Romney is such that (it is north of Linden)

Linden is such that (Romney is north of it)

Romney and Linden are such that (the former is north of the latter)

Mike is such that (he gave the package to Nancy)

the package is such that (Mike gave it to Nancy)

Nancy is such that (Mike gave the package to her)

Mike, the package, and Nancy are such that (the first gave the second to the third)

Tom is such that (he spoke of Ed to Sue)

Ed is such that (Tom spoke of him to Sue)

Sue is such that (Tom spoke of Ed to her)

Tom, Ed, and Sue are such that (the first spoke of the second to the third)

Sam is such that (he traveled to Atlanta by way of Chicago)

Atlanta is such that (Sam traveled to it by way of Chicago)

Chicago is such that (Sam traveled to Atlanta by way of it)

Sam, Atlanta, and Chicago are such that (the first traveled to the second by way of the third)

Ann nominated herself

Ann is such that (she nominated herself)

[x nominated x]_x Ann

Ralph tried the motor, and it started

the motor is such that (Ralph tried it, and it started)

[Ralph tried x and x started]_x the motor

[Ralph tried x ∧ x started]_x m

The analysis [Txy ∧ Sy]_xyrm is also correct, but a 2-place abstract is not needed in order to analyze pronouns since only the motor has a pronoun referring to it.

If the alarm is touched, it will go off

the alarm is such that (if it is touched, it will go off)

[if x is touched, x will go off]_x the alarm

[x will be touched → x will go off]_x a

Ralph fixed Sam’s car, and he drove it back to him

Ralph and Sam are such that (the former fixed the latters’s car, and he drove it back to him)

[x fixed y’s car, and x drove y’s car back to y]_xy Ralph Sam

[x fixed y’s car ∧ x drove y’s car back to y]_xyrs

[Fx(y’s car) ∧ Dx(y’s car)y]_xyrs

Ann and Bill each left a message for the other

Ann and Bill are such that (they each left a message for the other)

[x and y each left a message for the other]_xy Ann Bill

[x left a message for y ∧ y left a message for x]_xyab

The noun phrase a message is a quantifier phrase rather than an individual term so it must be left unanalyzed.

Al called Bill, and he called Carol

Al and Bill are such that (the former called the latter, and the former called Carol)

[x called y, and x called Carol]_xy Al Bill

[x called y ∧ x called Carol]_xyab

Al called Bill, and he called Carol

Al and Bill are such that (the former called the latter, and the latter called Carol)

[x called y, and y called Carol]_xy Al Bill

[x called y ∧ y called Carol]_xyab

The second interpretation can be indicated in spoken English by emphasizing the pronoun. The first interpretation could be indicated unambiguously by adding too to the end of the sentence.

Sam gave the book to Tom, but he didn’t read it

Sam, the book and Tom are such that (the first gave the second to the third, but the first didn’t read the second)

[x gave y to z, but x didn’t read y]_xyz Sam the book Tom

[x gave y to z ∧ ¬ x read y]_xyzsbt

Sam gave the book to Tom, but he didn’t read it

Sam, the book and Tom are such that (the first gave the second to the third, but the third didn’t read the second)

[x gave y to z, but z didn’t read y]_xyz Sam the book Tom

[x gave y to z ∧ ¬ z read y]_xyzsbt

It is hard to avoid this ambiguity in English without some rewording—e.g., resorting to the former or the latter instead of he or repeating one of the names.

Al washed his car, and so did Bill.

Al and Bill are such that (the former washed his car, and so did the latter).

[x washed his car, and so did y]_xy Al Bill

[x is such that (he washed his car) ∧ y is such that (he washed his car)]_xyab

[[z washed z’s car]_zx ∧ [z washed z’s car]_zy]_xyab

[[Wz(z’s car)]_zx ∧ [Wz(z’s car)]_zy]_xyab

Al washed his car, and so did Bill.

Al and Bill are such that (the former washed his car, and so did the latter).

[x washed his car, and so did y]_xy Al Bill

[x is such that (he washed x’s car) ∧ y is such that (he washed x’s car)]_xyab

[[z washed x’s car]_zx ∧ [z washed x’s car]_zy]_xyab

[[Wz(x’s car)]_zx ∧ [Wz(x’s car)]_zy]_xyab

The abstracts here serve two different purposes. The one with largest scope is used to analyze the patterns of co-reference while the two inside its body are designed to capture the function of so did. The ambiguity in the sentence arises because the sameness claimed for Al’s and Bill’s actions might suggest washing a car related to the washer in the same way (the first interpretation) or, indeed, washing the very same car (the second interpretation). In particular, it’s the difference between the idea of washing one’s own car—i.e., [Wz(cz)]_z—and washing the car of someone, x—i.e., [Wz(cx)]_z—someone who, in this case, is the first person to whom the predicate is applied. It is the function of the abstract with wider scope to capture this idea of a reference to the first person to whom the predicate is applied.

Since each abstract has many (indeed, infinitely many) alphabetic variants, the answers (ii) below are only examples.

In the original abstract for (g), [[Rcy]_ya ∧ Rby]_y, the variable y in Rcy falls in the scope of two abstractors for y. It is bound to the one with narrower scope, so the one with wider scope binds only the y in Rby. The pattern in (i) shows that the variable in the first abstract is thoroughly apparent from the point of the abstractor with wider scope: since the latter binds no variables in the first abstract, it does not matter whether that abstract uses the same variable as it does or a different one. In (f), on the other hand, the two abstractors must use different variables since one binds variables in the scope of the other.