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(4) Solve the validity of the following arguments using Venn diagrams:
[Exercise 36]"No empiricist is a rationalist. Positivists are empiricists. Therefore, no positivist is a rationalist.".
Base sets:
E={x/x is an Empiricist}
R={x/x is a rationalist}
P={x/x is a positivist}
Formalization:
1. E ∩ R= Ø
2. P⊆E
Combining both:
|= P ∩ R= Ø
Solution: the argument is valid since all the information contained in the conclusion is contained in the premises.
2. [Exercise 37]"Some mathematicians are rigorous. Some mathematicians make calculation errors. All mathematicians who make calculation errors are not rigorous. Therefore, all rigorous mathematicians do not make calculation errors."
3. [Exercise 38]"There are agnostic believers and non-agnostic believers. No atheist is a believer. All agnostics are atheists. Therefore, some atheist is neither a believer nor agnostic."
Base sets:
C={x/x is a believer}
G={x/x is agnostic}
A={x/x is atheist}
Formalization:
1. C ∩ G ≠ Ø
2. C - G ≠ Ø
3. A ∩ C = Ø
3. G - A = Ø
Combining the premises we get the following representation:
|= A - (C ∩ G) ≠ Ø
Solution: the argument is not valid since all the information contained in the conclusion is not contained in the premises.
4. [Exercise 39]"All dancers are egocentric. Some egocentrics like being watched, although there are others who don't. Those who like it are dancers and those who don't are too. Therefore, all egocentrics are dancers."
5. [Exercise 40]"Philosophers are lovers of wisdom. Some lovers of wisdom pursue the good. Therefore, some philosophers pursue the good."
Base sets:
F={x/x is a Philosopher}
A={x/x is a lover of wisdom}
P={x/x pursues the good}
Formalization:
1. F - A= Ø
2. A ∩ P ≠ Ø
Combining both propositions:
|= F ∩ P≠ Ø
Solution: the argument is not valid since all the information contained in the conclusion is not contained in the premises.