Auxiliaries and the λ -calculus Robert Levine Ohio State University levine.1@osu.edu Robert Levine 2019 5201 Auxiliaries and the λ -calculus 1 / 25
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 on the preceding slide should be f aux ( g )( y ) . . . ◮ . . . whereas what we want to wind up with is f aux ( g ( y )) . ◮ We have a syntax semantics mismatch and we’re going to need something extra to handle it. Robert Levine 2019 5201 Auxiliaries and the λ -calculus 2 / 25
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. Robert Levine 2019 5201 Auxiliaries and the λ -calculus 3 / 25
◮ 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. . . Robert Levine 2019 5201 Auxiliaries and the λ -calculus 4 / 25
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. Robert Levine 2019 5201 Auxiliaries and the λ -calculus 5 / 25
◮ 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 ; must • have • a • solution ; tkoe ; NP 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 . Robert Levine 2019 5201 Auxiliaries and the λ -calculus 6 / 25
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. Robert Levine 2019 5201 Auxiliaries and the λ -calculus 7 / 25
◮ 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? Robert Levine 2019 5201 Auxiliaries and the λ -calculus 8 / 25
◮ 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? Robert Levine 2019 5201 Auxiliaries and the λ -calculus 9 / 25
have ; a • solution ; have ; a - solution ; VP / NP NP have • a • solution ; must ; have ( a - solution ); λPλx. � P ( x ); VP VP / VP must • have • a • solution ; this • kind • of • equation ; λx. � ( have ( a - solution ))( x ); tkoe ; VP NP this • kind • of • equation • must • have • a • solution ; � ( have ( a - solution ))( tkoe ); S Robert Levine 2019 5201 Auxiliaries and the λ -calculus 10 / 25
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. Robert Levine 2019 5201 Auxiliaries and the λ -calculus 11 / 25
◮ 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 ? Robert Levine 2019 5201 Auxiliaries and the λ -calculus 12 / 25
Must not . . . . . . . . . answer ; that • letter ; answer ; VP / NP ι ( letter ); NP not ; answer • that • letter ; λQλy. ¬ Q ( y ); answer ( ι ( letter )); VP VP / VP not • answer • that • letter ; λQ [ λy. ¬ Q ( y )]( answer ( ι ( letter ))); VP must ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . not • answer • that • letter ; λPλv. � P ( v ); λy [ ¬ ( answer ( ι ( letter )))( y )]; VP 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 john ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . must • not • answer • that • letter ; j ; λv. � ¬ ( answer ( ι ( letter )))( v ); VP 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 Robert Levine 2019 5201 Auxiliaries and the λ -calculus 13 / 25
The auxiliary dependency eating ∗ eaten (9) John is lunch. ∗ eat ∗ eats ∗ eating eaten (10) John has lunch. ∗ eat ∗ eats ∗ eating ∗ eaten (11) John will lunch. eat ∅ ∗ eats eating ∗ eaten (12) John has been lunch. ∗ eat ∗ eats Robert Levine 2019 5201 Auxiliaries and the λ -calculus 14 / 25
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