Auxiliaries and the -calculus Robert Levine Ohio State University - - PowerPoint PPT Presentation

Auxiliaries and the λ-calculus

Robert Levine

Ohio State University levine.1@osu.edu

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Where we left off. . .

◮ We can draw a couple of general conclusions at this point. ◮ The most important one: even in very basic clause structure, we have

evidence of higher-order kinds of expressions. . .

◮ . . . not just functions which take individuals as arguments (the way VPs

such as read that book do). . .

◮ . . . but functions which themselves take other functions as their own

arguments.

◮ But we also can see that in some cases the semantic action of these

functions seems a bit wonky.

◮ Auxiliaries apparently take two arguments, so the last line of the proof

• n the preceding slide should be faux(g)(y). . .

◮ . . . whereas what we want to wind up with is faux(g(y)). ◮ We have a syntax semantics mismatch and we’re going to need

something extra to handle it.

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Something extra

◮ The basic problem is two-fold. On one hand, the auxiliary ‘gets in the

way’ between the VP and the subject. . .

◮ . . . and on the other, the auxiliary is forced to apply to arguments that

are semantically wrong.

◮ To illustrate: consider (1):

(1) This kind of equation must have a solution.

◮ The sense of (1) is that the proposition expressible as This equation

has a solution is necessarily true,

◮ there is no state of affairs such that the equation lacks a solution in

that state of affairs.

◮ In effect, must imposes this condition of necessary truth on

propositions; hence it should take an argument of type t. One argument of type e and a second argument of type e, t do not add up to a single semantic object of type t.

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◮ It is standard in semantics to notate this ‘necessarily true’ function as

.

◮ Using this notation, we can say that the auxiliary must interferes with

the correct compositional interpretation of (1) because it does not allow the VP to combine with the subject to form the proposition have(a-solution)(this-kind-of-equation)

◮ to which then applies. ◮ The property have(a-solution) is waiting to pick up the subject as

its argument to yield an object of type t. . .

◮ . . . but never gets it because gets there first. . . and can’t do

anything with a property-type argument!

◮ What happens when you have a crowd ahead of you blocking you

from a place that’s really reserved for you?. . .

◮ as in, say, a restaurant? ◮ Enter the mˆ

aitre d’y. . .

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The λ abstraction operator

◮ The mˆ

aitre d’y’s role is, among other things, to conduct you to a seat at your reserved table.

◮ In our case, the reserved table is a position (a ‘seat’) within a

semantic subexpession that applies to. We can think of as the crowd in front of you. . .

◮ . . . and the slot waiting for the subject, the argument position of the

functor correspnding to have a solution, is the table itself, where you are the only guest present.

◮ The waiting table with its seat corresponds to have(a-solution) . . . ◮ but with the line is there ahead of you, the picture is more like

have(a-solution) . . .

◮ . . . and now the mˆ

aitre d’y appears.

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◮ He or she has a card showing your name and the seat reserved for you. ◮ Let’s depict the situation as this:

(2) λx.[have(a-solution)(x)]

◮ When you (= this-kind-of-equation) show up at the back of the

line, this is what happens:

this • kind • of • equation; tkoe; NP must • have • a • solution; λx. have(a-solution)(x); NP\S this • kind • of • equation • must • have • a • solution; λx [ have(a-solution)(x)](tkoe); S this • kind • of • equation • must • have • a • solution; have(a-solution)(tkoe); S.

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Generalizing must

◮ We now know how we wound up getting have(a-solution)(tkoe)

by putting λx. have(a-solution)(x) together with tkoe . . .

◮ . . . but how did we get λx. have(a-solution)(x) in the first place? ◮ We start with must, which takes a VP (with a property semantics) as

an argument and combines with it to form a new property.

◮ It could be have a solution; it could be apologize to Anne; it could be

compete in a yodeling contest, or any number (literally!) of other VPs.

◮ How do we write the lexical entry for must so that

◮ regardless of which VP the auxiliary combines with, ◮ we wind up getting something which then takes a subect NP and makes

it the argument of the first VP’s semantics?

◮ That was a question for you to answer.

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◮ Here’s what we have going in:

must; ??; VP/VP ϕ; X; VP must • ϕ; λx. X(x); VP

◮ . . . and coming out. How did we get there?

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◮ Ask yourself: how can we get ?? to combine with X so that the result

is λx. X(x)?

◮ X, whatever it is, has to get ‘picked up’ and dropped into the ‘

’ slot in λx. (x)

◮ How can we do that using a trick that we already know?? ◮ How about

(3) must; λPλx. P(x); VP/VP

◮ with P a variable over properties, just as x is a variable over

individuals?

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have; have; VP/NP a • solution; a-solution; NP have • a • solution; have(a-solution); VP must; λPλx. P(x); VP/VP must • have • a • solution; λx. (have(a-solution))(x); VP this • kind • of • equation; tkoe; NP this • kind • of • equation • must • have • a • solution; (have(a-solution))(tkoe); S

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Negation

(4) John must not answer that letter.

◮ What’s the meaning here? For (4) to be true, we require that in all possible worlds

(or at least all such worlds or situations that would be relevant), the proposition that John answered the letter in question is false. . . (5) ¬answer(that-letter)(j)

◮ What do we want to say about not, then? First, what does it combine with?

(6)

• a. John does not work very hard.
• b. Anne should not have been shown that file.
• c. That kind of equation may not have a solution.

◮ So not shows up in company with VPs. ◮ What is its relationship to that VP? What’s the argument? ◮ Given that

◮ auxiliaries take VP arguments ◮ and that the result of combining not+VP appears following auxiliaries,

◮ what would the category of not have to be? ◮ So not is VP/VP.

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◮ Now what about the semantic part of the sign? ◮ Let’s go back to my earlier example:

(7)

• a. John must not answer that letter.
• b. ¬answer(that-letter)(j)

◮ Does the relationship between the syntax of the subject with respect

to negation, on the one hand, and the semantics of the subject with respect to negation on the other, remind you of anything?

◮ Solution:

(8) not; λQλy.¬Q(y); VP/VP

◮ Will this play nicely with the lexical specification for must?

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Must not . . .

answer; answer; VP/NP . . . . . . that • letter; ι(letter); NP answer • that • letter; answer(ι(letter)); VP not; λQλy.¬Q(y); VP/VP not • answer • that • letter; λQ[λy.¬Q(y)](answer(ι(letter))); VP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . not • answer • that • letter; λy[¬(answer(ι(letter)))(y)]; VP must; λPλv. P(v); VP/VP must • not • answer • that • letter; λP[λv. P(v)](λy[¬(answer(ι(letter)))(y)]); VP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . must • not • answer • that • letter; λv. λy[¬(answer(ι(letter)))(y)](v); VP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . must • not • answer • that • letter; λv.¬(answer(ι(letter)))(v); VP john; j; NP john • must • not • answer • that • letter; λv[¬(answer(ι(letter)))(y)](j); S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . john • must • not • answer • that • letter; ¬(answer(ι(letter)))(j); S

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The auxiliary dependency

(9) John is        eating ∗eaten ∗eat ∗eats        lunch. (10) John has        ∗eating eaten ∗eat ∗eats        lunch. (11) John will        ∗eating ∗eaten eat∅ ∗eats        lunch. (12) John has been        eating ∗eaten ∗eat ∗eats        lunch.

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(13) John should be∅        eating ∗eaten ∗eat ∗eats        lunch. (14) John should have∅ been        eating ∗eaten ∗eat ∗eats        lunch.

◮ The pattern is clear. But how to capture it? ◮ More specifically: how do we correlate the identity of functional types with

morphological subtypes of their argument categories?

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Labels for subtyping

◮ Suppose we say that the symbol S actually describes a ‘supertype’ to

which all strings associated with a semantic type t belong.

◮ We can notate this setup as Sα with α the name of the specific

subtype and Sα the set of strings ‘picked out’ by α.

◮ Suppose further that we write the set of values for α as Σα ◮ and posit that Σα = { fin, perf, prog, b(a)se, . . . } ◮ giving us Sfin, Sperf , Sprog, Sbse etc.

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◮ From these subcategories further syntactic (sub)types can be

• derived. . .

◮ . . . e.g. NP\Sfin, NP\Sperf , etc.. . . ◮ and from these, transitive, ditransitive and so on verbs. ◮ So:

eat; eat; (NP\Sbse)/NP eats; eat; (NP\Sfin)/NP ate; eat; (NP\Sfin)/NP eating; eat; (NP\Sprog)/NP eaten; eat; (NP\Sperf )/NP

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Deriving VP subtypes

◮ We can now obtain signs by proofs of the following sort:

eat; eat; (NP\Sbse)/NP lunch; lunch; NP eat • lunch; eat(lunch); NP\Sbse

◮ which we can write as VP bse

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Putting subtypes to work

◮ More generally, we write NP\Sα as VPα. ◮ So now, how should I write the lexical entry for will that is needed to

get John will eat lunch??

◮ will; λPλx.F P(x); VP fin/VP bse

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eat • lunch; eat(lunch); VP

bse

will; λPλx.F P(x); VP

fin/VP bse

will • eat • lunch; λx.F (eat(lunch))(x); VP

fin john; j; NP

john • will • eat • lunch; F (eat(lunch))(j); Sfin

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What else?

(15)

• a. John did

not n’t

• say anything to me.
• b. Did John say anything to you?
• c. Mary didn’t say anything, but John did.

◮ So do is an auxiliary.

(16) *John didn’t              be eating. have eaten will eat have been eating . . .              .

◮ But it doesn’t like other auxiliaries at all. ◮ How does it KNOW?

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More subtyping

◮ Suppose that S is specified for sets of subtypes, corresponding to

different kinds of information.

◮ We revise the lexical entries for auxiliaries slightly.

(17) will; λPλx.F P(x); VPfin, aux/VP

bse ◮ and so on for the other auxiliaries. ◮ So: how shall we write the lexical entry for do?

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◮ Before we can provide a complete entry for do, we need to decide on

its semantics. (18)

• a. John likes pizza.
• b. John does not like pizza.
• c. John does like pizza.

◮ What’s the relationship among these three?

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Yet more subtyping

◮ So: there is an emphatic version of do ◮ and a version which appears to do nothing but support a negated VP ◮ by providing a source of finite status for the S which is deducible from

that VP.

◮ This version of do apparently only selects negated VP arguments, ◮ which points us to the following lexical entry for not:

(19) not; λRλz.¬ R(z); VP

neg, α/VPα ◮ with α now a variable over sets of subtype specifications. ◮ Empty do simply adds its finite subtype value to the composition of

the S.

◮ Hence its semantics is nothing but the identity function on the

negated VP it combines with: (20) do; λQλw.Q(w); VPfin, aux/VP

neg, non-aux

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One last auxiliary mystery

◮ There is an obvious pattern in the linear ordering of auxiliaries: we

have (21) John          may will would . . .          have been reading.

◮ No other ordering is possible: modal + have + be + VP is the only

• rder allowed.

◮ Why? How can we best account for this pattern? What is the least

amount we have to say to enforce the order in (21)?

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