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chapter_2/exercises.md
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@ -1614,3 +1614,453 @@ Now let's consider:
$$ p \leftrightarrow q \equiv (p \to q) \wedge (q \to p) $$
$$ p \leftrightarrow q \equiv \neg(p \wedge \neg q) \wedge \neg(q \wedge \neg p) $$
---
**Exercise Set 2.3**
Page 99
Use modus ponens or modus tollens to fill in the blanks in the arguments of 1-5
so as to produce valid inferences.
1.
If $\sqrt{2}$ is rational, then $\sqrt{2} = \dfrac{a}{b}$ for some integers $a$
and $b$.
It is not true that $\sqrt{2} = \dfrac{a}{b}$ for some integers $a$ and $b$.
$\therefore$ ______.
2.
If $1 - 0.99999 \dots$ is less than every positive real number, then it equals
zero.
______.
$\therefore$ The number $1 - 0.99999 \dots$ equals zero.
3.
If logic is easy, then I am a monkey's uncle.
I am not a monkey's uncle.
$\therefore$ ______.
4.
If this graph can be colored with three colors, then it can be colored with four
colors.
This graph cannot be colored with four colors.
$\therefore$ ______.
5.
If they were unsure about the address, then they would have telephoned.
______.
$\therefore$ They were sure of the address.
Use truth tables to determine whether the argument forms in 6-11 are valid.
Indicate which columns represent the premises and which represent the
conclusion, and include a sentence explaining how the truth table supports your
answer. Your explanation should show that you understand what it means for a
form of argument to be valid or invalid.
6.
$$
p \to q \\
q \to p \\
\therefore p \vee q
$$
7.
$$
p \\
p \to q \\
\neg q \vee r \\
\therefore r
$$
8.
$$
p \vee q \\
p \to \neg q \\
p \to r \\
\therefore r
$$
9.
$$
p \wedge q \to \neg r \\
p \vee \neg q \\
\neg q \to p \\
\therefore \neg r
$$
10.
$$
p \vee q \to r \\
\therefore \neg r \to \neg p \wedge \neg q
$$
(This is the form of argument shown on pages 37 and 38.)
11.
$$
p \to q \vee r \\
\neg q \vee \neg r \\
\therefore \neg p \vee \neg r
$$
12. Use truth tables to show that the following forms of argument are invalid.
a.
$$
p \to q \\
q \\
\therefore p \\
\text{converse error}
$$
b.
$$
p \to q \\
\neg p \\
\therefore \neg q \\
\text{inverse error}
$$
Use truth tables to show that the argument forms referred to in 13-21 are valid.
Indicate which columns represent the premises and which represent the
conclusion, and include a sentence explaining how the truth table supports your
answer. Your explanation should show that you understand what it means for a
form of argument to be valid.
13. Modus tollens:
$$
p \to q \\
\neg q
\therefore \neg p
$$
14. Example 2.3.3(a)
15. Example 2.3.3(b)
16. Example 2.3.4(a)
17. Example 2.3.4(b)
18. Example 2.3.5(a)
19. Example 2.3.5(b)
20. Example 2.3.6
21 Example 2.3.7
Use symbols to write the logical form of each argument in 22 and 23, and then
use a truth table to test the argument for validity. Indicate which columns
represent the premises and which represent the conclusion, and include a few
words of explanation showing that you understand the meaning of validity.
22.
If Tom is not on team $A$, then Hua is on team $B$.
If Hua is not on team $B$, then Tom is on team $A$.
$\therefore$ Tom is not on team $a$ or Hua is not on team $B$.
23.
Oleg is a math major or Oleg is an economics major.
If Oleg is a math major, then Oleg is required to take Math 362.
$\therefore$ Oleg is an economics major or Oleg is not required to take
Math 362.
Some of the arguments in 24-32 are valid, whereas others exhibit the converse or
the inverse error. Use symbols to write the logical form of each argument. If
the argument is valid, identify the rule of inference that guarantees its
validity. Otherwise, state whether the converse or the inverse error is made.
24.
If Jules solved this problem correctly, then Jules obtained the answer $2$.
Jules obtained the answer $2$.
$\therefore$ Jules solved this problem correctly.
25.
This real number is rational or it is irrational.
This real number is not rational.
$\therefore$ This real number is irrational.
26.
If I go to the movies, I won't finish my homework.
If i don't finish my homework, I won't do well on the exam tomorrow.
$\therefore$ If I go to the movies, I won't do well on the exam tomorrow.
27.
If this number is larger than $2$, then its square is larger than $4$.
This number is not larger than $2$.
$\therefore$ The square of this number is not larger than $4$.
28.
If there are as many rational numbers as there are irrational numbers, then the
set of all irrational numbers is infinite.
The set of all irrational numbers is infinite.
$\therefore$ There are as many rational numbers as there are irrational numbers.
29.
If at least one of these two numbers is divisible by $6$, then the product of
these two numbers is divisible by $6$.
Neither of these two numbers is divisible by $6$.
$\therefore$ The product of these two numbers is not divisible by $6$.
30.
If this computer program is correct, then it produces the correct output when
run with the test data my teacher gave me.
This computer program produces the correct output when run with the test data my
teacher gave me.
$\therefore$ This computer program is correct.
31.
Sandra knows Java and Sandra knows C++.
$\therefore$ Sandra knows C++.
32.
If I get a Christmas bonus, I'll buy a stereo.
If I sell my motorcycle, I'll buy a stereo.
$\therefore$ If I get a Christmas bonus or I sell my motorcycle, then I'll buy a
stereo.
33. Give an example (other than Example 2.3.11) of a valid argument with a false
conclusion.
34. Give an example (other than Example 2.3.12) of an invalid argument with a
true conclusion.
35. Explain in your own words what distinguishes a valid form of argument from
an invalid one.
36. Given the following information about a computer program, find the mistake
in the program.
a. There is an undeclared variable or there is a syntax error in the first five
lines.
b. If there is a syntax error in the first five lines, then there is a missing
semicolon or a variable name misspelled.
c. There is not a missing semicolon.
d. There is not a misspelled variable name.
37. In the back of an old cupboard you discover a note signed by a pirate famous
for his bizarre sense of humor and love of logical puzzles. In the note he
wrote that he had hidden treasure somewhere on the property. He listed five
true statements (a-e below) and challenged the reader to use them to figure
out the location of the treasure.
a. If this house is next to a lake, then the treasure is not in the kitchen.
b. If the tree in the front yard is an elm, then the treasure is in the kitchen.
c. This house is next to a lake.
d. The tree in the front yard is an elm or the treasure is buried under the
flagpole.
e. If the tree in the back yard is an oak, then the treasure is in the garage.
Where is the treasure hidden?
38. You are visiting the island described in Example 2.3.14 and have the
following encounters with natives.
a. Two natives _A_ and _B_ address you as follows:
_A_ says: Both of us are knights.
_B_ says: _A_ is a knave.
What are _A_ and _B_?
b. Another two natives _C_ and _D_ approach you but only _C_ speaks.
_C_ says: Both of us are knaves.
What are _C_ and _D_?
c. You then encounter natives _E_ and _F_.
_E_ says: _F_ is a knave.
_F_ says: _E_ is a knave.
How many knaves are there?
d. Finally, you meet a group of six natives, _U_, _V_, _W_, _X_, _Y_, and _Z_,
who speak to you as follows:
_U_ says: None of us is a knight.
_V_ says: At least three of us are knights.
_W_ Says: At most three of us are knights.
_X_ says: Exactly five of us are knights.
_Y_ says: Exactly two of us are knights.
_Z_ says: Exactly one of us is a knight.
Which are knights and which are knaves?
39. The famous detective Percule Hoirot was called in to solve a baffling murder
mystery. He determined the following facts:
a. Lord Hazelton, the murdered man, was killed by a blow on the head with a
brass candlestick.
b. Either Lady Hazelton or a maid, Sara, was in the dining room at the time of
the murder.
c. If the cook was in the kitchen at the time of the murder, then the butler
killed Lord Hazelton with a false dose of strychnine.
d. If Lady Hazelton was in the dining room at the time of the murder, then the
chauffeur killed Lord Hazelton.
e. If the cook was not in the kitchen at the time of the murder, then Sara was
not in the dining room when the murder was committed.
f. If Sara was in the dining room at the time the murder was committed, then the
wine steward killed Lord Hazelton.
Is it possible for the detective to deduce the identity of the murderer from
these facts? If so, who did murder Lord Hazelton? (Assume there was only one
cause of death.)
40 Sharky, a leader of the underworld, was killed by one of his own band of four
henchmen. Detective Sharp interviewed the men and determined that all were lying
except for one. He deduced who killed Sharky on the basis of the following
statements:
a. Socko: Lefty killed Sharky.
b. Fats: Muscles didn't kill Sharky.
c. Lefty: Muscles was shooting craps with Socko when Sharky was knocked off.
d. Muscles: Lefty didn't kill Sharky.
Who did kill Sharky?
In 41-44 a set of premises and a conclusion are given. Use the valid argument
forms listed in Table 2.3.1 to deduce the conclusion from the premises, giving a
reason for each step as in Example 2.3.8. Assume all variables are statement
variables.
41.
a. $\neg p \vee q \to r$
b. $s \vee \neg q$
c. $\neg t$
d. $p \to t$
e. $\neg p \wedge r \to \neg s$
f. $\therefore \neg q$
42.
a. $p \vee q$
b. $q \to r$
c. $p \wedge s \to t$
d. $\neg r$
e. $\neg q \to u \wedge s$
f. $\therefore t$
43.
a. $\neg p \to r \wedge \neg s$
b. $t \to s$
c. $u \to \neg p$
d. $\neg w$
e. $u \vee w$
f. $\therefore \neg t$
44.
a. $p \to q$
b. $r \vee s$
c. $\neg s \to \neg t$
d. $\neg q \vee s$
e. $\neg s$
f. $\neg p \wedge r \to u$
g. $w \vee t$
h. $\therefore u \wedge w$

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chapter_2/notes.md
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@ -249,3 +249,56 @@ If $r$ and $s$ are statements:
$r$ is a **sufficient condition** for $s$ means "if $r$ then $s$."
$r$ is a **necessarily condition** for $s$ means "if not $r$, then not $s$."
---
Page 89
**Definition**
An **argument is a sequence of statements, and an **argument form** is a
sequence of statement forms. All statements in an argument and all statement
forms in an argument form, except for the final one, are called **premises** (or
**assumptions** or **hypotheses**). The final statement or statement form is
called the **conclusion**. The symbol $\therefore$, which is read "therefore,"
is normally placed just before the conclusion.
To say that an _argument form_ is **valid** means that no matter what particular
statements are substituted for the statement variables in its premises, if the
resulting premises are all true, then the conclusion is also true. To say that
an _argument_ is **valid** means that its form is valid.
---
Page 90
**testing an Argument for Validity**
1. Identify the premises and conclusion of the argument form.
2. Construct a truth table showing the truth values of all the premises and the
conclusion.
3. A row of the truth table in which all the premises are true is called a
**critical row**. If there is a critical row in which the conclusion is
false, then it is possible for an argument of the given form to have true
premises and a false conclusion, and so the argument form is invalid. If the
conclusion in _every_ critical row is true, then the argument form is valid.
---
Page 97
**Definition**
An argument is called **sound** if, and only if, it is valid _and_ all its
premises are true. An argument that is not sound is called **unsound**.
---
Page 97
**Contradiction Rule**
If you can show that the supposition that statement $p$ is false leads logically
to a contradiction, then you can conclude that $pr is true.

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chapter_2/test_yourself.md
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@ -116,3 +116,24 @@ $S$; $R$
**Solution**
$R$; $S$
---
**Test Yourself**
Page 99
1. For an argument to be valid means that every argument of the same form whose
premises _______ has a _______ conclusion.
are all true; true
2. For an argument to be invalid means that there is an argument of the same
form whose premises _______ and whose conclusion _______.
are all true; is false
3. For an argument to be sound means that it is _______ and its premises
_______. In this case we can be sure that its conclusion _______.
valid; are all true; is true