For the following arguments, a) Translate into standard form (if not already in standard form), b) symbolize to simplify, and then c) determine whether the syllogisms are valid or invalid, explaining why by reference to rules, AND by showing the appropriate Venn diagram:
Arg
All politicians are thieves
No students are thieves
No students are politicians
Arg
Only people who have lost a job can appreciate the difficulties the loss can cause. Some of the people in this room have lost jobs in the past, so there are people in this room who can appreciate the resulting troubles.
To answer these questions, we will follow the steps for each argument:
Argument 1:
a) Translate into standard form (if not already in standard form):
Premises:
1. All politicians are thieves.
2. No students are thieves.
Conclusion:
3. No students are politicians.
b) Symbolize to simplify:
Let's assign variables to each term to symbolize the statements:
P = politicians
T = thieves
S = students
Premises:
1. All politicians are thieves. - This can be symbolized as "All P are T."
2. No students are thieves. - This can be symbolized as "No S are T."
Conclusion:
3. No students are politicians. - This can be symbolized as "No S are P."
c) Determine whether the syllogism is valid or invalid:
To determine the validity of the syllogism, we can construct a Venn diagram:
___________________________
P | |
T | |
S | |
___________________________
Premise 1: All P are T.
___________________________
P | X |
T | X |
S | |
___________________________
Premise 2: No S are T.
___________________________
P | X |
T | X |
S | X |
___________________________
Conclusion: No S are P.
___________________________
P | X |
T | X |
S | X |
___________________________
Based on the Venn diagram, we can see that the conclusion is valid because there is no overlapping area between the student and politician circles.
Argument 2:
a) Translate into standard form (if not already in standard form):
Premises:
1. Only people who have lost a job can appreciate the difficulties the loss can cause.
2. Some of the people in this room have lost jobs in the past.
Conclusion:
3. There are people in this room who can appreciate the resulting troubles.
b) Symbolize to simplify:
Let's assign variables to each term to symbolize the statements:
P = people who have lost a job
A = people who can appreciate difficulties
R = people in this room
Premises:
1. Only people who have lost a job can appreciate the difficulties the loss can cause. - This can be symbolized as "All P are A."
2. Some of the people in this room have lost jobs in the past. - This can be symbolized as "Some R are P."
Conclusion:
3. There are people in this room who can appreciate the resulting troubles. - This can be symbolized as "Some R are A."
c) Determine whether the syllogism is valid or invalid:
To determine the validity of the syllogism, we can construct a Venn diagram:
___________________________
P | |
A | |
R | |
___________________________
Premise 1: All P are A.
___________________________
P | X |
A | X |
R | |
___________________________
Premise 2: Some R are P.
___________________________
P | X |
A | X |
R | X |
___________________________
Conclusion: Some R are A.
___________________________
P | X |
A | X |
R | X |
___________________________
Based on the Venn diagram, we can see that the conclusion is valid because there is a shared area between the room and appreciation circles.