Which one of the following sequents is valid?

Question

BlackTom AI · Accepted Answer

The prompt asks: Which one of the following sequents is valid? However, the provided data shows no answer options to compare. I will first restate what is given and then analyze the single sequent that appears in the data, treating it as a candidate to evaluate for validity.

Restated question and content:
- Question: Which one of the following sequents is valid?
- Provided sequent (appears as the candidate): P1, ¬P2, ¬P3 ⊢ (P1 ∧ P2) ⇒ (P1 ⇒ P3)
- No further answer choices are supplied to compare against.

Analysis of the given sequent's validity:
- In a sequent of the form Γ ⊢ φ, the sequent is valid if, in every valuation that makes all formulas in Γ true, the formula φ is also true.
- Here, Γ = {P1, ¬P2, ¬P3} and φ = (P1 ∧ P2) ⇒ (P1 ⇒ P3).
- Determine valuations that satisfy the premises Γ:
  - P1 must be true (P1 = T).
  - ¬P2 must be true, so P2 must be false (P2 = F).
  - ¬P3 must be true, so P3 must be false (P3 = F).
- Under this unique valuation, evaluate φ:
  - P1 ∧ P2 is (T ∧ F) which is F.
  - P1 ⇒ P3 is (T ⇒ F) which is F.
  - The conditional (P1 ∧ P2) ⇒ (P1 ⇒ P3) becomes F ⇒ F, which is T (an implication with a false antecedent is true, and even with a false consequent, F ⇒ F is true).
- Since the premises Γ are all true under this valuation and φ evaluates to True, the sequent holds in this valuation.
- There are no other valuations that satisfy Γ (P1, ¬P2, ¬P3) simultaneously, because these three literals fix all three propositional variables to specific truth values. Therefore, there is no counterexample valuation to falsify the sequent.

Thus, within the given data, the sequent P1, ¬P2, ¬P3 ⊢ (P1 ∧ P2) ⇒ (P1 ⇒ P3) is valid.

Important note about the missing options:
- Because the question requires selecting among multiple sequents and no answer options are provided, a definitive comparison against other candidates cannot be performed here.
- If you have the full list of answer choices, I can evaluate each option in turn for correctness and explain why each is valid or invalid relative to the others.

Summary of the key ideas:
- A sequent is valid if every valuation that makes the premises true also makes the conclusion true.
- The premises P1, ¬P2, ¬P3 lock the variables to P1 = T, P2 = F, P3 = F.
- Under that valuation, (P1 ∧ P2) is false, P1 ⇒ P3 is false, and thus the overall implication (false) ⇒ (false) is true.
- Therefore, the sequent is valid under the given premises.

Which one of the following sequents is valid?

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