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Here is an observation:

Things that exist are verifiable, i.e., there is some external object, within its scope of accessibility, that can construct its information. By ‘construct its information’, I mean that the inherent information, substance, or material constituting that thing impinges on that external object, and in a sense, the external object serves as an observer on that original thing, a reference frame or point of view.

On the other hand, if you define axioms, whose consequences imply the existence of something - something defined by the axioms of that formal system - we say that an object exists, within the scope of that deductive system. The ‘verification’ of said object is per derivation of the axioms; they are what speak to the existence of the object. Furthermore, this verification is trivial.

In both cases (which serve as possible frameworks to regard a more general question from), it seems as though if existence is a property of objects, then it must provide a semantic relation that maps between any two distinct objects. In other words, to say something exists, is to say that exists relationally: from the information-theoretic context, it exists in relation to the object ‘perceiving’ it; in the logical paradigm, it exists as a construction of basic notions in the axiom, so exists in relation to them.

Now, suppose it were not so.

Then, only the trivial semantic relation, f:X -> X remains. In other words, if an existing object has no “relation” to anything else, we could only know it or define it in some way, in relation to itself; and not to any other thing.

However, there are logical models, where the definition of an object, is not consistent with its mere postulation. That is, you may be able to claim the said object exists, without being able to give defining properties of what it is; or, you may be able to assert certain definitional properties of a thing, without being able to assert that it exists. This can lead to inconsistency in the logic, in scenarios where being able to define something would imply existence, and vice versa.

For instance, you can define the set of all sets that don't contain themselves.

Since it trivially implements identity, it should exist. That is, it should exist insofar as it is a definition one can formulate: and it is defined in relation to itself (a set of sets).

However, you can also construct naive set theory, and thus you derive a contradiction (that the set of all sets cannot exist, which is Russell’s paradox).

Thus, in general, postulating something into existence via the trivial (reflexive) semantic relation - that it exists in relation to itself - has as prerequisite the aforementioned ‘consistency’ (relation) between objects - the consistency between definition and mere existence, that if and only if something is definable, then it exists. This is because if something is defined in relation to itself, then it necessarily already exists; (unlike other definitions where we give a new name to an arrangement of other, pre-existing things, like, “a cat is a mammal with short hair”.)

However, the principle relating definability and existence is already a semantic relation between any two (distinct) objects. So even if we try to avoid taking relations as foundational, we end up rediscovering it embedded in our axiomatic system.

It follows that a trivial semantic is not enough to define an object consistently, before existence can be established.

And if it is not the trivial semantic due to being preceded by consistency (as a relation), then any object that exists must implement some semantic relation that is a map between different objects.

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  • "if existence is a property of objects, then it must provide a semantic relation that maps between any two distinct objects" What does it mean? IF existence is a property of objects, then we have "fictional" objects that do not exist in the real world, and "real" objects that exist. Jan 10 at 10:14
  • "For instance, you can define the set of all sets that don't contain themselves. Since it trivially implements identity, it should exist." Why? it is not enough to "define" something in order to conjure it into existence: see unicorn... try to "redefine" the total amount of money on your bank account. Jan 10 at 10:15
  • Abstract theories, like mathematical ones: we set up a theory writing axioms and we have that IF the theory is consistent we say that a model of the theory is every domain of objects and properties satisfying the axioms. "Common" reality: we do not "prove" that objects and facts exist: we have evidence either supporting or refuting their existence. Jan 10 at 10:18
  • Re "the trivial semantic relation f : X → X", Semantics in its essence is the relation between a word and an object (between a sentence and its meaning). Obviously, words (sentences) are also objects of some kind, but the semantic relation arise exactly when we use some object "in place of" another (different) one. Jan 10 at 10:48
  • I fear that you have awakened the sleeping Mauro.
    – Scott Rowe
    Jan 10 at 11:34

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I hope to see the question continually developed because I think there’s good substance here, but we haven’t gotten it in its clearest form as of yet.

That said, your question reminds of one thing I learned recently.

The following may be full of minor inaccuracies, I speak from a general impression I have received.

Set theory can be largely seen as a hardcore reductionism program whereby all logical (and eventually semantic and meaningful) concepts are reduced the structures that merge from the composition of a most basic conceptual element, that of ‘sets’, or, “putting some things together”.

Unfortunately, the abundant amount of work that has been done in set theory in the past century has revealed that this aspiration was not as simple as it seemed; there are paradoxes to be avoided; and it lead to a more sophisticated and particular theory that aims to be a definitional, construction-oriented system for defining basically everything (or, logical and mathematical notions, at a minimum) - for example, ZFC (Zermelo Fraenkel set theory with the axiom of choice, which consists of roughly 10 axioms, depending on how you count them).

Now from what I understand, those kinds of paradoxes that Russell encountered motivated him to develop type theory. Arguably, type theory introduced a little more ‘conceptual assumptions’ down at the bottom - instead of there being just one kind of ‘thing’ (a ‘set’), now there could be varying types of things (and this aimed to avoid paradoxes where you can’t discuss how many of such a thing there are, because no matter how many there were, there would always be the totality of those currently existing things, as another thing in itself.) This means that the notion of there being “different kinds of things” was no longer a posteriori, no longer emerged from more basic notions, but was taken as primary. For someone who wants to define everything in as totalistic a way as possible, this feels bothersome: how can we just assume something like that as fundamental? It doesn’t explain where it comes from, how it got there or came to be in the first place.

Type theory was developed further by Per Martin-Löf. He created yet another deductive system. One of the features of his system is that is also takes functions as primary. In set theory, functions are defined as collections of particular elements (the elements being, respectively, collections too). In Martin-Löf (or “intuitionistic”) type theory, against, they are taken as primary, not requiring any further definition or construction.

I’d like to tell more about intuitonistic type theory, but I don’t know any more (for now). But it reminded me of what you were talking about, and maybe something you would be interested in: a deductive system that takes “relationality” as inescapably primary, or primitive; and not requiring further definition, because it would be impossible to do so without already having such a concept to begin with.

Thus, I used to think of primitive notions as the most basic, fundamental things a person could conceive of as existing, if they rejected as many other assumptions as possible. But recently, I saw them a bit differently: they are the concepts you cannot escape, even if you try to; so even if it does not appeal to your philosophical sensibility, you find you are forced to, even if you don’t (yet) fully understand why.

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