Predicate formula of proposition, author lacks precision in explanations

Question

I have a problem: Consider the two following propositions:

1. All persons have a mother.
2. There is one mother of all persons.

Now consider the predicate formulas of both propositions:

1. $\forall x \exists y \Bigr[ M \bigl( x,y \bigl) \Bigr]$
2. $\exists y \forall x \Bigr[ M \bigl( x,y \bigl) \Bigr]$

The author of my logic textbook then claims the following:

“The predicate formula for 1. says: for every person x there is a person y such that x stands in the child-mother relation $M \bigl( x,y \bigl)$ with y. The predicate formula for 2 says: there is at least one person y such that for all persons x, x stands in the child mother relationship $M \bigl( x,y \bigl)$ with y.”

However I don’t understand the choice of positioning the connective “such that” in those two different places of the sentence. Intuitively, it seems as though the predicate form of proposition 2 could read as follows: “There is at least one person y for all persons x such that x stands in the child mother relationship $M \bigl( x,y \bigl)$ with y”.

Although the exact positioning of the words in the sentence differs it seems to me to say the exact same as the first predicate formula if I were to read it in this way, no?

Can someone explain to me what I am missing please? Is the author simply stating that this is the way the reader should intuitively read the predicate formula given the previous proposition we were given, which it translates, or is he saying this is the way to read this predicate formula, regardless of what set x and y are part of, and what the relation M stands for?

Answer 1

As you observe, natural language is very sensitive to position of a phrase within a sentence (for that matter, so is formal language). You cannot put the "such that" wherever you want when translating from FOL into English.

• "$\exists x\varphi$" always gets translated as "There is an $x$ such that $\varphi$."

• "$\forall x\varphi$" always gets translated as "For all $x$ it is the case that $\varphi$." (Of course we often abbreviate this to "For all $x$ $\varphi$," but early on I emphasize to students the importance of the phrase "it is the case that" as a direct parallel to the "such that" above.)

• We always translate "recursively" using the above rules, so e.g. "$\exists x\forall y\exists z\forall w\varphi$" always gets translated as the admittedly-ugly

  "[There is an $x$ such that] (for all $y$ it is the case that) [there is a $z$ such that] (for all $w$ it is the case that) $\varphi$"

  (parentheses added for clarity and/or amusement).

The text isn't ambiguous, you're just assuming a linguistic freedom you don't actually have. (That said, it's entirely possible that your text didn't do a good job of explaining how to perform this translation in the first place.)

Answer 2

It might easier to understand if you made the domain of quantification explicit, e.g. $P(x) = x$ is a person.

1. $\forall x: [P(x) \implies \exists y: [P(y) \land M(x,y)]]$

2. $\exists y: [P(y) \land \forall x: [P(x) \implies M(x,y)]]$