<<

. 44
( 107 .)



>>


you Choose one of
true
P∨Q P, Q that
Tarski™s World is true.
false

Tarski™s World Choose one of
true
P§Q P, Q that
you is false.
false

Choose some b
you
true
∃x P(x) that satis¬es
Tarski™s World the w¬ P(x).
false

Choose some b
Tarski™s World
true
∀x P(x) that does not
you satisfy P(x).
false

Replace ¬P
¬P either ” by P
and switch
commitment.

Replace P ’ Q
P’Q either ” by ¬P ∨ Q
and keep
commitment.

Replace P ” Q by
P”Q either ” (P ’ Q) § (Q ’ P)
and keep
commitment.




Game rules for the quanti¬ers
The game rules for the quanti¬ers are more interesting than those for the
truth-functional connectives. With the connectives, moves in the game in-
volved choosing sentences that are parts of the sentence to which you are
committed. With the quanti¬er rules, however, moves consist in choosing ob-
jects, not sentences.
Suppose, for example, that you are committed to the truth of ∃x P(x). This game rules for ∃




Section 9.4
238 / Introduction to Quantification


means that you are committed to there being an object that satis¬es P(x).
Tarski™s World will ask you to live up to this commitment by ¬nding such an
object. On the other hand, if you are committed to the falsity of ∃x P(x), then
you are committed to there being no object that satis¬es P(x). In which case,
Tarski™s World gets to choose: it tries to ¬nd an object that does satisfy P(x),
thus contradicting your commitment.
The rules for ∀ are just the opposite. If you are committed to the truth
game rules for ∀
of ∀x P(x), then you are committed to every object satisfying P(x). Tarski™s
World will try to ¬nd an object not satisfying P(x), thus contradicting your
commitment. If, however, you are committed to the falsity of ∀x P(x), then you
are committed to there being some object that does not satisfy P(x). Tarski™s
World will ask you to live up to your commitment by ¬nding such an object.
We have now seen all the game rules. We summarize them in Table 9.1.


You try it
................................................................

1. Open the ¬les Game World and Game Sentences. Go through each sentence
and see if you can tell whether it is true or false. Check your evaluation.

2. Whether you evaluated the sentence correctly or not, play the game twice
for each sentence, ¬rst committed to true, then committed to false.
Make sure you understand how the game works at each step.

3. There is nothing to save except your understanding of the game.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations

Exercises


9.4 If you skipped the You try it section, go back and do it now. This is an easy but important
 exercise that will familiarize you with the game rules for the quanti¬ers. There is nothing you
need to turn in or submit.

9.5 (Evaluating sentences in a world) Open Peirce™s World and Peirce™s Sentences. There are 30
‚ sentences in this ¬le. Work through them, assessing their truth and playing the game when
necessary. Make sure you understand why they have the truth values they do. (You may need to
switch to the 2-D view for some of the sentences.) After you understand each of the sentences,
go back and make the false ones true by adding or deleting a negation sign. Submit the ¬le
when the sentences all come out true in Peirce™s World.




Chapter 9
The four Aristotelian forms / 239



9.6 (Evaluating sentences in a world) Open Leibniz™s World and Zorn™s Sentences. The sentences
‚ in this ¬le contain both quanti¬ers and the identity symbol. Work through them, assessing
their truth and playing the game when necessary. After you™re sure you understand why the
sentences get the values they do, modify the false ones to make them true. But this time you
can make any change you want except adding or deleting a negation sign.

9.7 In English we sometimes say things like Every Jason is envied, meaning that everyone named
 “Jason” is envied. For this reason, students are sometimes tempted to write expressions like
∀b Cube(b) to mean something like Everything named b is a cube. Explain why this is not well
formed according to the grammatical rules on page 231.



Section 9.5
The four Aristotelian forms
Long before fol was codi¬ed, Aristotle studied the kinds of reasoning associ-
ated with quanti¬ed noun phrases like Every man, No man, and Some man,
expressions we would translate using our quanti¬er symbols. The four main
sentence forms treated in Aristotle™s logic were the following.

All P™s are Q™s
Some P™s are Q™s Aristotelian forms
No P™s are Q™s
Some P™s are not Q™s

We will begin by looking at the ¬rst two of these forms, which we have
already discussed to a certain extent. These forms are translated as follows.
The form All P™s are Q™s is translated as:

∀x (P(x) ’ Q(x))

whereas the form Some P™s are Q™s is translated as:

∃x (P(x) § Q(x))

Beginning students are often tempted to translate the latter more like the
former, namely as:
∃x (P(x) ’ Q(x))
This is in fact an extremely unnatural sentence of ¬rst-order logic. It is mean-
ingful, but it doesn™t mean what you might think. It is true just in case there
is an object which is either not a P or else is a Q, which is something quite



Section 9.5
240 / Introduction to Quantification


di¬erent than saying that some P™s are Q™s. We can quickly illustrate this
di¬erence with Tarski™s World.


You try it
................................................................

1. Use Tarski™s World to build a world containing a single large cube and
nothing else.

2. Write the sentence ∃x (Cube(x) ’ Large(x)) in the sentence window. Check
to see that the sentence is true in your world.

3. Now change the large cube into a small tetrahedron and check to see if
the sentence is true or false. Do you understand why the sentence is still
true? Even if you do, play the game twice, once committed to its being
false, once to its being true.

4. Add a second sentence that correctly expresses the claim that there is a
large cube. Make sure it is false in the current world but becomes true when
you add a large cube. Save your two sentences as Sentences Quanti¬er 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations
The other two Aristotelian forms are translated similarly, but using a
negation. In particular No P™s are Q™s is translated

∀x (P(x) ’ ¬Q(x))

Many students, and one of the authors, ¬nds it more natural to use the fol-
lowing, logically equivalent sentence:

¬∃x (P(x) § Q(x))

Both of these assert that nothing that is a P is also a Q.
The last of the four forms, Some P™s are not Q™s, is translated by

∃x (P(x) § ¬Q(x))

which says there is something that is a P but not a Q.
The four Aristotelian forms are the very simplest sorts of sentences built
using quanti¬ers. Since many of the more complicated forms we talk about
later are elaborations of these, you should learn them well.




Chapter 9
The four Aristotelian forms / 241



Remember

The four Aristotelian forms are translated as follows:
All P™s are Q™s. ∀x (P(x) ’ Q(x))
Some P™s are Q™s. ∃x (P(x) § Q(x))
No P™s are Q™s. ∀x (P(x) ’ ¬Q(x))
Some P™s are not Q™s. ∃x (P(x) § ¬Q(x))




Exercises


9.8 If you skipped the You try it section, go back and do it now. Submit the ¬le Sentences
‚ Quanti¬er 1.

9.9 (Building a world) Open Aristotle™s Sentences. Each of these sentences is of one of the four
‚ Aristotelian forms. Build a single world where all the sentences in the ¬le are true. As you
work through the sentences, you will ¬nd yourself successively modifying the world. Whenever
you make a change in the world, you had better go back and check that you haven™t made
any of the earlier sentences false. Then, when you are ¬nished, verify that all the sentences are
really true and submit your world.

9.10 (Common translation mistakes) Open Edgar™s Sentences and evaluate them in Edgar™s World.
 Make sure you understand why each of them has the truth value it does. Play the game if
any of the evaluations surprise you. Which of these sentences would be a good translation of
There is a tetrahedron that is large? (Clearly this English sentence is false in Edgar™s World,
since there are no tetrahedra at all.) Which sentence would be a good translation of There is
a cube between a and b? Which would be a good translation of There is a large dodecahedron?
Express in clear English the claim made by each sentence in the ¬le and turn in your answers
to your instructor.

9.11 (Common mistakes, part 2) Open Allan™s Sentences. In this ¬le, sentences 1 and 4 are the
‚| correct translations of Some dodecahedron is large and All tetrahedra are small, respectively.
Let™s investigate the logical relations between these and sentences 2 and 3.
1. Construct a world in which sentences 2 and 4 are true, but sentences 1 and 3 are false.
Save it as World 9.11.1. This shows that sentence 1 is not a consequence of 2, and
sentence 3 is not a consequence of 4.




Section 9.5
242 / Introduction to Quantification


2. Can you construct a world in which sentence 3 is true and sentence 4 is false? If so, do
so and save it as World 9.11.2. If not, explain why you can™t and what this shows.
3. Can you construct a world in which sentence 1 is true and sentence 2 is false? If so, do
so and save it as World 9.11.3. If not, explain why not.

Submit any world ¬les you constructed and turn in any explanations to your instructor.

9.12 (Describing a world) Open Reichenbach™s World 1. Start a new sentence ¬le where you will
‚ describe some features of this world using sentences of the simple Aristotelian forms. Check
each of your sentences to see that it is indeed a sentence and that it is true in this world.
1. Use your ¬rst sentence to describe the size of all the tetrahedra.
2. Use your second sentence to describe the size of all the cubes.
3. Use your third sentence to express the truism that every dodecahedron is either small,
medium, or large.

<<

. 44
( 107 .)



>>