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How is it that the union of two sets of sentences (that, individually, logically entail a sentence) logically entails a sentence while the intersection of the two sets does not logically entail a sentence?

visual of the question

Graham Kemp
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John B
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2 Answers2

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  1. If it is raining, then my car is wet.
  2. It is raining.
  3. If I am in the carwash, then my car is wet.
  4. I am in the carwash.

Set $S$ to be the first two propositions, $T$ to be the last two. Each of $S,T$ imply that my car is wet. Their union, i.e. all four propositions above, also implies that my car is wet. However $S\cap T=\emptyset$, which does not imply that my car is wet.

vadim123
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  • If Γ ⊨ φ and Δ ⊭ φ, then Γ ∪ Δ ⊨ φ. – John B Dec 23 '16 at 01:32
  • Thank you! could you please also advise on the following? I have been doing well but for some reason am not grasping sets for prop logic. If Γ ⊨ φ and Δ ⊭ φ, then Γ ∪ Δ ⊨ φ. I appreciate your time. – John B Dec 23 '16 at 01:34
  • doesn't an empty set(always 0) imply everything? – Rahul May 12 '21 at 08:48
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First of all, I have only been studying logic for a week, so feel free to correct me.

For your first question, let the two sets be Γ = {a,b} and Δ = {c,d} with a,b,c and d being propositional sentences (forgive me for being lazy); then Γ ⊨ φ tells us a∧b ⇒ φ, and Δ ⊨ φ means c∧d ⇒ φ. As vadim says above, Γ ∩ Δ = ∅ (empty set). Recall the definition of entailment: for Γ ∩ Δ to entail φ, every truth assignment that satisfies Γ ∩ Δ should also satisfy φ. Now, any truth assignment satisfies an empty set, so φ should hold under any truth assignment, i.e. φ is a tautology. But in our context no such information is mentioned, so the first proposition is false.

For the second question Γ ⊨ φ and Δ ⊭ φ, then Γ ∪ Δ ⊨ φ. WLOG, this time let Γ = {a,b} and Δ = {b,c}. Γ ⊨ φ means any truth assignment that satisfies a and b also satisfy φ, i.e. a∧b ⇒ φ. Δ ⊭ φ means some truth assignment that satisfies b and c does not satisfy φ. Now Γ ∪ Δ = {a,b,c}, so the question is about whether a∧b∧c ⇒ φ. Notice that any truth assignment that satisfies a∧b∧c must satisfy a∧b as well, so it also satisfies φ, and we have that Γ ∪ Δ entails φ. The fact that Δ ⊭ φ has nothing to do with it, since we know φ must be true as long as a∧b is true.

IMO, you just need to be aware that Γ ∪ Δ ≠ Γ ∨ Δ - this seems to be an easy mistake.

sheryl
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