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4.2: Transcribing from English into Logic

Transcribing into the language of predicate logic can be extremely difficult. Actually, one can do logic perfectly well without getting very good at transcription. But transcriptions into logic provide one of predicate logic's important uses. This is because, when it comes to quantification, English is often extremely confusing, ambiguous, and even downright obscure. Often we can become clearer about what is being said if we put a statement into logic. Sometimes transcribing into logic is a must for clarity and precision. For example, how do you understand the highly ambiguous sentence, 'All of the boys didn't kiss all of the girls.'? I, for one, am lost unless I transcribe into logic.

   Before we get started, I should mention a general point. Just as in the case of sentence logic, if two predicate logic sentences are logically equivalent they are both equally good (or equally bad!) transcriptions of an English sentence. Two logically equivalent sentences share the same truth value in all possible cases (understood as all interpretations), and in this sense two logically equivalent sentences "say the same thing." But if two predicate logic sentences say the same thing, then to the extent that one of them says what an English sentence says, then so does the other.

   We are going to be looking at quite a few examples, so let's agree on a transcription guide:

Transcription Guide

a: Adam                                 Px: x is a person
J: The lights will be on             Rx: x is a registered voter
Ax: x is an adult                      Vx: x has the right to vote
Bx:x is a boy                           Kxy: x kissed y
Cx: x is a cat                           Lxy: x love y
Dx: x is a dog                          Mxy: x is married to y
Fx: x can run a 3.45 mile          Oxy: x owns y
Gx: x is a girl                          Txy: x is a tail of y
Hx: x is at home

Take note of the fact that in giving you a transcription guide, I have been using open sentences to indicate predicates. For example, I am using the open sentence 'Px' to indicate the predicate 'is a person.' The idea of using an open sentence to indicate a predicate will soon become very useful.

   To keep us focused on the new ideas, I will often use subscripts on restricted quantifiers. However, you should keep in mind that complete transcriptions require you to rewrite the subscripts, as explained in the last section.

   Now let's go back and start with the basics. '(Vx)(Cx ⊃ Fx)' transcribes 'all cats are furry,' 'Every cat is furry,' 'Any cat is furry,' and 'Each cat is furry.' This indicates that

Usually, the words 'all', 'every', 'any', and 'each' signal a universal quantifier.

   Let's make a similar list for the existential quantifier. '(Ǝx)(Cx & Fx)' transcribes 'Some cat is furry', 'Some cats are furry,' 'At least one cat is furry', 'There is a furry cat,' and 'There are furry cats':

Usually, the expressions 'some', 'at least one', 'there is', and 'there are' signal an existential quantifier.

   These lists make a good beginning, but you must use care. There are no hard and fast rules for transcribing English quantifier words into predicate logic. For starters, 'a' can easily function as a universal or an existential quantifier. For example, 'A car can go very fast.' is ambiguous. It can be used to say either that any car can go very fast or that some car can go very fast.

   To make it clearer that 'a' can function both ways, consider the following examples. You probably understand 'A man is wise.' to mean that some man is wise. But most likely you understand 'A dog has four legs.' to mean that all dogs have four legs. Actually, both of these sentences are ambiguous. In both sentences, 'a' can correspond to 'all' or 'some'. You probably didn't notice that fact because when we hear an ambiguous sentence we tend to notice only one of the possible meanings. If a sentence is obviously true when understood with one of its meanings and obviously false when understood with the other, we usually hear the sentence only as making the true statement. So if all the men in the world were wise, we would take 'A man is wise.' to mean that all men are wise, and if only one dog in the world had four legs we would take 'A dog has four legs.' to mean that some dog has four legs.

   It is a little easier to hear 'A car can go very fast.' either way. This is because we interpret this sentence one way or the other, depending on how fast we take 'fast' to be. If 'fast' means 30 miles an hour (which is very fast by horse and buggy standards), it is easy to hear 'A car can go very fast.' as meaning that all cars can go very fast. If "fast' means 180 miles an hour it is easy to hear 'a car can go very fast.' as meaning that some car can go very fast.

   'A' is not the only treacherous English quantifier word. 'Anyone' usually gets transcribed with a universal quantifier. But not always. Consider

(3) If anyone is at home, the lights will be on.
(4) If anyone can run a 3:45 mile, Adam can.

We naturally hear (3), not as saying that if everyone is at home the lights will be on, but as saying that if someone is at home the lights will be on. So a correct transcription is

(3a) (Ǝx)PHx ⊃ J

Likewise, by (4), we do not ordinarily mean that if everyone can run a 3:43 mile, Adam can. We mean that if someone can run that fast, Adam can:

(4a) (Ǝx)PFx ⊃ Fa

   At least that's what one would ordinarily mean by (4). However, I think that (4) actually is ambiguous. I think 'anyone' in (4) could be understood as 'everyone'. This becomes more plausible if you change the '3:45 mile' to 'lo-minute mile'. And it becomes still more plausible after you consider the following example: 'Anyone can tie their own shoe laces. And if anyone can, Adam can.'

   Going back to (3), one would think that if (4) is ambiguous, (3) should be ambiguous in the same way. I just can't hear an ambiguity in (3). Can you?

   'Someone' can play the reverse trick on us. Usually, we transcribe it with an existential quantifier. But consider

(5) Someone who is a registered voter has the right to vote.

We naturally hear this as the generalization stating that anyone who is a registered voter has the right to vote. Thus we transcribe it as

(5a) (Vx)P(Rx ⊃ Vx)

   As in the case of (4), which uses 'anyone', we can have ambiguity in sentences such as (5), which uses 'someone'. If you don't believe me, imagine that you live in a totalitarian state, called Totalitarania. In Totalitarania, everyone is a registered voter. But voter registration is a sham. In fact, only one person, the boss, has the right to vote. As a citizen of Totalitarania, you can still truthfully say that someone who is a registered voter (namely, the boss) has the right to vote. (You can make this even clearer by emphasizing the word 'someone: 'someone who is a registered voter has the right to vote.') In this context we hear the sentence as saying

(5b) (Ǝx)P(Rx ⊃ Vx)

   Ambiguity can plague transcription in all sorts of ways. Consider an example traditional among linguists:

(6) All the boys kissed all the girls

This can easily mean that each and every one of the boys kissed each and every one of the girls:

(6a) (Vx)B(Vy)CKxy

But it can also mean that each of the boys kissed some girls so that, finally, each and every girl got kissed by some boy:

(6b) (Vx)B(Ǝy)GKxY & (Vy)G(Ǝx)BKxY

   If you think that was bad, things get much worse when multiple quantifiers get tangled up with negations. Consider

(7) All the boys didn't kiss all the girls.

Everytime I try to think this one through, I blow a circuit. Perhaps the most natural transcription is to take the logical form of the English at face value and take the sentence to assert that of each and every boy it is true that he did not kiss all the girls; that is, for each and every boy there is at least one girl not kissed by that boy:

(7a) (Vx)B~(Vy)CKxy,   or  (Vx)B(Ǝy)C~Kxy

But one can also take the sentence to mean that each and every boy refrained from kissing each and every girl, that is, didn't kiss the first girl and didn't kiss the second girl and not the third, and so on. In yet other words, this says that for each and every boy there was no girl whom he kissed, so that nobody kissed anybody:

(7b) (Vx)B~(Vy)C~Kxy,   or  (Vx)B~(Ǝy)CKxy,   or   ~(Ǝx)B(Ǝy)CKxy 

   We are still not done with this example, for one can also use (7) to mean that not all the boys kissed every single girl-that is, that some boy did not kiss all the girls, in other words that at least one of the boys didn't kiss at least one of the girls:

(7c) ~(Vx)B(Vy)CKxy,   or  (Ǝx)B~(Vy)CKxy ,  or    (Ǝx)B(Ǝy)C~Kxy 

   It's worth an aside to indicate how it can happen that an innocent-looking sentence such as (7) can turn out to be so horribly ambiguous. Modern linguistics postulates that our minds carry around more than one representation of a given sentence. There is one kind of structure that . represents the logical form of a sentence. Another kind of structure represents sentences as we speak and write them. Our minds connect these (and other) representations of a given sentence by making all sorts of complicated transformations. These transformations can turn representations of different logical forms into the same representation of a spoken or written sentence. Thus one sentence which you speak or write can correspond to two, three, or sometimes quite a few different structures that carry very different meanings. In particular, the written sentence (7) corresponds to (at least!) three different logical forms. (7a), (7b), and (7c) don't give all the details of the different, hidden structures that can be transformed into (7). But they do describe the differences which show up in the language of predicate logic.

   You can see hints of all this if you look closely at (7), (7a), (7b), and (7c). In (7) we have two universal quantifier words and a negation. But since the quantifier words appear on either side of 'kissed', it's really not all that clear where the negation is meant to go in relation to the universal quantifiers. We must consider three possibilities. We could have the negation between the two universal quantifiers. Indeed, that is what you see in (7a), in the first of its logically equivalent forms. Or we could have the negation coming after the two universal quantifiers, which is what you find in the first of the logically equivalent sentences in (7b). Finally, we could have the negation preceding both universal quantifiers. You see this option in (7c). In sum, we have three similar, but importantly different, structures. Their logical forms all have two universal quantifiers and a negation, but the three differ, with the negation coming before, between, or after the two quantifiers. The linguistic transformations in our hinds connect all three of these structures with the same, highly ambiguous English sentence, (7).

   Let's get back to logic and consider some other words which you may find especially difficult to transcribe. I am always getting mixed up by sentences which use 'only', such as 'Only cats are furry.' So I use the strategy of first transcribing a clear case (it helps to use a sentence I know is true) and then using the clear case to figure out a formula. I proceed in this way: Transcribe

(8) Only adults can vote.

This means that anyone who is not an adult can't vote, or equivalently (using the law of contraposition), anyone who can vote is an adult. So either of the following equivalent sentences provides a correct transcription:

(8a) (Vx)P(~Ax ⊃ ~Vx)
(8b) (Vx)P(Vx ⊃ Ax)

   This works in general. (In the following I used boldface capital P and Q to stand for arbitrary predicates.) Transcribe

(9) Only Ps are Qs

either as

(9a) (Vx)(~Px ~Qx)
(9b) (Vx)(Qx ⊃ Px)

Thus 'Only cats are furry' becomes (Vx)(Fx ⊃ Cx).'
   Nothing' and 'not everything' often confuse me also. We must carefully distinguish

(10) Nothing is furry: (Vx)~Fx, or ~(Ǝx)Fx


(11) Not everything is furry: ~(Vx)Fx, or (Ǝx)~Fx

(The alternative transcriptions given in (10) and (11) are logically equivalent, by the rules ~(Vx) and ~(Ǝx) for logical equivalence introduced in section 3-4.) 'Not everything' can be transcribed literally as 'not all x . . .'. 'Nothing' means something different and much stronger. 'Nothing' means 'everything is not . . . .' Be careful not to confuse 'nothing' with 'not everything.' If the distinction is not yet clear, make up some more examples and carefully think them through. 'None' and 'none but' can also cause confusion:

(12) None but adults can vote: (Vx)(~Ax ⊃ ~Vx)

(13) None love Adam: (Vx)~Lxe '

None but' simply transcribes as 'only.' When 'none' without the 'but' fits in grammatically in English you will usually be able to treat it as you do 'nothing'. 'Nothing' and 'none' differ in that we tend to use 'none' when there has been a stated or implied restriction of domain: "How many cats does Adam love? He loves none." In this context a really faithful transcription of the sentence 'Adam loves none.' would be '(Vx)C~Lax', or, rewriting the subscript, '(Vx)(Cx ⊃ ~Lax).

Perhaps the most important negative quantifier expression in English is 'no', as in

(14) No cats are furry.

To say that no cats are furry is to say that absolutely all cats are not furry, so that we transcribe (18) as

(15) (Vx)C~Fx, that is, (Vx)(Cx ⊃ ~Fx)

In general, transcribe

(16) No Ps are Qx.

(17) (Vx)P~Q, that is, (Vx)(P ⊃ ~Q)


4-4. Transcribe the following English sentences into the language of predicate logic. Use subscripts if you find them helpful in figuring out your answers, but no subscripts should appear in your final answers.

Transcription Guide

a: Adam                               Fx: x is furry
e: Eve                                  Px: x is a person
Ax: x is an animal                  Qx: x purrs
Bx: x is a blonde                    Lxy: x loves y
Cx: x is a cat                         Sxy: x is son of y
Dx: x is a dog                        Txy: x is tickling y

a) Anything furry loves Eve.
b) No cat is furry.
c) If anyone loves Adam, Eve does.
d) ve does not love anyone.
e) Nothing is furry.
f) Adam, if anyone, is blond.
g) Not all cats are furry.
h) Some cats are not furry.
i) No one is a cat.
j) No cat is a dog.
k) If something purrs, it is a cat.
l) Not everything blond is a cat.
m) A dog is not an animal. (Ambiguous)
n) Not all animals are dogs.
o) Only cats purr.
p) Not only cats are furry.
q) Any dog is not a cat.
r) No blonds love Adam.
s) None but blonds love Adam.
t) Some dog is not a cat.
u) Nothing furry loves anyone.
v) Only cat lovers love dogs. (Ambiguous?)
w) If someone is a son of Adam, he is blond.
x) No son of Adam is a son of Eve.
y) Someone who is a son of Adam is no son of Eve. (Ambiguous)
z) Each cat which loves Adam also loves Eve.
aa) Not everyone who loves Adam also loves Eve.
bb) Anyone who is tickling Eve is tickling Adam.
cc) None but those who love Adam also love Eve.

4-5. Give alternative transcriptions which show the ways in which the following sentences are ambiguous. In this problem you do not have to eliminate subscripts. (It is sometimes easier to study the ambiguity if we write these sentences in the compact subscript notation.)

a) Everyone loves someone.
b) Someone loves everyone.
c) Something is a cat if and only if Adam loves it.
d) All cats are not furry. e) Not anyone loves Adam.

4-6. In this section I discussed ambiguities connected with words such as 'a', 'someone', and 'anyone.' In fact, English has a great many sorts of ambiguity arising from the ways in which words are connected with each other. For example, 'I won't stay at home to please you.' can mean that if I stay at home, I won't do it in order to please you. But it can also mean that 1 will go out because going out will please you. 'Eve asked Adam to stay in the house.' can mean that Eve asked Adam to remain in a certain location, and that location is the house. It can also mean that Eve asked Adam to remain in some unspecified location, and that she made her request in the house.
   For the following English sentences, provide alternative transcripts showing how the sentences are ambiguous. Use the transcription guides given for each sentence.

a) Flying planes can be dangerous. (Px: x is a plane. Fx: x is flying. Dx: x can be dangerous. Ax: x is an act of flying a plane.)

b) All wild animal keepers are blond. (Kxy: x keeps y. Wx: x is wild. Ax: x is an animal. Bx: x is blond.)

c) Adam only relaxes on Sundays. (a: Adam. Rxy: x relaxes on day y. Lxy: x relaxes ("is lazy") all day long on day y. Sx: x is Sunday.)

d) Eve dressed and walked all the dogs. (e: Eve. Cxy: x dressed y. Dx: x is a dog. Wxy: x walked y.)

   Linguists use the expression Structural Ambiguity for the kind of ambiguity in these examples. This is because the ambiguities have to do with alternative ways in which the grammatical structure of the sentences can be correctly analyzed. Structural ambiguity contrasts with Lexical Ambiguity, which has to do with the ambiguity in the meaning of isolated words. Thus the most obvious ambiguity of 'I took my brother's picture yesterday.' turns on the ambiguity of the meaning of 'took' (stole vs. produced a picture). The ambiguity involved with quantifier words such as 'a', 'someone', and 'anyone' is actually structural ambiguity, not lexical ambiguity. We can see a hint of this in the fact that '(Ǝx)Hx ⊃ J' is logically equivalent to '(Vx)(Hx ⊃ J) and the fact that '(Vx)Hx ⊃ J' is logically equivalent to (Ǝx)(Hx ⊃ J), as you will prove later on in the course. 

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