Meaning and Language

Meaning and Language Part 1

Plan • We will talk about two different types of meaning, corresponding to two different types of objects: • Lexical Semantics: Roughly, the meaning of individual words • Compositional Semantics: How larger objects (clauses, sentences) come to mean what they do. Relatedly, how formal logic can be used as a tool to study language • However: These two fit together, as discussed in the reading (Partee) • That is, aspects of what we want to say about what words mean will interact with what we say about larger structures • Today: • Some distinctions • Basic sets and truth conditions • Working towards logic for language

Some Initial Points • Remember that for (content) words like cat, tree,horse, etc. there is an arbitrary connection between sound form and meaning:

Sound and Meaning • This pairing of sound and meaning is one component of language • “arbitrary” component: stressed by de Saussure • “predictable” component: logic, etc. • Rock bottom: basic connections in small units (morphemes,words) between sound and meaning • The full range of things that we associate with human language is found only when such connections are part of a generative system for creating larger units from smaller ones, i.e. the syntax (remember last week)

Outline • Basics of word meaning: traditional distinctions for sound/meaning connections (homophony, polysemy) • Words and sets (as in set theory) • Basic cases (nouns and adjectives) • Wednesday: Using formal logic to model meaning relations in language

Some Distinctions • First: cases in which the “one to one” mapping between sound forms and meanings is not so direct. • Homophony: A cases in which two words have the same sound form, but distinct and unrelated meanings • Bank-1 ‘side of a river’ • Bank-2 ‘financial institution’

Representation • In any case, with homophony we are dealing with distinct words; that is: • Bank-1 is to Bank-2 as cat is to dog or bank-1is to cat • This is equivalent to saying that in such cases, the identity in sound form is an accident • In other cases of the same sound form but differing meaning, this is not the case

Polysemy • We speak of polysemy ‘many meanings’ in cases in which we have the same word but with distinct yet related senses; one case: • Pool: water on the ground • Pool: swimming pool • In this case, there is no need to say that there are different words; perhaps really different senses of the same word

Polysemy, cont. • Sometimes with polysemy the intuition is that the word is basically ‘vague’, and that its fuller meanings are supplied by context • Something similar is found with verbs, where the context comes from the syntactic structure: • The whistle sirened lunch time. • The police car sirened the speeder to a stop. • Cases like this indicate that the basic meaning of words can be augmented with information from the syntactic structure • John shinned the ball. • Mary shinned the ball to John. • Etc. • The “core”meaning of the word shin or siren exists, but is augmented by what happens in the syntactic structure

Words and Sets • Let’s take an example of how we can represent meanings… • More interesting: how meanings of combinations of words are derived • We can think of the meaning of some words as relating to a system of categories, some more general, some more specific • This lends itself to representation in terms of sets • A set is, for our purposes, an abstract collection

Examples • Consider the relationship between dog and mammal: • All dogs are mammals. (true) mammals dogs

Examples, cont. • The set relationship is one of inclusion; the set denoted by dog is a subset of the set denoted by mammal • Other relationships are possible as well, both in terms of ‘some’ and ‘no’ • We will formalize an extension to this in the next lecture

‘Some’ and overlapping • It is not true that all snakes are poisonous: • All snakes are poisonous. (false) • But some are: • Some snakes are poisonous. (true) • In cases like this, the set denoted by snake and the set denoted by poisonous overlap: Poisonous things snakes

Non-overlapping: ‘No’ • It can also be the case that sets do not overlap, in addition to overlapping in very small ways • Consider the following: • No mammals are poisonous. • Ok, we want to know what no means, but is this a good example (is it true)?

Sets • So we need another example of sets that don’t overlap • No dogs are reptiles. (true) dogs reptiles

Truth Conditions • One way of approaching meanings is to look at the truth conditions of sentences • The truth conditions specify in precise terms the circumstances that obtain in order for a sentence to be true (or false) • Specifying the truth conditions is a necessary component of the study of meaning; if we can show that two sentences are true under different conditions, then we would like to say that they have different meanings

Some examples • Sometimes it seems like the specification of truth conditions is trivial: • The cat is on the mat. • The dog is on the mat. • Different truth conditions • But what about more complex cases? Consider: • The glass is half full. • The glass is half empty.

The ‘Glass’ Example • On the face of it, ‘half full’ and ‘half empty’ seem to have the same truth conditions. • But: Consider the following examples: • The glass is almost half full. (e.g. 48%) • The glass is almost half empty. (e.g. 53%) • These have different truth conditions • Assuming that ‘almost’ is the same in the two sentences, it must be the case that ‘half full’ and ‘half empty’ actually have different meanings • If these two phrases were not different in meaning, where else could the difference come from??

Other fractions • As a further point, consider what happens when we replace ‘half’ by other fractions: • The glass is three eighths full. • The glass is three eighths empty. • These do not mean the same thing • It looks as if ‘half full’ and ‘half empty’ mean different things, but sometimes can be true under the same circumstances

More on Adjectives • Some further cases from the study of adjectives illustrate • The relevance of our use of sets above • The interaction of lexical meaning with compositional meaning • Let’s take another simple example: • poisonous snake

Interpreting poisonous snake • One way of thinking of the adjective meaning with respect to the noun follows on what we were doing above • What we would like are some general rules that tell us how to interpret certain syntactic objects in terms of the semantics we are using • Rule (informal): When an adjective A modifies a noun N ([A N]), the interpretation of this object is the set defined by the intersection of A’s meaning with N’s meaning

Why a general rule? • Remember last week; we have grammars that generate combinations of Adjectives and Nouns: • Small book • Orange boy • Purring burrito • Stubborn cat • We’re not memorizing the meaning of these; there has to be some general principle determining the interpretation • The working hypothesis that we have with the rule on the last slide is that we use intersection

On the interpretation, cont. • This is just like the rule we saw above: snakes Poisonous things • With poisonous snake, we are indicating a member of the overlap between two sets • This can be indicated in a logical notation as well

Some notation • We need a notation for sets and their interaction • || X || = the set of things denoted by property X • Example: || red || = the set of red things • This can also be written as {x| x is red}, read as ‘the set of all things x such that x is red’ • What about how adjectives and nouns combine by the reasoning above? • We need notation for ‘and’; why? Because the things that are poisonous snakes are the set of things that are (1) poisonous AND (2) snakes

Putting the pieces together • So, for poisonous snake: • || poisonous || = {x|x is poisonous} • || snake || = {x|x is a snake} • || poisonous snake || = {x| x is poisonous AND x is a snake} • We can also use set notation for this, e.g.: • || poisonous ||  || snake || • I.e., this indicates that we interpret the modification of nouns by adjectives with intersection (“And”)

So… • Is it always so simple? Consider: • Reasoning 1: • Larry is a poisonous snake • Larry is a chess player. • Therefore: Larry is a poisonous chess player (valid…but this is more complicated than it looks. Think about the meanings…) • Reasoning 2: • Larry is a skillful artist. • Larry is a chess player • Therefore: Larry is a skillful chess player. (invalid!!)

Or • Some other examples: • Former bounty-hunter • Alleged thief • Beautiful dancer • Do these work in terms of intersection? Are former bounty-hunters bounty-hunters at all?

Consider… • One of the examples: • Larry is a poisonous snake • Larry is a chess player. • Therefore: Larry is a poisonous chess player • The phrase poisonous chess player is ambiguous…it can also mean that he’s not poisonous per se, but as a chess player, he is.

For example… • So, with poisonous chess player, it seems that some adjectives can be interpreted in either fashion. Here it’s more transparent: • Larry is a beautiful dancer. • Meaning1: He dances beautifully • Meaning2: He is beautiful, and he is a dancer (he might dance poorly) • Question: Do these differences involve different structures, or just a lexically ambiguous set of adjectives??

Further considerations • Could we have contexts like the following? • A: There are lots of beautiful dancers here. • B: Yes, but Mary is the only beautiful beautiful dancer. • If so, which adjective is the one with the ‘dances beautifully’ interpretation, and which carries the ‘is a beautiful person’ meaning? • Consider further: • John is the only ugly beautiful dancer. • John is the only beautiful ugly dancer. • Question (for thought): Does this mean that the difference is reducible to structure?

Meaning and Language