I'm going to turn now from particularly hard cases to general strategy. If you are transcribing anything but the shortest of sentences, don't try to do it all at once. Transcribe parts into logic, writing down things which are part logic and part English. Bit by bit, transcribe the parts still in English into logic until all of the English is gone.
Let's do an example. Suppose we want to transcribe
(18) Any boy who loves Eve is not a furry cat.
(18) says of any boy who loves Eve that he is not a furry cat; that is, it says of all things, x, of which a first thing is true (that x is a boy who loves Eve) that a second thing is true (x is not a furry cat). So the sentence has the form (
Vx)(Px ⊃ Q):
Vx)(x is a boy who loves Eve ⊃ x is not a furry cat)
Now all you have to do is to fashion transcriptions of 'x is a boy who loves Eve' and of 'x is not a furry cat' and plug them into (18a):
(18b) x is a boy who loves Eve: Bx & Lxe
(18c) x is not a furry cat: ~(Fx & Cx)
(Something which is not a furry cat is not both furry and a cat. Such a thing could be furry, or a cat, but not both.) Now we plug (18b) and (18c) into (18a), getting our final answer:
Vx)[(Bx & Lxe) ⊃ ~(Ǝx & Cx)]
Here is another way you could go about the same problem. Think of the open sentence 'Bx & Lxe' as indicating a complex one place predicate. The open sentence 'Bx & Lxe' presents something which might be true of an object or person such as Adam. For example, if the complex predicate is true of Adam, we would express that fact by writing in 'a' for 'x' in 'Bx & Lxe', giving 'Ba & Lae'. Now, thinking of 'Bx & Lxe' as a predi- ' cate, we can use the method of quantifier subscripts which we discussed in section 4-1. (18) is somewhat like a sentence which asserts that everything is not a furry cat. But (18) asserts this, not about absolutely everything, but just about all those things which have the complex property Bx & Lxe. So we can write (18) as a universally quantified sentence with the universal quantifier restricted by the predicate 'Bx & Lxe':
Vx)(Bx & Lxe)~(Fx & Cx)
Now you simply use the rule for rewriting subscripts on universal quantifiers, giving (18d).
In yet a third way of working on (18), you could first use the method of subscripting quantifiers before transcribing the complex predicates into logic. Following this route you would first write.
Vx)(x is a boy who loves Eve)(x is not a furry cat)
Now transcribe the English complex predicates as in (18b) and (lac), plug the results into (18f), giving (18e). Then you rewrite the subscript, giving (18d) as before. You have many alternative ways of proceeding.
Generally, it is very useful to think of complex descriptions as complex predicates. In particular, this enables us to use two place predicates to construct one place predicates. We really took advantage of this technique in the last example. 'Lxy' is a two place predicate. By substituting a name for 'y', we form a one place predicate, for example, 'Lxe'. 'Lxe' is a one place predicate which is true of anything which loves Eve.
Here is another useful way of constructing one place predicates from two place predicates. Suppose we need the one place predicate 'is married', but our transcription guide only gives us the two place predicate 'Mxy', meaning that x is married to y. To see how to proceed, consider what it means to say that Adam, for example, is married. This is to say that there is someone to whom Adam is married. So we can say Adam is married with the sentence '(Ǝy)May'. We could proceed in the same way . to say that Eve, or anyone else, is married. In short, the open sentence '(Ǝy)Mxy' expresses the predicate 'x is married'.
Here's another strategy point: When 'who' or 'which' comes after a predicate they generally transcribe as 'and'. As you saw in (la), the complex predicate 'x is a boy who loves Eve' becomes 'Bx & Lxe'. The complex predicate 'x is a dog which is not furry but has a tail' becomes 'Dx & (~Fx & (Ǝy)Tyx)'.
When 'who' or 'which' comes after, a quantifier word, they indicate a subscript on the quantifier: 'Anything which is not furry but has a tail' should be rendered as (
VX)(~Fx & (Ǝy)Tyx). When the quantifier word itself calls for a subscript, as does 'someone', you need to combine both these ideas for treating 'who1: 'Someone who loves Eve' is the subscripted quantifier '(Ǝx)Px & Lxe'.
Let's apply these ideas in another example. Before reading on, see if you can use only 'Cx' for 'x is a cat', 'Lxy' for 'x loves y', and 'Oxy' for 'x owns y' and transcribe
(19) Some cat owner loves everyone who loves themselves.
Let's see how you did. (19) says that there is something, taken from among the cat owners, and that thing loves everyone who loves themselves. Using a subscript and the predicates 'x is a cat owner' and 'x loves everyone who loves themselves', (19) becomes
(19a) (Ǝx) (x is a cat owner)(x loves everyone who loves themselves)
Now we have to fashion transcriptions for the two complex English predicates used in (19a). Someone (or something) is a cat owner just in case there is a cat which they own:
(19b) x is a cat owner: (Ǝy)(Cy & Oxy)
To say that x loves everyone who loves themselves is to say that x loves, not absolutely everyone, but everyone taken from among those that are, first of all people, and second, things which love themselves. So we want to say that x loves all y, where y is restricted to be a person, Py, and restricted to be a self-lover, Lyy:
(19c) x loves everyone who loves themselves: (
Vy)(Py & Lyy)Lxy
Putting the results of (19b) and (19c) into (19a), we get
(19d) (Ǝx)(Ǝy)(cy & Oxy)[(
Vy)(Py & Lyy)Lxy]
Discharging first the subscript of '(Ǝx)' with an '&' and then the subscript of '(
Vy)' with a '⊃', we get
(19e) (Ǝx)(Ǝy)(Cy & Oxy)[(
Vy)(Py & Lyy)Lxy]
(19f) (Ǝx)(Ǝy)(Cy & Oxy)[(
Vy)[(Py & Lyy)Lxy]}
This looks like a lot of work, but as you practice, you will find that you can do more and more of this in your head and it will start to go quite quickly.
I'm going to give you one more piece of advice on transcribing. Suppose you start with an English sentence and you have tried to transcribe it into logic. In many cases you can catch mistakes by transcribing your logic sentence back into English and comparing your retranscription with the original sentence. This check works best if you are fairly literal minded in retranscribing. Often the original and the retranscribed English sentences will be worded differently. But look to see if they still seem to say the same thing. If not, you have almost certainly made a mistake in transcribing from English into logic.
Here is an illustration. Suppose I have transcribed
(20) If something is a cat, it is not a dog.
(20a) (Ǝx)(Cx ⊃ ~Dx)
To check, I transcribe back into English, getting
(20b) There is something such that if it is a cat, then it is not a dog.
Now compare (20b) with (20). To make (20b) true it is enough for there to be one thing which, if a cat, is not a dog. The truth of (20b) is consistent with there being a million cat-dogs. But (20) is not consistent with there being any cat-dogs. I conclude that (20a) is a wrong transcription. Having seen that (20) is stronger than (20a), I try
Vx)(Cx ⊃ ~Dx)
Transcribing back into English this time gives me
(20d) Everything which is a cat is not a dog.
which does indeed seem to say what (20) says. This time I am confident that I have transcribed correctly.
(Is (20) ambiguous in the same way that (5) was? I don't think so!)
Here is another example. Suppose after some work I transcribe
(21) Cats and dogs have tails.
Vx)[(Cx & Dx) ⊃ (Ǝy)Txy]
To check, I transcribe back into English:
(21b) Everything is such that if it is both a cat and a dog, then it has a tail.
Obviously, something has gone wrong, for nothing is both a cat and a dog. Clearly, (21) is not supposed to be a generalization about such imaginary cat-dogs. Having noticed this, I see that (21) is saying one thing about cats and then the same thing about dogs. Thus, without further work, I try the transcription
(21c) (Vx)(Cx ⊃ (Ǝy)Txy) & (Vx)(Dx ⊃ (Ǝy)Txy)
To check (21c), I again transcribe back into English, getting
(21d) If something is a cat, then it has a tail, and if something is a dog, then it has a tail.
which is just a long-winded way of saying that all cats and dogs have tails-in other words, (21). With this check, 1 can be very confident that (21c) is a correct transcription.
Use this transcription guide for exercises 4-7 and 4-8:
a: Adam Fx: x is furry
e: Eve Px: x is a person
Ax: x is an animal Qx: x purrs
Bx: x is blond Lxy: x loves y
Cx: x is a cat Sxy: x is a son of y
Dx: x is a dog Txy: x is a tail of y
Oxy: x owns y
4-7. Transcribe the following sentences into English:
a) (Ǝx)(Ǝy)(Px & Py & Sxy)
b) ~(Ǝx)(Px & Ax)
Vx)[Qx ⊃ (Fx & Cx)]
d) (Ǝx)[Qx & ~(Fx & Cx)]
Vx)~[Pk & (Lxa & Lxe)]
Vx)[Px ⊃ ~(Lxa & Lxe)]
Vx)(Vy)[(Dx & Cy) ⊃ Lxy]
Vx)(Vy)[Dx ⊃ (Cy ⊃ Lxy)]
i) (Ǝx)[Pk & (Ǝy)(Ǝz)(Py & Szy & Lxz)]
j) (Ǝx)[Px & (Ǝy)(Ǝz)(Pz & Syz & Lxz)]
Vx)([Bx & (Ǝy)(Fy & Txy)] ⊃ (Ǝz)(Cz & Txz))
Vx)(Ǝy)Sxy ⊃ [(Ǝz)(Cz & Lxz) ≡ (Ǝz)(Dz & Lxz)])
4-8. Transcribe the following sentences into predicate logic. I have included some easy problems as a review of previous sections along with some real brain twisters. I have marked the sentences which seem to me clearly ambiguous, and you should give different transcriptions for these showing the different ways of understanding the English. Do you think any of the sentences I haven't marked are also ambiguous? You should have fun arguing out your intuitions about ambiguous cases with your classmates and instructor.
a) All furry cats purr.
b) Any furry cat purrs.
c) No furry cats purr.
d) None of the furry cats purr.
e) None but the furry cats purr. (Ambiguous?)
f) Some furry cats purr.
g) Some furry cats do not purr.
h) Some cats and dogs love Adam.
i) Except for the furry ones, all cats purr.
j) Not all furry cats purr.
k) If a cat is furry, it purrs. A furry cat purrs. (Ambiguous)
m) Only furry cats purr.
n) Adam is not a dog or a cat.
o) Someone is a son.
p) Some sons are blond.
q) Adam loves a blond cat, and Eve loves one too.
r) Adam loves a blond cat and so does Eve. (Ambiguous)
s) Eve does not love everyone.
t) Some but not all cats are furry.
u) Cats love neither Adam nor Eve.
v) Something furry loves Eve.
w) Only people love someone.
x) Some people have sons.
y) Any son of Adam is a son of Eve.
z) Adam is a son and everybody loves him.
aa) No animal is furry, but some have tails.
bb) Any furry animal has a tail.
cc) No one has a son.
dd) Not everyone has a son.
ee) Some blonds love Eve, some do not.
ff) Adam loves any furry cat.
gg) All blonds who love themselves love Eve.
hh) Eve loves someone who loves themself.
ii) Anyone who loves no cats loves no dogs.
jj) Cats love Eve if they love anyone. (Ambiguous)
kk) If anyone has a son, Eve loves Adam. (Ambiguous)
ll) If anyone has a son, that person loves Adam.
mm) Anyone who has a son loves Eve.
nn) If someone has a son, Adam loves Eve.
oo) If someone has a son, that person loves Adam.
pp) Someone who has a son loves Adam. (Ambiguous)
qq) All the cats with sons, except the furry ones, love Eve.
rr) Anyone who loves a cat loves an animal.
ss) Anyone who loves a person loves no animal.
tt) Adam has a son who is not furry.
uu) If Adam's son has a furry son, so does Adam.
vv) A son of Adam is a son of Eve. (Ambiguous)
ww) If the only people who love Eve are blond, then nobody loves Eve.
xx) No one loves anyone. (Ambiguous)
yy) No one loves someone. (Ambiguous)
zz) Everyone loves no one.
aaa) Everyone doesn't love everyone. (Ambiguous!)
bbb) Nobody loves nobody. (Ambiguous?)
ccc) Except for the furry ones, every animal loves Adam.
ddd) Everyone loves a lover. (Ambiguous)
eee) None but those blonds who love Adam own cats and dogs.
fff) No one who loves no son of Adam loves no son of Eve.
ggg) Only owners of dogs with tails own cats which do not love Adam.
hhh) None of Adam's sons are owners of furry animals with no tails.
iii) Anyone who loves nothing without a tail owns nothing which is loved by an animal.
jjj) Only those who love neither Adam nor Eve are sons of those who own none of the animals without tails.
kkk) Anyone who loves all who Eve loves loves someone who is loved by all who love Eve.
4-9. Transcribe the following sentences into predicate logic, making up your own transcription guide for each sentence. Be sure to show as much of the logical form as possible.
a) No one likes Professor Snarf.
b) Any dog can hear better than any person.
c) Neither all Republicans nor all Democrats are honest.
d) Some movie stars are better looking than others.
e) None of the students who read A Modem Formal Logic Primer failed the logic course.
f) Only people who eat carrots can see well in the dark.
g) Not only people who eat carrots can see as well as people who eat strawberries.
h) Peter likes all movies except for scary ones.
i) Some large members of the cat family can run faster than any horse.
j) Not all people with red hair are more temperamental than those with blond hair.
k) Some penny on the table was minted before any dime on the table.
l) No pickle tastes better than any strawberry.
m) John is not as tall as anyone on the basketball team.
n) None of the pumpkins at Smith's fruit stand are as large as any of those on MacGreggor's farm.
o) Professors who don't prepare their lectures confuse their students.
p) Professor Snarf either teaches Larry or teaches someone who teaches Larry.
q) Not only logic teachers teach students at Up State U.
r) Anyone who lives in Boston likes clams more than anyone who lives in Denver. (Ambiguous)
s) Except for garage mechanics who fix cars, no one has greasy pants.
t) Only movies shown on channel 32 are older than movies shown on channel 42.
u) No logic text explains logic as well as some professors do.
v) The people who eat, drink, and are merry are more fun than those who neither smile nor laugh.
chapter summary Exercise
In reviewing this chapter make a short summary of the following to ensure your grasp of these ideas:
a) Restricted Quantifiers
b) Rule ~
c) Rule ~ƎS,
e) Words that generally transcribe with a universal quantifier
f) Word that generally transcribe with an existential quantifier
g) Negative Quantifier Words
i) Give a summary of important transcription strategies