Introduction to Logic - Ch. 2

(Originally written September 5, 2006 in Book 9)

Introduction to Logic
Harry J. Gensler

Ch. 2 - Syllogistic Logic

Syllogistic logic - arguments whose validity depends on "all", "no", "some" and similar notions.

Aristotle developed this first type of logical reasoning.

Easier Translations:

English Language:
All logicians are charming.
Gensler is a logician.
Therefore, Gensler is charming.

Syllogistic Language:
all L is C
g is L.
Therefore, g is C.

Capital letters are used for categories. Small letters are used for particulars.

WFFs: Well-formed formulas - grammatical sentences having one of eight forms:
-all A is B
-some A is B
-x is A
-x is y
-no A is B
-some A is not-B
-x is not-A
-x is not-y

WFFs begin gin with a word (not a letter) use two capital letters:
(correct) all L is C
(incorrect) all l is c

WFFs beginning with a letter (not a word) begin with a small letter;
(correct) g is L
(incorrect) G is L

Use capital letters for general terms (terms that describe or put in a category)
B - a cute baby
C - charming
D - drives a Buick

Use small letters for singular terms (particulars)
b - the world's cutest baby
c - this child
d - david

English Language:
1) Will Gensler is a cute baby
2) Will Gensler is the world's cutest baby

Logic Language:
1) w is B
2) w is b

Syllogistic WFFs all have the verb 'is'. English sentences with a different verb should be altered to make 'is' the verb.
ex. 1 All dogs bark - All dogs is [are] barkers - All D is B
ex. 2 Al left the room - Al is a person who left the room - a is L.

Quiz: which of the following are WFFs?
1) no e is f (no)
2) g is H (yes)
3) J is K (no)
4) all M is not-Q (yes)
5) some L is m (no)
6) p is not-Q (yes)
7) R is not-S (yes)
8) not all T is U (yes)
9) some X is not-Y (yes)

Quiz: Translate these English sentences into WFFs
1) This is a sentence: t is S
2) This isn't the first sentence: t is not-f
3) No logical positivists believes in God. no L is G.
4) The book on your desk is green: b is G
5) All dogs hate cats: all D is H
6) Kant is the greatest philosopher: k is g
7) Ralph was born in Detroit: r is d
8) Detroit is the birthplace of ralph: d is r
9) Alaska is a state: a is S
10) Alaska is the biggest state: a is b
11) Carol is my only sister: c is s
12) Carol lives in Big Pine Key: c is b
13) The idea of goodness is itself good: i is G
14) All Michigan players are intelligent: all m is I
15) Michigan's team is awesome: m is A

The Star Test

A syllogism is an argument using syllogistic WFFs.

Syllogism - a vertical sequence of one or more WFFs in which each letter occurs twice and the letters 'form a chain'.

no P is B
some C is B
Therefore, some C is not P

A letter is distributed in a WFF if it occurs just after 'all' or anywhere after 'no' or 'not'.

The star test for syllogisms:

"Star the distributed letters in the premises and undistributed letters in the conclusion. Then the syllogism is VALID if and only if every capital letter is starred exactly once and there is exactly one star on the right-hand side" (Gensler, 10).

A valid argument must satisfy two conditions:
1) Each capital letter is starred in one and only one occurrence.
2) Exactly one right-hand letter (letter after 'is' or 'is not) is starred.

example:
all A* is B
some C is A
Therefore, some C* is B*

Gensler is a logician: g is L
Gensler is mean: g is M
Therefore, some logicians are mean: Therefore, some L* is M*

Quiz: Which of these are syllogisms?
1) all C is D
Therefore, some C is not-E
No
2) g is not l
Therefore, l is not g.
Yes
3) no Y is E
all G is Y
Therefore, no Y is E
no.
4) Therefore, all S is S
yes.
5) K is not L
all M is L
some N is M
Z is N (not a WFF)
Therefore, K is not Z
No.

Quiz: Underline the distributed letters in the following WFFs
1) w is not s
2) some C is B
3) No R is S
4) a is C
5) all P is B
6) r is not D
7) s is w
8) Some C is not P

Quiz: Valid or invalid?
1) No P is B
some C is not B
Therefore some C is P
Valid

2) x is W
x is not Y
Therefore, some W is not Y
invalid

3) no H is B
no H is D
Therefore, some B is not D
invalid

4) some J is not P
all J is F
therefore, some F is not P

5) g is not s
therefore s is not g

6) all L is M
g is L
therefore g is M

7) all L is M
g is not L
therefore, g is not M

8) some N is T
some C is not T
therefore, some N is not C

9) all C is K
S is K
Therefore, s is C

10) all D is A
therefore, all A is D

11) S is C
S is H
Therefore, some C is H

12) Some C is H
Therefore, some C is not H

13) a is b
b is c
c is d
Therefore a is d

14) no A is B
some B is C
some D is E
all D is E
therefore, some E is A

I must admit I am completely at a loss with the star test...

Quiz 2.3a: Using intuition
1) Valid 2)invalid 3)valid 4)valid 5)valid
6)valid 7)invalid 8)valid 9)valid 10)invalid
11)valid 12)valid 13)invalid 14)invalid 15)valid
16)valid 17)valid 18)valid 19)valid 20)invalid
21)valid 22)invalid 23)valid 24)valid 25) valid

c is H
c is not S
all I is S
c is I
Therefore, some H is not S

all B is CE
all CE is CS
no F is CS
no F is B

all B is CS

2.3b pg. 16

a- Alice
b-Bob
c-Carol
d-David
g-George
L-loves money
r-richest person
K-knew where the cash was
H-works for Herman
n-nastiest person
W-who stole
Z-hate Herman

1. a is not L
2. b is L
3. b is Not-r
4. c is K
5. d is H
6. d is not-n
7. all W is L
8. r is not W
9. all W is K
10. all H is Z
11. all Z is W
12. n is W

Arg. 1 Did Bob steal the money?
-Bob loves money (b is L) #2
-Bob is not the richest (b is not-r)#3
The richest person didn't seal (r is not-W)#8
-all who stole love money (W is L)
We cannot prove with data

Arg. 2 Did Carol steak?
-c is K Carol knew where it was
-all W is K (whoever sole knew where it was)
Can't prove

Arg. 3 Did David steal?
d is H (David works for Herman)
all H is Z (all who work for Herman hate Herman)
all Z is W (all who hate Herman stole)
Therefore, d is W (David Stole)

Arg. 4 Did more than one person steal?
yes
d is W
d is not-n
N is W
Therefore, at least n and d is W

Arg. 5 it would prove that one of the premises was false.