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 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 think of word 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)?

As far as I know… As far as I know, the statement ‘No mammals are poisonous’ is false The duck-billed platypus has a kind of poisonous thing on its leg

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

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 ||

So… Is it always so simple? Consider: Reasoning 1: Reasoning 2: 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) Reasoning 2: Larry is a skillful artist. Larry is a chess player Therefore: Larry is a skillful chess player. (invalid!!)