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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?

ryang
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CSch of x
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    The statement is true. What would be false would be to say: if a positive number is the sum of integers, that collection of integers has at most four elements. This just says: using at most four integers, you can build any positive number (obviously you don't NEED four, but certainly four will do the job). Ultimately, this is a linguistic question, rather than a math one, but I am translating it the way I think most mathematicians would. – Elchanan Solomon Oct 21 '21 at 19:58
  • But yes, it does depend on what is meant by “at most $4$ integers.” Does it mean it cannot be a sum of $5$ or more integers? Does it exclude the case of a sum of one integer? Does it mean the integers in the sum are all different? These can only be formalized by asking the English speaker what was meant by the words. – Thomas Andrews Oct 21 '21 at 20:02
  • @ThomasAndrews I think that by saying "at most 4" then the formalization should say that 5 or more is invalid, how would you formalize such proposition? – CSch of x Oct 21 '21 at 20:06
  • @fasttt The reason I would not formalize it that way is that (1) it is hard to formalize it that way, in first order logic, and (2) it doesn’t match a common language usage. There is a theorem (The Four Square Theorem) that can be written: “ All positive integers can be expressed as the sum of at most $4$ positive squares.” In that language, we would not be saying that $5$ is not possible, but that we only need $4$ squares at most. This example also includes the meaning that a sum of a single square is still a “sum of squares.” – Thomas Andrews Oct 21 '21 at 20:23
  • Under the “four square” analogy, this wouldn’t mean that the numbers are distinct, or even more than $1,$ so you’d formalize as: $$\forall n>0: \exists a,b,c,d: n=a\lor n=a+b\lor\n=a+b+c\lor n=a+b+c+d.$$ – Thomas Andrews Oct 21 '21 at 20:34
  • @ThomasAndrews What about this? $\forall n>0: \exists a,b,c,d \in \mathbb{Z}: n=a+b+c+d.$ ? does it signify the "at most" property? what does it say in "english" ? Thanks! – CSch of x Oct 21 '21 at 20:56
  • Well that doesn’t directly state what the English states, but one can easily prove it is equivalent. @fastttt – Thomas Andrews Oct 22 '21 at 02:11
  • @ThomasAndrews But how's ur answer states there there cannot be more than 4 ? u said "exists a,b,c,d" but not "exists a,b,c,d,e,f,...." – CSch of x Oct 22 '21 at 12:27
  • The point of the “four squares” example is that this language doesn’t imply we can’t represent the number as the sum of more than four integers. @fastttt – Thomas Andrews Oct 22 '21 at 16:45
  • @ThomasAndrews but i wanted to say "at most 4" so, how would you represent such query? – CSch of x Oct 22 '21 at 17:05
  • @ryang Correct! done :) – CSch of x Mar 23 '22 at 11:37

1 Answers1

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"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.

ryang
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  • Just to double-check, (1) is logically equivalent to $\forall n > 0 \ \exists a,b,c,d \ \big( n = a \vee n = a + b \vee n = a + b + c \vee n = a + b + c+ d\big)$ right? – Rob Mar 13 '22 at 02:25
  • Yes indeed; I’ve just added this observation to the answer. – ryang Mar 13 '22 at 09:33
  • This formula does not exclude the possibility that n can be a sum of 5 or more integers. Consider 4+5+6+7 = 1+2+3+4+5+7=22 – Tankut Beygu Mar 13 '22 at 15:48
  • @TankutBeygu The given statement indeed does not forbid $n$ from being a sum of 7 integers, and nobody has claimed otherwise. It is incidentally also trivially true and not useful (any integer number in fact requires at most 1 integer to be expressible as a sum). If I understand you correctly, you are fundamentally misunderstanding the exercise: this is a translation exercise, which is independent of whether the given statement is actually false or true. – ryang Mar 13 '22 at 16:55
  • @ryang Then, the answer should have been "the phrase 'at most' is not translatable into first-order predicate logic." – Tankut Beygu Mar 13 '22 at 17:32
  • @TankutBeygu “Exists at most four” is translatable in predicate logic. On the other hand, the given statement, which actually means “requires at most four”, has a different meaning, but is also translatable (I suggested several equivalent versions). P.S. If you were the downvoter, I hope this sufficiently clarifies? – ryang Mar 14 '22 at 08:03
  • @ryang What misguides you is the particularity of the phrase 'at most one'. This phrase means actually 'exactly one if not none'. 'Exactly n' is translatable into first-order predicate logic. See the answers for an explanation of this. – Tankut Beygu Mar 14 '22 at 08:44
  • @TankutBeygu Yes, in the answer I linked to, I gave examples of how to translate "exactly one" and "at most one". We seem to be talking at cross purposes this entire thread. Finally, this is a translation exercise, but not a literal-translation exercise, which is why the first step was to disambiguate what the author was meaning (trying) to say. Again, I feel that you are fundamentally misunderstanding the spirit of this exercise, and that besides your claim that 'at most' is not translatable into first-order predicate logic, nothing you've said contradicts my Answer or what I've said here. – ryang Mar 14 '22 at 09:02
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    @TankutBeygu Consider the Four Color Theorem: every map can be colored where no two areas that share a boundary have the same color. This statement is perfectly formalizable in first-order logic in a similar way that this Answer does … and that formalization does not rule out the possibilityof coloring a map with more than 5 colors where still no two bordering areas have the same color …. which isexactly what we want! The Four Color Theorem never makes the claim that with more than 4 colors you cannot do such a coloring. – Bram28 Mar 14 '22 at 13:51
  • @Bram28 It is a matter of one's expository style to add such non-technical descriptions as "so, four colours are enough for us" and the like; in strict terms, the analogy is faulty. What is significant here are the statements in which the phrase 'at most n' occurs essentially. Suppose you are given strictly expressed instructions for a number-theoretic problem; one instruction reads: "Put the balls into the boxes such that each box contains at most 4 balls." Would you think it is allowable to put 5 or 27, etc. balls? – Tankut Beygu Mar 14 '22 at 17:04
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    @TankutBeygu For that latter case, clearly not. Such are the ambiguities inherent in the English language: the same phrase can mean different things in different contexts. Backgrpund knowledge and common sense will often tell you what exactly is being meant. In the case of the OP’s question, my common sense told me that it wad to be interpreted like the Four Squares theorem or Four Color Theorem. But, obviously, I could be incorrect about that. So at this point we should probably ask the OP what exactly they had in mind or where they got this sentence from. – Bram28 Mar 14 '22 at 17:47
  • I don't deny that there is an ambiguity. Notice that the other usage has subjective connotations, as it were, a touch of anthropomorphism. For example., ". . . can be coloured using at most four colours", that is, "we can colour (contrary to your expectation) . . ." or "we need four colours, not more". That style is less pedantic, more communicative, but can be confusing. Some authors prefer such phrases as "minimum number of". – Tankut Beygu Mar 14 '22 at 19:12
  • TankutBeygu: the OP supplies a statement that is, strictly speaking, ambiguous, but whose meaning is nonetheless reasonably inferable; the OP would like to formalise this statement; we then do so by first disambiguating it. Unless you disagree with our best-guess interpretation, or with the fact that there's even a best-guess interpretation, you don't have an actual issue with my answer, and are merely making tangential points that (appear to me) keep sliding from one to another. – ryang Mar 15 '22 at 03:30