6.2.xa. Exercise answers

1. a.

Reagan’s vice president was the 41st president.

Reagan’s vice president = the 41st president

[λx (x’s vice president)] Reagan = [λx (the xth president)] 41

vr = pf
v of r is p of f

[p: λx (the xth president); v: λx (x ’s vice president); f: 41; r: Reagan]
  b.

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

Tom found a fly in his soupTom called the waiter

Tom found a fly in Tom’s soupTom called the waiter

[λxy (x found a fly in y)] Tom Tom’s soup ∧ [λxy (x called y)] Tom the waiter

Ft(Tom’s soup) ∧ Ctr

Ft([λx (x’s soup)] Tom) ∧ Ctr

Ft(st) ∧ Ctr
both F fits t ’n s of t and C fits t ’n r

[C: λxy (x called y); F: λxy (x found a fly in y)]; s: λx (x’s soup); r: the waiter; t: Tom]
  c.

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

[λxy (x found y)] Tom the book everyone had talked to Tom about ∧ [λxy (x bought a copy of y)] 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([λx (the book everyone had talked to x about)] Tom) ∧ Bt([λx (the book everyone had talked to x about)] Tom)

Ft(bt) ∧ Bt(bt)
both F fits t ’n b of t and B fits t ’n b of t

[B: λxy (x bought a copy of y); F: λxy (x found y); b: λx (the book everyone had talked to x about); t: Tom]

  d.

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

Wabash College is located in CrawfordsvilleCrawfordsville is the seat of Montgomery County

[λxy (x is located in y)] Wabash College CrawfordsvilleCrawfordsville = the seat of Montgomery County

Lbc ∧ c = [λx (the seat of x)] Montgomery County

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

[L: λxy (x is located in y); s: λx (the seat of x); b: Wabash; c: Crawfordsville; m: Montgomery County]
  e.

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

[λxyz (x and y set z)] Sue Tom the date of Sue and Tom’s wedding
∧ ¬ [λxyz (x and y decided on z)] 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([λx (the date of x)] Sue and Tom’s wedding)
∧ ¬ Dst([λx (the location of x)] Sue and Tom’s wedding)

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

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

Sst(d(wst)) ∧ ¬ Dst(l(wst))
both S fits s, t, ’n d of (w of s ’n t) and not D fits s, t, ’n l of (w of s ’n t)

[D: λxyz (x and y decided on z); S: λxyz (x and y set z); d: λx (the date of x); l: λx (the location of x); w: λxy (x and y ’s wedding); s: Sue; t: Tom]
2. a.

([λxy (x has spoken to y)] Ann Bill ∧ ¬ [λxy (x has spoken to y)] Ann ([λx (x’s father)] Carol)) → ¬ Bill = [λx (x’s father)] Carol

(Ann has spoken to Bill ∧ ¬ [λxy (x has spoken to y)] 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 BillAnn hasn’t spoken to Carol’s father) → Bill isn’t Carol’s father

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

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

  b.

(B([λx (x’s father)] Ann)([λx (x’s mother)] Bill) ∨ S([λx (x’s mother)] Ann)([λx (x’s father)] Bill)) → [λxy (x and y are cross-cousins)] Ann Bill

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

(Ann’s father is a brother of Bill’s motherAnn’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 fatherAnn 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

  c.

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

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

[λxyz (x persuaded y to accept z)] Ann Bill the best compromise between Bill’s proposal and Carol’s proposal
∧ [λxyz (x persuaded y to accept z)] 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 25 Aug 2005