Formalization of a statement that is false

Question

I have a hard time understanding how we formalize sentences from "pure English" to "pure mathematics" (i.e. logic statements, propositions etc..). Furthermore I don't get how we should formalize sentences which are false by their "core", for example:

1. "Every positive number is a sum of at most 4 integers".

My guess is:

$$ \forall n \in \mathbb{N} \exists a,b,c,d,e \in \mathbb{Z} . n =a+b+c+d+e \wedge ( (a=b) \vee (b=c) \vee (d=e) \vee (b=d) \vee (b=e) \vee (a=e) ...) $$

I tried to implement the "at most four integers" part, which is hard to understand. Also, this sentence is false at its core, I mean, it is true because every positive number is indeed a sum of 4 integers (false & positive + 0) but, the it states that "at most" which is not always true as we can get to every positive (or any number at this case) with 3 integers, 2 integers, 5, 6, etc... so why be specific on the four?

Or it doesn't really matter, because we treat this sentence as a "truth" and translate it no matter if it is false or true all the time?

Answer 1

"Every positive number is a sum of at most 4 integers"

Let the domain of discourse be $\mathbb Z.$

1. Disambiguate:

   “Every positive integer is expressible as a sum of integers, requiring at most four.”

   In other words, “Every positive integer is expressible as the sum of one to four integers.”

2. Translate literally: \begin{gather}\forall n{>}0\:\: \big(∃a\:\:n=a\:\:\lor\\∃a{,}b\:\: n=a+b\:\:\lor\\∃a{,}b{,}c\:\: n=a+b+c\:\:\lor\\∃a{,}b{,}c{,}d\:\: n=a+b+c+d\big).\tag1\end{gather}

   This is logically equivalent to \begin{gather}\forall n{>}0\:\: ∃a{,}b{,}c{,}d\\\big(n=a\:\:\lor\\n=a+b\:\:\lor\\n=a+b+c\:\:\lor\\n=a+b+c+d\big).\end{gather}

3. Alternatively, rephrase equivalently for easier translation:

   “Every positive integer is expressible as the sum of four integers.”

   Then translate: $$\forall n{>}0\:\: \exists a{,}b{,}c{,}d\quad n=a+b+c+d.\tag2$$

   (Although the above theorem is trivial, adding one word transforms it to Lagrange's Four-Square Theorem: “Every positive integer is expressible as the sum of four squared integers.”)

Note that statements $(1)$ and $(2)$ are equivalent, but not logically equivalent, to each other.