Question 1171517: For any TFL sentences 𝛼 and 𝛽 that are logically equivalent (i.e., whose truth values agree on every valuation of their sentence letters), does the following entailment hold:
𝛽 → 𝛼, 𝛼 ∨ 𝛽 ⊨ 𝛼 ∧ 𝛽
Could someone help me with this problem?
Thank you!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this entailment step-by-step.
**Given:**
* α and β are logically equivalent. This means they have the same truth values under every valuation (interpretation) of their sentence letters.
**Entailment to Prove:**
* β → α, α ∨ β ⊨ α ∧ β
**Analysis:**
1. **Logical Equivalence:**
* Since α and β are logically equivalent, we can say α ≡ β. This means that whenever α is true, β is true, and whenever α is false, β is false.
2. **β → α:**
* Because α ≡ β, the implication β → α is always true. If β is true, α is true, and if β is false, α is false. Thus the implication is always true.
3. **α ∨ β:**
* Since α ≡ β, α ∨ β will be true whenever either α or β (or both) are true. In fact, due to logical equivalence, α ∨ β is true when α is true, and also true when β is true.
4. **α ∧ β:**
* We want to show that α ∧ β is true in all cases where β → α and α ∨ β are true.
* Since β → α is always true, we only need to consider α ∨ β.
* If α ∨ β is true, then at least one of α or β is true.
* Because α ≡ β, if one is true, the other is also true.
* Therefore, both α and β are true.
* Hence, α ∧ β is true.
5. **Entailment:**
* We need to check if in every valuation where β → α and α ∨ β are true, α ∧ β is also true.
* β → α is always true.
* If α ∨ β is true, then both α and β are true (due to logical equivalence).
* Therefore, α ∧ β is true.
* Thus, the entailment holds.
**Conclusion:**
Yes, the entailment β → α, α ∨ β ⊨ α ∧ β holds.
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