Suppose that
𝑊
,
𝑊
1
and
𝑊
2
are wffs in the Propositional Calculus, such that the following sequent is valid:

⊢
∼
𝑊
1
⇒
∼
𝑊
2

Which one of the following sequents is valid?

Question

Suppose that 
𝑊
,
𝑊
1
 and 
𝑊
2
 are wffs in the Propositional Calculus, such that the following sequent is valid:

⊢
∼
𝑊
1
⇒
∼
𝑊
2

Which one of the following sequents is valid?

BlackTom AI · Accepted Answer

The question presents a sequent-soundness scenario in propositional calculus. It states that W, W1, and W2 are well-formed formulas and that the following sequent is valid: ⊢ ¬W1 ⇒ ¬W2. This is a form of contrapositive, which is logically equivalent to W2 ⇒ W1.

Option 1: W ⇒ W2 ⊢ W ⇒ W1. 
To assess this, consider what it means to derive W ⇒ W1 from the assumption W ⇒ W2. A natural approach is to use a conditional proof:
- Assume W. From W and the premise W ⇒ W2, you can infer W2 by modus ponens.
- Now use the derived W2 and the auxiliary fact W2 ⇒ W1 (which follows from ¬W1 ⇒ ¬W2 via contrapositive) to obtain W1 by another application of modus ponens.
- Since assuming W led to W1, you have established W ⇒ W1.
Thus, under the given logical relationship between W1 and W2 implied by ⊢ ¬W1 ⇒ ¬W2, the sequent W ⇒ W2 ⊢ W ⇒ W1 is valid.

Why this reasoning makes sense in context:
- The premise ⊢ ¬W1 ⇒ ¬W2 is equivalent to W2 ⇒ W1 by contrapositive, a standard logical equivalence that preserves validity.
- From W ⇒ W2 and W2 ⇒ W1, one can chain implications to get W ⇒ W1 through a straightforward hypothetical syllogism (if W then W2, and if W2 then W1, therefore if W then W1).

Why there aren’t alternative options to consider here:
- The provided data lists only one option to evaluate. Since the key fact is that ¬W1 ⇒ ¬W2 is valid, its contrapositive W2 ⇒ W1 provides the necessary bridge to derive W ⇒ W1 from W ⇒ W2 using a simple chain of implications. No other choices are given to critique, so the focus remains on verifying the validity of this specific sequent.

In summary, given the contrapositive relationship between W1 and W2, the derived chain W ⇒ W2 and W2 ⇒ W1 yields W ⇒ W1, making the sequent valid under the assumptions described.

MATH3066 (ND) 2025 Mid-Semester Test Practice Set C

Suppose that 𝑊 , 𝑊 1 and 𝑊 2 are wffs in the Propositional Calculus, such that the following sequent is valid: ⊢ ∼ 𝑊 1 ⇒ ∼ 𝑊 2 Which one of the following sequents is valid?

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