Analyzing the logical form of "All married couples fight"

Question

This is one of the example problems in Velleman's How to Prove book:

Analyze the logical forms of the following statements.

1. All married couples have fights.

Solution:

∀x∀y(M(x, y) → F(x, y)), where M(x, y) means “x and y are married to each other” and F(x, y) means “x and y fight with each other.”

But shouldn't the logical form should be like this:

∀x∀y((M(x, y) → F(x, y)) ∧ $ x \ne y $)

because one cannot be married to himself.

Answer 1

If one CAN be married to oneself, $M(x,x)$ will be true for that person.

Answer 2

I would say, no. The statement $\forall x \forall y (x \ne y)$ is false for any nonempty domain of discourse.

If you want to add an "acceptable couple" predicate $A$ (in your case $A(x,y)$ might be defined as $x \ne y$) it would look something like $\forall x \forall y ( A(x,y) \rightarrow (M(x,y) \rightarrow F(x,y)))$. This says "if the two are an acceptable couple, then, if they are married, they fight." If they are not an acceptable couple, the entire statement remains true.

On a side note, I see no reason why someone couldn't marry his/herself, claim the tax benefits, and then proceed to start fighting with his/herself.

Answer 3

The proposed alternative, $\forall x\forall y\left(\left(M(x,y)\implies F(x,y)\right)\land (x\not=y)\right)$, is simply a false statement: Examined carefully, it implies $\forall x\forall y (x\not=y)$, which patently fails when $x=y$.

An alternative that might be arguably correct is

$$\forall x\forall y\left(\left((x\not=y)\land M(x,y) \right)\implies F(x,y)\right)$$

That is, adding $x\not=y$ might specify what it means to be a married couple.

Answer 4

If you want to formalize the notion that no one can be married to themselves, it would be best to introduce another statement (an axiom) in addition to the answer given (itself an axiom) something like:

$$\forall x (\neg M(x,x))$$

You could also formalize other logical and legal requirements for $x$ and $y$ to be married (e.g. that they are both human beings, that they be of a certain age, etc.), but I doubt that this was the purpose of the exercise.