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Take from Possible world- an introduction to logic and its philosophy. p-21

Following quote provide us with necessary definition of what "logically necessary" or as far as i think "necessary truth" is.

"In saying that propositions of the several main kinds discussed so far are logically necessary truths we are, it should be noted, using the term "logically necessary" in a fairly broad sense. We are not saying that such propositions are currently recognized as truths within any of the systems of formal logic which so far have been developed. Rather, we are saying that they are true in all logically possible worlds."

Then the author assert -

"the truths of mathematics must also count as another main kind of logically necessary truth." p-21

"Among the most important kinds of necessarily true propositions are those true propositions which ascribe modal properties — necessary truth, necessary falsity, contingency, etc. — to other propositions. Consider, for example, the proposition

(1.15) It is necessarily true that two plus two equals four.

(1.15) asserts of the simpler proposition — viz., that two plus two equals four — that it is necessarily true. Now the proposition that two plus two equals four, since it is a true proposition of mathematics, is necessarily true. Thus in ascribing to this proposition a property which it does have, the proposition (1.15) is true" p-22

But my question is that how do we know for certain that mathematical truths like 2+2=4 are necessary true, that is they are true in all possible world as asserted by author? For e.g., what if the 2,4 etc don't carry their usual meaning in some other possible world supposed? Or is it supposed they carry their usual meaning?

P.S.-Pardon me, I am new to logic. If i may have wrote something erroneous.

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    $\begingroup$ Because they are only consequence of axioms, i.e. they are logic consequences of choices made, so cannot be negated. They live in a universe where they are true, indipendently from the universe they are looked at, or so I see it. $\endgroup$
    – b00n heT
    Jun 17, 2016 at 18:00
  • $\begingroup$ So does that mean, first we made the choice that for e.g., this is what it means to be a number, and then this is what it means to be 1,2,etc the addition and equality. And those are some object lying in some imaginary(maybe wrong term) universe let's say. Then no matter from which possible world/universe we observe that universe the fact 2+2=4 will remain true? @b00nheT $\endgroup$
    – Someone
    Jun 17, 2016 at 18:08
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    $\begingroup$ exactly, $2+2=4$ in that universe. $\endgroup$
    – b00n heT
    Jun 17, 2016 at 19:23

2 Answers 2

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A statement such as "$2+2=4$" does not have a truth value "on its own", without anything that restricts how the symbols in it can be interpreted.
In a formal system $S$ such as Peano Arithmetic, a statement is considered to be true if it is satisfied in every model of $S$. $2+2=4$ is a theorem of Peano Arithmetic, so in that context it is true.

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Your concern is correct: to be precise, an arithmetical theorem like $2+2=4$ is not necessarily true, if we equate "necessary truth" with "logically necessary".

What we have is that $2+2=4$ necessarily follows from (or is a logical consequence of) the axioms of arithemetic (like, e.g. Peano axioms).

Thus, if we call $\mathsf {PA}$ the set of Peano axioms, we have that:

$\mathsf {PA} \vDash 2+2=4$

that read as (see Robert's answer): $2+2=4$ is true in every "world" where the axioms of arithmetic hold.

We have also:

$\vDash \mathsf {PA} \to 2+2=4$.

This reads as: the conditional $\mathsf {PA} \to 2+2=4$ is valid, i.e. is a logical truth, i.e. is true in every possible "world".

You can see Logical Truth and Varieties of Modality.

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