3) The English sentence Grass is green is true if and only if the substance called grass has the property of being green in the real world.

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1 TRUTH CONDITIONAL SEMANTICS One distinctly new approach to the characterization of meaning in language was initiated in the nineteen sixties by importing ideas from the way logicians treated meaning. The point of departure was the attempt to understand meaning in terms of the truth value of a proposition. There is a long tradition in formal logic which holds that a proposition is either true or false. For example, consider (1). 1) A tiger is a mammal. The above statement, we know, is true because the animal called tiger belongs to the class of mammals. A sentence like (1) is called a proposition. For the time being, we can treat a proposition as equivalent to a declarative sentence which normally asserts (or denies) the truth of an action or a state of affairs. The truth/falsity of a proposition is always in relation to a world. What it means is this: it is conceivable that in a world different from the one we inhabit a tiger could a reptile. Therefore, ascertaining the truth value of a proposition presupposes a given world, i.e., a given set of statements that are true. One can say that the whole set of propositions characterizes that world. Now, it can be shown that knowing whether a statement is true or false is tantamount to understanding its meaning. It was a philosopher named Alfred Tarski who elucidated this. Tarski s famous example was (2). 2) Grass is green is true if and only if grass is green. When you hear for the first time a sentence like (2), it might sound trivial or even non-sense. This is because of an inherent problem with language. You are using the English language to talk about the meaning of a sentence in English. Notice that the first three words in (2) are within inverted commas. What (2) says can be paraphrased as (3). 3) The English sentence Grass is green is true if and only if the substance called grass has the property of being green in the real world. It is possible to replace the part in quotes in (2) by its equivalent in another language, say Malayalam and retain the rest unchanged, as in (2 ): 2 ) pullu pachcha aanu is true if and only if grass is green. (2 ) doesn t sound trivial because something is being said about the meaning of a Malayalam sentence in English. It is useful to make a distinction here. In (2) English is both the object language and metalanguage. Not so in (2 ). The part in single inverted commas in (2) as well as (2 ) is the object language, and the rest is in the metalanguage. It is customary in scientific discourses to draw this distinction between object language and metalanguage. Now, you will agree that this notion of truth contained in (2)/(2 ) is verifiable. One can check and confirm that grass is green. All right. How does it help in our pursuit of a theory of meaning? Truth as meaning:

2 Tarski s crucial insight was this: knowing the meaning of a sentence is equal to knowing how the world should be for the sentence (strictly, the proposition) to be true. Given the fairly uncontroversial stand that we use language to talk about the world, this is what we mean by meaning. It then follows that in principle, one can build a theory of meaning based on the truth conditions. This was the beginning of the truth conditional theory of meaning. Model theory The particular truth conditional theory of language that this course will present is known as Model Theoretic Semantics. This is borrowed from logic. We will introduce the technical details of the model theory gradually, trying to focus on the intuitive aspects at each step. Before we go into that there are a couple of things to be emphasized. One, the model theoretic semantics is an extensional theory, out and out. Recall from Unit xx the distinction between sense and reference. Frege used the terms Bedeutung and Sinn, which are usually rendered in English as reference and sense, respectively. Extension is another term for reference. When we say that model theoretic semantics is an extensional theory, what is asserted is that the meaning of a linguistic expression is what it refers to. We already noted that the meaning of a sentence is equated with its truth value True/False. Therefore, technically the extension of a proposition is a truth value. What about the constituents of a sentence? This question takes us directly into a second important assumption of model theory, namely, the meaning of a proposition is compositional. That is, the meaning of a sentence is composed out of the meanings of its constituents, in other words, the phrases that make up the sentence. Therefore, it becomes mandatory for this theory to spell out the extensions of phrases. Let us address this task. What is the extension of a name? Consider the name Noam Chomsky. We know we are using this name to refer to an individual who happens to be a professor of Linguistics at the Massachusetts Institute of Technology. That particular individual is the referent of the linguistic expression Noam Chomsky. It is important to bear in mind this distinction. The name, or what we call in grammar a proper noun is a linguistic object, the individual who answers to that name is an object in the world. The individual is the extension of the linguistic expression. (You can compare this, if you like, to the standard practice nowadays in software companies and similar establishments for the staff to hang around their neck a name plate.) At this point, you may wonder what we do with a name like James Bond. There is no individual in the real world who can be identified as its referent (i.e, extension). But, it is a linguistic expression as much as Noam Chomsky is. This is a problem with all fictional characters, and also with certain creatures appearing in fables like the dragon. There are no dragons or centaurs in the real world, but as linguistic expressions they do have a meaning. They have a sense, but do they have an extension? We will answer this question later. Next let us turn to common nouns. What is the extension of doctor? We want to adopt a simple procedure in determining the extension. Just as we did in the case of names, let us point to an object and ask the question Is this (a) doctor? if the answer is YES, we select that object and put it in a basket. Then we point to another object and ask the same question. In this second case the answer could be NO (assuming that we were pointing to an umbrella), then we do not select that object. In this fashion, we keep on asking the question with as many objects as there are in the world, and whenever

3 the answer is YES, we select and it and put it in the basket. When we have exhausted all objects and completed the exercise, whatever is there in the basket is the extension of the expression doctor. You know from this description that we are not talking here about a procedure we can actually perform and complete, but of a project that can in principle be carried out. Scientists are fond of this, they call it a thought experiment. The upshot of this is simply that the extension of doctor is nothing but the set of all doctors. A Digression into Set Theory As you have noticed, we are invoking set theory, a topic in your school mathematics. Set theory is actually a very powerful tool in the hands of a mathematician or logician. A notion of a set is very simple, in fact, nothing can be simpler. A set is a collection of objects. Period. Even though, in common parlance we use the word set in phrases like the set of rules and a set of IT professionals, a set need not necessarily consist of members possessing a common property. Any collection of objects is a set. Thus, (4a-e) are all examples of sets. 4) a. {Obama, Pluto, 8, Charminar} b. {0,1,2, 3,4} c. {all former students of IIT, Madras} d. {square root of 2} e. {} Here we have to make clear that what you have in (4a-e) is only a way of representing a set. There are four objects that comprise the set (4a) the objects that make up the sets are not the names as we have written them, but the objects themselves. That is, the man Obama who happens to the President of the US, the celestial object Pluto, the number 8 and the historical monument in the city of Hyderabad. Let us call this set A. There is nothing in common between the members of A, yet they form a set. Similarly, let us call (4b) as set B and so on. The members of B can be correctly described by a description the set of all natural numbers that are less than five. C is evidently a large set, and it is difficult, though not impossible, to write down all the members in the set C. Set D has only one member, it is called a singleton set. And finally, what about the set E? No members, yet it is a set. A set that has no members is a null set. A null set is denoted by the Greek letter φ (pronounced phi ). 5) A set is known by its membership. There are set theoretic operations like union, intersection, complement etc. which you were taught in school. It will be good if you could quickly revise those lessons. Let us now return to the topic of the extensions of linguistic expressions. The extension of a linguistic expression doctor can be expressed using set theory thus: the set of all x such that x is a doctor. Suppose we call this set S. Take any object in the set S and ask the question Is this a doctor? the answer is YES.

4 There are more difficult things in language like abstract nouns, say kindness. We will not worry about them at this point. Our humble attempt is to construct a model with words for which we can fix the extension without much trouble. Therefore let us proceed. We next look at simple adjectives. Take tall. What is its extension. The logician has the answer: set of all x such that x is tall. Notice that tall denotes a property. If an object in the world, say John has that property then we can put John in the set. If Bill is not tall, he will not be picked to be in the set, and so on. (Again, for our purposes, ignore the fact that tall is a gradable adjective). The extension of an adjective is a set. What about verbs? Take the verb sleep. Now you know the trick. The extension of sleep is the set of all x such that x sleeps. In ordinary language, sleep denotes the set of all sleepers. Notice an interesting result. We considered three words of English, doctor, tall and sleep. The English grammar tells us that they belong to different categories- a common noun, an adjective and an intransitive verb, respectively, yet their extensions are similar. Why? The answer is irrespective of the grammatical label, these three words denote property. Any x that has that property is in the extension of that expression. With this much we can give you a foretaste of how a semantics can be constructed using this framework. Recall that in the truth conditional approach to meaning, the meaning of a proposition is a truth value, either True or False. It is customary to use 1 for True and 0 for false, a practice we will be following henceforth. Now, consider the following sentences. 6) Sachin Tendulkar is a doctor. 7) Sachin Tendulkar is tall. 8) Sachin Tendulkar sleeps. We know that (6) and (7) are false and (8) is true. Therefore, the theory of meaning we are pursuing tells us that the extension of (6) and (7) is 0 and that of (8) is 1. This is intuitively clear to you, yet we need to know how we arrived at this result. There is an individual by name Sachin Tendulkar in the real world, who is a well-known cricketer. He is short in stature. (Of course, all living humans need to get sleep). With this knowledge all that we need to do is to check successively for these three sentences whether Sachin Tendulkar is in the set of doctors, in the set of tall people, and in the set of sleepers. The results follow. The illustration given is just to show that the truth conditional approach to meaning can handle simple sentences like (6). But we need to see whether this works in the case of more complex sentences. Secondly, what has been presented is an informal way of arriving at the extension of a sentence. We need a rigorous, formal mechanism that is capable of working out the extensions of sentences, keeping the principle of compositionality. Therefore, the model theory has two sets of rules, one for syntax and the other for semantics. Ordinary language sentences have to be first written in the language of logic. The particular language we use is called Predicate Calculus, PC for short. We will now introduce the PC for a fragment of English. Predicate Calculus

5 In any natural language, we have two basic categories terms and predicates. A term by definition refers to an object. The proper nouns we looked at are instances of terms. A term can be either a constant or a variable. For the time being we will consider only individual constants. The convention is to use lower case letters for constants, that is, n for the individual Noam Chomsky, s for Sachin Tendulkar etc. Coming to predicates, a predicate denotes a property. The common noun doctor does not denote an individual, but a property, any individual who has the property of being a doctor belongs to the set denoted by the predicate denoting expression doctor. Similarly, the adjective tall is a predicate, the intransitive verb sleep is a predicate. A sentence in a language consists of a subject and a predicate. We were all taught this by the grammar teacher in school. The subject is usually a term which combines with a predicate to give a sentence. From the point of view of logic, a predicate is an open formula. It has an empty slot. 9).. is a doctor. Traditional grammar tells us that is a doctor in (9) is the predicate. The real predicate part is the word doctor, as you would have grasped from the above discussion. is carries the tense, present tense in this case, and a is the indefinite article. These grammatical niceties we will have to ignore now, though in advanced semantics, these are of great significance. Thus, ignoring is and a, we can look at (9) as simply a predicate doctor with a blank slot preceding it. Fill that blank with a term, that is an individual, you get a sentence. If you decide to fill the blank with Sachin, you get (10) 10) Sachin is a doctor. The strike through is used to remind you that we are ignoring that part. Actually, in many languages it is the way (10) will get translated. Sachin doctor is a perfectly good grammatical sentence in Tamil. How do we translate (10) into PC? Upper case letters are used to denote predicates. Let us use D for the predicate in (10). Then the PC equivalent of (10) is (11). 11) D(s) Some more technical terminology is essential at this point. We said that a predicate is an open formula with a blank to be filled. In the mathematical language, a predicate is a function. (Recall the algebra you learned in school.) A function takes an argument as input and it outputs a value. In (11) s is the argument and the value of the function is 0. (zero is for false, and 1 is for true). The term argument is a technical term and it has nothing to do with the normal, dictionary meaning of the word argument. You can also say a function maps an argument to a value. We will be using these technical terms throughout this course. A predicate could be taking more than one argument. For example, take the predicate taller than. For a sentence to be meaningful, we have to mention two names, as shown by (12). 12) Amitabh Bachan is taller than Sachin Tendulkar.

6 In (12) there are two slots filled by Amitabh Bachan and Sachin Tendulkar, both are arguments of the predicate taller than. Therefore, we say taller than is a two-place predicate. Notice that filling one argument alone does not give us a structure that can be interpreted. We can now write the PC translation of (12). 13) T (a,s) There are predicates in natural languages which require three arguments. They are called three-place predicates. An example will be is between. 14) Pune is between Hyderabad and Mumbai. 15) B (p,h,m) Well Formed Formula The sentences of a natural language have to be first rewritten in the language of Predicate Calculus for them to lend themselves to semantic interpretation. When they are expressed in the language of PC, they are called formulas. There are formation rules for PC which we may call its syntax. A formula which is written as per these formation rules is called a well formed formula, or wff, for short. Exercise: Rewrite the following English sentences in PC. Give the code. We saw that an intransitive verb like sleep denotes a set, the set of all sleepers. A transitive verb, say like requires two arguments, the person who likes and the person or thing that is liked. Like is therefore a two-place predicate. There are verbs that require three arguments, give is the paradigmatic example. Verbs such as give are ditransitive. If you have done a course in syntax, you will know of this three-way classification of verbs in English. It is a simple matter to replace the labels intransitive-transitive-ditransitive by one, two, three-place predicates. However, you should be careful not to equate a predicate with a verb. Question: What are the reasons for saying that a predicate is not the same as a verb? In the next unit you will be introduced to a formal system of semantic interpretation using truth values.

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