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

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

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

4. 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. 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 S  N VP N  Pavarotti N  Pavarotti VP  Vi VP  Vi Vi  is boring Vi  is boring S  N VP S  N VP What is the interpretation of S? Put another way, what is [S] M ? What is the interpretation of S? Put another way, what is [S] M ? S N VP Vi is boring Pavarotti

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

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

7. The interpretation of S So, given that, we have: So, given that, we have: [ S N VP] M = true iff PAVAROTTI  [VP] M [ S N VP] M = true iff PAVAROTTI  [VP] M Now, what is [VP] M ? Now, what is [VP] M ? Since meaning is compositional and VP does not branch, [VP] M is the same as [Vi] M. Since meaning is compositional and VP does not branch, [VP] M is the same as [Vi] M. So, what is [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. 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. 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). So [is boring] M = F(is boring). S N VP Vi is boring Pavarotti [N] M = PAVAROTTI

8. The interpretation of S So far: So far: [ S N VP] M = true iff PAVAROTTI  [VP] M [ S N VP] M = true iff PAVAROTTI  [VP] M [VP] M = [Vi] M [VP] M = [Vi] M [Vi] M = [is boring] M [Vi] M = [is boring] M [is boring] M = F(is boring) [is boring] M = F(is boring) Now, what is 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. 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} On one particular model, perhaps F(is boring)= {PAVAROTTI, LOREN} In general: In general: F(is boring) = {x: x is boring in M} F(is boring) = {x: x is boring in M} S N VP Vi is boring Pavarotti [N] M = PAVAROTTI [is boring] M = F(is boring) = {x: x is boring in M}

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

10. Semantic rules of F1 Summarizing the rules we used so far: Summarizing the rules we used so far: [ S N VP] M = true iff [N] M  [VP] M [ S N VP] M = true iff [N] M  [VP] M [Pavarotti] M = F(Pavarotti) [Pavarotti] M = F(Pavarotti) [is boring] M = F(is boring) [is boring] M = F(is boring) F(Pavarotti) = the individual in M named by F as “Pavarotti” 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} F(is boring) = the set of individuals in M that are boring = {x: x is boring in M}

11. Saving ink and expressing a generalization Some of these rules are very specific. Rather than add a new rule for each individual and predicate… Some of these rules are very specific. Rather than add a new rule for each individual and predicate… [Bond] M = F(Bond) [Bond] M = F(Bond) [Loren] M = F(Loren) [Loren] M = F(Loren) [is hungry] M = F(is hungry) [is hungry] M = F(is hungry) [is cute] M = F(is cute) [is cute] M = F(is cute) …we can abstract out the pattern here and write a more general rule: …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) [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)

12. The role of F This perhaps also clarifies 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 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. F is responsible for assigning the interpretations to the terminal nodes. The semantic rules are responsible for assigning the interpretations to the combinations. The semantic rules are responsible for assigning the interpretations to the combinations.

13. Continuing with the semantic rules We can also generate trees with Neg that we need to assign an interpretation to as well. 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. 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 just just worked that out. We know what [S] M is, we just just 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. We know what we want [S] M to be—false when [S] M is true, and true when [S] M is false. S Neg N VP PavarottiVi is boring It is not the case that S

14. Neg S Goal: [ S Neg S] M = false if [S] M = true, true if [S] M = false. 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? 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. 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] M = {false}

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

16. It is not the case that Pavarotti is boring [S] M = [ S Neg S] M [S] M = [ S Neg S] M [ S Neg S] M = true iff [S] M  [Neg] M [ S Neg S] M = true iff [S] M  [Neg] M [Neg] M = {false} [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] 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 Neg N VP PavarottiVi is boring It is not the case that S

17. Transitive verbs The syntax of F1 also generates trees with transitive verbs, like likes. The syntax of F1 also generates trees with transitive verbs, like likes. S  N VP S  N VP VP  Vt N VP  Vt N Vt  likes Vt  likes 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. 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.

18. Transitive verbs Essentially, we want [likes Bond] M to be a set of those individuals that like Bond in M. 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. 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} [ VP likes Bond] M = {x: x likes Bond in M}

19. Transitive verbs A transitive verb relates two individuals. They stand in an (asymmetrical) relationship. 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. 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) = {, } For example, if P likes L, L likes B and that’s all the liking in this situation, then F(likes) = {, } We could express this as follows, to use a (metalanguage) shorthand: We could express this as follows, to use a (metalanguage) shorthand: [likes] M = { : x likes y in M } [likes] M = { : x likes y in M }

20. Transitive verbs And then, we define a rule that will interpret the VP in a sentence with a transitive verb: And then, we define a rule that will interpret the VP in a sentence with a transitive verb: [ VP Vt N] M = {x :  [Vt] M } [ VP Vt N] M = {x :  [Vt] M } If [N] M = Bond, [ VP Vt N] M is the set containing those individuals who like Bond in M. If [N] M = Bond, [ VP Vt 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 = {, }, then [ VP Vt N] M1 = {L}, and [S] M1 = true. For example Loren likes Bond: If in a particular model M1, [likes] M1 = {, }, then [ VP Vt N] M1 = {L}, and [S] M1 = true. In general, [S] M = true iff F(Loren)  {x:  F(likes)} = true iff  F(likes). In general, [S] M = true iff F(Loren)  {x:  F(likes)} = true iff  F(likes).

21. Sentence coordination We also need a way to interpret or and and. 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. Two options: New rule for ternary branching and symmetric relations. Or recast as binary branching. S Neg S N VP PavarottiVi is boring It is not the case that S N VP LorenVi is hungry SConj or

22. Thoughts on coordination Like transitive verbs, or and and express a kind of relation (between truth values, rather than between individuals). 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. 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). 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. 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. 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.

23. 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: 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 S  S ConjP ConjP  Conj S ConjP  Conj S

24. Revised structure for or: Thus: Thus: S Neg S N VP PavarottiVi is boring It is not the case that S N VP LorenVi is hungry SConj or ConjP

25. Or For or we need to consider pairs of sentences. We want S 1 or S 2 to be false when S 1 is false and S 2 is false, and true under any other circumstance. For or we need to consider pairs of sentences. We want S 1 or S 2 to be false when S 1 is false and S 2 is false, and true under any other circumstance. Goal: Goal: [ S S 1 [ ConjP or S 2 ]] M = true iff [S 1 ] M  [S 2 ] M. [ S S 1 [ ConjP or S 2 ]] M = true iff [S 1 ] M  [S 2 ] M. The combination occurs in two stages, first with S 2, to yield a property then applied to S 1. The combination occurs in two stages, first with S 2, to yield a property then applied to S 1.

26. Or On the model of transitive verbs, suppose that F(or) is a set of relations between true values: On the model of transitive verbs, suppose that F(or) is a set of relations between true values: F(or) = {,, } F(or) = {,, } And a rule of combination just like that for [ VP Vt N]: And a rule of combination just like that for [ VP Vt N]: [ ConjP Conj S] M = {x :  [Conj] M } [ ConjP Conj S] M = {x :  [Conj] M } Does it work? Does it work? What’s F(and)? What’s F(and)? What would be involved in adding if? What would be involved in adding if?

27. Semantic rules of F1 Summarizing the rules we used so far: Summarizing the rules we used so far: [ S N VP] M = true iff [N] M  [VP] M [ S N VP] M = true iff [N] M  [VP] M [ S S 1 Conj S 2 ] M = true iff {[S 1 ] M, [S 2 ] M }  [Conj] M [ S S 1 Conj S 2 ] M = true iff {[S 1 ] M, [S 2 ] M }  [Conj] M [ S Neg S] M = true iff [S] M  [Neg] M [ S Neg S] M = true iff [S] M  [Neg] M [X] M = F(X) where X is a terminal node [X] M = F(X) where X is a terminal node F(It is not the case that) = {false} F(It is not the case that) = {false} F(or) = {{true, true}, {false, true}} F(or) = {{true, true}, {false, true}} F(and) = {{true, true}} F(and) = {{true, true}} Note the change for and, or, not (ultimately assigned by F) Note the change for and, or, not (ultimately assigned by F)

28. Full summary of F1 S  N VP [ S N VP] M = true iff [N] M  [VP] M S  Neg S [ S Neg S´] M = true iff [S´] M  [Neg] M S  S ConjP [ S S ConjP] M = true iff [S] M  [ConjP] M ConjP  Conj S [ ConjP Conj S] M = {x :  [Conj] M } VP  Vt N [ VP Vt N] M = {x :  [Vt] M } VP  Vi [ [X] ] M = [X] M for any X [X] M = F(X) where X is a terminal node N  Pavarotti, … F(Pavarotti) = PAVAROTTI Vi  is boring, … F(is boring) = {x: x is boring in M} Vt  likes F(likes) = { : x likes y in M } Conj  and, or F(and) = { }, F(or) = {,, } Neg  it is not… F(iintct) = {false}

29. 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 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. 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.

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

31. One step further…? If we have these rules: If we have these rules: [A B] M = true iff [A] M  [B] M [A B] M = true iff [A] M  [B] M (where [B] M is a set of [A] M -type things) (where [B] M is a set of [A] M -type things) [A B] M = {x:  [B] M } [A B] M = {x:  [B] M } (where [B] M is a set of pairs, the second member being an [A] M -type thing) (where [B] M is a set of pairs, the second member being an [A] M -type thing) [ [A] ] M = [A] M [ [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? 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: One option: Redefine F(is boring) as, e.g., {,,…} Redefine F(is boring) as, e.g., {,,…} Define {true} as true and {false} as false. Define {true} as true and {false} as false. Redefine F(likes) as, e.g., { >, >,…} Redefine F(likes) as, e.g., { >, >,…} See how it works? But it’s confusing… See how it works? But it’s confusing…

32. Exploring the option… The option: The option: Redefine F(is boring) as, e.g., {,,…} Redefine F(is boring) as, e.g., {,,…} Define {true} as true and {false} as false. Define {true} as true and {false} as false. Redefine F(likes) as, e.g., { >, >,…} Redefine F(likes) as, e.g., { >, >,…} 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. 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. 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. Is-boring(x) = true iff x is boring. Ah. It would be less confusing if we just wrote it as a function. 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) F(is boring) = the function f such that f(x)=true iff X is boring (in M) Or, using the -notation we saw before: Or, using the -notation we saw before: F(is boring) = x[x is boring in M] 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. See where the notation comes from? 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. See where the notation comes from?

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

34. What this buys us Defining things in terms of functions allows us to reduce our semantic rules to just two: Defining things in terms of functions allows us to reduce our semantic rules to just two: Functional application: [   ] M = [  ] M ([  ] M ) or [  ] M ([  ] M ), whichever is defined. Functional application: [   ] M = [  ] M ([  ] M ) or [  ] M ([  ] M ), whichever is defined. Pass up: [   ] M = [  ] M Pass up: [   ] M = [  ] 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.” 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. 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. We will also consider an alternative version in terms of “events” and “states” in future classes.

35.                       