LX CAS 502 Semantics
3a. A formalism for meaning (contd) 3.2, 3.6
Recap
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F1 = Rules for generating and interpreting a small fragment of English.
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Syntax: Phrase structure rules
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Reviewed on the next slide Idea: All and only sentences generated by the PS rules are part of the language (F1, approximating English). Goals:
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Interpretation: [ ]M
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Assign an interpretation to every node in the structure Arrive at the interpretation compositionally
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Interpretation is assigned with respect to a model (effectively, the facts about the world: The players [U] and their properties [F]).
F1: The syntax
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Using the syntax of F1
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Phrase Structure rules (the syntax):
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To be revised N ! Pavarotti, Loren, Bond Vi ! is boring, is hungry, is cute Vt ! likes Conj ! and, or Neg ! it is not the case that
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Starting with S, we can rewrite it using the rules of the syntax until we get to a structure such as this one.
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S N Pavarotti VP Vi is boring
S ! N VP S ! S conj S S ! neg S VP ! Vt N VP ! Vi
S ! N VP N ! Pavarotti VP ! Vi Vi ! is boring S ! N VP
What is the interpretation of S? Put another way, what is [S]M?
The interpretation of S
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The interpretation of S
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We developed a semantic rule that tells us what the interpretation of [S N VP] is:
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So far:
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S
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[S N VP]M = true iff [N]M " [VP]M [N]M = [Pavarotti]M
S N Pavarotti VP Vi is boring
[S N VP]M = true iff [N]M " [VP]M
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Great, are we done? Well, we would be, if we knew what [N]M and [VP]M were. Whats [N]M? Since meaning is compositional and N does not branch, [N]M is the same as [Pavarotti]M. So, whats [Pavarotti]M?
N Pavarotti
VP Vi is boring
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Whats [Pavarotti]M? We have a semantic rule that tells us that:
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[Pavarotti]M = F(Pavarotti)
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That is, the interpretation of a name is the individual from the model M that the pointing (or naming) function F designates. F(Pavarotti) in this model is the individual PAVAROTTI. So [Pavarotti]M = PAVAROTTI. So [N]M = PAVAROTTI.
[Pavarotti]M = F(Pavarotti) = PAVAROTTI
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The interpretation of S
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[N]M = PAVAROTTI
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The interpretation of S
So far:
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[N]M = PAVAROTTI
So, given that, we have:
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[S N VP]M = true iff PAVAROTTI " [VP]M
S N Pavarotti VP
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Now, what is [VP]M? Since meaning is compositional and VP does not branch, [VP]M is the same as [Vi]M. So, what is [Vi]M? Since meaning is compositional and VP does not branch, [Vi]M is the same as [is boring]M. We have a semantic rule that tells us that [is boring]M is the set of individuals from the model M that the function F designates. So [is boring]M = F(is boring).
[S N VP]M = true iff PAVAROTTI " [VP]M [VP]M = [Vi]M M = [is boring]M [Vi] [is boring]M = F(is boring)
S N Pavarotti VP Vi is boring
Vi is boring
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Now, what is F(is boring)? It will depend on the modelwho are the boring individuals in this particular model? F(is boring) will be a set of individuals that are boring in this model. On one particular model, perhaps F(is boring)= {PAVAROTTI, LOREN} In general:
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F(is boring) = {x: x is boring in M}
[is boring]M = F(is boring) = {x: x is boring in M}
The interpretation of S
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[N]M = PAVAROTTI
Semantic rules of F1
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Now, were basically done.
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F(is boring) = {x: x is boring in M} [is boring]M = F(is boring) [is boring]M = {x: x is boring in M} [Vi]M = [is boring]M [Vi]M = {x: x is boring in M} [VP]M = [Vi]M [VP]M = {x: x is boring in M} [S N VP]M = true iff PAVAROTTI " [VP]M [S N VP]M = true iff PAVAROTTI " {x: x is boring in M}
S N Pavarotti VP Vi is boring
Summarizing the rules we used so far:
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As desired. Picking the particular model where {x: x is boring in M} = {PAVAROTTI, LOREN}, [S]M = true.
[is boring]M = F(is boring) = {x: x is boring in M}
N VP]M = true iff [N]M " [VP]M [Pavarotti]M = F(Pavarotti) ! [is boring]M = F(is boring) ! F(Pavarotti) = the individual in M named by F as Pavarotti ! F(is boring) = the set of individuals in M that are boring = {x: x is boring in M}
Saving ink and expressing a generalization
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The role of F
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Some of these rules are very specific. Rather than add a new rule for each individual and predicate
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[Bond]M = F(Bond) [Loren]M = F(Loren) [is hungry]M = F(is hungry) [is cute]M = F(is cute)
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we can abstract out the pattern here and write a more general rule:
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[X]M = F(X) where X is a terminal node (has no children, does not appear on the LHS of a PS rule in the syntax)
This perhaps also clarifies the role of F. F is essentially the thing that translates the object language (English, say) into the metalanguage in terms of the model. F is responsible for assigning the interpretations to the terminal nodes. The semantic rules are responsible for assigning the interpretations to the combinations.
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Continuing with the semantic rules
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Neg S
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We can also generate trees with Neg that we need to assign an interpretation to as well.
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Goal: [S Neg S#]M = false if [S#]M = true, true if [S#]M = false. What interpretation must we assign to [Neg]M to arrive at this result? Lets try to make this look like is hungry in a certain sense. A property of truth values, in this case the property of being false. [Neg]M = {false}
Notice that we have written one of the S nodes as S#. This is like painting one blue and one redwe just want to be able to refer to each one separately. As far as the rules are concerned, it is just a normal S.
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We know what [S#]M is, we just just worked that out. Neg We know what we want [S]M to befalse when [S#]M is true, and true when [S#]M is false.
S S#
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N VP It is not the case that Pavarotti Vi is boring
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Neg S
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It is not the case that Pavarotti is boring
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Goal: [S Neg S#]M = false if [S#]M = true, true if [S#]M = false. [Neg]M = {false}
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So [Neg] M is a set of truth values (like [is hungry]M is a set of individuals).
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Now we can define an interpretation rule very much like our previous [S N VP]M rule. [S Neg S#]M = true iff [S#]M " [Neg]M
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[S]M = [S Neg S#]M [S Neg S#]M = true iff [S#]M " [Neg]M [Neg]M = {false} [S#]M = true iff PAVAROTTI " {x: x is boring in M} S [S]M = true iff [PAVAROTTI " {x: x is boring in M}] S# Neg [S]M = true iff N VP PAVAROTTI $ It is not {x: x is boring in M} the case that Pavarotti Vi is boring
Transitive verbs
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Transitive verbs
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The syntax of F1 also generates trees with transitive verbs, like likes.
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Essentially, we want [likes Bond]M to be a set of those individuals that like Bond in M. However, we need a definition for [likes] M (we already have one for [Bond]M). It should be something that creates a set of individuals that depends on the individual next to it in the structure. [VP likes Bond]M = {x: x likes Bond in M}
S ! N VP VP ! Vt N Vt ! likes
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We want to be able to evaluate [S N VP] M the same way whether VP is built from a transitive verb or an intransitive verb. That is, we want [VP]M to be a predicate, a set of individuals in either case.
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Transitive verbs
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Transitive verbs
And then, we define a rule that will interpret the VP in a sentence with a transitive verb:
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A transitive verb relates two individuals. They stand in an (asymmetrical) relationship. Suppose that this is expressed in the model as a set of pairs that are involved in the relationship.
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[VP Vt N] M = {x : < x, [N] M > " [Vt] M }
For example, if P likes L, L likes B and thats all the liking in this situation, then F(likes) = { <P,L>, <L,B> }
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If [N]M = Bond, [VP Vt N] M is the set containing those individuals like who Bond in M.
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We could express this as follows, to use a (metalanguage) shorthand:
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[likes] M = { <x,y> : x likes y in M }
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For example Loren likes Bond: If in a particular model M1, [likes] M1 = {<P,L>, <L,B>}, then [VP Vt N]M1 = {L}, and [S]M1 = true. In general, [S]M = true iff F(Loren) " {x: <x, F(Bond)> " F(likes)} = true iff <F(Loren), F(Bond)> " F(likes).
Sentence coordination
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Thoughts on coordination
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We also need a way to interpret or and and. Two options: New rule for ternary branching and symmetric relations. Or recast as binary branching. S S Neg N S VP Vi Conj or N Loren S VP Vi is hungry
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It is not the case that Pavarotti
is boring
Like transitive verbs, or and and express a kind of relation (between truth values, rather than between individuals). The relation expressed by or and and is symmetrical, order does not seem to affect the relation. But some transitive verbs are like this too (e.g. resemble). And we might want to consider if a kind of coordinatorbut for if, order does matter. Lets consider symmetry an accidental property, due to the definition of the word in question (according to F), and not a property inherent in a new type of semantic combination.
Breaking the structural symmetry
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Revised structure for or:
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In order to reduce symmetrical and and or to a binary-branching (and therefore necessarily asymmetrical) structure, we modify the syntax slightly: S ! S ConjP ConjP ! Conj S
Thus:
S S Neg N S VP Vi ConjP Conj or N Loren S VP Vi is hungry
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It is not the case that Pavarotti
is boring
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Or
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Or
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For or we need to consider pairs of sentences. We want S1 or S2 to be false when S 1 is false and S2 is false , and true under any other circumstance. Goal: [S S1 [ConjP or S2 ]]M = true iff [S1]M % [S2]M. The combination occurs in two stages, first with S2, to yield a property then applied to S1 .
On the model of transitive verbs, suppose that F(or) is a set of relations between true values:
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F(or) = {<true, true>, <true, false>, <false, true>}
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And a rule of combination just like that for [VP Vt N]: ! [ConjP Conj S]M = {x : < x, [S] M > " [Conj] M } Does it work? Whats F(and)? What would be involved in adding if?
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Semantic rules of F1
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Full summary of F1
S ! N VP S ! Neg S S ! S ConjP ConjP ! Conj S VP ! Vt N VP ! Vi N ! Pavarotti, Vi ! is boring, Vt ! likes Conj ! and, or Neg ! it is not [S N VP]M = true iff [N]M " [VP]M [S Neg S]M = true iff [S]M " [Neg]M [S S ConjP]M = true iff [S]M " [ConjP]M [ConjP Conj S]M = {x : < x, [S]M > " [Conj]M } [VP Vt N]M = {x : < x, [N]M > " [Vt]M } [ [X] ]M = [X]M for any X [X]M = F(X) where X is a terminal node F(Pavarotti) = PAVAROTTI F(is boring) = {x: x is boring in M} F(likes) = { <x,y> : x likes y in M } F(and) = {<true, true>}, F(or) = {<true, true>, <true,false>, <false,true>} F(iintct) = {false}
Summarizing the rules we used so far:
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[S N VP]M = true iff [N]M " [VP]M [S S1 Conj S2] M = true iff {[S1] M, [S2]M} " [Conj]M [S Neg S#]M = true iff [S#]M " [Neg]M [X]M = F(X) where X is a terminal node F(It is not the case that) = {false} F(or) = {{true, true}, {false, true}} F(and) = {{true, true}}
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Note the change for and, or, not (ultimately assigned by F)
What we have
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One step more general
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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.
Looking over the rules that we have, there are basically just two kinds:
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[S N VP]M = true iff [N]M " [VP]M [S S ConjP] M = true iff [S]M " [ConjP]M [S Neg S#] M = true iff [S#] M " [Neg] M [VP Vt N]M = {x: <x,[N]M> " [Vt] M } [ConjP Conj S]M = {x: <x,[S]M> " [Conj]M } [A B]M = true iff [A]M " [B]M
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More generally:
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(where [B]M is a set of [A]M-type things) (where [B]M is a set of pairs, the second member being an [A]M-type thing)
[A B]M = {x: <x,[A]M>} " [B]M
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[ [A] ]M = [A]M
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This will cover our other rules and make it easier to extend our syntax as well.
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One step further?
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Exploring the option
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If we have these rul...