Predicate formula of proposition, author lacks precision in explanations

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?

• I actually think this is a reasonable question, and in my experience this sort of error is quite common. So I disagree with the downvote and vote to close. (The close vote especially confuses me: what other context is desired?) Commented Jun 28 at 21:07
• @Noah: agreed! It seems it's common on math.SE for some people to close and downvote questions that are not precisely phrased requests for proofs or calculations. I find this strange. There are many other kinds of questions that naturally come up when you're learning mathematics and this should be a place to discuss those. Commented Jun 28 at 21:43

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.)

• Ah alright thank you very much! The author doesn’t specify that there is a specific way to always translate this, instead only saying this reads: “[…]” not explaining he is using a axiomatic rule, nor does he mention this or these rules previously. Commented Jun 28 at 21:09
• @ryang i thought they weren’t? The comment i just responded to above, stated that symbolic language had a specific set of rules that make it false to translate symbol proposition 2 by « all persons have a mother ». Commented Aug 2 at 16:23
• @NoahWurtz Noah's answer makes explicit reference neither Verbal 1 nor 2; yes, Verbal 2 certainly means Symbolic 2; while Verbal 1 is intended to mean Symbolic 1, it could also mean Symbolic 2, though admittedly this is a bit of a stretch. Commented Aug 3 at 2:56
• @ryang But didn’t he say: ∃xφ " always gets translated as "There is an x such that φ. ∀xφ " always gets translated as "For all x it is the case that φ." Which means that Verbal 1 could not be syntactically written as Symbolic 2? Forgive me if I’m wrong but it seems that you are contradicting what Noah said earlier? Commented Aug 3 at 7:22
• @NoahWurtz There is no contradiction between what you're quoting and what I said. My point is just about the potential ambiguity of Verbal 1, and is peripheral to the crux of this discussion. Commented Aug 3 at 8:10

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

• please excuse me for not answering but I wasn’t able to understand your statement well, and given that Noah’s answer satisfied me I left the thread. Commented Aug 3 at 7:27
• In "standard FOL," the domain of discourse is implicitly assumed to be non-empty. This allows you to introduce new free variables by universal specification/instantiation. In mathematical proofs, this is usually not allowed as it can create confusion. It is better to make the existence of any domain of quantification explicit as a named set or other domain that may or may not be empty. Commented Aug 4 at 2:13