself a cat, it is a set, so the set of cats is not a member of itself. The set of non-cats however is a member of itself; likewise the set of all sets is a member of itself.
But take a closer look at the set of all the sets that are not members of themselves. If it is a member of itself then it is a set which is not a member of itself; equally, if it is not a member of itself, then it is a member of itself. This paradox was discovered by Bertrand Russell so it is called Russell’s paradox. As with the liar paradox, many logicians and philosophers have proposed solutions over the years, but none of them seem to be conclusive. However, they keep working at it.
Chapter 6 is ‘Necessity and Possibility: What Will be Must be?’.
We often claim that something must be so, as when we say ‘it must be going to rain’. We can also say something could be so, as in ‘it’s possible it’ll rain tomorrow’. Necessity and possibility are the modes in which things can be true or false. They are in fact related. To say that something must be so is to say it is not possible for it not to be so. Likewise to say something is possible is to say it is not necessarily the case that it is not so. Priest notes that the truth-value of a modal statement can be different from the truth-value of a factual statement about the same thing. For example it is true to say it is possible that I shall get up at 4.00am tomorrow morning, but it does not follow that I will.
He provides quite an interesting and detailed discussion of fatalism, the doctrine (or dogma, if you like) that ‘whatever will be, will be’ meaning that there is nothing we can do to alter the course of future events. Apart from the fact that there is ample evidence to show that we can alter the course of future events, there is another refutation of fatalism which is delightfully simple. This is to point out that the sentence ‘whatever will be, will be’ is only a tautology, and tautologies are only vacuously true, no different from saying ‘the future is whatever will happen’. Tautologies only elucidate the meanings of words, symbols or other linguistic expressions, and it is impossible to derive statements about the course of future events from the definitions of words. To say ‘whatever will be, will be’ tells us nothing about what we can or cannot do, or what we will or will not be doing in the future.
We are not yet halfway through Priests’s book so, bearing in mind time constraints, I shall have to give briefer summaries of the remainder. Chapter 7 is ‘Conditionals: What’s in an If?’. A conditional is a sentence of the form ‘if a, then c’. ‘a’ is called the antecedent; ‘c’ is the consequent. For example, ‘if you miss the bus, you will be late’. If this is true, you can infer that it is false that you will miss the bus and not be late, other things being equal. We can also infer that if you weren’t late, then (unless you had a lift from someone) you didn’t miss the bus. Priest says that conditionals are ‘fundamental to much of our reasoning’, but they are ‘deeply puzzling’ for various reasons. Consider one puzzle he gives at p53:- There is an election with two candidates, Smith and Jones. But suppose we say:-
‘If Smith dies before the election, Jones will win. If Jones wins the election, Smith will retire and take her pension. So if Smith dies before the election, she will retire and take her pension.’ What’s wrong with this argument? I think the answer is simple enough: the second statement needs the proviso ‘other things being equal’. This includes her remaining alive, so the conclusion that she could die but still take her pension is ruled out because it does not follow. I think Priest has given rather a contentious example.
Chapter 8 is ‘The Future and the Past: Is Time Real?’.
Time seems to flow. What is happening now, at this very second, becomes the past in a fraction of a second. Also, we never experience the future until it arrives and becomes the present – whereupon it immediately becomes the past. Philosopher John McTaggart Ellis McTaggart (1866-1925) argued quite seriously that time is unreal, that it is an illusion. His argument was that there would be no time if there were no past and future. Past and future are contradictory, so nothing in reality can correspond to them. They are contradictory because they are incompatible. If an event is past it is not future, and vice versa. Obviously an event cannot be past and future, and because a sentence about an event has in principle every possible tense, contradiction is unavoidable. Priest does not agree with McTaggart’s argument, presenting quite a complex counter-argument. I would simply take a short-cut and say that just like solipsism, McTaggart’s thesis is untestable and is therefore neither true nor false.
Chapter 9 is ‘Identity and Change: Is Anything Ever the Same?’.
What is to be said about the identity of objects that change through time? We naturally think that objects retain their identity, even with change. A cupboard painted a different colour is still the same cupboard. On the other hand, what about radical change, such as multiple transplant operations, a complete revamp of your appearance and a change of name? Are you a different person, or are you really the same person all along?
In logic there is a distinction between the ‘is’ of predication and the ‘is’ of identity. The ‘is’ of predication denotes the attribute(s) of something or other, which to a greater or lesser extent can change. We often say that if two objects share exactly the same attributes they are ‘the same’, as with identical twins or kittens. But this is misleading – they are still different individuals (even the kittens). The ‘is’ of identity by contrast is not about attributes, it conveys an entirely different kind of information, as in ‘Jones is the person who won the race’. That sentence is still true even if, after the race, he has multiple transplant operations, undergoes plastic surgery and changes his name.
A general point about identity is that for any objects x and y, if x = y then x has the same attributes that y has and vice versa. This is called Leibniz’s Law, after the 17th century philosopher who first formulated it. Of course, there can be complications. Attributes can change, as we have seen, but identity can remain the same. People may not realize that the Morning Star and the Evening Star are one and the same thing. If an amoeba divides to become two amoebas, we now have two amoebas instead of one (and then they become four, and so on). As Priest says, if two things are the same, it does not necessarily follow that they are always going to be the same.
Chapter 10 is ‘Vagueness: How do you Stop Sliding down a Slippery Slope?’.
Problems about identity continue. Things wear out and parts get replaced. Obviously, when I replace the clutch on my bike it remains the same bike – likewise, when I replace the wheels, the saddle etc., it is the same bike. But suppose, as the years go by, I replace every part of my bike. Also suppose I keep all the old parts and put them back together again to recreate the original bike. Do I now have two bikes that share the same identity in some way?
Another example, a 5 year old child is still a child one second later. He is still a child one second after that, one more second after that, and so on and so on. On this basis he is still a child after 630 million seconds have passed, when he’s 25...... These problems are called ‘sorites paradoxes’, after the Greek word for heap. There are plenty of other examples. As Priest says, ‘these are some of the most annoying paradoxes in logic’. They arise when the predicate (such as ‘is a child’) is vague, ‘when its applicability is tolerant with respect to very small changes’.
One answer to these problems is a recent development in logic called ‘fuzzy logic’. Just as being a child fades out gradually over time, the truth-value of the sentence ‘Jack is a child’ also fades over time from true to false. Truth (or falsity) changes by degrees. The idea is that these degrees can be ‘measured’ by numbers between 1 and 0, 1 being complete truth and 0 being complete falsity. In this way a sentence can have a truth-value denoted by a number (such as 4.5, 3.2, 8.1 etc.). Obviously the detail of all this (pp73-77) is quite complex. But it is not clear how fuzzy logic can solve the problems we have been looking at. Priest expresses reservations too, saying that whilst the original problem was that we could not define an exact moment when Jack ceases to be a child, with fuzzy logic we cannot define that point either. Where does the truth-value of the sentence ‘Jack is a child’ change from ‘1’ to ‘0.9’, for example? Or ‘0.99’? As Priest says, ‘we haven’t really solved the most fundamental problem about vagueness: we have just relocated it’ (p77).
For the last three chapters I must be very brief.
Chapter 11 is ‘Probability: The Strange Case of the Missing Reference Class’.
This is not the place for a full discussion of Probability Theory, a vast subject in itself which is largely mathematical in nature. However, it does have common borders with logic and philosophy which Priest summarizes. Basically he reiterates some of his earlier comments on inductive reasoning where we derive general conclusions from limited information. Thus we say ‘all ravens are black’ upon the basis of data about all the ravens we have ever observed. We draw conclusions that fit the observable facts and accept them in the absence of evidence to the contrary – a procedure which generally works, although the example I have just given is a very simple one.
Probability is where you assign a number to a statement which measures how likely it is that it is true. Providing you have made a sufficiently large number of observations, you could say it is 80% likely that I shall go to bed at 11.00pm give or take 5 minutes either way. In more complex cases, a critical question is which reference class you use for your data. For example if you are analyzing the behavioural characteristics of mammals, should you include data on human beings (or domesticated animals such as cats) whose behaviour is significantly different from other mammals? Should we distinguish between farm animals such as cows or pigs and completely wild animals such as lions and tigers?
Chapters 12 and 13 deal with God.
Chapter 12 looks at the ‘Argument from Design’ where it is argued that the universe is not completely chaotic, it is sufficiently orderly for the evolution of life, including ourselves. Is this a reason to believe in a divine creator? How likely is it that that the universe could be orderly without some sort of creator? We are considering probability again. So the question is ‘the God hypothesis’ versus ‘the no God hypothesis’, but how can this be determined one way or the other in terms of probability? Priest’s conclusion (p90) is that it is more likely that there is no creator. Other philosophers might argue that the entire question is simply indeterminate.
Chapter 13 considers 17th century French philosopher Pascal’s wager that it is more prudent to wager that there is a God than to wager there isn’t one. If you wager there isn’t a God the risk is that if there is a God after all, you’ll be punished and will go to Hell. But if you wager there is a God and there isn’t one, you’ll be none the worse. So the sensible thing is to wager that there is a God and to behave accordingly. Priest analyses the various possibilities in considerable detail, although his discussion seems inconclusive.
APPENDIX (a) - SYLLOGISMS
Examples of Valid and Invalid Syllogisms:-
‘All A’s are B’s, all B’s are C’s, therefore all A’s are C’s.’
For example, all cats are mammals, all mammals are warm-blooded animals, therefore all cats are warm-blooded animals.
‘All A’s are B’s, no B’s are C’s, therefore no A’s are C’s.’
For example, all cats are mammals, no mammals are reptiles, therefore no cats are reptiles.
There are many other valid syllogisms of various patterns.
‘All A’s are B’s, some B’s are C’s, therefore some A’s are C’s.’
On this basis we could say that human beings are bipeds, some bipeds are birds, therefore some human beings are birds. The premises are true, but the conclusion is false.
‘All A’s are B’s, all B’s are C’s, therefore all A’s are C’s and some A’s are D’s.’
It might be true that some A’s are D’s, but this does not follow from the premises as stated which only tell us about A’s, B’s and C’s.
‘All A’s are C’s, all B’s are C’s, therefore all A’s are B’s.’
On this basis we could say ‘all cats are mammals, all dogs are mammals, therefore all cats are dogs’.
There are plenty of other examples, often much less obvious. They are very common.
Note that an argument can be logically valid but its conclusion can nevertheless be false if one or more of its premises is false. For example ‘all men have X-ray eyes, Bob is a man, therefore Bob has X-ray eyes’. The argument is logically valid but the first premise is false because no men have X-ray eyes. Therefore the conclusion is false.
APPENDIX (b) – TRUTH TABLES
Using ‘T’ for true and ‘F’ for false, a sentence can have a ‘truth-value’ of either T or F. Sentence A and Sentence B can each have the truth-value T or F. They can be conjoined by the word ‘and’ to make ‘A and B’, called sentence C. Is C true or false?
If Sentence A is true and Sentence B is false, Sentence C is false.
If A is false and B is true, C is false.
If A is false and B is false, C is false.
If A is true and B is true, C is true.
We can put this into a tabular format called a truth-table:-
A B C
T F F
F T F
F F F
T T T
We can see that C only has the truth-value T when sentences A and B both have the value T.
A truth-table for negation is where if ‘A’ is true, ‘not-A’ is false, and if ‘A’ is false, ‘not-A’ is true:-
In the truth-table for disjunction, ‘A or B’ is true if A or B or both are true but false if both A and B are false:-
A B A or B
T F T
F T T
F F F
T T T
APPENDIX (c) – A Note on Negation and Double Negation
If a sentence P is true then its negation not-P is false. It follows that double negation, ‘not(not-P)’ is true and is equivalent to P.
To say ‘P and not-P’ is self-contradictory and therefore false. On this basis, it follows that ‘not (P and not-P)’ is true. This means that two contradictory sentences cannot both be true.
It follows in turn that two contradictory cannot both be false. Applying the double negative, we have:
‘not (not-P and not not-P)’.