Why can't we have a tighter definition of what makes something necessary?

Question

Why can’t we define 'necessary' tighter, by making it include empirical evidence of such a statement?

Everything that is agreed to be necessarily can be empirically verified. For example, if we put two twos together, we get 4, and we can empirically verify this by putting two sets of 2 sweets together and getting four.

Even for logical principles such as Ockham’s Razor it is possible for them to be empirically verified by looking at the evolution of different theories (for example the decline of Caloric due to its unnecessary multiplication of entities).

The definition for necessary truth would therefore be: (a tautology/analytic truth/sound deductive argument/intuition) + empirical evidence.

There have numerous arguable “sleight of the hand tricks” regarding necessary statements (such as the "God exists necessarily"/The Ontological argument, we arguably don't have empirical evidence for God's existence), but a lot of these arguments seem to succeed in proving necessary truth, however this is entirely unfounded as far as empirical evidence goes.

Obviously verificationism/logical positivism specified for a statement to be meaningful it must be empirically verifiable, but this definition would elevate this to necessary truth itself.

This would make far more sense as when we talk about necessary truth, we are saying it is more than contingent, and therefore to make it necessary we should at least try and prove its contingency before doing so.

Answer 1

Hmm. the Verification Principle cannot be demonstrated to be logically necessary, so your elevation of it to the category of "necessary" is a weakening of the category, not a tightening. Verification has also been repeatedly demonstrated to NOT be empirically valid -- we can easily "verify" things that are not true. Confirmation bias exists and is a continual challenge for all of us, BECAUSE we are innately inclined to think using the "verification principle" and it is invalid. Popper's "refutably" challenge, plus the assistance of others to think outside our own boxes to find the refutations of our preconceptions, is how empiricism is done.

Also, your presumption that our world follows basic arithmetic is false. Your empirical test does not verify 2+2=4 is necessary. Apply falsification testing to your presumption, not confirmation bias. Take 2 crickets, and two crows, and put them in a small room with no obstacles together, and you will not have 4 items in that room in just a minute or two. Put two rabbits in a hutch with two other rabbits, come back in two months, and you will have either many rabbits if you provided them sufficient food and water, or no rabbits if you did not. The logic assumptions you presume require that A=A, and Identity is not stable for physical objects in our world, so A=A is only a temporary approximation for them.

Or, just in the world of mathematics, take 2% and 2%, add them together, and you get 3.96%. Arithmetic does not apply to all logic.

This there are, in actuality, infinite possible logics, and whether one or another approximates the behavior of some aspects of our physical world, is a contingent question, not a necessary fact.

Answer 2

Welcome.

Since you're using mathematics, let's continue.

Peano's Axioms allows us to state rigorously the conditions for the existence of natural numbers. It is necessary therefore that we recognize they exist either as an actual or potential infinity by the inductive principle. This, of course, is the sort of proof we can give to a young child. If anyone makes the claim they have the largest number, use the standard algorithm and add 1. This of course works because we are generating numbers using the successor function, so whether or not you commit to recognizing infinity as a thing or you merely fall back to seeing infinity as describing a processes, what everyone can agree on is that given PA, it is necessary that there are an infinite number of naturals.

But how would you go about proving that empirically? Well, from a radical constructivist perspective (SEP), you would have to empirically construct every number for it to exist. Yet, there are physical constraints to construction including those of time and space. Even if all of the computers in the world were dedicated to a single distributed machine sharing in the task of constructing natural numbers, necessarily we could reason that we could not succeed.

So there are limits to empirical verification rooted in the very practical consideration of the constraints that physical reality provides. In fact, there are constraints in the logical world too, such as Russell's paradox and Gödel's incompleteness theorems. So, you're on the right track to say that metaepistemologically speaking, our truths are fortified by combining empirical and rational methods, and today, most empistemologists would accept that the original debate between empiricism and rationalism has been satisfactorily dissolved. Mathematical physics is a contemporary example that one can point towards to show that knowledge is strengthened by an appeal to theories of truth that accommodate both logical coherence and physical correspondence.

It should be said, however that we can tighten up our understanding of necessity better in a different way; we can understand the various manners in which we use the term 'necessary' or 'logically implies'. Bumble's answer here addresses the intersectionality of natural language and implication. These sorts of ways are an exploration of metaphysical explanation (SEP) and are best undertaken through philosophical discourse much to the chagrin of logical postivists.