Is the existence (existence or subsistance) of a thing an analytic proposition?
If I talk about a thing x, any thing, and say "x exists", would it be an analytic proposition? I think it would be because you can't really talk about a thing without it existing in some sense, and thus the existence of the thing, I think, is contained in the thing's definition, thus making "x exists" an analytic proposition.
Am I correct?
No, existence does not seem to me analytic.
Your argument for the analyticity of existence relies on extraction from context, like this:
a exists "in some sense" => a exists
If this were a valid inference, we would have also, by analogy:
a doesn't exist "in some sense" => a doesn't exists
Now, for any controversial object, e.g. Batman, there will be a sense (s1) in which it exists, and another sense (s2) in which it doesn't exist. So we have:
Batman exists in sense s1 => Batman exists
But also
Batman doesn't exist in sense s2 => Batman doesn't exist
And from these two, we have
Batman both exists and doesn't exist
Which is absurd.
To prevent such absurd conclusions, we need to disallow the extraction from context. So that, for example
That Batman exists "in some sense" does not imply that Batman (simply) exists
By the way, Kant, who inserted the analytic/synthetic distinction into the philosophical dictionary, was also of the clear opinion that judgments about existence were synthetic.
For the real object ... is not analytically contained in my conception, but forms a synthetical addition to my conception (which is merely a determination of my mental state) ... Whatever be the content of our conception of an object, it is necessary to go beyond it, if we wish to predicate existence of the object. (Critique of Pure Reason, "Of the Impossibility of an Ontological Proof of the Existence of God")
It's traditional to translate "x exists" into classical logic as "x is something"or "(Ey)x=y". In this case, the translation of "something exists" is "something is something" or "(Ex)(Ey)x=y", which is a classical tautology. So classically, it is analytic.
But it's intuitively possible for nothing to have existed. Because of this, there are various free logics, in which "something exists" is not a tautology. So the answer is controversial - it depends on which logic you take to be correct.
Kant's analytic-synthetic division was attacked by Quine in his paper "Two Dogmas of Empiricism" if you want to know more. I recommend reading it to get a handle on the distinction.
He conflated Kant's analytic a priori and synthetic a priori (true by definition and true by reason alone but not just true by definition respectively) to mean the same thing — to both be analytic statements. He goes further to state that this distinction is a 'dogma' and gives counterexamples to show how the distinction can break down. One example he gives is how "the morning star" and "the evening star" have acquired their meaning by recourse to experience and also by logic alone. That is the planet Venus is BOTH analytic and synthetic. He also cites the number of planets in the solar system as another example. The other 'dogma' he cites is the way we talk about disconfirmations about a theory about the world in relation to the entire theory or web of knowledge (a type of coherentism). The major weight behind his conclusions is that it could imply that all statements are subject to revision. Even those based purely on logic. Examples include quantum logic which was a new form of logic invented or discovered that went beyond the truth-falsity of two-valued truth values. And statements here might not just be true or false, they could be 'both' like the Schrodinger cat thought experiment that is supposed to give some crude insight into quantum mechanics. But basically particles can be in a superposition etc.
Is existence a predicate? Does it state something about a subject of a sentence? Or is this just a tautology as others have mentioned.
Existence, if you accept, is a property. Therefore a predicate. Everything "exists" in or can be represented in either an abstract or empirical way. Whatever ontology and metaphysical view you take will determine which categories you decide will be appropriate here. And this all depends on your definition of "exist" and "reality". If you accept we can perceive reality (or whatever limited version we can access), then when we make a claim about an object in our reality we are making an existence claim about it. Whether we are mistaken about this we can try to find out by using science, observation, or reason. Pure use of human reason is often only appropriate for understanding the formal sciences such as mathematics as the others depend on empirical methodologies to verify their validity. We don't really accept the existence of entities and phenomena outside of the scientific method unless you are a mystic or believe in deities.
The statement "The current King of France is Bald" is false. Not because he is bald, but because he does not "exist" in any empirical way. The existential quantifying that is embedded in natural language explicated by Frege originally is of use here (when we are talking about non-existent entities). Some philosophers, however, I believe Wittgenstein was one of the first, claim that some statements lack meaning as opposed to being meaningless: Unsinnig is the term he used to describe it. "The chair exists", "Two is a number", "Socrates is", and "There are objects" all equally lack meaning because they do not state anything about the subject of the sentence beyond that it exists. Only that there is an object. He thought we could only talk about how things are rather than what things are. This was explored in his Tractatus.
This builds upon relational predicate logic (e.g. x stands in relation to y) and can be a way we think about how our language is used to form linkages with all things we know about or claim to know about. The dog stands in relation to the fridge, the fridge to the floor, the floor to splinters of wood, splinters of wood to atoms, atoms to subatomic particles, subatomic particles to strings (we think might be the case), and so on.
Also, very loosely, philosophers use the term "analytic" in different ways. Most would think that an "analytic statement" to be along the lines of what I outlined (for analytic philosophers at least). Others may think that an "analytic statement" is more-so a style of philosophical thinking that includes a close analysis of logic, linguistics, semantics, syntax, and so on.
References:
Quine, W. V. O. (1951). "Two Dogmas of Empiricism". The Philosophical Review. 60 (1): 20–43. doi:10.2307/2181906. JSTOR 2181906. Reprinted in his 1953 From a Logical Point of View. Harvard University Press.
"Wittgenstein's Logical Atomism (Stanford Encyclopedia of Philosophy)". https://plato.stanford.edu/entries/wittgenstein-atomism/#1
