Forms of ambiguous sentences

Question

The sentence "Every artist is friendlier to some pianist than to some master sewer" is ambiguous.

One reading is given by
∀x(A(x)→∃y∃z((P(y)∧M(z))∧F(x,y,z)))

Another reading is
∃x∃y((P(x)∧M(y))∧∀z(A(z)→F(z,x,y)))

Please tell me another one as I cannot figure it out for the life of me. TIA.

Answer 1

This is similar to the classic "Everybody loves somebody" versus "Somebody is loved by everybody."

$$\forall x\,\exists y\, L(x,y)\text{ vs. }\exists y\,\forall x\,L(x,y)$$

You are adding a third variable, and distinct domains for each, but the same principle applies.

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"Every artist is friendlier to some pianist than to some master sewer."

$$\forall x~(A(x)\to\,\exists y\,\exists z~(P(y)\land M(z)\land F(x,y,z)))$$

"There is some pianist and some master sewer where any artist is friendlier to that pianist than to that master sewer."

$$\exists y\,\exists z~(P(y)\land M(z)\land\forall x~(A(x)\to F(x,y,z)))$$