5. Observe that some dodecahedron is not large. Express this.
6. Notice that some dodecahedron is small. Express this fact.
7. Observe that some dodecahedron is not small. Express this.
8. Notice that some dodecahedron is neither large nor small. Express this.
9. Express the observation that no tetrahedron is large.
10. Express the fact that no cube is large.
Now change the sizes of the objects in the following way: make one of the cubes large, one
of the tetrahedra medium, and all the dodecahedra small. With these changes, the following
should come out false: 1, 2, 4, 7, 8, and 10. If not, then you have made an error in describing
the original world. Can you ļ¬gure out what it is? Try making other changes and see if your
sentences have the expected truth values. Submit your sentence ļ¬le.
9.13 Assume we are working in an extension of the ļ¬rst-order language of arithmetic with the
Ć‚ additional predicates Even(x) and Prime(x), meaning, respectively, āx is an even numberā and
āx is a prime number.ā Create a sentence ļ¬le in which you express the following claims:
1. Every even number is prime.
2. No even number is prime.
3. Some prime is even.
4. Some prime is not even.
5. Every prime is either odd or equal to 2.
[Note that you should assume your domain of discourse consists of the natural numbers, so
there is no need for a predicate Number(x). Also, remember that 2 is not a constant in the
language, so must be expressed using + and 1.]
Translating complex noun phrases / 243
9.14 (Name that object) Open Maigretā™s World and Maigretā™s Sentences. The goal is to try to ļ¬gure
Ć‚ out which objects have names, and what they are. You should be able to ļ¬gure this out from
the sentences, all of which are true. Once you have come to your conclusion, assign the six
names to objects in the world in such a way that all the sentences do indeed evaluate as true.
Submit your modiļ¬ed world.
Translating complex noun phrases
The ļ¬rst thing you have to learn in order to translate quantiļ¬ed English
expressions is how to treat complex noun phrases, expressions like āa boy
living in Omahaā or āevery girl living in Duluth.ā In this section we will
learn how to do this. We concentrate ļ¬rst on the former sort of noun phrase,
whose most natural translation involves an existential quantiļ¬er. Typically,
these will be noun phrases starting with one of the determiners some, a, and
an, including noun phrases like something. These are called existential noun existential
phrases, since they assert the existence of something or other. Of course two
of our four Aristotelian forms involve existential noun phrases, so we know
the general pattern: existential noun phrases are usually translated using ā,
frequently together with ā§.
Letā™s look at a simple example. Suppose we wanted to translate the sen-
tence A small, happy dog is at home. This sentence claims that there is an
object which is simultaneously a small, happy dog, and at home. We would
translate it as
āx [(Small(x) ā§ Happy(x) ā§ Dog(x)) ā§ Home(x)]
We have put parentheses around the ļ¬rst three predicates to indicate that
they were all part of the translation of the subject noun phrase. But this is
not really necessary.
Universal noun phrases are those that begin with determiners like every, universal
each, and all. These are usually translated with the universal quantiļ¬er.
Sometimes noun phrases beginning with no and with any are also translated
with the universal quantiļ¬er. Two of our four Aristotelian forms involve
universal noun phrases, so we also know the general pattern here: universal
noun phrases are usually translated using ā, frequently together with ā’.
Letā™s consider the sentence Every small dog that is at home is happy. This
claims that everything with a complex property, that of being a small dog
at home, has another property, that of being happy. This suggests that the
244 / Introduction to Quantification
overall sentence has the form All Aā™s are Bā™s. But in this case, to express the
complex property that ļ¬lls the āAā position, we will use a conjunction. Thus
it would be translated as
āx [(Small(x) ā§ Dog(x) ā§ Home(x)) ā’ Happy(x)]
In this case, the parentheses are not optional. Without them the expression
would not be well formed.
In both of the above examples, the complex noun phrase appeared at
the beginning of the English sentence, much like the quantiļ¬er in the fol
translation. Often, however, the English noun phrase will appear somewhere
noun phrases in
non-subject positions else in the sentence, say as the direct object, and in these cases the fol
translation may be ordered very diļ¬erently from the English sentence. For
example, the sentence Max owns a small, happy dog might be translated:
āx [(Small(x) ā§ Happy(x) ā§ Dog(x)) ā§ Owns(max, x)]
which says there is a small, happy dog that Max owns. Similarly, the English
sentence Max owns every small, happy dog would end up turned around like
āx [(Small(x) ā§ Happy(x) ā§ Dog(x)) ā’ Owns(max, x)]
You will be given lots of practice translating complex noun phrases in the
exercises that follow. First, however, we discuss some troublesome cases.
1. Translations of complex quantiļ¬ed noun phrases frequently employ
conjunctions of atomic predicates.
2. The order of an English sentence may not correspond to the order of
its fol translation.
Conversational implicature and quantiļ¬cation
You will ļ¬nd that translating quantiļ¬ed phrases is not diļ¬cult, as long as
quantiļ¬ers are not ānestedā inside one another. There are, however, a couple
of points that sometimes present stumbling blocks.
One thing that often puzzles students has to do with the truth value of
sentences of the form
āx (P(x) ā’ Q(x))
Translating complex noun phrases / 245
in worlds where there are no objects satisfying P(x). If you think about it,
you will see that in such a world the sentence is true simply because there
are no objects that satisfy the antecedent. This is called a vacuously true vacuously true
Consider, for example, the sentence
āy(Tet(y) ā’ Small(y))
which asserts that every tetrahedron is small. But imagine that it has been
asserted about a world in which there are no tetrahedra. In such a world the
sentence is true simply because there are no tetrahedra at all, small, medium,
or large. Consequently, it is impossible to ļ¬nd a counterexample, a tetrahedron
which is not small.
What strikes students as especially odd are examples like
āy(Tet(y) ā’ Cube(y))
On the face of it, such a sentence looks contradictory. But we see that if it is
asserted about a world in which there are no tetrahedra, then it is in fact true.
But that is the only way it can be true: if there are no tetrahedra. In other
words, the only way this sentence can be true is if it is vacuously true. Letā™s
call generalizations with this property inherently vacuous. Thus, a sentence of inherently vacuous
the form āx (P(x) ā’ Q(x)) is inherently vacuous if the only worlds in which it
is true are those in which āx Ā¬P(x) is true.
You try it
1. Open Dodgsonā™s Sentences. Note that the ļ¬rst sentence says that every
tetrahedron is large.
2. Open Peanoā™s World. Sentence 1 is clearly false in this world, since the
small tetrahedron is a counterexample to the universal claim. What this
means is that if you play the game committed to the falsity of this claim,
then when Tarskiā™s World asks you to pick an object you will be able to
pick the small tetrahedron and win the game. Try this.
3. Delete this counterexample and verify that sentence 1 is now true.
4. Now open Peirceā™s World. Verify that sentence 1 is again false, this time
because there are three counterexamples. (Now if you play the game com-
mitted to the falsity of the sentence, you will have three diļ¬erent winning
moves when asked to pick an object: you can pick any of the small tetra-
hedra and win.)
246 / Introduction to Quantification
5. Delete all three counterexamples, and evaluate the claim. Is the result
what you expected? The generalization is true, because there are no coun-
terexamples to it. It is what we called a vacuously true generalization,
since there are no objects that satisfy the antecedent. That is, there are
no tetrahedra at all, small, medium, or large. Conļ¬rm that all of sentences
1ā“3 are vacuously true in the current world.
6. Two more vacuously true sentences are given in sentences 4 and 5. How-
ever, these sentences are diļ¬erent in another respect. Each of the ļ¬rst three
sentences could have been non-vacuously true in a world, but these latter
two can only be true in worlds containing no tetrahedra. That is, they are
7. Add a sixth generalization to the ļ¬le that is vacuously true in Peirceā™s
World but non-vacuously true in Peanoā™s World. (In both cases, make sure
you use the unmodiļ¬ed worlds.) Save your new sentence ļ¬le as Sentences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations
In everyday conversation, it is rare to encounter a vacuously true gener-
alization, let alone an inherently vacuous generalization. When we do ļ¬nd
either of these, we feel that the speaker has misled us. For example, suppose a
professor claims āEvery freshman who took the class got an A,ā when in fact
no freshman took her class. Here we wouldnā™t say that she lied, but we would
certainly say that she misled us. Her statement typically carries the conver-
sational implicature that there were freshmen in the class. If there were no
implicature freshmen, then thatā™s what she would have said if she were being forthright.
Inherently vacuous claims are true only when they are misleading, so they
strike us as intuitively as false.
Another source of confusion concerns the relationship between the follow-
ing two Aristotelian sentences:
Some Pā™s are Qā™s
All Pā™s are Qā™s
Students often have the intuition that the ļ¬rst should contradict the second.
After all, why would you say that some student got an A if every student got
an A? If this intuition were right, then the correct translation of Some Pā™s
are Qā™s would not be what we have suggested above, but rather
Translating complex noun phrases / 247
āx (P(x) ā§ Q(x)) ā§ Ā¬āx (P(x) ā’ Q(x))
It is easy to see, however, that the second conjunct of this sentence does not
represent part of the meaning of the sentence. It is, rather, another example
of a conversational implicature. It makes perfectly good sense to say āSome
student got an A on the exam. In fact, every student did.ā If the proposed
conjunction were the right form of translation, this ampliļ¬cation would be
1. All Pā™s are Qā™s does not imply, though it may conversationally suggest,
that there are some Pā™s.
2. Some Pā™s are Qā™s does not imply, though it may conversationally sug-
gest, that not all Pā™s are Qā™s.
9.15 If you skipped the You try it section, go back and do it now. Submit the ļ¬le Sentences
Ć‚ Vacuous 1.
9.16 (Translating existential noun phrases) Start a new sentence ļ¬le and enter translations of the
Ć‚ following English sentences. Each will use the symbol ā exactly once. None will use the symbol
ā. As you go, check that your entries are well-formed sentences. By the way, you will ļ¬nd that
many of these English sentences are translated using the same ļ¬rst-order sentence.
1. Something is large.
2. Something is a cube.
3. Something is a large cube.
4. Some cube is large.