For the following argument, a) Translate into standard form (if not already in standard form), b) symbolize to simplify, and then c) determine whether the syllogism is valid or invalid, explaining why by reference to rules, AND by showing the appropriate Venn diagram:
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.
a) To translate the argument into standard form, we need to identify the premises and the conclusion:
Premise 1: Only people who have lost a job can appreciate the difficulties the loss can cause.
Premise 2: Some of the people in this room have lost jobs in the past.
Conclusion: There are people in this room who can appreciate the resulting troubles.
b) To symbolize and simplify the argument, let's assign variables to each component and use logical operators to represent the relationships:
Let:
P: People who have lost a job
Q: Appreciation of difficulties
R: People in this room
Premise 1: Only P → Q
Premise 2: R → P
Conclusion: R → Q
By simplifying the premises, we can see that if someone is in this room (R), then they have lost a job (P), and by extension, they can appreciate the difficulties (Q).
c) To determine the validity of the syllogism, we can analyze the argument using the rules of deduction.
The argument follows the form of a valid syllogism known as the "Barbara" syllogism, which states:
All S is P.
All M is S.
Therefore, all M is P.
In this case:
All people in this room (R) are people who have lost a job (P).
Some people who have lost a job (P) can appreciate the difficulties (Q).
Therefore, we can conclude that some people in this room (R) can appreciate the difficulties (Q).
To further illustrate this, we can create a Venn diagram:
P
/ |
Q --- R
In the diagram, the circle labeled "P" represents people who have lost a job, the circle labeled "Q" represents individuals who can appreciate difficulties, and the region where the two circles overlap represents the people in this room who fulfill both conditions (R, P), indicating that they can appreciate the difficulties caused by job loss (Q).