I think it might be a good idea to remind ourselves now and then about how to be a good thinker. One of the basics for that is, obviously, logic. So I’m going to start making some excerpts available here from an excellent book by Graham Priest titled Logic: A Very Short Introduction (Oxford University Press). It is excellent, brief, concise, and accessible. Everybody should buy a copy, preferably from your local independent bookstore, which you can locate here if you don’t know where the closest one to you is.
Validity: What Follows from What?
Most people like to think of themselves as logical. Telling someone ‘You are not being logical’ is normally a form of criticism. To be illogical is to be confused, muddled, irrational.
But what is logic? In Lewis Carroll’s Through the Looking Glass, Alice meets the logic-chopping pair Tweedledum and Tweedledee. When Alice is lost for words, they go onto the attack:
‘I know what you are thinking about’, said Tweedledum: ‘but it isn’t so, nohow.’
‘Contrariwise,’ continued Tweedledee, ‘if it was so, it might be; and if it were so, it would be: but as it isn’t, it ain’t. That’s logic.’
What Tweedledee is doing — at least, in Carroll’s parody — is reasoning.
And that, as he says, is what logic is about.
We all reason. We try to figure out what is so, reasoning on the basis of what we already know. We try to persuade others that something is so by giving them reasons. Logic is the study of what counts as a good reason for what, and why. You have to understand this claim in a certain way, though. Here are two bits of reasoning — logicians call them inferences:
1. Rome is the capital of Italy, and this plane lands in Rome; so the plane lands in Italy.
2. Moscow is the capital of the USA; so you can’t go to Moscow without going to the USA.
In each case, the claims before the ’so’ — logicians call them premises — are giving reasons; the claims after the ’so’ — logicians call them conclusions — are what the reasons are supposed to be reasons for. The first piece of reasoning is fine; but the second is pretty hopeless, and wouldn’t persuade anyone with an elementary knowledge of geography: the premise, that Moscow is the capital of the USA, is simply false.
Notice, though, that if the premise had been true — if, say, the USA had bought the whole of Russia (not just Alaska) and had moved the White House to Moscow to be nearer the centers of power in Europe — the conclusion would indeed have been true. It would have followed from the premises; and that is what logic is concerned with. It is not concerned with whether the premises of an inference are true or false. That’s somebody else’s business (in this case, the geographer’s). It is interested simply in whether the conclusion follows from the premises. Logicians call an inference where the conclusion really does follow from the premises valid.
So the central aim of logic is to understand validity. You might think this a rather dull task — an intellectual exercise with somewhat less appeal than solving crossword puzzles. But it turns out that this is not only a very hard matter; it is one that cannot be divorced from a number of important (and sometimes profound) philosophical questions. We will see some of these as we go along. For the moment, let us get a few more of the basic facts about validity straight.
To start with, it is common to distinguish between two different kinds of validity. To understand this, consider the following three inferences:
1. If the burglar had broken in through the kitchen window, there would be footprints outside; but there are no footprints; so the burglar didn’t break in through the kitchen window.
2. Jones has nicotine-stained fingers; so Jones is a smoker.
3. Jones buys two packets of cigarettes a day; so someone left footprints outside the kitchen window.
The first inference is a very straightforward one. If the premises are true, so must the conclusion be. Or, to put it another way, the premises couldn’t be true without the conclusion also being true. Logicians call an inference of this kind deductively valid.
Inference number two is a bit different. The premise clearly gives a good reason for the conclusion, but it is not completely conclusive. After all, Jones could simply have stained his hands to make people think that he was a smoker. So the inference is not deductively valid. Inferences like this are usually said to be inductively valid.
Inference number three, by contrast, appears pretty hopeless by any standard. The premise seems to provide no kind of reason for the conclusion at all. It is invalid — both deductively and inductively. In fact, since people are not complete idiots, if someone actually offered a reason like this, one would assume that there is some extra premise that they had not bothered to tell us (maybe that someone passes Jones his cigarettes through the kitchen window).
Inductive validity is a very important notion. We reason inductively all the time; for example, in trying to solve problems such as why the car has broken down, why a person is ill, or who committed a crime. The fictional logician Sherlock Holmes was a master of it. Despite this, historically, much more effort has gone into understanding deductive validity — maybe because logicians have tended to be philosophers or mathematicians (in whose studies deductively valid inferences are centrally important), and not doctors or detectives.
We will come back to the notion of induction later in the book. For the present, let’s think some more about deductive validity. (It is natural to suppose that deductive validity is the simpler notion, since valid inferences are more cut-and-dried. So it’s not a bad idea to try to understand this first. That, as we shall see, is hard enough.) Until further notice ‘valid’ will simply mean ‘deductively valid’.
So what is a valid inference? One, we saw, where the premises can’t be true without the conclusion also being true. But what does that mean? In particular, what does the can’t mean? In general, ‘can’t’ can mean many different things. Consider, for example: ‘ Mary can play the piano, but John can’t'; here we are talking about human abilities. Compare: ‘You can’t go in here: you need a permit’; here we are talking about what some code of rules permits.
It is natural to understand the ‘can’t’ relevant to the present case in this way: to say that the premises can’t be true without the conclusion being true is to say that in all situations in which all the premises are true, so is the conclusion. So far so good; but what, exactly, is a situation? What sorts of things go into their makeup, and how do these things relate to each other? And what is it to be true? Now, there’s a philosophical problem for you, as Tweedledee might have said.
These issues will concern us by and by; but let us leave them for the time being, and finish with one more thing. One shouldn’t run away with the idea that the explanation of deductive validity that I have just given is itself unproblematic. (In philosophy, all interesting claims are contentious.) Here is one problem. Assuming that the account is correct, to know that an inference is deductively valid is to know that there are no situations in which the premises are true and the conclusion is not.
Now, on any reasonable understanding of what it is to be a situation, there are an awful lot of them: situations about things on the planets of distant stars; situations about events before there were any living beings in the cosmos; situations described in works of fiction; situations imagined by visionaries. How can one know what holds in all situations? Worse, there would appear to be an infinite number of situations (situations one year hence, situations two years hence, situations three years hence, . . .). It is therefore impossible, even in principle, to survey all situations. So if this account of validity is correct, and given that we can recognise inferences as valid or invalid (at least in many cases) we must have some insight into this, from some special source. What source?
Do we need to invoke some sort of mystic intuition? Not necessarily. Consider an analogous problem. We can all distinguish between grammatical and ungrammatical strings of words of our native language without too much problem. For example, any native speaker of English would recognize that ‘This is a chair’ is a grammatical sentence, but ‘A chair is is a’ is not. But there would appear to be an infinite number of both grammatical and ungrammatical sentences. (For example, ‘One is a number’, ‘Two is a number’, ‘Three is a number’, . . . are all grammatical sentences. And it is easy enough to produce word salads ad libitum).
So how do we do it? Perhaps the most influential of modern linguists, Noam Chomsky, suggested that we can do this because the infinite collections are encapsulated in a finite set of rules that are hard-wired into us; that evolution has programmed us with an innate grammar. Could logic be the same? Are the rules of logic hard-wired into us in the same way?
To be continued.