Which one of the following is a valid sequent in the Propositional Calculus:

Question

BlackTom AI · Accepted Answer

The question asks to assess whether a given sequent is valid in the Propositional Calculus.
First, restating the presented sequent: from the premises ¬(P ∧ Q) and P, derive ¬Q.
Consider the logical relationship: if P is true and it is not the case that P ∧ Q holds, what can we conclude about Q?
- From ¬(P ∧ Q) and P, we know that it is not the case that both P and Q are true simultaneously.
- Since P is assumed true, the only way for ¬(P ∧ Q) to hold is for Q to be false; otherwise P ∧ Q would be true, contradicting ¬(P ∧ Q).
- Therefore, ¬Q logically follows from the premises, making the sequent valid under classical logic.
Now, addressing the format: there is no list of multiple-choice options provided for evaluation in the question data. Nevertheless, the single sequent presented, viewed as a sequent with premises ¬(P ∧ Q) and P and conclusion ¬Q, stands as a valid entailment in propositional calculus.
For completeness, let me outline potential points of confusion that might arise with similar-looking statements:
- If one omits P as a premise, ¬(P ∧ Q) alone does not entail ¬Q; from ¬(P ∧ Q) we cannot infer the falsity of Q in all cases, only that not both are true together.
- If one tried to derive Q from these premises, that would be incorrect because the conclusion contradicts the given entailment: from P and ¬(P ∧ Q), Q cannot be forced to be true.
- The reasoning relies on the distributive property of conjunction and negation, specifically that P ∧ Q being false while P is true necessitates Q being false.
In summary, the sequent ¬(P ∧ Q), P ⊢ ¬Q is valid because the premises jointly force Q to be false, given that P holds and the conjunction P ∧ Q cannot hold.

Which one of the following is a valid sequent in the Propositional Calculus:

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