Strategies for translating statements into formal logic language I've seen many posts asking for help translating a specific statement (posed in natural language) into a formal logic language, and the answers are typically straightforward translations without much insight into the underlying process. There is a sense of "intuition" that one develops with practice, however, for beginners who lack intuition, I would like to know what strategies people use for translating statements into various languages, particularly into first order logic.
As an example, how would you go about translating any one of the following:
$(1)$ All people who eat ice cream obsessively and refuse to exercise have lacked an authoritative figure to teach them proper discipline.
$(2)$ Every even number greater than or equal to $4$ is the sum of two primes.
$(3)$ No student failed Chemistry, but at least one student failed History.
 A: Let's start with one of your examples. I'll go with "Every even number greater than or equal to $4$ is the sum of two primes". For simplicity, we'll use the predicate $E$ for evenness and $P$ for primeness. (Exercise for the reader: express the "evenness" and "primeness" predicates without using anything other than arithmetical and logical symbols.)
Looking at the part before the "is", "Every even number greater than or equal to $4$ ..." clearly implies a universal quantifier. However, since we're also talking about prime numbers, our domain of discourse is likely not just even numbers greater than or equal to $4$, but all numbers. So we need to rephrase in a way that says "every number". The natural way is "Every number that's even and greater than or equal to $4$...", but that's still not quite right--the quantifier is being restricted by the relative clause. Instead, we word it as "For every number, if it is even and greater than or equal to $4$, then...".
Looking at the part after the "is", "...is the sum of two primes." Well, we could just define the predicate $S$ meaning "is the sum of two primes" and call it a day, but that seems a little unsatisfying. Let's put it in terms of the primeness predicate $P$. What does it mean for a number to be a sum of two primes? Well, that means there are two primes whose sum is that number. Hmm, an existential quantifier has snuck in. Now, once again we can't say "there are two primes", because we have to quantify over numbers. So we reword to "There are two numbers that are prime", and the full statement is "there are two numbers that are prime and whose sum is that number."
So let's substitute these back into the original statement. "For every number, if it is even and greater than or equal to $4$, then there are two numbers that are prime and whose sum is that number." Now it's ready to be translated into logic. The existential is easy enough--it's just saying that there are two numbers $y$ and $z$ that are both prime and sum to another number $x$. So that translates to $\exists y\exists z[P(y)\land P(z)\land y+z = x]$. For the universal, it's saying that for any $x$, if it is even and greater than or equal to $4$, then the existential part is true. This translates as $\forall x ([E(x)\land x\ge 4] \rightarrow [\mathrm{existential\,part}])$. Putting those together gives
$$
\forall x([E(x)\land x\ge 4]\rightarrow \exists y \exists z[P(y)\land P(z)\land y+z=x])
$$ 

The third example is simpler, but it's a good way to look at how negation interacts with the quantifiers. The most sensible domain of discourse is students, and we'll use $C$ and $H$ to represent failing chemistry and history respectively. "No student failed chemistry" is worded in quantifiers as "There does not exist a student who failed chemistry", which translates into logic as $\neg[ \exists x C(x)]$. Similarly, "At least one student failed History" translates to "There is a student who failed History", which in logic is $\exists y H(y)$. The original sentence is saying these are both true, so translated into logic this is
$$
\neg[ \exists x C(x)]\land[\exists y H(y)]
$$
Now, suppose instead we defined $C$ and $H$ as passing Chemistry and History, respectively. Then failing would be $\neg C$ and $\neg H$, so our statement would be
$$
\neg[ \exists x \neg C(x)]\land[\exists y \neg H(y)].
$$
So how does negation affect quantifiers? It reverses them: $\neg[\exists x A(x)]\equiv \forall x[\neg A(x)]$ and $\neg[\forall x A(x)]\equiv \exists x[\neg A(x)]$. This means our original statement could instead be written as
$$
[\forall x C(x)]\land\neg[\forall y H(y)],
$$
or in English, "Every student passed Chemistry, but not every student passed History." I think you'll agree this is indeed equivalent to the sample sentence.
