Understanding triple mixed quantifiers

Question

I'm having a hard time understanding mixed quantifiers of this form: $$\forall x\exists y\forall z(...)$$ and similarly $$\exists x\forall y\exists z(...)$$

It really hurts my head to think about these. For all x, there exists a y, such that for all z?

Let's take as an example, a relation M, thought of as x being married to y, for the sake of visualizing this, even if it would make no sense in the real world. Stating: $$\forall x\exists y\forall z(M(x,y) \land M(x,z) \land M(y,z))$$ seems to mean that for all people x, there exists a person y, such that for all people z, x is married to y, and x is married to z, and y is married to z. What would that even mean? How can one unwind this expression, to more clearly see what is being said?

I have tried to somehow think about each quantifier one by one, but that seems to make no sense.

Thanks in advance.

Answer 1

For a mathematical example, see the "standard" epsilon-delta definition of limit :

$\forall \epsilon, \exists \delta, \forall x \ldots$.

For a natural language example, see Jan von Plato, Elements of Logical Reasoning (2013), page 118 :

A sequence of quantifiers such as $∀x∃y∀z$ is quite natural.

There is for every person some problem on which every attempt will fail.

The three-place relation here is: person $x$ fails on problem $y$ in attempt $z$. This example also illustrates the choice of a domain. In this case the domain seems to contain, at least, persons, problems, and attempts. There is a threeplace relation that we can write as $Fail(x, y, z)$, in which $x$ is a person, $y$ a problem, and $z$ an attempt. These have to go in the right places in the relation.

The usual way to do it is to introduce predicates such as $Person(x), Problem(y)$, and $Attempt(z)$, and to write the formalization as in:

$Person(x) \land Problem(y) \land Attempt(z) \rightarrow Fail(x, y, z)$

The condition for $Fail(x, y, z)$ can be lifted only if $x, y, z$ are appropriately chosen.

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Note

Regarding your example, you can rewrite it as a more "natural" one as follows :

$∀x∃y[Father(y,x)∧∀z((z \ne x ∧ Father(y,z)) \rightarrow Brother(x,z))]$

which is equivalent to :

$∀x∃y∀z[Father(y,x)∧((z \ne x ∧ Father(y,z)) \rightarrow Brother(x,z))]$