CAS LX 502

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CAS LX 502

• 8b. Formal semantics
• A fragment of English

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Infinite use, finite means

• A fundamental property of language is its
  recursive naturewe can create unboundedly many
  new sentences, and understand what they mean.
• Infinite use of finite means, one of the main
  reasons to suppose that our knowledge of language
  is systematic, that language is not a collection
  of habits and analogy, but must be described by a
  grammar.

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Infinite use, finite means

• In the domain of syntax, the task is primarily to
  describe/explain why some arrangements of words
  count as sentences of English, others dont, and
  more broadly, how this system relates to those
  underlying other languages, and how this system
  can arise.

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Syntax

• The generally accepted view of syntax breaks
  sentences down into hierarchical parts. There are
  nouns, there are verbs, there are units made of
  verbs and nouns. New sentences can be created by
  mixing and matching these components together.
• S Pat AuxP will VP eat NP the sandwich
• S The students AuxP have VP risen PP in
  protest

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Semantics

• Were not here to study syntax, were here to
  study semantics, but were going to delve a bit
  into both.
• The syntactic system that defines what are good
  sentences of English provides hierarchical
  structures, but we know not only what sequences
  of words might be classified as English but we
  know what those sequences of words mean.
• Just as there must be a grammar that defines what
  sequences of words are English, there must also
  be a grammar that tells us how the meanings of
  the parts contribute to the meaning of the whole.

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F1

• To that end, we are going to create a
  mini-grammar of English, a fragment. This
  grammar will provide both the syntactic structure
  of a small number of English sentences and the
  rules by which we can understand their meaning.
  By doing this, we can start to understand what is
  involved in the grammar of semantics more
  generally.

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F1

• Rewrite rules (the syntax)

S ? N VP N ? Pavarotti, Loren, Bond
S ? S conj S Vi ? is boring, is hungry, is cute
S ? neg S Vt ? likes
VP ? Vt N Conj ? and, or
VP ? Vi Neg ? it is not the case that
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Using the syntax of F1

• We start with S (we are building a sentence).

S
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Using the syntax of F1

• We start with S (we are building a sentence).
• Several different rules can apply. We can either
  rewrite S as N VP, or as S conj S, or as neg S.
  Lets pick N VP.

S
VP
N
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Using the syntax of F1

• We start with S (we are building a sentence).
• Several different rules can apply. We can either
  rewrite S as N VP, or as S conj S, or as neg S.
  Lets pick N VP.
• Now, N can be rewritten as Pavarotti, Loren, or
  Bond.

S
VP
N
Bond
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Using the syntax of F1

• We start with S (we are building a sentence).
• Several different rules can apply. We can either
  rewrite S as N VP, or as S conj S, or as neg S.
  Lets pick N VP.
• Now, N can be rewritten as Pavarotti, Loren, or
  Bond.
• And VP can be rewritten either as Vt N or Vi.

S
VP
N
Bond
Vi
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Using the syntax of F1

• We start with S (we are building a sentence).
• Several different rules can apply. We can either
  rewrite S as N VP, or as S conj S, or as neg S.
  Lets pick N VP.
• Now, N can be rewritten as Pavarotti, Loren, or
  Bond.
• And VP can be rewritten either as Vt N or Vi.
• And Vi can be rewritten as is boring, is hungry,
  or is cute.

S
VP
N
Bond
Vi
is hungry
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Using the syntax of F1

• With this little grammar, we can already create
  an unbounded number of sentences.
• It is not the case that Bond is boring or Loren
  is hungry.

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Using the syntax of F1

• It is not the case that Bond is boring or Loren
  is hungry.

S
Neg
S
S
S
Conj
It is notthe case that
N
N
VP
VP
or
Bond
Vi
Loren
Vi
is boring
is hungry
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Using the syntax of F1

• It is not the case that Bond is boring or Loren
  is hungry.

S
S
S
Conj
N
S
VP
or
Neg
N
VP
Loren
Vi
It is notthe case that
Bond
Vi
is hungry
is boring
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Compositionality

• A fundamental assumption about how it is that we
  can know what novel sentences mean is that
  meaning is compositional.
• The meaning of the whole is derived from the
  meaning of the parts and how the parts are
  arranged.
• The syntax gives us the parts and how they are
  arranged, now we must approach the question of
  how the meaning is assigned to the parts and from
  there to the whole.

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Enter M

• Here, we turn to M, the evaluation function.
• We already talked about the first steps
• MltU,Fgt
• PavarottiM F(Pavarotti) Pavarotti
• LorenM F(Loren) Loren
• BondM F(Bond) Bond
• is boringM F(is boring) Loren, Pavarotti
• is hungryM F(is hungry) Bond, Pavarotti
• is cuteM F(is cute) Loren, Bond

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M
S

• We can write the denotation of the terminal nodes
  using those rules.

VP
N
Bond
Vi
BondM F(Bond) Bond
is hungry
is hungryM F(is hungry) Bond, Pavarotti
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M
S

• We can write the denotation of the terminal nodes
  using those rules.
• And, on the principle of compositionality, we can
  assume the that nodes above share the same
  denotation (where there is no combination
  involved)

VP
N
Bond
Vi
BondM F(Bond) Bond
is hungry
is hungryM F(is hungry) Bond, Pavarotti
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M
S

• Now, to determine the meaning of the S as a
  whole, we want to combine the denotation of N and
  VP such that the S is true just in case (here),
  Bond is hungry.
• That is, true just in case NM is in the set
  VPM.

VP
N
Bond
Vi
BondM F(Bond) Bond
is hungry
is hungryM F(is hungry) Bond, Pavarotti
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M
S

• We can define a semantic rule for interpretation
  that says just that
• S N VPM true iffNM ? VPM,otherwise
  false.

VP
N
Bond
Vi
BondM F(Bond) Bond
is hungry
is hungryM F(is hungry) Bond, Pavarotti
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M
S

• Thus, we end up with an interpretation of this
  sentence that goes like this
• SM true iffF(Bond) ? F(is hungry), otherwise
  false.
• Given this particular model, that boils down to
• SM true iff Bond ? Bond, Pavarotti,
  otherwise false.
• (True in this situation)

VP
N
Bond
Vi
BondM F(Bond) Bond
is hungry
is hungryM F(is hungry) Bond, Pavarotti
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A semantic rule for every structural rule

• Our goal is to design a semantics for F1 that can
  provide an interpretation (truth conditions) for
  any structure that the syntax can provide.
• So, we also need rules for structures like S conj
  S, neg S, Vt N.

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Neg S

• As for Neg S, we want it to be false whenever S
  is true, and true whenever S is false.
• Neg SM false if SM true, true if SM
  false.
• However, this is not quite enoughwe want to have
  an interpretation for every node in the tree.
  This gives us an interpretation of S Neg S, but
  what is the interpretation of Neg?

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Neg

• What Neg does is takes the truth value of the S
  it is next to and reverses it.
• It is a functionit takes the truth value of the
  S it is next to as an argument, and returns a
  truth value (the opposite one).
• it is not the case thatM true ?
  false false ? true

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Neg S

• S Neg It is not the case that S Pavarotti is
  boring.
• NegM It is not the case thatM true ?
  false false ? true
• SM true iff NM ? VPM, otherwise false
  true iff PavarottiM ? ViM, otherwise false
  true iff PavarottiM ? is boringM, otherwise
  false F(Pavarotti) ? F(is boring), otherwise
  false

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Neg S

• S Neg It is not the case that S Pavarotti is
  boring.
• And, so S Neg SM NegM ( SM ).
• Resulting in
• SM false if F(Pavarotti) ? F(is
  boring),otherwise true.

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And

• For dealing with and and or, we also want to
  define a function. We want S1 and S2 to be true
  when S1 is true and S2 is true, and false under
  any other circumstance.
• S S1 Conj S2M ConjM ( lt S1M, S2M gt )
• andM lt true, true gt ? true lt true, false
  gt ? false lt false, true gt ? false lt false,
  false gt ? false

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Or

• For dealing with and and or, we also want to
  define a function. We want S1 or S2 to be false
  when S1 is false and S2 is false , and true under
  any other circumstance.
• S S1 Conj S2M ConjM ( lt S1M, S2M gt )
• orM lt true, true gt ? true lt true, false
  gt ? true lt false, true gt ? true lt false,
  false gt ? false

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Revisiting verbs

• Earlier, we defined a meaning for is boring by
  explicitly listing the set of boring individuals.
  This relies on a specific model/situation. We
  want to be more general than that, so that our
  interpretation rules work in any model.
• is boringM x x is boring in M
• is boringM x ? U x ? F(is boring)

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Generalizing

• We also do not yet have a general statement of
  how to evaluate S N VPM.
• VP Vi is boringM x x is boring in M
• S N VPM true iff NM ? VPM, otherwise
  false

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Transitive verbs

• The one piece of the model that we have not
  addressed yet are transitive verbs, like likes.
• S ? N VP
• VP ? Vt N
• Vt ? likes
• We want to be able to evaluate S N VPM the same
  way whether VP is built from a transitive verb or
  an intransitive verb. That is, we want VPM to
  be a predicate, a set of individuals.

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Transitive verbs

• Essentially, we want likes BondM to be a set of
  those individuals that like Bond in M.
• However, we need a definition for likesM (we
  already have one for BondM). It should be
  something that creates a set of individuals that
  depends on the individual next to it in the
  structure. A function again.

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Transitive verbs

• Like and, likes relates two things, although
  likes relates two individuals, and and relates
  two sentences.
• So, we build a two-place predicate, in the same
  way
• likesM ltx,ygt x likes y in M
• For example, if P likes L, L likes B and thats
  all the liking in this situation, then likesM
  ltP,Lgt, ltL,Bgt

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Transitive verbs

• And then, we define a rule that will interpret
  the VP in a sentence with a transitive verb
• VP Vt NM x lt x, NM gt ? VtM
• So if NM Bond, then VP Vt NM is the set
  containing those individuals who like Bond in M.

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S ? N VP S N VPM true iff NM ? VPM, otherwise false
S ? S Conj S S S1 Conj S2M ConjM ( lt S1M, S2M gt )
S ? Neg S S Neg SM NegM ( SM ).
VP ? Vt N VP Vt NM x lt x, NM gt ? VtM
VP ? Vi
N ? Pavarotti, PavarottiM F(Pavarotti)
Vi ? is boring, is boringM x x is boring in M
Vt ? likes likesM ltx,ygt x likes y in M
Conj ? and, andM ltlttrue,truegt,truegt, lttrue,falsegt,falsegt,
Neg ? it is not the case that iintctM lttrue,falsegt, ltfalse,truegt
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What we have

• We have created a little fragment describing a
  (very small) subset of English, generating
  structural descriptions of syntactically valid
  sentences and providing the means to determine
  the truth conditions of these sentences.
• We did this by formulating a set of syntactic
  rewrite rules, each accompanied by a semantic
  rule of interpretation, such that every syntactic
  step can be interpreted compositionally.

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One step more general

• Looking over the rules that we have, we can
  actually go a step further in generalizing our
  semantic rules (helpful as we expand our
  fragments coverage).
• There are basically two kinds of rules we have
  Those that combine meanings of adjacent (sister)
  nodes in the syntactic structure, and those that
  define intrinsic (non-compositional) meanings.

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Semantic type

• The entire semantics that we are creating here
  depends on two types of things, individuals and
  truth values.
• We can label individuals as being of type e
  (traditional, think entity), and truth values
  as being of type t.
• In these terms, names like Bond are of type ltegt,
  and sentences like Bond is hungry are of type lttgt.

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Characteristic functions

• For predicates like is hungry, we have considered
  these to be sets of individuals (e.g., those that
  are hungry in the model).
• We can look at those same individuals in a
  slightly different way, using the characteristic
  function of the set.
• A characteristic function is a function that,
  given an argument, will return true iff the
  argument was a member of the set, and false
  otherwise. The same information content as the
  set.

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Predicates as functions

• So, without losing information, we can view
  predicates from the perspective of their
  characteristic functions and define is hungry to
  instead be a function that, given an individual,
  will return true if the individual is hungry in
  the model.
• is hungryM x ? true if x is hungry in M x
  ? false otherwise

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Semantic type

• Predicates like is hungry can then be said to
  have semantic type lte,tgt. That is, a function
  from individuals to truth values.
• Similarly, it is not the case that can be taken
  to be of type ltt,tgt, a function from truth values
  to truth values.

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Transitive verbs

• For transitive verbs, what we want is a relation
  between two individuals, resulting in a truth
  value. The way we have it set up now, a verb like
  likes will combine with the object to form a
  simpler predicate like likes Bond, at which point
  it acts just like is boring.
• So, here, we want likes to take an argument of
  type ltegt and return a predicate of type lte,tgt.
  So, we define it as a function of type lte,lte,tgtgt.

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Transitive verbs

• That is, we can define likesM as something like
  this
• likesM x ? f where f is a function from
  individuals to truth values and f(y) true iff y
  likes x in M, otherwise false.
• That is, likesM is a function from individuals
  to functions (from individuals to truth values)
  semantic type lte,lte,tgtgt.

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Why were doing this

• Once we have defined things in terms of semantic
  type, and in terms of functions and arguments, we
  can collapse a number of our semantic
  interpretation rules into more general rules.
• Functional applicationa bM aM (bM ) or
  bM (aM), whichever is defined.
• Pass upb aM aM

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