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 analyze the given arguments and determine their validity or invalidity, we need to follow the steps below:
Argument 1:
a) Translate into standard form (if not already in standard form):
- All politicians are thieves.
- No students are thieves.
- No students are politicians.
b) Symbolize to simplify:
- Let P represent politicians.
- Let T represent thieves.
- Let S represent students.
The argument in standard form becomes:
- All P are T.
- No S are T.
- No S are P.
c) Determine the validity using Venn diagrams and rules:
To represent the argument using Venn diagrams, we draw intersecting circles to represent the categories mentioned: P for politicians, T for thieves, and S for students.
In this argument, the first premise states that "All politicians are thieves." This means that we can shade the region where the circle for politicians (P) overlaps with the circle for thieves (T).
The second premise states that "No students are thieves." Therefore, we can shade the region where the circle for students (S) intersects with the circle for thieves (T), indicating that they have no overlap.
Finally, the conclusion states that "No students are politicians." Since we already shaded the intersection between the circles for S and T, the remaining part of the circle for students (S) represents the region that is not politicians (P).
By observing the Venn diagram, we can see that there is no overlap between the circles for S and P, supporting the conclusion of the argument. Thus, the argument is valid.
Argument 2:
a) Translate into standard form (if not already in standard form):
- Only people who have lost a job can appreciate the difficulties the loss can cause.
- Some people in this room have lost jobs in the past.
- Therefore, there are people in this room who can appreciate the resulting troubles.
b) Symbolize to simplify:
- Let J represent people who have lost a job.
- Let A represent people who can appreciate the difficulties of job loss.
- Let R represent people in this room.
The argument in standard form becomes:
- Only J can A.
- Some R are J.
- Therefore, some R are A.
c) Determine validity using Venn diagrams and rules:
Again, we will represent the argument using Venn diagrams, with circles for J, A, and R.
The first premise states that "Only people who have lost a job can appreciate the difficulties the loss can cause." In the Venn diagram, this means that the circle for J should be entirely contained within the circle for A.
The second premise states that "Some people in this room have lost jobs in the past." This suggests that there should be an overlap between the circles for R and J.
The conclusion states that "Therefore, some people in this room are those who can appreciate the resulting troubles." This implies that there should be an overlap between the circles for R and A.
By examining the Venn diagram, we can see that the overlap between the circles for R and A exists, supporting the conclusion of the argument. Therefore, the argument is valid.
Remember that the validity of an argument is determined by whether the conclusion follows logically from the premises and the relationships between the categories as shown in the Venn diagram.