# If A entails C, and B entails C, why doesn’t (A and B) necessarily entail C?

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. • @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). Jan 2, 2022 at 21:30

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`
• 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. Jan 15, 2022 at 9:24