Logic complementation

Question

$G$$(x)$: $x$ is a game

$M$$(x)$: $x$ is a movie

$F$$($$x$, $y$$)$: $x$ is more fun than $y$

Using the predicate symbols shown and appropriate quantifiers, write English language statement as a predicate wff(well formed formulae).

No game is more fun than every movie.

I tried do it in opposite way. First tried to find out the complement of this statement and write wff.Complementing it again will give me my result.

So the complement of the given question would be (I thought)

Some game is not more fun than some movie.

Then I constructed wff and again complemented. The result came out to be

All game are more fun than every movie.

Which is definitely not equivalent as No game is more fun than every movie.

It means my initial step of finding complement was wrong which is

Some game is not more fun than some movie.

So my question is why it is incorrect ?

Which one will be right?

How to handle this type of statements?

Does this statement implies

Some game is not more fun than some movie $->$ some game is more fun than some movie

I think its completely wrong because all games can be less funnier than some movies.

Please provide me the right ideas for complementing statements.

Help appreciated :)

Answer 1

No game is more fun than every movie.

No game: $\neg~\exists x: G(x)\wedge \ldots$

Every movie: $\forall y: M(y)\to\ldots $

Combining with More Fun.

$$\neg~\exists x~\forall y~\Big( G(x)\wedge \big(M(y)\to F(x,y)\big)\Big)$$

We may push the negation to the interior using duality rules, giving the equivalent expression:

$$\forall x~\exists y~\Big( G(x)\to \big(M(y)\wedge \neg F(x,y)\big)\Big)$$

Which reads: "every game is not more fun than some movie", or more plainly "each game has a movie it is less fun than."

Which indeed means that "no game is more fun than every movie."

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Now look at your proposed complement:

Some game is not more fun than some movie.

In the same way we find that this is:

$$\exists x~\exists y~\big(G(x)\wedge M(y)\wedge \neg F(x,y)\big)$$

And that is not the complement of the first statement.

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tl;dr

The complement of "no game is stuff" is just "some game is stuff".   Don't touch the stuff.

Answer 2

The "complement" of your sentence no game is more fun than every movie should be there is a game that is more fun than every movie. Symbolically, we could write no game is more fun than every movie as $$\forall g \in G(\exists m \in M(\neg F(g,m)),$$ where $G$ is the set of games, $M$ is the set of movies, and $F(g,m)$ is the relation "$g$ is more fun than $m$". The negation of this will be $$\exists g \in G(\forall m \in M(F(g,m)),$$ meaning that there exists a game which is more fun than every movie.

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One reason why some game is not more fun than some movie is not the complement of no game is more fun than every movie, is that the two should not be true at the same. However, if no game is more fun than every movie is true, then it can still be true that there exists some game which is not more fun than some given movie. These two are not contradictory.