Determine the validity of the next arguements 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. Com 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
I cant draw circles here for you. See this reference.
http://plato.stanford.edu/entries/diagrams/
To determine the validity of the arguments using Euler Circles, we can follow these steps:
1. Draw two overlapping circles to represent the two terms in each argument.
(a) For argument (a), label one circle as "A" and the other as "B".
(b) For argument (b), label one circle as "B" and the other as "A".
2. Based on the statements provided, place the information inside the circles to represent the relationships between the terms.
(a) In argument (a), "No A is B" means that there is no overlap between circles A and B. "Some C is A" indicates that there is an overlapping region between circles A and C.
C A
| /
| /
-----
(b) In argument (b), "All B is A" means that circle B is completely contained within circle A. "All C is A" indicates that circle C is also completely contained within circle A.
A
/|\
/ | \
B | C
3. Check if the conclusion of each argument is valid based on the Euler Circles.
(a) For argument (a), "Some C is not B" is true since there is an overlapping region between circles C and A, but no overlap between circles C and B. Thus, the conclusion is valid.
(b) For argument (b), the conclusion is not provided.
Now, let's translate the statements into logical statements using basic connectives for argument (a) and evaluate the validity using truth tables:
(a)
1. No A is B - Translated as "For all x, if x is A, then x is not B"
2. Some C is A - Translated as "There exists an x such that x is C and x is A"
3. Some C is not B - Translated as "There exists an x such that x is C and x is not B"
Truth table for argument (a):
| A | B | C | No A is B | Some C is A | Some C is not B |
|---|---|---|-----------|-------------|-----------------|
| T | T | T | F | F | F |
| T | T | F | F | F | F |
| T | F | T | T | T | T |
| T | F | F | T | T | T |
| F | T | T | F | F | F |
| F | T | F | F | F | F |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
Since there are cases where assumption statements are true and the conclusion statement is false, the argument is invalid.
I hope this helps!