Homework 4 and Class Notes

multiple letter identifiers

constants of type e : U
variables of type e : x-z
variables of type <e,t> : P-Q


# Definitions

define Penn : penn_e
define Sarah : sarah_e
define Irina : irina_e
define Amy : amy_e

define every : LP.LQ.{x | P(x)} << {x | Q(x)}
define two : LP.LQ.[|{y | P(y)} ^^ {y | Q(y)}| >= two_i]
define the : LP.Ix.P(x)

define president : Ly.Lz.president-of(z,y)
define likes : LyLx[likes(x,y)]
define bearded : Lx.bearded(x)
define student : Lx.student(x)
define shaves : Lx.shaves(x)
define reads : LyLx[reads(x,y)]
define woman: Lx.woman(x)
define books : Lx.book(x)

# Rules

use rule function application
use rule non-branching nodes
use rule lambda abstraction
use rule predicate modification

exercise tree

title Homework 4

directions You'll need to remember that this program represents things a little differently: "shaves(x)" instead of "x shaves", "student(x)" instead of "x is a student", and so on. Transitve verbs are represented by two arguments in parentheses: "likes(x,y)" for "x likes y". \\ \\ Also, individuals such as "Amy" and "Penn" are marked with "_e", as a note to the program that they're individuals.
instructions Every student shaves.

[.1 [.2 every [.5 bearded student]] shaves]

instructions Every student who is bearded shaves.

[.1 [.2 every [.5 student [.7 (who) [.9 (is) bearded]]]] shaves]

instructions Every student likes the president of Penn.
instructions \\ \\Note: the ι here means, basically, "the". ιx.P(x) is shorthand for "the unique x such that P(x) is true".

[.1 [.2 every student] [.3 likes [.7 the [.9 president [.11 (of) Penn]]]]]

exercise tree
title In-class examples, 3/20: Part I, simple movement.

directions Note that, for complicated bookkeeping reasons, "t-number-48" has the meaning "g(48)" rather than just "t-48". That shouldn't affect the derivations in any significant way.
instructions Irina, Sarah likes.

[.S Irina [. 37 [.S Sarah [.VP likes t_37]]]]

instructions Sarah likes herself.

[.S Sarah [. 48 [.S t_48 [.VP likes t_48]]]]

instructions Every woman likes herself.

[.S [.NP every woman] [. 37 [.S t_37 [.VP likes t_37]]]]

instructions Amy likes every student.

[.S [.NP every student] [. 23 [.S Amy [.VP likes t_23]]]]

exercise tree
title In-class examples, 3/20: Part II, double movement.


directions Here, the "_i" after "two" indicates that it's a number. (Mentally replace "two_i" with "2".)

instructions Every student reads two books.
instructions \\ \\Meaning: "What every student has to do is pick two books and read them."

[.S [.NP every student] [. 1 [.S [.NP two books] [. 2 [.S t_1 [.VP reads t_2]]]]]]

instructions Every student reads two books.
instructions \\ \\Meaning: "There are two books, such that every student has to read them."

[.S [.NP two books] [. 2 [.S [.NP every student] [. 1 [.S t_1 [.VP reads t_2]]]]]]