Last night I felt like I messed something up, but I couldn’t figure it out during class and everything seemed correct, so I ignored my intuitions. Bad idea. It dawned on me that we came to the conclusion that the statement in problem 2 on page 127 was consistent after we tested the negation for validity and found it (the negation) invalid, we decided the statement was consistent. So far, so good, but our original intuition was that the statement was inconsistent (which it is). Hmmm, something went badly awry. My intuitions that I had somehow really bollixed things up were right. But what? The symbolization!
I gave you ∃FG. -∃(G ? K) which, negated, is invalid.
The proper symbolization should be ∃FG. -∃(G ? K)-(FJ)
My original, erroneous symbolization of the right hand conjunct says “all who take either Latin or chemistry take either logic or Greek” and not that “all who take either Latin or chemistry take BOTH logic and Greek.
Is ∃FG. -∃(G v K)-(FJ) consistent? Well, let’s negate and test the negation for validity.
-(∃FG. -∃(G v K)-(FJ))
-∃FG v ∃(G v K)-(FJ)
∃FG → ∃(G v K)-(FJ) Now we have an existential conditional that can be tested directly for validity
Does FG imply (G v K)-(FJ)?
FG → (G v K)-(FJ)
TTTT → (T v ?)-(T⊥)
T YES, so the existential conditional is VALID which means the original is INCONSISTENT
Sorry ‘bout that.