meaning and language part 1. plan we will talk about two different types of meaning, corresponding...
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Meaning and Language
Part 1
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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
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Some Initial Points
• Remember that for (content) words like cat, tree, horse, etc. there is an arbitrary connection between sound form and meaning:
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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)
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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
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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’
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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
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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
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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
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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
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Examples
• Consider the relationship between dog and mammal:– All dogs are mammals. (true)
dogsdogs
mammals
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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
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‘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:
snakes
Poisonousthings
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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)?
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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
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Sets
• So we need another example of sets that don’t overlap– No dogs are reptiles. (true)
dogs reptiles
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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
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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.
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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??
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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
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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
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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
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On the interpretation, cont.
• This is just like the rule we saw above:
snakesPoisonousthings
•With poisonous snake, we are indicating a member of the overlap between two sets
•This can be indicated in a logical notation as well
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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
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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 ||
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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)
– Reasoning 2:• Larry is a skillful artist.• Larry is a chess player• Therefore: Larry is a skillful chess player. (invalid!!)