# Are the truths of arithmetic logically necessary? [duplicate]

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?

• 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. Dec 5, 2022 at 12:00
• 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. 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. 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