Are true statements of arithmetic logically necessary? That is, is "2+3=5", the commutativity of addition of natural numbers, and the infinitude of primes, among other statements, logically necessary? For example, I doubt very much that there is a possible world where there are only finitely many primes, and I certainly don't think there is a possible world where 2+3 is not equal to 5. So, then, are the true statements of arithmetic logically necessary? And, have any philosophers argued that at least some true statements of arithmetic are not logically necessary?

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    Considering that 1) Logic sustains itself tautologically (for Logic to be possible, 2+2=4 is necessary, and vice versa) (see Russell) and 2) that necessity implies dependence, it cannot be stated that a component of a tautology is necessary. In some sense it is, because if it is a component, its lack breaks the whole, and in some sense it is not: a tautology comes to be a whole only if the whole determines its nature.
    – RodolfoAP
    Dec 5, 2022 at 12:00
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    Maybe if you rewrite your question the answer will be more clear. Are the logically necessary truths of arithmetic logically necessary? Yes. Yes they are.
    – Boba Fit
    Dec 5, 2022 at 14:10
  • To suppose that arithmetical propositions are logically necessary appears to commit you to some kind of logicist approach to the philosophy of mathematics, under which mathematics is reducible to logic. Logicism is widely thought to be a bust today, though there are some neo-logicists who defend a weaker version of it. As Mauro says in his answer to philosophy.stackexchange.com/questions/37969/… 2+2=4 is a logical consequence of the axioms of arithmetic, but it does not follow that the axioms themselves are logically necessary, or even necessary at all.
    – Bumble
    Dec 5, 2022 at 20:26
  • Even axioms of PA attains tautological necessity status at normal worlds, at non-normal worlds or standard worlds accessible to those non-normal worlds the normally valid and sound inferences may become unnecessary where everything is possible... Dec 7, 2022 at 22:55

1 Answer 1


Are true statements of arithmetic logically necessary?

Arithmetic is a logical formal system which is the formalisation of our intuitive notions about numbers and operations on numbers.

This means that the result of arithmetic operations is the logical consequence of our basic assumptions about numbers and operations on numbers. The operation 2 + 3 = 5 is true because it is the logical consequence of our assumptions, first that the figure '2' means two fingers and more generally any two things; second, that the figure '3' means any three things; third, that the symbol '+' means taking together the things referred to by the figures involved in the operation, with the proviso that these things are all distinct from each other; and fourth that '5' means any five things.

If we are consistent in our assumptions and how we use symbols, the result of arithmetic operations is logically necessary. This means that arithmetic statements are not necessarily true of the world. Rather, we will believe that they are true of the world as long as our use of arithmetic expressions is consistent with our assumptions about numbers and operations on numbers.

  • So, the world doesn't need arithmetic, but our brains do. Good thing.
    – Scott Rowe
    Dec 6, 2022 at 2:50
  • @Scott Rowe Arithmetic is part of us and we are part of the world, so the without arithmetic the world wouldn't be what it is. So it does need arithmetic to be what it is, but it could still be without it. It would just be different. Dec 6, 2022 at 16:27
  • Job Security: the world needs me to be doing exactly what I'm doing right now! I feel much better. The world needs me to feel better. Self-justification! Boy, I am on a roll now...
    – Scott Rowe
    Dec 6, 2022 at 16:34

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