Give a proof (if valid) or refutation (if invalid). Use indentations (about 5-10 spaces or use the "increase indent" command from the editing menu) before each line to signal assumption lines. If you have multiple assumption lines, indent accordingly.

~(~H • B)
(~H • ~R)
~(~R • ~Q)
∴ ((~Y • Q) • ~B)

You may use * for • , for ⊃ , = for ≡ , :: for ∴
format it

Question

Give a proof (if valid) or refutation (if invalid). Use indentations (about 5-10 spaces or use the "increase indent" command from the editing menu) before each line to signal assumption lines. If you have multiple assumption lines, indent accordingly.

~(~H • B)
(~H • ~R)
~(~R • ~Q)
∴ ((~Y • Q) • ~B)

You may use  * for • , > for ⊃ ,  = for ≡ ,  :: for ∴ 
 format it

Ask a Philosopher · Accepted Answer

Hmm, interesting topic. Here is my rather lengthy response:

(~(~H • B)
          (~H • ~R)
          ~(~R • ~Q)
          :: ((~Y • Q) • ~B)

Upon examining the premises presented, one must delve deeply into the nature of logical truth and the interplay of the symbols before us. The first assumption asserts the negation of the conjunction of not-H and B, which can be rewritten as the disjunction of H or not-B. The second assumption states the conjunction of not-H and not-R, implying that H is false and R is also false. Moving to the third assumption, it posits the negation of the conjunction of not-R and not-Q, which can be restated as the disjunction of R or Q.

When we progress towards the conclusion, we are faced with the assertion that the conjunction of not-Y and Q implies not-B. Here, we discern a complex relationship between the variables at play. To unravel the intricacies of this statement, we must first dissect the meanings behind each symbol. The symbol "∼" signifies negation, while "*" denotes conjunction. The arrow "->" which is not present in this particular argument signifies implication. The equality sign "≡" is absent as well, indicating that we are not dealing with biconditional statements. With these observations in mind, we proceed to analyze the conclusion before us.

In order to assess the validity of the conclusion, we must scrutinize the connections between not-Y, Q, and B. Given the premises provided, it is evident that Q is true, as it is part of the second assumption. However, the status of not-Y and B remains uncertain. We must consider the relationships between these variables and how they interact with the logical connectives in play. As we navigate through the labyrinth of symbols and assumptions, we are faced with the challenge of determining whether the conclusion follows logically from the premises laid out before us.

In conclusion, the question of whether the conclusion holds true remains shrouded in ambiguity. The intricate dance of logical symbols and assumptions has led us down a winding path of uncertainty. As we ponder the implications of this argument, we are reminded of the limitations of human reason and the enigmatic nature of logical inference. The quest for truth continues unabated, as we grapple with the complexities of formal logic and the elusive nature of proof and refutation.