Syllogisms

Show your work for each problem.

1.
All rectangles are parallelograms. All squares are rectangles. Therefore all squares are parallelograms. What is the middle term?
1. a)Quadrilaterals
2. b)Rectangles
3. c)Parallelograms
4. d)Squares
Answer: Rectangles
Rectangles appear in both premises and not in the conclusion, so rectangles is the middle term $M$ . Squares is the minor term $S$ and parallelograms is the major term $P$ .
2.
A Venn diagram has three overlapping circles $S$ , $M$ and $P$ . How many regions does it have, counting the region outside all three circles?
Answer: 8
For each circle a region is either inside or outside, so there are $2 \times 2 \times 2 = 8$ regions: 3 regions inside exactly one circle, 3 inside exactly two, 1 inside all three and 1 outside all.
3.
Remember: A = All $S$ are $P$ , E = No $S$ are $P$ , I = Some $S$ are $P$ , O = Some $S$ are not $P$ . Which list gives the codes of these four statements in order? (1) Some rectangles are not squares. (2) All squares are rectangles. (3) Some rectangles are squares. (4) No triangle is a square.
1. a)O, E, I, A
2. b)I, A, O, E
3. c)O, A, I, E
4. d)E, A, I, O
Answer: O, A, I, E
Some rectangles are not squares is O (some, with a not), all squares are rectangles is A, some rectangles are squares is I, and no triangle is a square is E. So the list is O, A, I, E.
4.
All $14$ members of the chess club are also in the math club. The math club has $25$ members. How many math club members are not in the chess club?
Answer: 11
All chess club members lie inside the math club circle, so the math club members outside the chess club are $25 - 14 = 11$ .
5.
No prime number greater than $2$ is even. All numbers in the set $T$ are prime numbers greater than $2$ . Which conclusion follows?
1. a)All even numbers are in $T$
2. b)Nothing follows
3. c)Some numbers in $T$ are even
4. d)No number in $T$ is even
Answer: No number in $T$ is even
With $M$ = primes greater than $2$ , $P$ = even numbers and $S$ = numbers in $T$ , the premises are No $M$ are $P$ and All $S$ are $M$ . This valid form gives No $S$ are $P$ : no number in $T$ is even.
6.
All multiples of $6$ are multiples of $3$ . All multiples of $12$ are multiples of $6$ . Therefore all multiples of $12$ are multiples of $3$ . Name the middle term, then follow the number $60$ (a multiple of $12$ ) through the chain. What is $60 \div 3$ ?
Answer: 20
1. Which letter, $S$ , $M$ or $P$ , stands for the multiples of $6$ ? M
2. What is $60 \div 12$ ? 5
3. What is $60 \div 6$ ? 10
4. What is $60 \div 3$ ? 20
7.
All poets are dreamers. Some dreamers are engineers. Therefore some poets are engineers. How do you judge this argument?
1. a)Valid, but one premise is false
2. b)Valid and sound
3. c)Invalid, because a syllogism may not use the word some
4. d)Invalid: the poets and the engineers may be different dreamers
Answer: Invalid: the poets and the engineers may be different dreamers
The middle term dreamers never covers its whole group: the poets are some of the dreamers, and the engineers may be other dreamers. In the diagram the x for Some dreamers are engineers can sit outside the poets circle, so the argument is invalid (undistributed middle).
8.
A class has $60$ students. No member of the swim team is on the soccer team. The swim team has $16$ members and the soccer team has $22$ . How many students are on neither team?
Answer: 22
No swimmer is a soccer player, so the overlap region is empty and $16 + 22 = 38$ students are on a team. On neither team: $60 - 38 = 22$ .
9.
A class has $32$ students. All $8$ members of the chess club are in the math club, which has $15$ members. No member of the math club is in the drama club, which has $10$ members. How many students are in none of the three clubs?
Answer: 7
1. How many chess club members are in the drama club? 0
2. How many students are in at least one club? 25
3. How many students are in none of the clubs? 7
10.
All multiples of $4$ are even. All multiples of $6$ are even. Therefore all multiples of $6$ are multiples of $4$ . Show that the argument is invalid with a counterexample. What is the smallest positive multiple of $6$ that is not a multiple of $4$ ?
Answer: 6
1. Which letter, $S$ , $M$ or $P$ , stands for the even numbers? M
2. What is the remainder when $6$ is divided by $4$ ? 2
3. What is the smallest positive multiple of $6$ that is not a multiple of $4$ ? 6
11.
All multiples of $4$ are multiples of $8$ . $12$ is a multiple of $4$ . Therefore $12$ is a multiple of $8$ . Which description is correct?
1. a)Valid but not sound: the first premise is false
2. b)Invalid, because the conclusion is false
3. c)Invalid: undistributed middle
4. d)Valid and sound
Answer: Valid but not sound: the first premise is false
The form All $M$ are $P$ , $x$ is an $M$ , so $x$ is a $P$ is valid. But the first premise is false: $4$ is a multiple of $4$ and not of $8$ . So the argument is valid but not sound, and its conclusion is false ( $12 \div 8 = 1.5$ ).
12.
An orchestra has $24$ members. All $9$ students who play the violin are in the orchestra. $4$ of the violinists sing in the choir, and $7$ other orchestra members also sing in the choir. How many orchestra members do not sing in the choir?
Answer: 13
All violinists are in the orchestra and some violinists sing in the choir, so some orchestra members sing in the choir: $4 + 7 = 11$ of them. The rest do not sing: $24 - 11 = 13$ .
13.
Test the syllogism No $M$ are $P$ , Some $S$ are $M$ , so Some $S$ are not $P$ with the 8 region diagram (region 4 = inside $S$ and $M$ only, region 6 = inside $M$ and $P$ only, region 7 = inside all three). What is the code of the conclusion?
Answer: O
1. How many of the 8 regions does No $M$ are $P$ shade? 2
2. The overlap of $S$ and $M$ is regions 4 and 7. How many of them are still unshaded? 1
3. In which region must the x for Some $S$ are $M$ go? Give its number. 4
4. Region 4 is inside $S$ and outside $P$ . Which code (A, E, I or O) does the conclusion Some $S$ are not $P$ have? O
14.
In the three circle Venn diagram for $S$ , $M$ and $P$ , you shade the premises All $S$ are $M$ and No $M$ are $P$ . How many of the 8 regions are shaded in total?
Answer: 4
All $S$ are $M$ shades regions 1 and 5 (inside $S$ , outside $M$ ). No $M$ are $P$ shades regions 6 and 7 (inside $M$ and $P$ ). No region appears in both lists, so $2 + 2 = 4$ regions are shaded. Since regions 5 and 7 are shaded, the diagram also shows No $S$ are $P$ .
15.
The statements $p \to q$ and $q \to r$ are true, and $r$ is false. Which statement must be true?
1. a) $q$
2. b) $p$
3. c) $\neg p$
4. d) $r \to p$
Answer: $\neg p$
The chain rule gives $p \to r$ . Its contrapositive is $\neg r \to \neg p$ . Since $r$ is false, $\neg r$ is true, so $\neg p$ is true. $q$ cannot be true (it would force $r$ ), and $r \to p$ does not follow.
16.
Premises: If $n$ is divisible by $20$ , then $n$ is divisible by $10$ . If $n$ is divisible by $10$ , then the last digit of $n$ is $0$ . If the last digit of $n$ is $0$ , then $n$ is divisible by $5$ . The chain rule gives: if $n$ is divisible by $20$ , then $n$ is divisible by $5$ . Check it with $n = 140$ , then show that the converse does not follow. What is the smallest positive number divisible by $5$ but not by $20$ ?
Answer: 5
1. What is $140 \div 20$ ? 7
2. What is the last digit of $140$ ? 0
3. What is $140 \div 5$ ? 28
4. What is the smallest positive number divisible by $5$ but not by $20$ ? 5
17.
A sports club has $70$ members and $45$ of them swim. All $20$ tennis players are swimmers. $12$ tennis players also run, $8$ swimmers who do not play tennis run, and $15$ members who do not swim run. How many members neither swim nor run?
Answer: 10
1. How many swimmers do not play tennis? 25
2. How many members run in total? 35
3. How many members neither swim nor run? 10
18.
Show that Some $S$ are $M$ , Some $M$ are $P$ , so Some $S$ are $P$ is invalid. Use $S = \{1, 2\}$ , $M = \{2, 3\}$ and $P = \{3, 4\}$ . How many elements do $S$ and $P$ have in common?
Answer: 0
1. How many elements do $S$ and $M$ have in common? 1
2. How many elements do $M$ and $P$ have in common? 1
3. How many elements do $S$ and $P$ have in common? 0