Either arithmetic and logic are both necessary or logic and geometry are both necessary. If all knowledge is experiential, then even logic is not necessary. So, not all knowledge is experiential.
-Consider assuming 'E' for an indirect proof of the given argument. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for indirect proof?
A) L
B) ∼L
C) L $ \lor $ (A • G)
D) L • (A $ \lor $ G)
E) L • (A • G)

Question

Quizplus · Accepted Answer

Assuming 'E', we can use indirect proof to show that not all knowledge is experiential. The first premise states that either arithmetic and logic are both necessary or logic and geometry are both necessary. This can be represented as (A&L)∨(L&G). However, since we are assuming that all knowledge is experiential, we know that even logic is not necessary. This means that we can eliminate the possibility of (A&L) being necessary, and are left with L&G as the only necessary knowledge, or simply L. Thus, the immediate consequence of the premises with assumption 'E' is ∼L.

Either Arithmetic and Logic Are Both Necessary or Logic and Geometry