1Anotes 2122 - All lectures of the 1st year philosophy course PDF

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All lectures of the 1st year philosophy course...

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1A Logic: How do I think formally?

Adam Rieger

Contents 1 Introduction 1.1 Arguments: some revision . . . . . . . . . . . . . . . . . . . . 1.2 Formal validity . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Why study logic? . . . . . . . . . . . . . . . . . . . . . . . . . 2 The 2.1 2.2 2.3 2.4

language of propositional Sentence functors . . . . . . Truth functors . . . . . . . . An artificial language . . . . Complex sentences: scope .

logic . . . . . . . . . . . . . . . .

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3 Truth and validity for the formal language 15 3.1 A truth table for any formula . . . . . . . . . . . . . . . . . . 15 3.2 Semantic entailment . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Arguments in natural language . . . . . . . . . . . . . . . . . 22 4 The 4.1 4.2 4.3 4.4

tableau method The tableau rules . . . . . . . . . . . . . Testing for consistency . . . . . . . . . . Testing for validity . . . . . . . . . . . . Tableaux for natural language arguments

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1 1.1

Introduction Arguments: some revision

In a previous section of this course you have encountered arguments. An argument, in the sense we shall be concerned with, has a number of premises and a conclusion. Roughly speaking, the conclusion is what the argument is trying to establish, and the premises give reasons for believing the conclusion. Here’s an example: (Argument 1) Either Prof Plum or Miss Scarlett is the murderer. But Prof Plum is not the murderer. Therefore, Miss Scarlett is the murderer.

To see what’s going on more clearly, we can write it out as a sequence, with the premises first, and the conclusion at the end (sometimes called ‘logic book style’, or ‘standard form’): (Argument 1) Either Prof Plum or Miss Scarlett is the murderer. But Prof Plum is not the murderer. Therefore, Miss Scarlett is the murderer. The argument above is, in a certain sense, good. But here’s another argument: (Argument 2) Either Prof Plum or Miss Scarlett is the murderer. But Prof Plum is the murderer. Therefore, Miss Scarlett is the murderer. This argument should strike you as bad. What makes the first argument good, and the second one bad? In the first, the conclusion follows from the premises. There is no way the conclusion can fail to be true, if the premises are true. 1

In the second argument, by contrast, the conclusion might be false, even though the premises are all true. This leads us to make the following important definition: an argument is valid if it is impossible for its premises to be true but its conclusion false. This is the most important thing in logic, so let’s put it in a box: An argument is valid if it is impossible for its premises to be true and its conclusion false. In a valid argument the premises are said to entail the conclusion.

1.2

Formal validity

Now for another example:

Photo: Jens Lelie

(Argument 3) The deer went down path A or path B. But it didn’t go down path A. Therefore the deer went down path B. Another valid argument. This argument has an interesting history. The Ancient Greek philosopher Chrysippus claimed to have seen a dog, following a trail in a forest in pursuit of a deer some distance ahead. Suddenly, the dog encountered a fork in the path. It sniffed at the left path, and, having not detected any scent there, ran quickly down the right path without sniffing. The dog, Chrysippus claimed, had worked out that the deer must have gone down the right-hand path by reasoning, not by smell. Dogs can do logic! (Controversial! And especially for the ancients, who doubted even that dogs were conscious.) Now here’s the important point: Arguments 1 and 3 have exactly the same structure. It’s not a coincidence that both the arguments are valid: it’s not 2

too hard to see that any argument that shares this structure, or form, will be valid.1 And that’s what logic is all about: we are going to try to classify which structures give rise to valid arguments. Logic is sometimes described as the study of formal validity, that is, validity which is due to form. The key idea of formal logic is that arguments can be valid because of their form or structure. To bring out the form, it’s convenient to use some symbols. For example, both the valid arguments we’ve had so far can be symbolized as follows: P ∨Q ¬P ∴ Q. The symbol ∨ stand for ‘or’, and the ¬ symbol is read as ‘not’.

1.3

Why study logic?

You’ve already, I hope, got some idea as to why studying arguments is a good idea: it’s helpful both in everyday life, and in Philosophy. Let me give you a a couple more. Firstly, we’re going to be putting language under the microscope. This is, again, generally useful, and particularly for a philosopher. For example, we’ll have to think carefully about what makes certain kinds of sentences true, and we’ll reveal certain kinds of ambiguity that one might easily not notice. The artificial languages developed by formal logicians are completely free of such ambiguity; philosophers therefore frequently find them useful to express themselves with complete clarity. Secondly, one way of looking at what logic attempts to do is turn (good) human reasoning into completely precise rules. This is clearly connected to the project of artificial intelligence, since the rules, once discovered, can be programmed into a computer. Can a computer, then, do everything that a human can, in terms of reasoning? In more advanced logic, there are very interesting results, in particular the incompleteness theorems of G¨odel, which show there certain limits to rule-based reasoning; for any particular formal system, there are mathematical truths which are beyond it. This is problematic for a certain natural view in the philosophy of mind, that the human mind is a piece of ‘software’ running on the hardware of the brain; it 1 This particular structure is known as disjunctive syllogism. Over the years people have given names to particular argument forms, often in Latin; don’t worry too much about them, but you may encounter them later on in philosophical work.

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seems that either the human mind is limited in certain unexpected ways, or that the hardware/software model cannot be correct.2

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The language of propositional logic

So our project is to look at the form of arguments. You will also have seen arguments like this one: All logicians are shortsighted. Adam is a logician. So Adam is shortsighted. This argument is formally valid (remember, this just means valid in virtue of its structure). But we won’t be discussing it in this course. The reason is that the notion of structure required to analyse this is quite complicated: it involves names (like ‘Adam’), what are called predicates (like ‘is shortsighted’), and special words like ‘all’ and ‘some’, called quantifiers. The logic required for this is predicate logic and will be studied in a more advanced course. For the moment we shall consider propositional logic.3 The distinguishing feature of this logic is its coarse-grained notion of structure: in propositional logic the smallest unit is a complete sentence. Thus, we will just be interested in complete sentences, and the way sentences are joined together to make other, longer and more complex, sentences.

2.1

Sentence functors

Well, how do we join sentences together to make new ones? We use things called sentence functors. These are simply strings of words with one or more gaps, such that when you plug the gaps with declarative4 sentences, you get another declarative sentence. We’ll show where the gaps are by using the letters ‘A’, ‘B’, etc, which will stand for the sentences we’ll fill the gaps with. Here are some examples of sentence functors: 2

If you want to investigate further, try searching for ‘G¨ odel incompleteness theorems’ and ‘Lucas-Penrose argument’. 3 We could also have called it sentential logic, and in fact this term is sometimes, though not frequently, used. We will not worry here about the difference between a sentence and a proposition. 4 A declarative sentence is one which says something is the case, and is thus true or false. Non-declaritives such as questions and orders, (‘What time is it?’ ‘Shut the door!’) are not true or false, and are ignored by mainstream logic.

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Only a buffoon could possibly believe that A Michel Barnier has repeated several times to Boris Johnson that A Nicola Sturgeon knows that A That A strongly suggests that B A because B etc etc.

2.2

Truth functors

There are lots and lots of sentence functors. But we’re only interested in a very small and particular set of them: the truth functors.

Suppose you and I play a little game. I tell you that A is a true sentence, but I don’t tell you what A is. Now, I give you a challenge: tell me whether the following is true or false (in the jargon, tell me the truth value): (*) Nicola Sturgeon knows that A. Well, you can’t do it! The game is unfair. Nicola knows some truths (eg, that Edinburgh is the capital of Scotland), but not all of them (eg, she doesn’t know that I had muesli for breakfast this morning). Whether (*) comes out true or false is going to depend on which truth gets plugged in for A. Suppose, though, we play the same game, with a different sentence functor: (**) It is not the case that A. Again, I tell you just that A is some true sentence (but not what it is). Can you tell me anything about the truth value of (**)? Well, this time you’re in a better position: for when A is plugged in, (**) will surely come out false. For example, if A is the sentence ‘Grass is green’, you will get 5

It is not the case that grass is green. But A could have been any other true sentence — it doesn’t matter at all that we chose ‘Grass is green’ as our example. It could equally well have been ‘Snow is white’, ‘2 + 2 = 4’ or whatever. We’d still get a false result. Similarly, if A is some false sentence, then what we get if we stick it into the functor is always going to be a true sentence. We can therefore construct a truth table: A It is not the case that A T F F T This simply records how the truth value — that is, the truth or falsity — of the complex sentence depends on the truth value of A. If A is true, ‘It is not the case that A’ comes out false; and vice versa. But we couldn’t do a truth table for the ‘Nicola Sturgeon knows that A’ functor, because we wouldn’t know whether to put T or F in the top line. A sentence functor with a truth table is called a truth functor. We’ll also often consider 2-place truth functors, such as ‘A and B’. The truth tables for these will have four lines, because there are four possible combinations of truth values for A and B. Thus: A T T F F

2.3

B A and B T T F F T F F F

An artificial language

From now on, we shall only be concerned with truth functors, that is, those sentence functors which have truth tables. Our notion of form is fixed by this: we will study the way longer sentences are built out of shorter ones using truth-functors. To explore this efficiently, we are going to develop a very simple artificial language. It’s going to have just the following: some letters ‘P ’, ‘Q’, ‘R’, . . . which stand for complete sentences; brackets ‘(’ and ‘)’; and some special symbols ‘¬’, ‘∧’, ‘∨’, ‘→’, and ‘↔’. Each of the special symbols (called connectives) will have a truth table. These connectives correspond roughly (we’ll talk about how roughly in a minute) to English sentence functors using the words ‘not’, ‘and’, ‘or’, ‘if . . . then . . . ’, and ‘if and only if’. 6

Not Here’s the truth table for the ¬ connective: A ¬A T F F T So ¬ just has the effect of reversing the truth value. As we’ve seen above, this is the same effect as the English truth functor ‘It’s not the case that . . . ’. More usually in English the word ‘not’ is buried somewhere inside the sentence. So, for example, to get a perfectly good symbolization of I have not visited Mull. we let P stand for ‘I have visited Mull’, and then write down ¬P ¬A is called the negation of A. And The connective ∧ has the following truth table: A T T F F

B A∧B T T F F T F F F

For many purposes, it behaves the same way as English ‘and’. We’ll also use it to symbolize ‘but’ and ‘although’. Thus all of the following can be represented in the formal language as P ∧ Q: It’s delicious and it tastes of fish. It’s delicious but it tastes of fish. It’s delicious although it tastes of fish. 7

Clearly we’re losing some subtlety here. We can’t expect our very primitive language to have all the expressive power of English. But this doesn’t matter as long as everything works as regards the truth values; the point is that the three sentences above are all true or false together (though on occasion, one or other of them might be a strange thing to say — for example, if you’re talking about a fish). Since we’re ultimately interested in validity, and so only in truth and falsity, not subtle shades of meaning, we can ignore the differences between ‘and’, ‘but’, etc. Another point involving ‘and’: this word is often used in English to connect words together, rather than sentences: Sven and Inga are Norwegian. But we can get round this by the following paraphrase:5 Sven is Norwegian ∧ Inga is Norwegian. One should be aware, however, that this sometimes doesn’t work: for example we cannot do the same thing with Sven and Roald are brothers. since we lose the information that they are brothers of each other. A ∧ B is called the conjunction of A and B . Or The symbol ∨ has the following truth table: A T T F F

B A∨B T T F T T T F F

It usually translates English ‘or’. There’s a simple but important point here. Suppose I’m thinking of taking up a new hobby, and I tell you I’m going to take up knitting or I’m going to have paragliding lessons. 5

I’m using here a kind of semi-logical language, where the sentences are still in English but we use the logical symbol to connect them.

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This much is clear: what I have said is false if both the constituent sentences are false, and true if one is true and the other is false. Question: suppose in the end I do both (possibly at the same time?). Did I lie? What do you think? Did I rule out doing both in saying the above sentence? In fact English usage does not seem to settle the matter. We seem sometimes to use ‘or’ in such a way that it allows ‘or both’ (called the inclusive or) and sometimes in such a way that it seems to suggest ‘but not both’ (the exclusive or). For example, if a menu says ‘Main course: spaghetti or pizza’ you may well get a raised eyebrow from the waiter if you ask for both. So an exclusive or is intended here. On the other hand, consider ‘If you’re under 18 or a foreigner, you can’t vote in the election’. Here it’s clearly an inclusive or — the ‘or both’ is understood. (If you’re under 18 and foreign, you certainly can’t vote in the election!) That the English language can get by with this kind of horrible ambiguity looks, from our position of logical sophistication, rather surprising. But, anyway, this is just the kind of thing that would muck things up for us. So we make a decision that our symbol will be inclusive or, hence the truth table above. A ∨ B is called the disjunction of A and B. If . . . then . . . The connective → has this truth table: A T T F F

B A→B T T F F T T F T

The relationship between this symbol and English is a bit more problematic than with the other symbols we have encountered. For most purposes, P → Q is a good translation of ‘If P then Q’, or ‘If P , Q’ (the word ‘then’ doesn’t always appear). Think of it this way: the only thing which is ruled out if I say something of the form ‘If P then Q’ is P being true but Q false. 9

For example, consider the (true) sentence If Fiona is in Glasgow, then she’s in Scotland. If you try in succession Glasgow, Abu Dhabi, and Auchtermuchty as locations for Fiona, you will find you will get the truth values TT, FF, and FT respectively.

So all these combinations of truth values are possible, with a true ‘if. . . then. . . ’ sentence. What you will never get, however, is TF. That would mean ‘Fiona is in Glasgow’ is true, and ‘Fiona is in Scotland’ is false: but there’s no way Fiona could be in Glasgow, without also being in Scotland. And this is just how it should be if you symbolize the sentence using →. Often in English ‘if’ occurs inside a sentence, rather than at the beginning. For example: I’ll go out on Saturday if it’s a nice day.

We treat this the same way as If it’s a nice day, then I’ll go out on Saturday. 10

These both rule out the same thing, namely my staying in if Saturday is nice. (We take the view that if I go out even though Saturday is horrible I haven’t gone against my word, even if I might have been misleading.) So, letting P stand for ‘I go out on Saturday’ and Q for ‘Saturday is a nice day’, we can symbolize each as Q → P . Now let’s consider ‘only if’. Suppose instead that I had said I’ll go out on Saturday only if it’s a nice day. What does this rule out? Strictly, only my going out when Saturday turns out to be horrible. So the right symbolization is P → Q. Rather surprisingly, the insertion of ‘only’ has the dramatic effect of reversing the direction of the arrow. It must be admitted that there are cases where using → gives very strange results. The problem comes from the fact that, by the truth table for →, P → Q is true whenever P is false (whether or not Q is). Suppose, for example, that it is (as a matter of fact) true that I will lecture next Thursday. Then If I don’t lecture next Thursday, the universe will end. will come out true if it is symbolized as P → Q, since P is false. This is odd, as, intuitively, the sentence looks false. (This is sometimes known as a paradox of material implication.) This opens up a fascinating area of philosophical logic known as the theory of conditionals. (Statements of the form ‘If A then B’ are often called conditionals. A is called the antecedent and B the consequent of the conditional.) There is, in fact, no agreement amongst philosophers about the meaning of the word ‘if’. (There are whole books and conferences about this, believe it or not.) Using → is by far the simplest way of handling conditionals, and we shall use it, while remaining aware that it may sometimes lead to peculiarities. If and only if The double arrow ↔ has this truth table: A T T F F

B A↔B T T F F T F F T 11

Notice that A ↔ B is true when A and B have the same truth value (that is, either they’re both true, or they’re both false), and false when the truth values of A and B differ. The usual English translation for ↔ is the rather awkward ‘if and only if’. For some reason the English language doesn’t provide a single word corresponding to this useful connective! (Another strange thing we’ve discovered about English.) So we logic types have invented one, ‘iff’, which is in standard use by philosophers and mathematicians. (We put things in emails like ‘I’ll come iff Fred can give me a lift’.) Like the English language, we could have lived without the ↔ connective, because we could always write (P → Q)∧(Q → P ) instead of P ↔ Q. (These formulae have the same truth tables — the jargon is that they...