PLIN0009 Semantic Theory Spring 2020 Lecture Notes 1

1 What this module is about

This module is an introduction to truth-conditional semantics with a focus on two impor- tant topics in this area: compositionality and quantiication.

The framework adopted here is often called formal semantics and/or model-theoretical semantics, and it is characterized by its essential use of tools and concepts developed in mathematics and logic in order to study semantic properties of natural languages.

Although no textbook is required, I list some introductory textbooks below for your refer- ence.

• L. T. F. Gamut (1991) Logic, Language, and Meaning. The University of Chicago Press. • Irene Heim & Angelika Kratzer (1998) Semantics in Generative Grammar. Blackwell. • Thomas Ede Zimmermann & Wolfgang Sternefeld (2013) Introduction to Semantics: An Essential Guide to the Composition of Meaning. De Gruyter Mouton. • Pauline Jacobson (2014) Compositional Semantics: An Introduction to the Syntax/Semantics. Oxford University Press. • Daniel Altshuler, Terence Parsons & Roger Schwarzschild (2019) A Course in Semantics. MIT Press.

There are also several overview articles of the ield by Barbara H. Partee, which I think are enjoyable.

• Barbara H. Partee (2011) Formal semantics: origins, issues, early impact. In Barbara H. Partee, Michael Glanzberg & Jurģis Šķilters (eds.), Formal Semantics and Pragmatics: Discourse, Context, and Models. The Baltic Yearbook of Cognition, Logic, and Communica- tion, vol. 6. Manhattan, KS: New Prairie Press. • Barbara H. Partee (2014) A brief history of the syntax-semantics interface in Western Formal Linguistics. Semantics-Syntax Interface, 1(1): 1–21. • Barbara H. Partee (2016) Formal semantics. In Maria Aloni & Paul Dekker (eds.), The Cambridge Handbook of Formal Semantics, Chapter 1, pp. 3–32. Cambridge University Press.

There is also a video interview with Partee, which I highly recommend: https://vimeo. com/20664367

2 Truth-conditional semantics

Let us irst review the idea of truth-conditions. We will call the aspect of meaning perti- nent to truth-conditions truth-conditional meaning.

While we focus on truth-conditional meaning in this module, it is not meant to deny the existence of other types of meaning that natural language is capable of expressing, or to suggest that such meanings are not amenable to the approach we adopt here. In particu-

1 lar, non-truth-conditional meanings like scalar implicatures and presuppositions are quite extensively studied in the framework of formal semantics.

One of the reasons why we focus on truth-conditional meaning is because our intuitions about them, which constitute our primary data, are relatively clear. The clarity of data is very important for us, as we want to build a theory of natural language semantics and test its predictions against empirical data. If the data are unclear, we can’t learn much from them.

A particularly important fact about truth-conditions is that you, as a native speaker of En- glish (or any other language, for that matter), ‘know’ the truth-conditions of any grammat- ical declarative sentence in English (or whatever language you are a native speaker of). Let us state this fact as follows:

A native speaker has truth-conditional intuitions about any grammatical declar- ative sentence, i.e. he or she ‘knows’ in what situations it is true and in what situations it is false.

I put know in scare-quotes here because this knowledge, just like most other linguistic knowledge we talk about in linguistics, is often only implicitly known and native speakers are not necessarily consciously aware of it.

In order to see that you have truth-conditional intuitions, let us look at an example.

(1) There is a circle in a square.

If you are given a situation depicted below, you know that the sentence is true.

On the other hand, in the following situation, the sentence is false.

Of course these are not the only situations that make this sentence true or false, and in fact there are ininitely many situations where the sentence is true and ininitely many situa- tions where it is false. However, it is not dificult to see that the sentence is true whenever there is a circle and there is a square and the circle is inside the square, and and is false

2 whenever this is not the case. We take this to be the truth-conditions of (1).

Because writing such lengthy statements all the time would be too cumbersome, we usu- ally abbreviate descriptions of truth-conditions using ‘iff’, which stands for for ‘if and only if’.¹ For example, we write the truth-conditions of (1) as follows:

(2) “There is a circle in a square” is true iff there is a circle in a square.

One important caveat: Admittedly, sometimes your truth-conditional intuitions might not be very clear. For example, when is “There are many tall buildings in this city” true and when is it false? In order to determine this, you need to know what this city refers to, and what counts as a ‘tall building’ and what counts as ‘many’. But, if you are given the sentence and a particular situation, you (often) can tell whether the sentence is true or false in that situation. In this module we primarily look at sentences for which your truth-conditional intuitions are relatively clearcut.²

3 Object language vs. metalanguage

We have just looked at one example sentence and its truth-conditions. Generally, the truth- conditions of any declarative sentence S in English can be written in the following schema:

(3) “S” is true iff S.

This might look trivial but it is not. In particular, it is important to distinguish the object language—the language we are analysing—and metalanguage—the language in which we state our analysis. The sentence in “ ” in the above statement is part of the object lan- guage, while the rest belongs to the metalanguage. In our examples so far, we are using English both as our object language and as our metalanguage, so the truth-conditions de- ceivingly look trivial, but consider the following Czech sentence instead:

(4) Vlk zmrzl, zhltl hrst zrn.

Suppose your job as a semanticist is to igure out the truth-conditions of this sentence. You ask your native speaker informants, and ind out that the speakers judge it to be true whenever there is a wolf that froze and swallowed a handful of grains. Then, your analysis of the sentence can be stated as: ¹This notation comes from mathematical logic. ‘S if T ’ means whenever T is true S is also true. ‘S only if T ’ means whenever S is true T is also true. Taken together S if and only if T ’ means S and T are always both true or both false, or in other words, they are equivalent. ²Expressions such as tall and many give rise to dificult but important issues for truth-conditional semantics: Sentences containing do not seem to have clear truth-conditions. Such expressions are said to be vague. This is a very good topic for a Long Essay. Here are some suggested readings on this. The book by Van Deemter is a accessible and well-written non-technical introduction and especially recommended. • Kees van Deemter (2014) Not Exactly: In Praise of Vagueness. Oxford University Press. • Hans Kamp & Galit W. Sasoon (2016) Vagueness. In Maria Aloni & Paul Dekker (eds.), The Cambridge Handbook of Formal Semantics, Chapter 14, pp. 389–441. Cambridge University Press. • Hans Kamp & Barbara H. Partee (1995) Prototype theory and compositionality. Cognition, 57: 129–191.

3 (5) “Vlk zmrzl, zhltl hrst zrn” is true iff a wolf froze and swallowed a handful of grains.

This is not at all trivial, because it can be a correct or incorrect description of the truth- condition of the sentence. The distinction between object language and meta-language is very important to keep in mind.

4 The Compositionality Principle

We take truth-conditional intuitions to be part of linguistic knowledge, on a par with other linguistic intuitions like grammaticality judgments (which have to do with well-formedness of linguistic expressions). One of the primary goals of truth-conditional semantics is to ig- ure out how the truth-conditional intuitions arise.

It is instructive compare semantics and syntax here. Syntax tries to construct a theory that explains why and how native speakers acquire the grammaticality judgments that they have. Likewise, semantics tries to construct a theory that explains why and how native speakers acquire the truth-conditional judgements that they have.

What makes truth-conditional semantics not so trivial is the fact that native speakers have truth-conditional intuitions about all grammatical declarative sentences. This is impor- tant because it entails that it’s impossible to simply memorize all pairs of declarative sen- tences and their truth-conditions, because there would be ininitely many pairs to mem- orize. It is easy to demonstrate that there are ininitely many sentences in a given natural language.

(6) a. It’s raining. b. he thinks it’s raining. c. she knows that he thinks it’s raining. d. He thinks that she knows that he thinks it’s raining. e. She knows that he thinks that she knows that he thinks it’s raining.

We can repeat this process indeinitely to create more sentences. At some point it will be- come very hard to understand such sentences, but nonetheless they will stay grammatical, and also should have some truth-conditions.

Since one cannot remember the truth-conditions of ininitely many sentences, there must be a way to compute them. What does it mean to compute truth-conditions?

Let’s take an analogy with syntax again. The syntactic component of grammar combines expressions to produce more complex expressions, such as noun phrases, verb phrases, sentences, etc. Since it allows such complex expressions to be embedded under other complex expressions of the same kind, as illustrated in (6), it generates ininitely many grammatical expressions. This property is called recursion and is considered to be one of the deining characteristics of human language. Note that the syntax itself is (hypothesized to be) a inite mechanism, but produces ininitely many expressions.

The idea we will pursue in semantics is similar. That is, the semantic component of gram- mar does essentially the same thing as its syntactic counterpart, except that instead of combining expressions, it combines meanings. For instance, it takes the meaning of the

4 expression a and the meaning of the expression man and combine them to produce the meaning of the syntactically complex expression a man.

We furthermore assume that the semantics combines meanings of various expressions, simple or complex, in predictable ways. If this is so, then from the meaning of a and the meaning of man, we should be able to predict the meaning of a man. Let us state this as a principle:

(7) The Compositionality Principle: The meaning of a syntactically complex ex- pression is determined solely by the meanings of its parts and how they are arranged.

This is a very important principle, because a semantic system obeying it can deal with ininitely many expressions. Before explaining why, let’s consider an example, say, the sentence A man saw Mary. We can ind out its truth-conditions buy asking a native speaker (who could be you). Then, according to the Compositionality Principle, the truth-conditions of this sentence is determined only by the meanings of its parts, namely a, man, saw, and Mary, and how these expressions are arranged.

This sounds like a truism, but it is not so hard to imagine an artiicial language whose semantics doesn’t obey the Compositionality Principle. In such a language, the meaning of a man, for example, might have nothing to do with the meaning of a or the meaning of man. Or take a more abstract ‘language’ in which sequences of mouse clicks are ‘grammatical sentences’. Suppose that one click means ‘Select the ile that the cursor is currently on’, and two clicks means ‘Open the ile the cursor is currently on’. The sentence with two clicks is syntactically decomposable into two individual clicks, but its meaning cannot be analysed as two instances of the meaning of one click. In languages like this that do not obey the Compositionality Principle, there is no general way of computing the meaning of a complex expression based on the meanings of its parts. In fact, what would three clicks mean in the above language of mouse clicks? Another way of stating the same thing is that in such a non-compositional language the meanings of complex expressions need to be memorized. Natural languages are not like this.

If natural language obeys the Compositionality Principle, there must be general ways to combine two grammatical expressions (as long as the result is a grammatical expression; semantics doesn’t need to deal with ungrammatical sentences). And if we only need a inite number of such ways to combine meanings to account for all sentences in a given natural language, then we will have a inite semantic system that computes the meanings of any grammatical sentence in that language. One of the goals of truth-conditional semantics is to do exactly this.

We will not be able to actually construct a complete semantic theory in this module (and I don’t think anyone has succeeded in doing so yet), but as proof of concept, we will see how it could be done by building a theory for a limited number of constructions in English.

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