Phi 270
Fall 2013
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6.2.xa. Exercise answers

1. In each case, the English restatement appears first, followed by the partial symbolization.
  a. i.

Romney is such that (it is north of Linden)

[x is north of Linden]x Romney

    or

Linden is such that (Romney is north of it)

[Romney is north of x]x Linden

    ii.

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

[x is north of y]xy Romney Linden

  b. i.

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

[x gave the package to Nancy]x Mike

    or

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

[Mike gave x to Nancy]x the package

    or

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

[Mike gave the package to x]x Nancy

    ii.

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

[x gave y to z]xyz Mike the package Nancy

  c. i.

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

[x spoke of Ed to Sue]x Tom

    or

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

[Tom spoke of x to Sue]x Ed

    or

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

[Tom spoke of Ed to x]x Sue

    ii.

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

[x spoke of y to z]xyz Tom Ed Sue

  d. i.

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

[x traveled to Atlanta by way of Chicago]x Sam

    or

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

[Sam traveled to x by way of Chicago]x Atlanta

    or

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

[Sam traveled to Atlanta by way of x]x Chicago

    ii.

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

[x traveled to y by way of z]xyz Sam Atlanta Chicago

2. a.

Ann nominated herself

Ann is such that (she nominated herself)

[x nominated x]x Ann

[Nxx]xa
reduced form: Naa

N: [ _ nominated _ ]; a: Ann

  b.

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

[Trx ∧ Sx]xm
reduced form: Trm ∧ Sm

S: [ _ started]; T: [ _ tried _ ]; m: the motor; r: Ralph

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.

  c.

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

[Tx → Gx]x a
reduced form: Ta → Ga

T: [ _ will be touched]; G: [ _ will go off]; a: the alarm

  d.

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

[Fx(cy) ∧ Dx(cy)y]xyrs
reduced form: Fr(cs) ∧ Dr(cs)s

D: [ _ drove _ back to _ ]; F: [ _ fixed _ ]; c: [ _’s car]; r: Ralph; s: Sam

  e.

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

[Mxy ∧ Myx]xyab
reduced form: Mab ∧ Mba

M: [ _ left a message for _ ]; a: Ann; b: Bill

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

3. a. i.

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

[Cxy ∧ Cxc]xyab

    ii.

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

[Cxy ∧ Cyc]xyab

C: [ _ called _ ]; a: Al; b: Bill; c: Carol

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.

  b. i.

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

[Gxyz ∧ ¬ Rxy]xyzsbt

    ii.

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

[Gxyz ∧ ¬ Rzy]xyzsbt

G: [ _ gave _ to _ ]; R: [ _ read _ ]; b: the book; s: Sam; t: Tom

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

  c. i.

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

[[Wz(cz)]zx ∧ [Wz(cz)]zy]xyab

    ii.

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

[[Wz(cx)]zx ∧ [Wz(cx)]zy]xyab

W: [ _ washed _ ]; c: [ _’s car]; a: Al; b: Bill

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.

4.

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

  a. i. ii. [Fy]y
  b. i. ii. [Fx → Gx]x
  c. i. ii. [Txyx]yx
  d. i. ii. [fzx]xz

 

  e. i. ii. [Sxyz]xyz

 

  f. i. ii. [[Ryx]ya ∧ Rbx]x
  g. i. ii. [[Rcx]xa ∧ Rbz]z
 

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.

Glen Helman 01 Aug 2013