Is every painting like by someone who didnt paint it, or, does someone like all paintings he or she did not paint?

Question

I am given the following predicate:

$\exists x \forall y [(P(y) \wedge \neg M(x, y))\rightarrow L(x, y)]$ where $P(y)$ means $y$ is a painting, $M(x, y)$ means $x$ paints $y$, and $L(x, y)$ means $x$ likes painting $y$.

My goal is to translate this into English.

My interpretation of this is along the lines of: "every painting is liked by someone who didnt paint it"

I have seem another interpretation which says that "somebody likes all paintings he or she did not paint.".

These two interpretations are different and I would like to know whether my interpretation is correct and if it is not, what is the correct reasoning behind the answer?

Answer 1

Your interpretation is incorrect, and I think it is mainly due to the order of the quantifiers ($\forall, \exists$).

To illustrate this clearer, I'll make the following examples:

$\exists y \forall x(P(y) \land L(x,y))$: there is a painting that everyone liked.

$\forall x \exists y (P(y) \land L(x,y))$: everyone has a painting they like.

The difference in order of the quantifiers creates two distinct situations.

Your interpretation is closer to $\forall y \exists x (P(y) \land \neg M(x,y) \land L(x,y))$

Answer 2

Your answer is incorrect. The order of the quantifiers matters: compare "$\forall x\exists y(x<y)$" and "$\exists y\forall x(x<y)$." In $\mathbb{R}$ with the usual notion of $<$, the former is true ("For every number there is a number which is bigger") but the latter is false.

It will help at first to be very methodical in translating sentences like these, going left to right and giving each symbol its standard English phrasing. In particular, in our case before going straight to natural-sounding English we should consider the following:

"There is some $x$ such that for every $y$, if $y$ is a painting and $x$ did not paint $y$ then $x$ likes $y$."

Note that this makes it clear that there is a single person who has the relevant relationship to all paintings at once. We can then tweak this rather awkward sentence to get something more natural:

"There is some person who likes every painting they did not paint."

Note that this does not mean that this person $x$ likes only the paintings they did not paint. While natural language often includes such implicit "only-if"s, formal logic doesn't.