Basic Semantic Concepts Carl Pollard Department of Linguistics - - PowerPoint PPT Presentation

Basic Semantic Concepts

Carl Pollard

Department of Linguistics Ohio State University

September 1, 2016

Carl Pollard Basic Semantic Concepts

Expressions

The grammar of each human language recursively specifies what the language’s expressions are, including:

what they sound like (phenogrammatics, roughly similar to what is traditionally called phonology) what they mean (semantics) their potential for combining with other expressions to form larger expressions (tectogrammatics, roughly similar to what is traditionally called syntax, e.g. S, NP, N, etc.)

An expression is said to express its meaning. To do linguistic semantics, it’s not enough just to say what some sentences mean; you also have to show how the grammar determines the meaning of each sentence (or

• ther complex expression) from the meanings of the smaller

expressions that were combined to form it (compositionality).

Carl Pollard Basic Semantic Concepts

Static vs. Dynamic Theories of Meaning

In reality, what an expression expresses depends in part on the context in which it is uttered. And part of the meaning of an expression is the way that uttering it changes the context for subsequent utterances. Semantic theories that take into account the interaction between context and utterance interpretation are called dynamic. To have a dynamic theory, you have to model contexts. Semantic theories which ignore the role of context are called static. We’ll start out static and ramp things up to dynamic in due course.

Carl Pollard Basic Semantic Concepts

Senses

Following Frege, we call (static) linguistic meanings senses. Senses are external to language and to the minds of language users (though perhaps there are mental representations of them). Following Montague, we assume different syntactic categories of expressions express different kinds of senses. For example:

senses of declarative sentences are propositions. (We’ll discuss these in detail soon.) senses of proper nouns are individuals, also called

• entities. (This assumption is not uncontroversial, but we’ll

adopt it for now.) senses of intransitive verbs, common nouns, and predicative adjectives are unary properties. senses of transitive verbs are binary properties.

Properties are understood to be functions that map their arguments to propositions.

Carl Pollard Basic Semantic Concepts

Senses and their Extensions

We distinguish between a sense and its extension.

With Frege, we assume the extension of a proposition is its truth value (so propositions are the kind of thing that have a truth value). With (roughly) Kripke, we assume the extension of an individual is the individual itself. We take the extension of a unary property to be the set of things that have that property. There’s a system to this, which we’ll come to soon.

What extension a sense has in general depends on contingent fact, or, informally, on how things are, whereas senses themselves are independent of contingent fact.

Carl Pollard Basic Semantic Concepts

Sense and Reference

The reference of an expression is the extension of its sense, so this too can depend on how things are. For example: The reference of a declarative sentence is whatever the truth value of the proposition that it expresses happens to be. the reference of an intransitive verb (or common noun or predicative adjective) is the set of individuals that happen to have the property it expresses. the reference of a proper noun is the same as the individual it expresses, and therefore is independent of how things

• are. (This is a simplification, but we need a dynamic

theory to fix it.)

Carl Pollard Basic Semantic Concepts

Possible Worlds

Most (not all) semantic theories take explicit account of the way that extensions (and therefore reference) can depend on how things are, or might be. Ways that things are or might be are called (possible) worlds, or just worlds. So a semantic theory that take these into account is called a possible worlds semantics, and the model-theoretic interpretation of the theory explicitly represents them. By a world, we understand not just a snapshot at a particular time, but a whole history, stretching as far back and as far forward as things go. One of the worlds, called the actual world, or just actuality, is the way things really are (again, stretching as far back and as far forward as things go).

Carl Pollard Basic Semantic Concepts

Different Ways of Conceptualizing Worlds

In tractarian theories (named after Wittgenstein’s (1918) Tractatus Logico-Philosophicus), worlds are certain sets of propositions, namely the maximal consistent ones. (Examples: Wittgenstein, C.I. Lewis, Robert Adams, Alvin Plantinga, William Lycan) In kripkean theories (based on Kripke’s (1963) semantics

• f modal logic), worlds are taken to be theoretical
• primitives. This remains the prevalent view in modal logic.

Montagovian theories are kripkean theories in which propositions are taken to be sets of worlds. (Examples: Richard Montague, David Kaplan, David Lewis, Robert Stalnaker, inter multa alia) Usually when linguists speak of possible worlds semantics, they have the Montagovian conception in mind and aren’t aware of the other (and older) options.

Carl Pollard Basic Semantic Concepts

Agnostic Possible Worlds Semantics

Agnostic possible worlds semantics is a logically weak version that is neutral among all these positions: it could be strengthened into either a tractarian or a kripkean (including montagovian) theory.

Carl Pollard Basic Semantic Concepts

The Extension of a Sense at a World

We don’t speak of a sense as simply having an extension, but rather as having that extension at a given world. In particular, we don’t speak of a proposition as simply being true or false, but rather as being true or false at a given world. In other words, we assume there is a relation between propositions and worlds, called being true at, and we say p is true at w (written p@w) if the ordered pair p, w is in this relation. As we’ll see in due course, for any sense s (not just propositions), the extension of s at a world w can be defined in terms of the @ relation. Some versions of possible worlds semantics (e.g. tractarian and montagovian) specify what the @ relation is, while the agnostic version does not (but asserts axioms about it).

Carl Pollard Basic Semantic Concepts

The Reference of an Expression at a World

When we say that an expression has reference r at a world w, we mean that the sense it expresses has the extension r at w. In particular, when we say that a sentence is true (or false) at w, we mean that the proposition it expresses is true (or false) at w.

Carl Pollard Basic Semantic Concepts

Entailment (1/2)

For two propositions p and q, we say p entails q provided, no matter how things are, if p is true when things are that way, then so is q. In terms of possible worlds: p entails q if and only if, for every world w, if p@w, then also q@w. Obviously entailment is a preorder (relexive and transitive). Two propositions are called (truth-conditionally) equivalent if they entail each other. Equivalence is obviously an equivalence relation (reflexive, transitive, and symmetric).

Carl Pollard Basic Semantic Concepts

Entailment (2/2)

As with truth (at a world), the use of the term ‘entailment’ is extended from propositions to the declarative sentences that express them. (And likewise for ‘equivalent’.) So ‘S1 entails S2’ means that the proposition expressed by S1 entails the proposition expressed by S2. Native speaker judgments about entailments between sentences (or better, in-context utterances of sentences) are important (some would say, the most important) data in testing semantic hypotheses.

Carl Pollard Basic Semantic Concepts

Bolzano’s Notion of Proposition (1/2)

Something similar to the notion of proposition used here seems to have first been suggested by the mathematician/philosopher Bernard Bolzano (Wissenschaftslehre, 1837)—his term was Satz an sich ‘proposition in itself’: They are expressed by declarative sentences. They are the ‘primary bearers of truth and falsity’. (A sentence is only secondarily, or derivatively, true or false, depending on what proposition it expresses.) They are the the ‘objects of the attitudes’, i.e. they are the things that are known, believed, doubted, etc.

Carl Pollard Basic Semantic Concepts

Bolzano’s Notion of Proposition (2/2)

They are nonlinguistic. They are nonmental. They are not located in space and time. Sentences in different languages, or different sentences in the same language, can express the same proposition. Two distinct propositions can entail each other.

Carl Pollard Basic Semantic Concepts

Kinds of Propositions

A proposition p is called: a necessary truth, or a necessity, iff it is true at every world. a possibility iff it is true at some world. a truth iff it is true at the actual world. contingent iff it is true at some world and false at some world. a falsehood iff it is false at the actual world. a necessary falsehood, or an impossibility, or a contradiction, iff it is true at no world. a fact of w iff it is true at w.

Carl Pollard Basic Semantic Concepts

Interdependence of Context and Utterance Meaning

Those aspects of the circumstances of an utterance involved in the determination of its meaning are called its context. For example, what entity is expressed by a use of the name Chris depends on the context. Likewise, what proposition is expressed by a use of the declarative sentence she kicked him depends on the context. Conversely, each utterance helps create the context involved in determining the meaning of the next utterance:

• a. He sat down. A farmer walked in.
• b. A farmer walked in. He sat down.

Carl Pollard Basic Semantic Concepts

Dynamic and Static Semantic Theories (1/2)

This interdependence between context and utterance meaning is called dynamicity, and semantic theories that take dynamicity into account are called dynamic. Dynamicity plays a central rule in (for example) anaphora, (in-)definiteness, presupposition, conventional implicature, contrast, topicality, focus, ellipsis, and the relationship between questions and answers. Dynamic theories must formally model contexts.

Carl Pollard Basic Semantic Concepts

Dynamic and Static Semantic Theories (2/2)

Semantic theories that steer clear of dynamicity, by ignoring context or pretending that the context is held fixed, are called static. Usually (and here), dynamic semantic theories are built on the foundation of a static theory. As long as we ignore context, the distinction between expression and utterance is not so important, and we will not always make it terminologically.

Carl Pollard Basic Semantic Concepts

First Steps in Modelling Senses and Extensions

To model senses inside set theory, we first assume there is a set P of propositions, and a set I of individuals. Additionally, in order to model the notion of senses having extensions at worlds, we assume there is a set W of worlds. We’ll model truth values as members of the set 2, with 1 and 0 corresponding to true and false respectively.

Carl Pollard Basic Semantic Concepts

Modelling Different Kinds of Senses

Following Bolzano, the senses expressed by declarative sentences are assumed to be propositions. Following Kripke, the senses expressed by names are assumed to be individuals. The meanings of ‘logic words’ such as and, implies, and it is not the case that will be modelled as operations on propositions. Most common nouns, predicate adjectives, and intransitive verbs are assumed to express properties of individuals (such as being a dog, being hungry, or barking), and these in turn are modelled as functions from individuals to propositions, i.e. members of I ⇒ P. Verbs which take more than one ‘grammatical argument’ (roughly, subject and/or complement(s)) are modelled as functions which map multiple arguments of various kinds to propositions.

Carl Pollard Basic Semantic Concepts

Technical Preminary: Currying and Uncurrying (1/2)

It has long been realized (at least since Cantor) that a function which takes two arguments can be thought of a function which maps a single argument to a function taking a single argument. More precisely, for any sets A, B, and C, and any function f : (A × B) → C there is a function curry2(f): A → B ⇒ C defined as follows: for each x ∈ A, (curry2(f))(x) is the function that maps each y ∈ B to f(x, y). Note that curry2 itself is not a function. Rather, for each A, B, and C, there is a distinct function curry2,A,B,C that maps (A × B) ⇒ C to A ⇒ B ⇒ C. Each of these functions is bijective, with inverse uncurry2,A,B,C : (A ⇒ B ⇒ C) → ((A × B) ⇒ C) given by (uncurry2(g))(x, y) = (g(x))(y).

Carl Pollard Basic Semantic Concepts

Technical Preminary: Currying and Uncurrying (2/2)

We can generalize (un)currying from n = 2 to n ≥ 2 by (un)currying n − 1 times. For example, for f : (A × B × C) → B, we have curry3(f) = def curry2(curry2(f)) : A → (B ⇒ C ⇒ D) where ((curry3(f)(x))(y))(z) = f(x, y, z) for all x ∈ A, y ∈ B, and z ∈ C. Semanticists like currying because usually in grammatical composition of an expression, at each step there are two constituents, with one of them a semantic argument of the

• ther.

Carl Pollard Basic Semantic Concepts

Sense Sets

Aside from individuals and propositions, other senses are modelled by (curried) functions drawn from certain sets. These sets, together with I and P, are called sense sets. The set of sense sets is defined recursively:

• 1. As a technicality, we choose some singleton set (call it U),

whose member is used for vacuous meanings (especially, dummy pronouns).

• 2. I is a sense set.
• 3. P is a sense set.
• 4. If A and B are sense sets, so is A ⇒ B.
• 5. Nothing else is a sense set.

Carl Pollard Basic Semantic Concepts

Examples of Different Kinds of Senses (1/2)

dummy pronoun (weather it) U name (Pedro): I common noun (farmer): I ⇒ P typical intransitive verb (yells): I ⇒ P typical predicative adjective (ajar): I ⇒ P sentential-subject predicative adjective (obvious): P ⇒ P sentential adverb (obviously): P ⇒ P weather verb (rains): U ⇒ P weather predicative adjective (windy): 1 ⇒ P transitive verb (sees): I ⇒ I ⇒ P ditransitive verb (give): I ⇒ I ⇒ I ⇒ P

Carl Pollard Basic Semantic Concepts

Examples of Different Kinds of Senses (2/2)

attitude verb (knows): I ⇒ P ⇒ P transitive S-complement verb (informs): I ⇒ I ⇒ P ⇒ P subject-control verb (tries): I ⇒ (I ⇒ P) ⇒ P

• bject-control verb (orders): I ⇒ I ⇒ (I ⇒ P) ⇒ P

quantificational NP (something): (I ⇒ P) ⇒ P determiner (every): (I ⇒ P) ⇒ (I ⇒ P) ⇒ P declarative complementizer (that): P ⇒ P sentential conjunction (and): P ⇒ P ⇒ P attributive adjective (former): (I ⇒ P) ⇒ (I ⇒ P) relativizer (that): (I ⇒ P) ⇒ (I ⇒ P) ⇒ (I ⇒ P)

Carl Pollard Basic Semantic Concepts

Extension Sets

For each sense set A, the corresponding extension set Ext(A) is defined recursively:

• 1. Ext(U) = U
• 2. Ext(I) = I
• 3. Ext(P) = 2
• 4. Ext(A ⇒ B) = A ⇒ Ext(B)

Carl Pollard Basic Semantic Concepts

A-Properties and A-Sets

For any sense set A: the set A ⇒ P is called the set of A-properties. the corresponding extension set Ext(A ⇒ P) = A ⇒ 2 is called the set of A-sets. Technically, the members of A ⇒ 2 are not really sets of A’s, but rather characteristic functions of such sets. But there is an obvious one-to-one correspondence between (actual) sets of A’s and functions from A to 2 (which maps each such set to its characteristic function).

Carl Pollard Basic Semantic Concepts

A-Quantifiers

For any sense set A: the set (A ⇒ P) ⇒ P is called the set of A-quantifiers. So an A-quantifier is a property of A-properties. The corresponding extension set Ext((A ⇒ P) ⇒ P) = (A ⇒ P) ⇒ 2 is the set of (characteristic functions of) sets of A-properties.

Carl Pollard Basic Semantic Concepts

A-Determiners

For any sense set A: the set (A ⇒ P) ⇒ (A ⇒ P) ⇒ P is called the set of A-(semantic) determiners. The corresponding extension set

Ext((A ⇒ P) ⇒ (A ⇒ P) ⇒ P) = (A ⇒ P) ⇒ (A ⇒ P) ⇒ 2

is the set of (characteristic functions of) binary relations between A-properties.

Carl Pollard Basic Semantic Concepts

The Extension of a Sense at a World (1/3)

Recall that we assume there is a binary relation @ of being true at between propositions and worlds. Let’s denote the curried version of the characteristic function of that relation by @P. So this function is in the set P ⇒ W ⇒ 2. Let’s write it infix-style, i.e. if p is a proposition and w is a world, we write p@Pw instead of @P(p)(w) So p@Pw is the extension of p at w. More generally, for each sense type A, we will recursively define the function @A ∈ A ⇒ W ⇒ Ext(A) that maps each sense and world to the sense’s extension at that world.

Carl Pollard Basic Semantic Concepts

The Extension of a Sense at a World (2/3)

To be more precise: what we are defining recursively is the function that maps each sense set A to the function @A from A to W ⇒ Ext(A). The codomain of @A, namely W ⇒ Ext(A), is called the set of A-intensions. And so, for each sense a ∈ A, the function a@A ∈ W → Ext(A) is called the intension associated with a. In Montague semantics, for senses which are propositions

• r properties, it turns out that the sense and its intension

are the same thing. For example, Montague semantics defines P to be W ⇒ 2, and @P is defined to be the identity function on P. This is a bug, not a feature.

Carl Pollard Basic Semantic Concepts

The Extension of a Sense at a World (3/3)

For each sense type A, the function @A is defined recursively: A = P: @P is assumed to be given in advance. A = U: u@Uw = u, where u is the only member of U. A = I: for any individual a and any world w, a@Iw = a. A = B ⇒ C, where B and C are sense types: for any function f ∈ B ⇒ C and any world w, f@B⇒Cw is the function from B to Ext(C) that maps each b ∈ B to (f(b))@Cw. Note: To avoid notational clutter, the subscript on @ will usually be omitted when it can be inferred from the context.

Carl Pollard Basic Semantic Concepts