Meaning and Truth

Semantics is hard and so it’s quite likely you’ll find this the hardest chapter in the book. This is where you scale the summit of linguistics, and I hope you’ll find the effort worth it; the view from the top is pretty impressive.

One of the reasons semantics is so hard is that its subject matter is hard to define. As we’ve seen, phonetics is about the sounds of language, phonology is about how those sounds pattern in linguistic systems, morphology is about the structure of words and syntax is about the structure of sentences. In all these areas, the subject matter is pretty apparent, and in fact we didn’t spend any time worrying about it in the previous chapters.

Semantics is about meaning. But what is meaning? What is it about the phoneme sequence /kæt/ that means furry feline? Why does /dɒg/ mean canine mammal? One thing we can certainly say is that these meanings – or whatever they are – are conventional and arbitrary. The English word dog (or /dɒg/) means nice canine, but if you’re French it’s chien, if you’re German it’s Hund, if you’re Welsh it’s ci and so on. These are just the conventions of the different speech communities. The relation between the phonological shape of a word and what it means is arbitrary; this idea is known as the arbitrary nature of the linguistic sign (and was put forward just over a hundred years ago by the Swiss linguist Ferdinand de Saussure). This idea is connected to duality of patterning (introduced in Chapter 3), in that the second level of patterning (the one involving meaningful elements) is arbitrarily connected to the first (combining phonemes).

But that doesn’t help with really understanding meaning. The main reason this is so difficult is that meaning relates linguistic stuff (words, morphemes, sentences) to non-linguistic stuff. That’s what talking about stuff involves. But how? One completely impractical answer is to say that, since meaning is the relation between language and non-language, semantics involves studying everything except linguistics. If that’s right, we might as well end this chapter here and log on to Wikipedia. But all the knowledge in the world (or Wikipedia), for example everything zoologists and biologists know about cats and dogs, doesn’t really tell you what the words cat and dog mean. Still less does it tell you how the sentences The cat bit the dog and The dog bit the cat mean what they do and why and how they mean different things. The basic problem is that any attempt to say what meaning is involves language and so we’re defining language in terms of language. Is there any way to break the circle?

The best, although nonetheless highly problematic, idea about meaning is to say that meaning can, in many cases, be reduced to reference. This is clearest in the case of proper names: J.K. Rowling, for example, refers to the well-known author. So we could say that that’s the meaning of this expression. The intuitive idea here, which is quite appealing at least at first, is that words label things. This approach seems to break the circle – we’re relating words to things.

So, Clover refers to a certain specific cat, Sim to another one, Wonky-Head to another one and so on. The NP the cat, refers to some known cat; a cat to any individual cat, known or unknown, and, maybe, cat to all possible cats. The problem for this approach concerns non-existent entities, which we can quite happily – and meaningfully – talk about. What does unicorn refer to? If there are no unicorns, it doesn’t refer to anything. But the word is not meaningless, and neither does it mean the same as nothing, which seemingly refers to, well, nothing.

Maybe we could say that at least words like unicorn refer to concepts in our minds (and perhaps words like cat do too). Fine, except we don’t really know what concepts are. Are there concepts for which there are no words, perhaps waiting (where?) to be referred to when we come up with the right word? Are there words which designate impossible concepts (certainly there are phrases, like a round square)? We just don’t really know. Linguists, philosophers and psychologists have speculated about these fascinating questions but with little real result. One thing at least seems clear: it’s ok talking about cats as a concept as we know that they exist independently of our concept of them (or at least we think they do. . .). But abstract concepts, such as justice,sincerity or three (numbers are highly abstract concepts!) are very hard for most of us to think about without thinking of the word. Are the words and the concepts really separate from one another? Is our set of concepts just our vocabulary? If so, then we’re locked back in the linguistic circle.

A way out of this is to say that the intuitive notion of meaning really has two parts to it: reference, the thing(s) a word labels in the world, and sense, an intrinsic property of the word that gives it its power of referring to something. Then we can say that the reference of cat is furry felines, and the sense of the word is what makes it mean furry felines, clearly somewhat related to but not identical to the concept of cat. And this way you don’t have to be a zoologist to understand what the word cat means.

There are two advantages of this approach. First, we can say that a word like unicorn has no reference (in our world), because there aren’t any unicorns. But it still has sense: if there were some unicorns, it would refer to them and not to other stuff (or nothing). And in worlds of our imagination populated by unicorns, it refers to them.

A second advantage is that it allows us to understand how two words or expressions can refer to exactly the same thing and yet, in an obvious intuitive way, mean something different. So, for example, J.K. Rowling and the author of the Harry Potter books both refer to the same person, but they mean different things because they have different senses (and you can imagine a world in which someone else had written the Harry Potter books and then these two expressions wouldn’t refer to the same person).

So let’s conclude that words, and at least some phrases, have reference (that’s what they label in the world) and sense (what makes the label stick). What about sentences? At first this question looks even harder. But two things can be said here, and they point together towards a very interesting approach to semantics.

The first thing about sentences is that – for the most part – their meaning derives from the meaning of their parts. Let’s look again at a sentence from the previous chapter:

This sentence obviously means something involving cats, mice and eating, whatever the senses and references of these words. Also, the syntax of the sentence is relevant; keeping the same words but putting them in a different order changes the meaning of the sentence, as (2) shows:

This sentence still involves cats, mice and eating, but it clearly relates to a different situation from that described by (1). So, sentence meaning depends on the words that make up the sentences and how they are put together: sentence meaning is compositional.

The second thing will give us a basis for the rest of our discussion of semantics. If you understand what a sentence means, you understand what the world would have to be like for that sentence to be true. So, if I say It’s raining, you can look out of the window, or up at the sky, or whatever and say Yes, it is or No, it’s not. If you’ve understood the sentence, you know what the world would have to be like for the sentence to be true (and, by implication, if the world isn’t that way, you know the sentence is false). So, to know the meaning of a sentence is to know the conditions that would make that sentence true. In this way, we can identify the meaning of a sentence with its truth conditions. This approach to semantics is known as truth-conditional semantics, and we’ll explore it a little bit in the remainder of this chapter.

We can link this idea about sentence meaning to sense and reference by saying that the reference of a sentence is True or False, while the sense of a sentence is what makes it true or false. This in turn depends on the meanings (references and senses) of the words that make the sentence up, and on how they are combined. This approach works best, and was developed for, declarative sentences. Other types of sentence, such as interrogatives as in Did the cat eat a mouse? don’t seem to fit so well. To cut a long story short, we can integrate this kind of interrogative into the truth-conditional approach by taking it to mean something like ‘Either the cat ate a mouse or the cat didn’t eat a mouse, which is true?’ You might note that an appeal is being made here to a more knowledgeable interlocutor, who is presumed to be able to provide the answer – I’ll come back to the question of speaker-hearer interaction, and, briefly, the nature of interrogatives, in the next chapter.

So let’s look in a bit more detail at how all of this works. But before looking at compositional, truth-conditional semantics any further we need to take a quick detour into formal logic, for reasons that will become apparent.

Logic

Logic is the study of deductions, inferences and related matters. The laws of logic are sometimes called ‘the laws of thought’. Our intuitions about logic allow us to say that the reasoning in (3) is cogent while that in (4) is not:

1. • All cats are mortal.

   • Clover is a cat.

   • Therefore Clover is mortal.

1. • All cats are mortal.

   • Fido is a dog.

   • Therefore Paris is the capital of France.

This much is pretty clear. But what has this kind of thing got to do with semantics? The connection lies in the notion of truth. Logically sound inferences like that in (3) are truth-preserving: if the premises (the first two lines) are true, then the conclusion (in the last line) just has to be true; if you deny it you’re being ridiculous. So the rules of logical inference are really also about preserving truth. Now, if the meaning of a sentence has to do with what makes it true, then it’s clear that logic and semantics are closely connected. In fact, as linguists, we can use two and half millennia of logic (since the time of Aristotle) to give ourselves a leg-up with our notion of truth, and therefore of meaning.

Logical truth is not factual truth, as the inference in (5) shows:

1. • If the moon is made of green cheese, then everyone is happy.

   • The moon is made of green cheese.

   • Therefore everyone is happy.

This is a valid inference, even though both the second premise (in the second line) and the conclusion are factually false. The logic is nonetheless sound: if the second premise were true, the conclusion would hold.

In fact, the validity of logical inferences depends on the interaction and meanings of certain logical words, such as if, then and not. We don’t really need to worry about the other parts in examples like (3) and (5) since logic isn’t about facts. For example, any inference of the following form is valid, where p and q stand for any sentence (or in logic, any proposition):

1. • If p, then q

   • p

   • Therefore q

In (6), we are displaying what is known as the logical form of the inference. Logical form is a key concept in logic because of the generalisation that if an inference of a certain logical form is valid or true, then all inferences or sentences of that form are valid and true. So formal logic is mainly about encoding valid inferences by putting them into logical form.

Again, what has this got to do with semantics? The connection is that truth, as we have suggested, is a core notion in semantics. We can identify the meaning of a sentence with its truth conditions, as we saw. Our theory of meaning is therefore going to involve truth (especially if sentences refer to the True or the False). Logic is all about truth, too. Logical notations which display truth relations among sentences can be thought of as expressing certain aspects of meaning. So we can, in a way, hijack logic and logical form in the service of truth-conditional semantics. So now let’s do a bit of logic.

There are two main kinds of formal logic: propositional logic and predicate logic. We have to look at propositional logic first, as we need that in order to understand predicate logic.

Propositional logic, as the name implies, deals with relations among propositions (roughly sentences – we’ll come back to the differences between sentences and propositions in the next chapter). Propositional logic is mainly concerned with the connectives, the ‘logical words’ like if and and, which connect propositions. The connectives are known as truth-functional because they always affect the truth of the propositions they connect in the same systematic way.

There are five truth-functional connectives in standard propositional logic: negation, conjunction, disjunction, implication and equivalence. Let’s look at each of these, and introduce the symbol for each one, in turn, using one of the standard logical notations.

First, negation. Actually this isn’t strictly speaking a connective, as it doesn’t connect two propositions, but just affects one. The symbol for negation is ¬. So ¬p (read ‘not-p’) has the opposite truth value to p. Whenever p is true, ¬p is false, and vice versa. We can summarise this in a simple truth table as in (7):

1. (7)

   p	¬p
   true	false
   false	true

This actually tells us the truth-conditional meaning of ¬, and comes close to the meaning of not in English: if it’s raining is true (look out the window!), it’s not raining is false and vice versa.

Conjunction is symbolised &. The conjunction of two propositions, p & q, is true just where (in Logic-Speak, if and only if) both p is true and q is true; otherwise p & q is false. The truth table for conjunction is:

1. (8)

   p	q	p & q
   t	t	t
   f	t	f
   t	f	f
   f	f	f

The meaning of & is close to English and; if I say Clocks are big and machines are heavy, this is true just in case (if and only if) Clocks are big is true and Machines are heavy is true. If either or both of these sentences is false, then the sentence which conjoins them with and is false.

Next, disjunction, symbolised v. The disjunction of two propositions, p v q, is true if and only if either p or q is true on its own. The truth table for disjunction is:

1. (9)

   p	q	p v q
   t	t	t
   f	t	t
   t	f	t
   f	f	f

Actually, there are two kinds of disjunction, inclusive and exclusive. The table in (9) is the truth table for inclusive disjunction, which approximates to ‘and/or’. You can see that the first line of (9), where both p and q are independently true, makes p v q true; this is just the same as conjunction, as you can see by checking the first line of (8).

Exclusive disjunction, on the other hand, corresponds roughly to ‘either/or’. It is sometimes written ⊕. The exclusive disjunction of p and q, p ⊕ q, is true if and only if either p or q is true, but not both together. The truth table is (10), which you can compare with (9) to see that the first line is different:

1. (10)

   p	q	p ⊕ q
   t	t	f
   f	t	t
   t	f	t
   f	f	f

Next comes implication. This one is a bit trickier than the last three, as it’s not quite as intuitive (for reasons we’ll touch on in the next chapter). Implication is written →. So p → q (‘if p then q’) is true where either p is false or q is true. The truth table is:

1. (11)

   p	q	p → q
   t	t	t
   f	t	t
   t	f	f
   f	f	t

Take the sentence If it’s raining, then I’ll take an umbrella. Obviously, this is true when it’s raining and I take my umbrella (first line of (11)). It’s also true if I take my umbrella and it isn’t raining – not so obviously in this case (second line of (11)). On the other hand, it’s false just when it’s raining and I don’t take my umbrella (third line). It’s also, rather vacuously, true if I don’t take my umbrella and it’s not raining.

The second line of (11) is conceptually important. It really says that nothing follows from a false proposition. False propositions are of no use to logicians as inference is about the preservation of truth.

The final connective is equivalence (also known as the biconditional). This means ‘exactly when’, ‘only when’, ‘just when’ or, more formally, ‘if and only if’ (abbreviated as ‘iff’). It’s written as ↔. Here’s the truth table:

1. (12)

   p	q	p ↔ q
   t	t	t
   f	t	f
   t	f	f
   f	f	t

Comparing this truth table with the one in (11), you can see that the only difference is the second line. (This, by the way, is what makes the non-lawn-mowing trick just described work with implication, but not with equivalence).

In a way, when we give these truth-functional definitions of the logical connectives, we are defining part of the meaning of words like not, or, and and if by giving their logical form. So here we see the connections among logic, truth and meaning and how logic can be a tool for semantics.

Now let’s go back to our earlier deductions. We can restate (6) as follows:

1. (13)

   p → q

   p

   ∴ q

You should be able to see that, on the basis of the truth table in (11), (13) holds, whatever truth value of p and q you start from. You can also work out some neat deductions. For example, whenever p → q is true, its contrapositive ¬q → ¬p is also true. This follows from the truth table in (7) and the one for implication in (11).

So propositional logic formalises some of our intuitions about valid inferences. As such, it expresses aspects of logical form. Hence, if we think that truth is central to understanding meaning, it expresses some aspects of meaning.

But propositional logic is pretty limited. It doesn’t capture anything like enough of our intuitions regarding logical deductions or the truth conditions of sentences. To see this, look again at the inference in (3):

1. • All cats are mortal.

   • Clover is a cat.

   • Therefore Clover is mortal.

We can write this in propositional logic as (p & q) → r (i.e. ‘if both all cats are mortal (p) and Clover is a cat (q), then Clover is mortal (r)’; as in school algebra, you do the calculation inside the brackets first). You can work out, using the truth tables for conjunction and implication, that if p and q are both true, then r must be too. But exactly the same holds for the non-inference in (4):

1. (4)

   All cats are mortal.	(p)
   Fido is a dog.	(q)
   Therefore Paris is the capital of France.	(r)

Again, (4) has the form (p & q) → r in propositional logic and so the truth of r will follow from the truth of both p and q. But of course there’s a big intuitive difference between (3) and (4). The truth of the conclusion in (3) seems to follow inevitably from the truth of the two premisses (as Aristotle noted twenty-five centuries ago), but this isn’t so in (4).

This is where predicate logic comes in. Predicate logic looks inside propositions and breaks them down into predicates (roughly corresponding to verbs and adjectives, or VPs and APs, in syntax) and arguments (roughly corresponding to nouns or NPs). A simple sentence like (14) would be written as (15a) in propositional logic and as (15b) in predicate logic:

1. 1. a. p

   2. b. W(c)

In order to see how predicate logic can tell us the difference between (3) and (4), we need to pay attention to one crucial word that shows up in the first line of both: the word all. This is an example of a quantifier, a word that expresses a quantity of something. So all cats expresses a quantity of cats (the whole lot), for example. The other really important quantifier in predicate logic is some: some cats also expresses a quantity of cats, but a different (and seemingly smaller) quantity than all cats.

Quantification, including words like all and some in NPs, is central to predicate logic and very important for semantics, as we shall see. The main reason for this is that quantified expressions, like all cats in (3) and (4), are expressions which don’t involve individuals (unlike Clover, J.K. Rowling, Priscilla, Bellatrix, Mary, etc.), but rather a quantity or group of individuals of some kind – hence the name. Quantified expressions have special logical properties. To see this, compare the next two sentences:

1. 1. a. Clover is wise and Clover is not wise.

   2. b. Some cats are wise and some cats are not wise.

(16a) is a logical contradiction; it’s always false. In propositional logic, we write it as p & ¬p. If you apply the truth tables for conjunction and negation, you’ll see that this just has to always come out false (pragmatic considerations which we’ll look at in the next chapter might allow you to say (16a) in a certain situation without contradicting yourself, but let’s leave that aside for now). So propositional logic does a good job of telling us about (16a). But it treats (16b) exactly the same way, and yet (16b) doesn’t have to be false at all (pragmatics or not). It doesn’t express a contradiction; instead it expresses a proposition that may or may not be true, depending on how the world is.

A further point, which will help us see the difference between (3) and (4), concerns sentences like (17):

We could try writing (17) as (18) in predicate logic, on the model of (15b):

Here W is still ‘are wise’ and a stands for ‘all cats’. But, as we said, ‘all cats’ isn’t (aren’t?) an individual we can pick out and so representing it in predicate-logic notation as a is misleading. If you think about it, what (17) really says is something like ‘any cat you can find will be wise’, or ‘if something is a cat, then it’s wise’ or even (entering Logic-Speak now) ‘for everything in the universe, if it is a cat then it is wise’. Predicate logic, but not propositional logic, allows us to express this. Let’s see how.

This is where the quantifiers come in. In predicate logic, we write (19) as (20):

As with the phonological rules and PS-rules seen in earlier chapters, (20) is like a chemical formula or an algebraic equation expressing a great deal of information in a very concise way. To see what it says, we need to go through the symbols slowly, systematically and step by step. So let’s do that.

The first symbol, the backwards capital E, ∃, is the existential quantifier. It means ‘There is at least one’, or, more simply, ‘some’. The x following it is a variable like in school algebra, except in school algebra it means ‘any number’, while here it means ‘any individual’. Then (note the brackets; square and round brackets are used just to distinguish them), we have the predicate C for ‘is a cat’. After that, we have another x, which means ‘any individual’ again. But: whatever value x gets here, it must get everywhere in the formula (or, more precisely, everywhere inside the square brackets following the quantifier). So if we ‘fill in’ x with one value (say, Clover, a good example of an individual) in one place in the formula, all the other x’s have to be filled in the same way, so if one x is Clover, all the other x’s inside the square brackets must be Clover. Next we have good old conjunction; here it conjoins C(x) and W(x), and it works just the same way as in propositional logic, that is it follows the truth table in (8), where we treat C(x) and W(x) as if they were propositions like p and q. Then we have W, still meaning ‘is/are wise’ and then another x requiring the same value as the others.

So we translate (20) into Logic-Speak as ‘There is at least one something or other, such that that something or other is both a cat and is wise’. That’s the Logic-Speak version, which sticks very close to the formula and as such is somewhat stilted, to put it mildly. In slightly more everyday English, we could restate this as ‘Something is both a cat and wise’ or ‘There is at least one wise cat’. And here we’re getting pretty close to (19).

Why go to all this trouble to translate ‘some’ or ‘some cats’ this way? The reason is that using quantifiers in predicate logic like this allows us to see the difference between (16a) and (16b). (16a) comes out as:

Conjunction and negation are just the same as in propositional logic. Since ‘W(c)’, the combination of a predicate and an argument, is a proposition, (21) is exactly equivalent to p & ¬p and hence a contradiction. That seems right: Clover can’t be both wise and non-wise at the same time – he has to be one or the other (of course, he might be wise at one time but not at another, as he gets older, perhaps, but I’m leaving aside the complications of tense here).

Given the way we’ve just translated (19) using the existential quantifier, (16b) comes out in predicate logic as:

The first part of (22) is the same as (20), and we just took that apart. We saw that it means ‘There is at least one wise cat’, roughly. The second part, after the conjunction symbol &, is almost the same as the first part, except for two crucial details. First, the variable is y, not x, which means ‘something or other’ still, being a variable, but a potentially different something or other from the x one in the first conjunct. The other difference is the negation in front of W(y). What the second part of (22) says, in Logic-Speak, is ‘There is another something or other which is both a cat and not wise’, or ‘There is at least one cat which is not wise’. Putting this together with what we said about (20), we get the whole proposition, in Logic-Speak: ‘There is at least one something or other which is a cat and is wise, and there is at least one other something or other which is a cat and is not wise’. Closer to everyday English, ‘There is at least one wise cat and there is at least one non-wise cat’. This is clearly not a contradiction and is a decent gloss of (16b). So we see that using the existential quantifier and its associated variables in formulae like (20) and (22) and treating NPs like some cats in this way can bring out very clearly the different logical and semantic properties of sentences like (16a) and (16b).

What about a statement like ‘All cats are wise’, as in (17)? In predicate logic, we use the other quantifier, the universal quantifier. This is written with an upsidedown A, ∀, and means roughly ‘all’ or ‘every’. So we write (17) as:

Let’s go through (23) in the same way as we did with (20). The universal quantifier ∀ means roughly ‘every’, so ∀x means ‘for every something or other’. C(x) means ‘that something or other is a cat’ and W(x) means ‘that same something or other is wise’ (remember that occurrences of the same variable inside the square brackets must have the same value). The arrow → is the symbol for implication, ‘if-then’, just as in propositional logic. Since C(x) and W(x) are equivalent to propositions, C(x) → W(x) means ‘if x is a cat then x is wise’. So the whole thing means, in Logic-Speak, ‘For every something or other, if it is a cat, then it is wise’. Or: ‘if anything is a cat, then it is wise’, or ‘every cat you find will be wise’. So you can see that this pretty much captures what (17) means.

Now, at last, we are in a position to see how predicate logic can tell us the difference between (3) and (4). Let’s go back to (3), along with its predicate-logic translation:

1. (24)

   All cats are mortal.	∀x [C(x) → M(x)]
   Clover is a cat.	C(c)
   Therefore Clover is mortal.	M(c)

In predicate logic, the validity of the inference comes from simply substituting values for the variables. We put c (for the individual argument ‘Clover’) as the value of the variable x in the second line because C(x) and M(x) are in the brackets immediately following the universal quantifier. Then in the third line we have to put it in as the value of the second x. Implication does the rest (you can check this with the truth table in (11)), and so the inference must, inexorably, hold.

Now let’s look at (4) in predicate-logic terms:

1. (25)

   All cats are mortal.	∀x [C(x) → M(x)]
   Fido is a dog.	D(f)
   Therefore Paris is the capital of France.	Cof(p)

(Here D is ‘is a dog’, f is ‘Fido’, Cof ‘is the capital of France’ and p ‘Paris’, not to be confused with the p of propositional logic). Predicate logic clearly shows us that there are no logical connections between the statements in (25) and so no inference to make. So the difference between (3) and (4) emerges very nicely, thanks to the use of quantifiers, predicates and variables. These are the basic elements that predicate logic adds to propositional logic, and here we can see their usefulness.

Compositional Semantics

One might reasonably think that propositional and predicate logic are all very well, but what they can really tell us about the semantics of real languages (rather than some kind of Logic-Speak) is pretty limited. It’s easy to see what they can tell us about the meanings of words like not, and, if, some and all, but there’s rather more to language than that (although you’d be surprised how much you can do with propositional and predicate logic; we’ve only scratched the surface here).

In particular, there’s syntax, complete with recursive PS-rules, as we saw in the previous chapter. Can we connect the logical, truth-based semantics we’ve seen here to what we’ve seen in syntax?

The answer is yes, although here I can only show this in a very limited and simplified way. To see how this works, let’s take a very simple sentence:

As we saw in the last chapter, this sentence has a structure like (27), generated by the relevant PS-rules (in particular the rule S → NP VP, see (24) of Chapter 4):

(27)

This sentence has the logical form, in predicate logic, in (28), following the conventions for writing predicates and their arguments that we introduced in the last section:

Two questions arise if we really want to do a truth-based semantics for (28). First, how do we assign a truth value to (28)? Second, how do we relate the syntax in (27) to the logical expression in (28)?

The answers to both questions involve type theory, the theory of logical types. This is really a precise, and in some ways simpler, version of the school-grammar idea of defining syntactic categories in terms of vague semantic notions that we briefly mentioned in Chapter 4 (‘a noun is the name of a person, place or thing’). Type theory recognises two basic types: entities and truth values. These are written <e> and <t>.

As we’ll see, more complex types can be built up from these two basic ones. Sentences are of type <t>, as they correspond (usually) to propositions and so have truth values; they are true or false (as we’ve seen, they refer to the True or the False).

We can answer our first question – how do we assign a truth value to an expression like (28) – by introducing a standard way of expressing denotations. Denotations are a cover term for the different logical meanings of the different types; in this case, since we’re looking at the denotation of a proposition and propositions are of type <t>, i.e. they refer to truth values, the denotation is a truth value. We write this as in (29):

This looks pretty uninformative, but that’s because we’re using approximations to English both for the expression we’re interpreting (‘Sleep(c)’) and the way we’re expressing the truth conditions. Put it this way:

In (30), I’ve used Welsh to express the truth conditions of ‘Sleep(c)’ and it at least looks a bit more informative than (29). The truth conditions can in principle be stated in various ways. Welsh is not usually used (I used it here for illustration); English often is (for convenience). It’s also quite common to use an enriched version of predicate logic. Another interesting possibility is to take enriched predicate logic to represent the Language of Thought, the language of our mental belief systems. (Since if something is true then you believe it; our belief systems are part of our cognitive abilities that are related to but distinct from language, and an awful lot of our talk is about expressing our beliefs and trying to alter those of others). We don’t know much – or anything really – about the Language of Thought, so I’ll leave this fascinating possibility aside.

So we can see a way to state the truth conditions of a logical expression like (28). But how do we get from the syntactic structure in (27) to (28)? Here again we can use type theory to get us started. What we can try to do is to convert syntactic categories like S, NP and VP (and so on) into semantic types like <e> and <t>. It’s easy at first: S expresses a proposition, so it’s type <t>. NPs (and Ns) express an entity – Clover is clearly one of these – so NP is of type <e>. But what about VP?

This is where the key idea in compositional semantics comes in. What does the syntactic object [_VP sleeps ] in (27) correspond to in the logical expression in (28)? Obviously, the predicate Sleep, you might think. Well, not exactly; a predicate must have an argument, and Sleep, written just like that, doesn’t have one. If we say that [_VP sleeps ] corresponds to Sleep(c) then we’re saying, incorrectly, that [_VP sleeps ] somehow means ‘Clover sleeps’, which it obviously doesn’t. Instead, we need to put in a variable for c in (28): Sleep(x).

Now we’re getting somewhere. A simple VP like [_VP sleeps ] corresponds to the predicate-logic expression Sleep(x). But what does Sleep(x) mean? On its own, strictly speaking, nothing. Any expression containing a free variable, i.e. an x with no quantifier around, is uninterpretable. In the absence of a quantifier, semanticists ‘borrow’ a device from mathematics and use the λ-operator (λ is the Greek letter lambda) in order to express the intuition behind what the predicate on its own means. Although this looks very impressive, it’s really just a way to save the day for the variable, which, technically, can then be bound by λ. So we get the formula λx[Sleep(x)], which just means ‘those x’s which sleep’ (in Logic-Speak ‘the x such that x sleep’). This formula gives the denotation of the VP in our example, so we can say:

So our VP here means ‘those things which sleep’.

Now we can see a way to put things together. The VP sleeps means ‘those x which sleep’. You can think of an expression like this as a way to divide everything in the universe into two classes: things which sleep and things which don’t. In the first class, we have Clover, me, colourless green ideas, lions and indeed all mammals. In the second class, we have New York (according to Frank Sinatra), rocks, numbers and stars. So the meaning of [_VP sleep ] takes an entity and tells you whether or not it sleeps. In other words, it maps entities to truth values: for any entity (me, Clover, New York, etc.) it says whether or not that entity sleeps. So VPs (or, really VP denotations, [[VP]]) map entities <e> to truth values <t>. So we say that they take an <e> and give you a <t>; they are of type <e, t>.

So we can combine our syntactic representation and these ideas about type theory, decorating the tree with logical fairy lights which give the denotation of each node:

(32)

Remember that we said that Clover and NPs like that (proper names, for example), are of type <e>, an entity. Syntactically combining NP and VP to make S means semantically combining an expression of type <e> (NP) with one of type <e, t> (VP) to get one of type <t> (S). So, an entity (type <e>), such as Clover, denoted by an NP, combines with a predicate of type <e, t>, denoted by a VP, to give an S, of type <t>. There’s an operation akin to cancellation of the two <e>’s going on here. Given the denotations of the syntactic elements specified in this way, this amounts to saying that ‘Clover sleeps’ is true if and only if the denotation of ‘Clover’, a noun and an NP, the entity Clover, is ‘in’ the denotation of the VP, ‘those things that sleep’. Put another way, our sentence is true if Clover is among the sleeping entities and false otherwise. This is a pretty good rendering of what the sentence Clover sleeps means. So this gets the meaning of the sentence nicely. The syntactic combination of NP and VP corresponds to the semantic operation of putting the NP-denotation c (representing Clover) ‘in’ the VP-denotation λx [Sleep(x)]. This corresponds to substituting c for the variable in λx[Sleep(x)] and knocking out the λ (which was only there to bind the variable anyway), so we get Sleep(c) of type <t>. And that has the truth conditions I just stated.

There’s a beautiful and elegant mathematical basis for all this, but this isn’t the place to go into those details. Just one last thing: calling NPs and Ns type <e> is something of an oversimplification. It works for proper names like Clover and J. K. Rowling but not for all NPs. To see this, try a sentence with a quantifier in it:

We can give (33) the constituent structure in (34), taking all to be a D:

(34)

This is fairly straightforward (see PS-rule (28ii) of Chapter 4). The predicate logic expression for (34) is:

How do we connect (34) and (35)? This has been a major conundrum for semantic theory for many years. I won’t give the whole story here but just point two things out. First, as (35) indicates, we actually have to treat simple common nouns like mammal as predicates, having the logical form Mammal(x), etc., and so being, like VPs, of type <e, t>. A common noun tells you whether something is one or not: mammal divides the world into mammals and non-mammals, just as sleep divides the world into sleepers and non-sleepers.

So all has to convert mammals into an <e>. But actually it’s worse: we have to do something with the implication → too. To cut a long story short, quantifiers like all relate predicates to one another: what all says is that anything which belongs to the <e,t> mammal also belongs to the <e,t> sleep. This involves assigning a rather complex type to all, so I’ll spare you the details of how this is done.

This approach has some nice results. If simple common nouns are really of type <e,t>, then, just as the meaning of sleep is λx[Sleep(x)], the meaning of mammal is λx[Mammal(x)], and the meaning of life is given below:

*

By now, we’ve seen how everything fits together, at least in outline. We’ve gone from the writhing and wriggling of the organs of speech in phonetics, to how the phonological brain imposes patterns on the phonetic brawn, to the duality of patterning which permits a small number of phonemes to make a very large number of morphemes and words, to how the rules of syntax can make an infinite number of sentences out of those morphemes and words, to how a truth-based, logical semantics can interpret the syntactic structures and give them meanings, construed in a very precise way as truth conditions. This is how we can say – and understand – anything. Not bad, eh?

But this book is about making noises and influencing people. You might have noticed that people (and societies, cultures – that sort of thing) have been rather absent in our discussions of PS-rules, logical forms, type theory and so on. People use language to mean things, influence other people and do stuff (like building spaceships). And they do it very cleverly indeed! Time for a shift of gears, to look at how people manipulate syntax, and, in particular, logic, when they want to influence each other.