Determine the validity of the next arguments by using Euler circles, then translate the statements into logical statements using basic connectives, and using truth tables, determine the validity of the arguments. Compare your answers.

(a)No A is B (b) All B is A
Some C is A All C is A
____________ _____________

Some C is not B All C is B

Hi,

1)

The area is part of C and not part of B hence the statement some C is not B is valid

Let r be "The object belongs to set C"

(p=>~q)Λ(pΛr) => (rΛ~q)

The statement is valid because in all cases the statment is true (note that all Ts in the final column)

2)


The diagram shows the arguement is not valid since the diagram satisfies all B is A and all C is A without the conclusion all C is B (in fact you couldn't concluded even some C is B as the diagram shows that it is possible that no C is B)

In statement form we have

(q=>p)Λ(r=>p) => (r=>q)

graphic
View Full Image

As you can see there is a case when the staement isn't true (counterexample I guess). This is when an element was a member of A and C but not B. This case didn't contradict either of the statements All B is A and All C is A but it did contradict the conclusion that all C is B.

I hope this has helped. Any questions then please ask. All the best,
justanswer dot com/questions/1s1wr-truth-tables-are-related-to-euler-circles-arguments-in-the#ixzz0osmCruQD

To determine the validity of the arguments using Euler circles, we need to draw circles representing each category mentioned in the statements and analyze the relationships between them.

(a) No A is B:
Draw two circles representing categories A and B. Since there is no overlap between the circles, we conclude that there is no "A" that is also "B." This statement is valid.

Some C is A:
Add a circle representing category C and draw an overlapping region with circle A indicating that there is some commonality between C and A. This statement is valid.

Therefore, the argument is valid:

No A is B
Some C is A
____________
Some C is not B

(b) All B is A:
Draw two circles representing categories A and B. Since all of B falls within A, this statement is valid.

All C is A:
Add a circle representing category C and draw it completely within circle A indicating that all of C is part of A. This statement is valid.

Therefore, the argument is valid:

All B is A
All C is A
_____________
All C is B

To translate the statements into logical statements using basic connectives, we can use the following symbols:

- No A is B: ¬(A ∧ B)
- Some C is A: C ∧ A
- Some C is not B: C ∧ ¬B
- All B is A: B → A
- All C is A: C → A
- All C is B: C → B

To determine the validity of the arguments using truth tables, we need to analyze all possible combinations of truth values for the statements involved.

(a) Argument:
No A is B
Some C is A
____________
Some C is not B

By translating the statements into logical statements, the argument becomes:

¬(A ∧ B)
C ∧ A
__________
C ∧ ¬B

Using truth tables, we can evaluate 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 | F |
| T | F | T | T | T | T |
| T | F | F | T | F | F |
| F | T | T | T | F | T |
| F | T | F | T | F | T |
| F | F | T | T | F | T |
| F | F | F | T | F | T |

Since there are cases where the premises are true, but the conclusion is false (e.g., the first row of the truth table), the argument is invalid.

(b) Argument:
All B is A
All C is A
_____________
All C is B

By translating the statements into logical statements, the argument becomes:

B → A
C → A
__________
C → B

Using truth tables, we can evaluate the validity of the argument:

| A | B | C | B → A | C → A | C → B |
|---|---|---|-------|-------|-------|
| T | T | T | T | T | T |
| T | T | F | T | T | T |
| T | F | T | T | T | F |
| T | F | F | T | T | T |
| F | T | T | T | F | T |
| F | T | F | T | F | T |
| F | F | T | T | F | T |
| F | F | F | T | F | T |

In this case, all rows of the truth table result in true conclusions. Therefore, the argument is valid.

Comparing the answers:
- Using Euler circles:
- (a) The argument is valid.
- (b) The argument is also valid.
- Using logical statements and truth tables:
- (a) The argument is invalid.
- (b) The argument is valid.

There is a discrepancy between the answers obtained using Euler circles and logical statements with truth tables for argument (a). This discrepancy could be due to the limitations of representing complex relationships using Euler circles, whereas logical statements and truth tables provide a more precise analysis.