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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.

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  • $\begingroup$ Apply the PNF operations: the first one is $∀x∃y∃z(A(x)→(P(y)∧M(z)∧F(x,y,z)))$ while the second one is: $∃y∃z∀x(A(x)∧P(y)∧M(z)→F(x,y,z))$ that in turn is $∃y∃z∀x(A(x)→P(y)→M(z)→F(x,y,z))$ $\endgroup$ Dec 7, 2023 at 7:20
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    $\begingroup$ The second one says that the pianist and the master sewer are the same whatever artist we choose. $\endgroup$ Dec 7, 2023 at 7:21
  • $\begingroup$ Also note that a comparison of the statements is easier when both are written with the same tokens for the predicates' variables, as @MauroALLEGRANZA did. ie: $A(x), P(y), M(z), F(x,y,z)$ . $\endgroup$ Dec 8, 2023 at 12:55

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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.


"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."

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

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