Determine the validity of the next arguments by using Euler circles, then translate the statements into logical statements using the basic connectives, and using truth tables, determine the validity of the arguments. Compare your answers.
(a No A is B
Some C is A
Some C is not B
(b) All B is A.
All C is A.
All C is B
http://plato.stanford.edu/entries/diagrams/
To determine the validity of the arguments using Euler circles, we can visually represent the statements using circles and determine if the statements are consistent with each other.
(a) Argument:
1. No A is B
2. Some C is A
3. Some C is not B
Using Euler circles, we can draw two circles representing sets A and B. The first statement says that there is no overlap between A and B. The second statement says that there is some overlap between C and A. The third statement says that there is some overlap between C and the complement of B (the area outside B).
Based on the Euler circles, it is possible to satisfy all three statements simultaneously. So, the argument is valid using Euler circles.
To translate the statements into logical statements using the basic connectives, we can assign variables to the terms:
A: A is true
B: B is true
C: C is true
Using the basic connectives, we can rewrite the statements as:
1. ¬A ∧ B
2. C ∧ A
3. C ∧ ¬B
We can now create a truth table to determine the validity of the argument:
| A | B | C | ¬A ∧ B | C ∧ A | C ∧ ¬B |
|---|---|---|--------|-------|--------|
| T | T | T | F | T | F |
| T | T | F | F | F | T |
| T | F | T | T | T | T |
| T | F | F | T | F | T |
| F | T | T | T | F | F |
| F | T | F | T | F | T |
| F | F | T | T | F | T |
| F | F | F | T | F | T |
By checking all possible combinations of truth values for the statements, we can see that there are some cases where all three statements are true simultaneously. Therefore, the argument is valid using truth tables as well.
(b) Argument:
1. All B is A.
2. All C is A.
3. All C is B.
Using Euler circles, we can draw two circles representing sets A and B. The first statement says that all of B is contained within A. The second statement says that all of C is contained within A. The third statement says that all of C is contained within B.
Based on the Euler circles, it is possible to satisfy all three statements simultaneously. So, the argument is valid using Euler circles.
To translate the statements into logical statements using the basic connectives, we can assign variables to the terms:
A: A is true
B: B is true
C: C is true
Using the basic connectives, we can rewrite the statements as:
1. A → B
2. A → C
3. B → C
We can now create a truth table to determine the validity of the argument:
| A | B | C | A → B | A → C | B → C |
|---|---|---|-------|-------|-------|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | T | F | T | T | F |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
By checking all possible combinations of truth values for the statements, we can see that there is at least one case where all three statements are true simultaneously. Therefore, the argument is valid using truth tables as well.
In both cases, the arguments are valid using both Euler circles and truth tables.