phi_270_text_4_1

Tommy ate his vegetables ∨ Tommy didn’t get any dessert

Tommy ate his vegetables ∨ ¬ Tommy got dessert

V ∨ ¬ D
either V or not D

D: Tommy got dessert; V: Tommy ate his vegetables

¬ (Mike heard either the phone or the doorbell)

¬ (Mike heard the phone ∨ Mike heard the doorbell)

¬ (P ∨ D)
not either P or D

D: Mike heard the doorbell; P: Mike heard the phone

Mike wasn’t home ∨ Mike wasn’t answering the phone

¬ Mike was home ∨ ¬ Mike was answering the phone

¬ H ∨ ¬ P
either not H or not P

H: Mike was home; P: Mike was answering the phone

The package was sent ∧ either the package is still on its way or it’s been lost in the mail

The package was sent ∧ (the package is still on its way ∨ the package has been lost in the mail)

S ∧ (W ∨ L)
both S and either W or L

L: the package has been lost in the mail; S: the package was sent; W: the package is still on its way

Neither the House nor the Senate had acted on the bill ∧ the White House expressed confidence that the bill would pass

¬ either the House or the Senate had acted on the bill ∧ the White House expressed confidence that the bill would pass

¬ (the House had acted on the bill ∨ the Senate had acted on the bill) ∧ the White House expressed confidence that the bill would pass

¬ (H ∨ S) ∧ W
both not either H or S and W

H: the House had acted on the bill; S: the Senate had acted on the bill; W: the White House expressed confidence that the bill would pass

¬ Sam will pass through without either stopping by or calling

¬ (Sam will pass through ∧ ¬ Sam will either stop by or call)

¬ (Sam will pass through ∧ ¬ (Sam will stop by ∨ Sam will call))

¬ (P ∧ ¬ (S ∨ C))
not both P and not either S or C

C: Sam will call; P: Sam will pass through; S: Sam will stop by

Davis will take you or give you directions ∨ Edwards will take you or give you directions

(Davis will take you ∨ Davis will give you directions) ∨ (Edwards will take you ∨ Edwards will give you directions)

(D ∨ G) ∨ (E ∨ V)
either either D or G or either E or V

D: Davis will take you; E: Edwards will take you; G: Davis will give you directions; V: Edwards will give you directions

We’ll have a can without an opener ∨ we’ll have an opener without a can

(we’ll have a can ∧ we won’t have an opener) ∨ (we’ll have an opener ∧ we won’t have a can)

(we’ll have a can ∧ ¬ we’ll have an opener) ∨ (we’ll have an opener ∧ ¬ we’ll have a can)

(C ∧ ¬ O) ∨ (O ∧ ¬ C)
either both C and not O or both O and not C

C: we’ll have a can; O: we’ll have an opener

¬ either Jan or Ken had matches or a lighter

¬ (Jan had matches or a lighter ∨ Ken had matches or a lighter)

¬ ((Jan had matches ∨ Jan had a lighter) ∨ (Ken had matches ∨ Ken had a lighter))

¬ ((M ∨ L) ∨ (K ∨ G))
not either either M or L or either K or G

G: Ken had a lighter; K: Ken had matches; L: Jan had a lighter; M: Jan had matches

Both Ann and Bill were in town ∧ neither Ann nor Bill knew the other was in town

(Ann was in town ∧ Bill was in town) ∧ ¬ either Ann or Bill knew the other was in town

(Ann was in town ∧ Bill was in town) ∧ ¬ (Ann knew Bill was in town ∨ Bill knew Ann was in town)

(A ∧ B) ∧ ¬ (K ∨ N)
both both A and B and not either K or N

A: Ann was in town; B: Bill was in town; K: Ann knew Bill was in town; N: Bill knew Ann was in town

Tom will handle both the scheduling and the publicity ∨ Dick will handle both the scheduling and the publicity ∨ Harry will handle both the scheduling and the publicity

(Tom will handle the scheduling ∧ Tom will handle the publicity) ∨ (Dick will handle the scheduling ∧ Dick will handle the publicity) ∨ (Harry will handle the scheduling ∧ Harry will handle the publicity)

(T ∧ P) ∨ (D ∧ B) ∨ (H ∧ L)
(both T and S) or (both D and C) or (both T and S)

B: Dick will handle the publicity; D: Dick will handle the scheduling; H: Harry will handle the scheduling; L: Harry will handle the publicity; P: Tom will handle the publicity; T: Tom will handle the scheduling

The scheduling will be handled by either Tom, Dick, or Harry ∧ the publicity will be handled by either Tom, Dick, or Harry

(the scheduling will be handled by Tom ∨ the scheduling will be handled by Dick ∨ the scheduling will be handled by Harry) ∧ (the publicity will be handled by Tom ∨ the publicity will be handled by Dick ∨ the publicity will be handled by Harry)

(T ∨ D ∨ H) ∧ (P ∨ B ∨ L)
both (T or D or H) and (P or B or L)

B: the publicity will be handled by Dick; D: the scheduling will be handled by Dick; H: the scheduling will be handled by Harry; L: the publicity will be handled by Harry; P: the publicity will be handled by Tom; T: the scheduling will be handled by Tom

It was the butler ∨ it was the nephew

It was either the butler or the nephew

¬ (the alarm worked ∨ the sprinkler worked)

¬ (either the alarm or the sprinkler worked)

Neither the alarm nor the sprinkler worked

¬ the part arrived ∨ ¬ the part was the problem

The part didn’t arrive ∨ the part wasn’t the problem

Either the part didn’t arrive or it wasn’t the problem

Ann has a large car ∨ ¬ (Bill will ride with us ∧ Carol will ride with us)

Ann has a large car ∨ ¬ Bill and Carol will ride with us

Ann has a large car ∨ Bill and Carol won’t both ride with us

Either Ann has a large car or Bill and Carol won’t both ride with us

(it rained over night ∨ there was a heavy dew) ∧ it is wet

It rained over night or there was a heavy dew ∧ it is wet

It rained over night or there was a heavy dew but, either way, it is wet

(AAA ∧ Co. will profit from the deal ∧ ZZZ Inc. will profit from the deal) ∨ (the deal will fall through ∧ ¬ (AAA ∧ Co. will profit from the deal ∨ ZZZ Inc. will profit from the deal))

AAA ∧ Co. and ZZZ Inc. will both profit from the deal ∨ (the deal will fall through ∧ ¬ (either AAA ∧ Co. or ZZZ Inc. will profit from the deal))

AAA ∧ Co. and ZZZ Inc. will both profit from the deal ∨ (the deal will fall through ∧ neither AAA ∧ Co. nor ZZZ Inc. will profit from the deal)

AAA ∧ Co. and ZZZ Inc. will both profit from the deal ∨ the deal will fall through and neither AAA ∧ Co. nor ZZZ Inc. will profit from it)

Either AAA ∧ Co. and ZZZ Inc. will both profit from the deal, or the deal will fall through and neither will profit from it

	j.	`Both Ann and Bill were in town` ∧ `neither Ann nor Bill knew the other was in town` (`Ann was in town` ∧ `Bill was in town`) ∧ ¬ `either Ann or Bill knew the other was in town` (`Ann was in town` ∧ `Bill was in town`) ∧ ¬ (`Ann knew Bill was in town` ∨ `Bill knew Ann was in town`) (A ∧ B) ∧ ¬ (K ∨ N) `both` `both` A `and` B `and` `not` `either` K `or` N A: `Ann was in town`; B: `Bill was in town`; K: `Ann knew Bill was in town`; N: `Bill knew Ann was in town`
	k.	`Tom will handle both the scheduling and the publicity` ∨ `Dick will handle both the scheduling and the publicity` ∨ `Harry will handle both the scheduling and the publicity` (`Tom will handle the scheduling` ∧ `Tom will handle the publicity`) ∨ (`Dick will handle the scheduling` ∧ `Dick will handle the publicity`) ∨ (`Harry will handle the scheduling` ∧ `Harry will handle the publicity`) (T ∧ P) ∨ (D ∧ B) ∨ (H ∧ L) (`both` T `and` S) `or` (`both` D `and` C) `or` (`both` T `and` S) B: `Dick will handle the publicity`; D: `Dick will handle the scheduling`; H: `Harry will handle the scheduling`; L: `Harry will handle the publicity`; P: `Tom will handle the publicity`; T: `Tom will handle the scheduling` Note: this sentence is ambiguous and could also be interpreted as equivalent to the following one.
	l.	`The scheduling will be handled by either Tom, Dick, or Harry` ∧ `the publicity will be handled by either Tom, Dick, or Harry` (`the scheduling will be handled by Tom` ∨ `the scheduling will be handled by Dick` ∨ `the scheduling will be handled by Harry`) ∧ (`the publicity will be handled by Tom` ∨ `the publicity will be handled by Dick` ∨ `the publicity will be handled by Harry`) (T ∨ D ∨ H) ∧ (P ∨ B ∨ L) `both` (T `or` D `or` H) `and` (P `or` B `or` L) B: `the publicity will be handled by Dick`; D: `the scheduling will be handled by Dick`; H: `the scheduling will be handled by Harry`; L: `the publicity will be handled by Harry`; P: `the publicity will be handled by Tom`; T: `the scheduling will be handled by Tom`

2.	a.	`both` A `and` `either` B `or` C
	b.	`either` `both` A `and` B `or` C
	c.	¬ (A ∨ ¬ B)
	d.	(A ∨ B) ∧ (A ∨ C)

3.	a.	`It was the butler` ∨ `it was the nephew` `It was either the butler or the nephew`
	b.	¬ (`the alarm worked` ∨ `the sprinkler worked`) ¬ (`either the alarm or the sprinkler worked`) `Neither the alarm nor the sprinkler worked`
	c.	¬ `the part arrived` ∨ ¬ `the part was the problem` `The part didn’t arrive` ∨ `the part wasn’t the problem` `Either the part didn’t arrive or it wasn’t the problem`
	d.	`Ann has a large car` ∨ ¬ (`Bill will ride with us` ∧ `Carol will ride with us`) `Ann has a large car` ∨ ¬ `Bill and Carol will ride with us` `Ann has a large car` ∨ `Bill and Carol won’t both ride with us` `Either Ann has a large car or Bill and Carol won’t both ride with us` Note: `both` is introduced here to help distinguish this sentence from A ∨ (¬ B ∧ ¬ C)
	e.	(`it rained over night` ∨ `there was a heavy dew`) ∧ `it is wet` `It rained over night or there was a heavy dew` ∧ `it is wet` `It rained over night or there was a heavy dew but, either way, it is wet` Note: `either way` here serves to indicate that the scope of the disjunction has ended and that the final clause is unhedged and `but` reinforces this by marking the contrast between the indefinite disjunction and the definite final clause.
	f.	(`AAA ∧ Co. will profit from the deal` ∧ `ZZZ Inc. will profit from the deal`) ∨ (`the deal will fall through` ∧ ¬ (`AAA ∧ Co. will profit from the deal` ∨ `ZZZ Inc. will profit from the deal`)) `AAA ∧ Co. and ZZZ Inc. will both profit from the deal` ∨ (`the deal will fall through` ∧ ¬ (`either AAA ∧ Co. or ZZZ Inc. will profit from the deal`)) `AAA ∧ Co. and ZZZ Inc. will both profit from the deal` ∨ (`the deal will fall through` ∧ `neither AAA ∧ Co. nor ZZZ Inc. will profit from the deal`) `AAA ∧ Co. and ZZZ Inc. will both profit from the deal` ∨ `the deal will fall through and neither AAA ∧ Co. nor ZZZ Inc. will profit from it`) `Either AAA ∧ Co. and ZZZ Inc. will both profit from the deal, or the deal will fall through and neither will profit from it`