Consider the following sequent and an attempted proof:

Which one of the following accurately describes the error in the attempted proof?

Question

Consider the following sequent and an attempted proof:

Which one of the following accurately describes the error in the attempted proof?

BlackTom AI · Accepted Answer

First, let me restate what we have: the question asks to identify the error in an attempted proof of a sequent, and the provided answer choice claims that Existential Elimination is incorrectly applied at line (6) because the constant symbol a appears in the conclusion at line (5).

Option analysis:
- Assessing the proposed issue with Existential Elimination (∃E): The standard rule for ∃E says that from ∃x P(x), one may assume P(a) for a fresh constant a (an eigenvariable) and derive some conclusion C, but this C must not depend on the particular choice of a; moreover, the constant a must be new and must not appear in any open assumptions or in the final conclusion unless allowed by the system. A common precise formulation is: if from P(a) you can derive C, and a is introduced as a new constant not appearing in ∃x P(x) or any undischarged assumption, then you may conclude C provided a does not appear in C.
- The reasoning in the option states that ∃E is incorrectly applied at line (6) because a appears in the conclusion at line (5). This aligns with the eigenvariable condition: if the constant a is used in deriving the conclusion, and that conclusion depends on a, then you cannot discharge the existential correctly with ∃E. In other words, if C contains a, then C cannot be derived by ∃E from ∃x P(x) using the temporary assumption P(a), because the eigenvariable would be present in the final result, violating the restriction that the eigenvariable should not appear in the conclusion.

Why this explanation makes sense: The core mistake commonly made in ∃E proofs is either reusing the eigenvariable outside its scope or allowing the variable to appear in the conclusion. If line (5) (the conclusion) contains the constant a that was introduced for the existential, the derivation is invalid under the ∃E rule’s eigenvariable condition, so identifying this as the error is plausible.

What to watch out for in other options (if they existed):
- If an option claimed the error was due to a different rule being misapplied (e.g., Universal Elimination, or a misstep in a conditional proof), you would check whether the specific constraints of that rule were violated in the given derivation.
- If an option claimed a syntactic or semantic misstep unrelated to eigenvariable restrictions (like an unjustified instantiation or an incorrect use of a previously proven lemma), you would evaluate whether such a misstep would actually reproduce the given line structure and dependencies.

Note on missing options: The prompt provides no alternative answer choices to compare against the proposed one. Consequently, we can only assess the internal consistency of the proposed error description with the standard ∃E eigenvariable condition. The core idea—that presenting a as a constant in the conclusion violates the eigenvariable restriction of ∃E—is a well-established criterion in natural deduction systems.

Consider the following sequent and an attempted proof: Which one of the following accurately describes the error in the attempted proof?Single choice

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