Thread: Determine if valid or invalid by use of truth tables, standard forms...

Determine if valid or invalid by use of truth tables, standard forms, Euler diagrams or logical manipulation
" no odd number has a factor of 2. some prime numbers are odd. therefore some prime numbers have a factor of 2"

P-it is odd
q-it has a factor of 2
r-it is prime

p \to not q
r \to p
therefore r and q

or
p and not q
r and p
therefore r and q

i'm not sure which is right and i'm not sure how to prove it. please help!

This is useful for truth tables.
You can change the input to do the others.

P-it is odd
q-it has a factor of 2
r-it is prime

p \to not q
r \to p
therefore r and q

or
p and not q
r and p
therefore r and q

The first assumption should be an implication and the second one should be a conjunction. However, these propositions must be encoded in predicate logic and not in propositional logic. First, please allow me to redefine notation.

O(n) - n is odd
F2(n) - n has a factor of 2
P(n) - n is prime

Then the propositions are written as follows.

∀n (O(n) -> ~F2(n))
∃n (P(n) /\ O(n))
--------------------
∃n (P(n) /\ F2(n))

I assume that you need to determine whether this argument is valid in virtue of its syntactic form only, i.e., for any interpretation of O(n), F2(n) and P(n). (If we fix the interpretation above, then the argument is valid simply because the conclusion is true: there exists n = 2 that is prime and has a factor of 2.)

Consider some interpretation where F2(n) is always false, e.g., F2(n) is n < n, and let interpretations of P and O be as above. What can you cay about the truth of the two assumptions and the conclusion?

Note, on the other hand, that the two premises do imply ∃n (P(n) /\ ~F2(n)).