Does this reformulation of the the analytic / synthetic distinction overcome Quine's objections?

Question

He seemed to be asserting that synonymity cannot possibly be defined in a non-circular way. We attempt to show this is false on the basis of defining the Quine/Carnap example in a non-circular way.

https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism#Analyticity_and_circularity

Quine's main objection seemed to be that it is impossible to define synonymity in a non-circular way. He mentions some form of the word: "synonym" 93 times. I show how to define his example in a non-circular way thus at least overcoming this aspect of his objection.

When we specify the relevant conceptual classes

M1 is the class of all adult male humans
M2 is the class of all married adult male humans
B is the class of all unmarried adult male humans
U is the class of all unmarried adult male humans
B = M1 - M2
U = M1 - M2

B and U are verified to be synonymous on the basis that they specify the exact same conceptual class.

TWO DOGMAS OF EMPIRICISM by W. V. Quine The Philosophical Review, Vol. 60, No. 1 (Jan., 1951), pp. 20-43 https://pdfs.semanticscholar.org/675b/0ac190985cb2a91f9c8b505af25bd5b10833.pdf?_ga=2.94076253.153026009.1586290224-1393250255.1585101085

Does this reformulation of the the analytic / synthetic distinction overcome Quine's objections?

The way that I divide analytic from synthetic may be unconventional. Every aspect of knowledge that can be represented in language and encoded as strings of characters is {analytic knowledge}. Every aspect of knowledge that can only be perceived as sensations through the sense organs is {empirical knowledge}. I discard the use of the term synthetic.

This distinction between analytic and empirical seems unequivocal.

By defining the distinction this way we avoid all of the prior difficulties of specifying the meaning the word: "meaning". As long as sense data from the sense organs is not used as a basis for evaluating the truth of the expression the expression is not synthetic.

This would seem to address Quine's objections by possibly reformulating the analytic/synthetic distinction so that the original objections would no longer apply to this new reformulation. This reformulation would seem to have no undecidable boundary conditions.

By reformulating the analytic / synthetic distinction to make it unequivocal we now derive a definite basis for foundationalism when it is restricted to the analytic side of this newly reformulated analytic / synthetic distinction.

Now we attempt to show that there is at least one example of analytic knowledge that definitely meets the original analytic/synthetic distinction:

An “analytic” sentence, such as “Ophthalmologists are doctors,” has historically been characterized as one whose truth depends upon the meanings of its constituent terms (and how they’re combined) alone. https://plato.stanford.edu/entries/analytic-synthetic/

The semantic meaning of this expression proves that it is true:
Successor(Successor(0)) > Successor(0)

That its truth only depends on its semantic meaning proves that there are expressions of language that are proved to be true entirely on the basis of their semantic meaning. The above example seems to prove that there are at least some expressions of language that meet the original analytic / synthetic distinction.

Analytical_Knowledge
Is knowledge of the relations between abstract objects. The semantic meaning of an analytic expression is entirely specified by the relations that it represents. Abstract objects and their relations are always expressed as relations between expressions of language.

Because analytic knowledge is relations between abstract objects expressed using language every analytic expression can be verified as true only on the basis that it has all of the required relations.

The above Peano axiom example shows the relation between the abstract objects of {two} and {one}. There are many different ways that this relation can be encoded in language. Each one of these ways has the same semantic meaning. 二大于一

An expression of language is analytic as long as it can be verified as true entirely based on its linguistic compositional meaning., [and does not require sense data from the sense organs].

Now we focus on one single aspect of Quine's's original objection:

He seemed to be asserting that synonymity cannot possibly be defined in a non-circular way. We attempt to show this is false on the basis of defining the Quine/Carnap example in a non-circular way.

https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism#Analyticity_and_circularity

Quine's main objection seemed to be that it is impossible to define synonymity in a non-circular way. He mentions some form of the word: "synonym" 93 times. I show how to define his example in a non-circular way thus at least overcoming this aspect of his objection.

marital_status(bill, married).
marital_status(sam, single).
bachelor(X) :- \+ marital_status(X, married).

?- bachelor(bill).
false

?- bachelor(sam).
true

The above simple Prolog shows how to define bachelor(X) as synonymous with not married(X) without any cycles that the Wikipedia article about Quine's objection indicated would be required.

Quine, W. V. (1951) TWO DOGMAS OF EMPIRICISM
The Philosophical Review, Vol. 60, No. 1 (Jan., 1951), pp. 20-43
https://pdfs.semanticscholar.org/675b/0ac190985cb2a91f9c8b505af25bd5b10833.pdf?_ga=2.94076253.153026009.1586290224-1393250255.1585101085

The above paper goes on and on in very tedious great depth of its author's difficulty of defining synonymity between two terms. This is actually trivial. We simply define the meaning of one term and then define the meaning of the synonymous term on the basis of the definition of the first term as the Prolog has shown above.

Here is another way of saying the same thing
To define the synonymity of Bachelor(x) and ~Married(x) we fully elaborate all of the details of Married(x) using Rudolf Carnap (1952) Meaning Postulates then define the otherwise meaningless finite string: "Bachelor(x)" using the negation of the previously defined term Married(x).

To make this match the ordinary understanding of "Bachelor" we can augment its definition:

Bachelor(x) ≡ (~Married(x) ∧ Male(x) ∧ Human(x) ∧ Adult(x))
Bachelor derives all of its semantic meaning from terms that have been previously defined thus converting what Quine perceived as circularity into an inheritance hierarchy.

Answer 1

Actually, the claim 'Dogs generally bark' is synthetic, not analytic. How would one ever know that they generally bark without hearing them bark, and frequently (for the 'generally' part). In fact, how would one even have the concept of barking without first hearing barking, or that it is dogs that bark without hearing barking from dogs?

Answer 2

One cannot get around Quine's objection to analyticity simply by appeal to stipulated definitions. For one thing, the vast majority of words in a natural language such as English don't have stipulated definitions. Carnap is not a deity who hands down definitions on tablets of stone that we are obliged to use. Lexicographers do not stipulate definitions when they compile dictionaries; they merely document the way that English words are conventionally used by English speakers. Stipulative definitions typically only arise in technical disciplines where a term of art is needed. Nobody can stipulate the meaning of 'bachelor' - it is the property of the entire community of English speakers, and they will use it as they see fit.

Consider that the uses of words frequently drift over time. Three hundred years ago, to be a 'gentleman' a man needed to be landowner, but now he does not. The world did not wake up one day and decide to change the definition; its use just evolved slowly. As Wittgenstein noted, meanings are often more a matter of family resemblance than of necessary and sufficient conditions. I have even seen the word 'bachelor' used to refer to a married man. A society magazine that I picked up in a doctor's surgery featured an article about the ten richest bachelors in the UK, and one of them was married. The author justified his inclusion on the grounds that he was going through a divorce and would be unmarried soon. One could say that this was a misuse of the term, or that the meaning was being extended to cover men who are either unmarried or soon will be. But if that use were to catch on with other English speakers, lexicographers would eventually have to update their dictionaries. Natural meanings just evolve naturally.

The situation is not much different with technical disciplines. One might suppose that if scientists define a bunch of technical terms in a particular way then the relationships between them are analytic. But scientific knowledge, like everthing else, evolves over time and frequently requires adjustments to the terminology. An example that Quine gives is that in classical physics momentum was defined to be mass times velocity, but in the light of special relativity we now define momentum to be mass times velocity times the Lorentz factor, and our reasons for doing so are ultimately empirical. Our definitions are eligible for revision in the light of empirical discoveries just as much as the equations are. It would be pointless to claim that it used to be analytic and hence a priori knowable that p = mv, and now it is analytic and a priori knowable that p = γmv. The fact is we used to think that p = mv was true and in the light of empirical data we now think it is merely a limiting case. There is nothing analytic or a priori about it. Maybe the same will happen in future with other scientific terms. Who can say? We cannot anticipate all possible future discoveries.

The only place where one might plausibly retain analyticity is in mathematics. If a triangle is given the conventional definition then it is analytic that a triangle has three sides. Even here, Quine has his reservations. For Quine, mathematics pays its epistemological debts by the contribution it makes to science. The reason we are in practice highly unlikely to want to revise mathematical definitions is because in the holistic scheme of things, our beliefs about mathematics are well-protected from revision, though perhaps not entirely immune.

Answer 3

The problem is that, "X is stipulated to refer to Y," is itself synthetic. We could've stipulated a different reference for it. Quine is ultimately saying less that "there are no analytic truths" and more that "since all meaning is synthetic, analytic truth is at best a subset of synthetic truth, not opposite to it," so then plug in the paradox of analysis on its own terms and voila, what reason is there to dwell on "analytic truths" as such?

Kant, of course, anticipated all this; despite saying there is no analytic aposteriority in itself, he at one point speaks of giving something analytically and a posteriori that was first given synthetically a priori, and as his discussion of analysis of the concept and word "water" shows, he also anticipated Kripke's theory about, "Water is H2O," but so anyway the point is that a truth being analytic turns out to be somewhat vague and trivial and maybe even irrelevant to major epistemological problems.

With respect to the successor operation example, the dream was to start from the simplest possibility and "logically construct" the natural numbers therefrom. But there is a circle, here: if we define succession as a unary operation, we have smuggled the number 1 into our system through the back door. But anyway, we might say, "When 1 is defined as the successor of 0, then by definition 1 + 1 = the successor of 1 = our definition of 2," but if numbers have their own reality over and above this relation, then our definitions, though formally clever, are inadequate to the reality. (This was the point of Kant's claim that arithmetic is synthetic: not that adding 1 to 1 might somehow somewhere = something besides 2 in particular, but rather that the concept of numbers as more than variations on 1 + 1 + ... will never appear analytically, so 1 + 1 = 1 + 1, and there is no resolution of the additive symbolism. To wit, we would've solved the Continuum Hypothesis by now if arithmetic was analytic as we would just analyze 2^aleph0 and presto, we would see which other aleph this was by "definition" (keep in mind that exponentiation is in part the successor of another operation, in a series meant to collapse at 'zeration,' so the same logic that "defines" numbers from successions of 0 would then "define" the alephs likewise, so that transfinite exponentiation should be completely reducible to transfinite succession).)

EDIT: or consider that 2 + 2 = 2*2 = 2^2 = ... = 4. As far as I know, even if we use transfinite levels of these operations, (2, 2) is always 4. This seems to be true given how the operations relate in general, since "x + x = x * 2, x * x = x^2," etc., so 2 of course itself never varies in the series as such, and we might say this is true by definition. But it seems vaguely possible that a hyperoperator indexed by the initial ordinal of an inaccessible cardinal might actually break the symmetry, here; at least, I don't want to just "stipulate" not.