6.1.xa. Exercise answers

1. a.

Ann introduced Bill to Carol

[ _ introduced _ to _ ] Ann Bill Carol

Iabc
I fits a, b, ən c

I: [ _ introduced _ to _ ]; a: Ann; b: Bill; c: Carol

  b.

Ann gave the book to either Bill or Carol

Ann gave the book to Bill ∨ Ann gave the book to Carol

[ _ gave _ to _ ] Ann the book Bill ∨ [ _ gave _ to _ ] Ann the book Carol

Gakb ∨ Gakc
either G fits a, k, ən b or G fits a, k, ən c

G: [ _ gave _ to _ ]; a: Ann; b: Bill; c: Carol; k: the book

  c.

Ann gave the book to Bill and he gave it to Carol

Ann gave the book to Bill ∧ Bill gave the book to Carol

[ _ gave _ to _ ] Ann the book Bill ∧ [ _ gave _ to _ ] Bill the book Carol

Gakb ∧ Gbkc
both G fits a, k, ən b and G fits b, k, ən c

G: [ _ gave _ to _ ]; a: Ann; b: Bill; c: Carol; k: the book

  d.

Tom had the package sent to Sue, but it was returned to him

Tom had the package sent to Sue ∧ the package was returned to Tom

[ _ had _ sent to _ ] Tom the package Sue ∧ [ _ was returned to _ ] the package Tom

Htps ∧ Rpt
both H fits t, p, ən s and R fits p ən t

H: [ _ had _ sent to _ ]; R: [ _ was returned to _ ]; p: the package; s: Sue; t: Tom

  e.

Georgia will see Ed if she gets to Denver before Saturday

Georgia will see Ed ← Georgia will get to Denver before Saturday

[ _ will see _ ] Georgia Ed ← [ _ will get to _ before _ ] Georgia Denver Saturday

Sge ← Ggds
Ggds → Sge
if G fits g, d, ən s then S fits g ən e

G: [ _ will get to _ before _ ]; S: [ _ will see _ ]; d: Denver; e: Ed; g: Georgia; s: Saturday

  f.

If the murderer is either the butler or the nephew, then I’m Sherlock Holmes

the murderer is either the butler or the nephew → I’m Sherlock Holmes

(the murderer is the butler ∨ the murderer is the nephew) → I = Sherlock Holmes

(the murderer = the butler ∨ the murderer = the nephew) → i = s

(m = b ∨ m = n) → i = s
if either m is b or m is n then i is s

b: the butler; i: I; m: the murderer; n: the nephew; s: Sherlock Holmes

  g.

Neither Ann nor Bill saw Tom speak to either Mike or Nancy

¬ (Ann saw Tom speak to either Mike or Nancy ∨ Bill saw Tom speak to either Mike or Nancy)

¬ ((Ann saw Tom speak to Mike ∨ Ann saw Tom speak to Nancy) ∨ (Bill saw Tom speak to Mike ∨ Bill saw Tom speak to Nancy))

¬ (([ _ saw _ speak to _ ] Ann Tom Mike ∨ [ _ saw _ speak to _ ] Ann Tom Nancy) ∨ ([ _ saw _ speak to _ ] Bill Tom Mike ∨ [ _ saw _ speak to _ ] Bill Tom Nancy))

¬ ((Satm ∨ Satn) ∨ (Sbtm ∨ Sbtn))
not either either S fits a, t, ən m or S fits a,t, ən n or either S fits b,t, ən m or S fits b,t, ən n

S: [ _ saw _ speak to _ ]; a: Ann; b: Bill; m: Mike; n: Nancy; t: Tom

  h.

Tom will agree if each of Ann, Bill, and Carol asks him

Tom will agree ← each of Ann, Bill, and Carol will ask Tom

Tom will agree ← ((Ann will ask Tom ∧ Bill will ask Tom) ∧ Carol will ask Tom)

[ _ will agree] Tom ← (([ _ will ask _ ] Ann Tom ∧ [ _ will ask _ ] Bill Tom) ∧ [ _ will ask _ ] Carol Tom)

Gt ← ((Aat ∧ Abt) ∧ Act)
((Aat ∧ Abt) ∧ Act) → Gt
if both both A fits a ən t and A fits b ən t and A fits c ən t then G fits t

A: [ _ will ask _ ]; G: [ _ will agree]; a: Ann; b: Bill; c: Carol; t: Tom

The function of each here is to indicate individual requests rather than a request made by Ann, Bill, and Carol as a group; the latter idea might be analyzed using a single four-place predicate [ _, _, and _ will ask _ ] in place of the series of two-place predications used above.

  i.

Reagan’s vice president was the 41st president.

Reagan’s vice president = the 41st president

[ _’s vice president] Reagan = [the _th president] 41

vr = pf
v of r is p of f

p: [the _th president]; v: [ _ ’s vice president]; f: 41; r: Reagan

  j.

Tom found a fly in his soup and he called the waiter

Tom found a fly in his soup ∧ Tom called the waiter

Tom found a fly in Tom’s soup ∧ Tom called the waiter

[ _ found a fly in _ ] Tom Tom’s soup ∧ [ _ called _ ] Tom the waiter

Ft(Tom’s soup) ∧ Ctr

Ft([ _’s soup] Tom) ∧ Ctr

Ft(st) ∧ Ctr
both F fits t ən s of t and C fits t ən r

C: [ _ called _ ]; F: [ _ found a fly in _ ]; s: [ _’s soup]; r: the waiter; t: Tom

  k.

Tom found the book everyone had talked to him about and he bought a copy of it

Tom found the book everyone had talked to him about ∧ Tom bought a copy of the book everyone had talked to him about

Tom found the book everyone had talked to Tom about ∧ Tom bought a copy of the book everyone had talked to Tom about

[ _ found _ ] Tom the book everyone had talked to Tom about ∧ [ _ bought a copy of _ ] Tom the book everyone had talked to Tom about

Ft(the book everyone had talked to Tom about) ∧ Bt(the book everyone had talked to Tom about)

Ft([the book everyone had talked to _ about] Tom) ∧ Bt([the book everyone had talked to _ about] Tom)

Ft(bt) ∧ Bt(bt)
both F fits t ən b of t and B fits t ən b of t

B: [ _ bought a copy of _ ]; F: [ _ found _ ]; b: [the book everyone had talked to _ about]; t: Tom

  l.

Wabash College is located in Crawfordsville, which is the seat of Montgomery County

Wabash College is located in Crawfordsville ∧ Crawfordsville is the seat of Montgomery County

[ _ is located in _ ] Wabash College Crawfordsville ∧ Crawfordsville = the seat of Montgomery County

Lbc ∧ c = [the seat of _ ] Montgomery County

Lbc ∧ c = sm
both L fits b ən c and c is s of m

L: [ _ is located in _ ]; s: [the seat of _ ]; b: Wabash; c: Crawfordsville; m: Montgomery County

  m.

Sue and Tom set the date of their wedding but didn’t decide on its location

Sue and Tom set the date of their wedding
∧ Sue and Tom didn’t decide on the location of their wedding

Sue and Tom set the date of Sue and Tom’s wedding
∧ ¬ Sue and Tom decided on the location of Sue and Tom’s wedding

[ _ and _ set _ ] Sue Tom the date of Sue and Tom’s wedding
∧ ¬ [ _ and _ decided on _ ] Sue Tom the location of Sue and Tom’s wedding

Sst(the date of Sue and Tom’s wedding)
∧ ¬ Dst(the location of Sue and Tom’s wedding)

Sst([the date of _ ] Sue and Tom’s wedding)
∧ ¬ Dst([the location of _ ] Sue and Tom’s wedding)

Sst(d(Sue and Tom’s wedding)) ∧ ¬ Dst(l(Sue and Tom’s wedding))

Sst(d([ _ and _’s wedding] Sue Tom))
∧ ¬ Dst(l([ _ and _’s wedding] Sue Tom))

Sst(d(wst)) ∧ ¬ Dst(l(wst))

both S fits s, t, ən d of (w of s ən t) and not D fits s, t, ən l of (w of s ən t)

D: [ _ and _ decided on _ ]; S: [ _ and _ set _ ]; d: [the date of _ ]; l: [the location of _ ]; w: [ _ and _ ’s wedding]; s: Sue; t: Tom

2. a.

[ _ is west of _ ] Crawfordsville Indianapolis
∧ [ _ is south of _ ] Crawfordsville Lafayette

Crawfordsville is west of Indianapolis ∧ Crawfordsville is south of Lafayette

Crawfordsville is west of Indianapolis and south of Lafayette

  b.

[ _ has met _ ] Ann Bill → [ _ has met _ ] Bill Ann

Ann has met Bill → Bill has met Ann

If Ann has met Bill then he has met her

  c.

[ _ introduced _ to _ ] Alice Clarice Boris
∧ [ _ introduced _ to _ ] Alice Doris Boris

Alice introduced Clarice to Boris ∧ Alice introduced Doris to Boris

Alice introduced Clarice and Doris to Boris

  d.

[ _ wrote to _ ] Alice Boris
∧ [ _ asked _ to write _ about _ ] Alice Boris Alice Boris

Alice wrote to Boris ∧ Alice asked Boris to write Alice about Boris

Alice wrote to Boris ∧ Alice asked Boris to write her about himself

Alice wrote to Boris and asked him to write her about himself

  e.

g = c → (f = s ∧ p = t)

Green Bay = the city → (football = the sport ∧ the Packers = the team)

Green Bay is the city → (football is the sport ∧ the Packers are the team)

Green Bay is the city → football is the sport and the Packers are the team

If Green Bay is the city, then football is the sport and the Packers are the team

  f.

([ _ has spoken to _ ] Ann Bill ∧ ¬ [ _ has spoken to _ ] Ann ([ _’s father] Carol)) → ¬ Bill = [ _’s father] Carol

(Ann has spoken to Bill ∧ ¬ [ _ has spoken to _ ] Ann Carol’s father) → ¬ Bill = Carol’s father

(Ann has spoken to Bill ∧ ¬ Ann has spoken to Carol’s father) → ¬ Bill is Carol’s father

(Ann has spoken to Bill ∧ Ann hasn’t spoken to Carol’s father) → Bill isn’t Carol’s father

Ann has spoken to Bill but not to Carol’s father → Bill isn’t Carol’s father

If Ann has spoken to Bill but not to Carol’s father, then Bill isn’t Carol’s father

  g.

(B([ _’s father] Ann)([ _’s mother] Bill) ∨ S([ _’s mother] Ann)([ _’s father] Bill)) → [ _ and _ are cross-cousins] Ann Bill

([ _ is a brother of _ ] Ann’s father Bill’s mother ∨ [ _ is a sister of _ ] Ann’s mother Bill’s father) → Ann and Bill are cross-cousins

(Ann’s father is a brother of Bill’s mother ∨ Ann’s mother is a sister of Bill’s father) → Ann and Bill are cross-cousins

Ann’s father is a brother of Bill’s mother or Ann’s mother is a sister of Bill’s father → Ann and Bill are cross-cousins

If Ann’s father is a brother of Bill’s mother or Ann’s mother is a sister of Bill’s father, then Ann and Bill are cross-cousins

  h.

Pab(m([ _’s proposal] Bill)([ _’s proposal] Carol))
∧ Pac(m([ _’s proposal] Bill)([ _’s proposal] Carol))

Pab([the best compromise between _ and _ ] Bill’s proposal Carol’s proposal)
∧ Pac([the best compromise between _ and _ ] Bill’s proposal Carol’s proposal)

[ _ persuaded _ to accept _ ] Ann Bill the best compromise between Bill’s proposal and Carol’s proposal
∧ [ _ persuaded _ to accept _ ] Ann Carol the best compromise between Bill’s proposal and Carol’s proposal

Ann persuaded Bill to accept the best compromise between his and Carol’s proposals ∧ Ann persuaded Carol to accept the best compromise between Bill’s proposal and hers

Ann persuaded each of Bill and Carol to accept the best compromise between their proposals

Glen Helman 17 Jul 2012