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The original question is in Greek letters Γ and Δ, each representing a set of sentences, and φ representing an individual sentence (atomic proposition). The question is from Introduction to Logic by Stanford University available on Coursera. enter image description here

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  • @Arno Notice that when φ is a compound proposition of the form equivalent to P → Q, it is preserved across intersections. Surprisingly, this very instructive detail is explicitly stated only in Chang and Keisler's quite "unfriendly" Model Theory (3rd edition, p. 15. Amsterdam: North-Holland, 1990).
    – Tankut Beygu
    Jan 2, 2022 at 21:30

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You have a ∩ (denoting intersection), not a ∧ (denoting AND) here.

Thus, in the third statement, it could be that sentences present in both Γ and Δ do not necessarily entail the formula φ. Γ and Δ could have an empty intersection.

For example, given two distinct formulas ψ1 and ψ2,let Γ be {(φ AND ψ1 )} and let Δ be {(φ AND ψ2)}.

Γ |= φ and Δ |= φ, but Γ ∩ Δ is the empty set, so Γ ∩ Δ |\= φ

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    There is one additional step needed here, which is that φ needs to be given a bit more precisely. For example, if φ were a logical truth then the empty set would in fact entail it.
    – Paul Ross
    Jan 15, 2022 at 9:24
  • This is great help! I think the confusion between intersection and And is exactly the reason why I didn’t understand it. Now it is crystal clear! Thank you!
    – James Young
    Jan 24, 2022 at 15:15

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