CAS LX 502 Semantics

CAS LX 502Semantics 3a. A formalism for meaning (cont’d) 3.2, 3.6

Recap • “F1” = Rules for generating and interpreting a small fragment of English. • Syntax: Phrase structure rules • Reviewed on the next slide • Idea: All and only sentences generated by the PS rules are part of the language (F1, approximating English). • Interpretation: [ ]M • Goals: • Assign an interpretation to every node in the structure • Arrive at the interpretation compositionally • Interpretation is assigned with respect to a model (effectively, the facts about the world: The players [U] and their properties [F]).

F1: The syntax • Phrase Structure rules (the syntax): • To be revised…

Using the syntax of F1 • Starting with S, we can “rewrite it” using the rules of the syntax until we get to a structure such as this one. • 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? S VP N Pavarotti Vi is boring

The interpretation of S • We developed a semantic rule that tells us what the interpretation of [S N VP] is: • [S N VP]M = true iff [N]M [VP]M • Great, are we done? Well, we would be, if we knew what [N]M and [VP]M were. • What’s [N]M? • Since meaning is compositional and N does not branch, [N]M is the same as [Pavarotti]M. • So, what’s [Pavarotti]M? S VP N Pavarotti Vi is boring

The interpretation of S • So far: • [S N VP]M = true iff [N]M [VP]M • [N]M = [Pavarotti]M • What’s [Pavarotti]M? • We have a semantic rule that tells us that: • [Pavarotti]M = F(Pavarotti) • 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. S VP N Pavarotti Vi is boring [Pavarotti]M =F(Pavarotti) =PAVAROTTI

The interpretation of S [N]M = PAVAROTTI • So, given that, we have: • [S N VP]M = true iff PAVAROTTI  [VP]M • 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 VP N Pavarotti Vi is boring

The interpretation of S [N]M = PAVAROTTI • So far: • [S N VP]M = true iff PAVAROTTI [VP]M • [VP]M = [Vi]M • [Vi]M = [is boring]M • [is boring]M = F(is boring) • Now, what is F(is boring)? • It will depend on the model—who 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: • F(is boring) = {x: x is boring in M} S VP N Pavarotti Vi is boring [is boring]M =F(is boring) ={x: x is boring in M}

The interpretation of S [N]M = PAVAROTTI • Now, we’re basically done. • 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} • As desired. Picking the particular model where {x: x is boring in M} = {PAVAROTTI, LOREN}, [S]M = true. S VP N Pavarotti Vi is boring [is boring]M =F(is boring) ={x: x is boring in M}

Semantic rules of F1 • Summarizing the rules we used so far: • [S 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 • Some of these rules are very specific. Rather than add a new rule for each individual and predicate… • [Bond]M = F(Bond) • [Loren]M = F(Loren) • [is hungry]M = F(is hungry) • [is cute]M = F(is cute) • …we can abstract out the pattern here and write a more general rule: • [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)

The role of F • 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.

Continuing with the semantic rules • We can also generate trees with Neg that we need to assign an interpretation to as well. • Notice that we have written one of the S nodes as S. This is like painting one blue and one red—we just want to be able to refer to each one separately. As far as the rules are concerned, it is just a normal S. • We know what [S]M is, we justjust worked that out. • We know what we want[S]M to be—false when[S]M is true, and true when[S]M is false. S S Neg N VP It is notthe case that Pavarotti Vi is boring

Neg S • Goal: [SNeg 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? • Let’s 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}

Neg S • Goal: [SNeg S]M = false if [S]M = true, true if [S]M = false. • [Neg]M = {false} • So [Neg]M is a set of truth values (like [is hungry]M is a set of individuals). • 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

It is not the case thatPavarotti is boring • [S]M = [S NegS]M • [SNeg S]M = true iff [S]M [Neg]M • [Neg]M = {false} • [S]M= true iff PAVAROTTI {x: x is boring in M} • [S]M = true iff[PAVAROTTI{x: x is boring in M}] • [S]M = true iff PAVAROTTI {x: x is boring in M} S S Neg N VP It is notthe case that Pavarotti Vi is boring

Transitive verbs • The syntax of F1 also generates trees with transitive verbs, like likes. • S  N VP • VP  Vt N • Vt  likes • We want to be able to evaluate [SN 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.

Transitive verbs • 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}

Transitive verbs • 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. • For example, if P likes L, L likes B and that’s all the liking in this situation, then F(likes) = { <P,L>, <L,B> } • We could express this as follows, to use a (metalanguage) shorthand: • [likes]M = { <x,y> : x likes y in M }

Transitive verbs • And then, we define a rule that will interpret the VP in a sentence with a transitive verb: • [VP Vt N]M = {x : < x, [N]M >  [Vt]M } • If [N]M = Bond, [VPVt N]M is the set containing those individuals who like Bond in M. • 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 iffF(Loren)  {x: <x, F(Bond)>  F(likes)}= true iff <F(Loren), F(Bond)>  F(likes).

Sentence coordination • 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 Conj S N S or VP Neg N VP Loren Vi It is notthe case that Pavarotti Vi is hungry is boring

Thoughts on coordination • 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 coordinator—but for if, order does matter. • Let’s 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 • 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

Revised structure for or: • Thus: S S ConjP S Conj S Neg N N VP or VP It is notthe case that Pavarotti Vi Loren Vi is boring is hungry

Or • For or we need to consider pairs of sentences. We want S1 or S2 to be false when S1 is false and S2 is false , and true under any other circumstance. • Goal: • [SS1[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.

Or • On the model of transitive verbs, suppose that F(or) is a set of relations between true values: • F(or) = {<true, true>, <true, false>, <false, true>} • 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? • What’s F(and)? • What would be involved in adding if?

Semantic rules of F1 • Summarizing the rules we used so far: • [S N VP]M = true iff [N]M  [VP]M • [SS1 Conj S2]M = true iff {[S1]M, [S2]M} [Conj]M • [SNeg 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}} • Note the change for and, or, not (ultimately assigned by F)

Full summary of F1

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.

One step more general • Looking over the rules that we have, there are basically just two kinds: • [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 } • More generally: • [A B]M = true iff [A]M [B]M • (where [B]M is a set of [A]M-type things) • [A B]M = {x: <x,[A]M>} [B]M • (where [B]M is a set of pairs, the second member being an [A]M-type thing) • [ [A] ]M = [A]M • This will cover our other rules… and make it easier to extend our syntax as well.

One step further…? • If we have these rules: • [A B]M = true iff [A]M [B]M • (where [B]M is a set of [A]M-type things) • [A B]M = {x: <x,[A]M> [B]M } • (where [B]M is a set of pairs, the second member being an [A]M-type thing) • [ [A] ]M = [A]M • It feels as if we still have a kind of specific rule: the first looks kind of like a “special case” of the second. But how can we reduce them to one rule? • One option: • Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…} • Define {true} as true and {false} as false. • Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…} • See how it works? But it’s confusing…

Exploring the option… • The option: • Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…} • Define {true} as true and {false} as false. • Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…} • What we have to do is, for properties: redefine the set so that there is a pair for each individual, with true or false depending on whether the individual has the property. • But, wait. What we just defined is in fact a function. The first member of the pair is the argument, the second is the return value. • Is-boring(x) = true iff x is boring. • Ah. It would be less confusing if we just wrote it as a function. • F(is boring) = the function f such that f(x)=true iff X is boring (in M) • Or, using the -notation we saw before: • F(is boring) = x[x is boring in M] • This is the same thing as a set of pairs, the first of which is an individual, and second of which is true if the individual is boring in M and false otherwise. But thinking of it as a function is more graspable. It’s something that needs an individual and provides a truth value. Type <e,t>. See where the notation comes from?

Exploring the option… • As for transitive verbs: • Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…} • What we want is a function that applies to the object and returns a property. • A property is a function that applies to an individual and returns a truth value. • F(likes) = y[x[x likes y in M]] • F(is boring) = x[x is boring in M] • Well, that’s much more compact. • So, combining likes with Bond yields: • [likes Bond]M= y[x[x likes y in M]](Bond)= x[x likes Bond in M](the property of liking Bond)

What this buys us • Defining things in terms of functions allows us to reduce our semantic rules to just two: • Functional application:[ab]M = [a]M ([b]M ) or [b]M ([a]M), whichever is defined. • Pass up:[ba]M = [a]M • This will be the basis of F2, which we will define fully next time and then move on to the connection with “theta-roles.” • By the time we’re done, there will be one more semantic rule, to interpret “modification” relations like adjectives and adverbs. • We will also consider an alternative version in terms of “events” and “states” in future classes.

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CAS LX 502 Semantics