Marcus analyzed this problem in 1961, which she called the "familiar example,"

(27) 9 eq the number of planets

is said to be a true identity for which substitution fails in

(28) ◻(9 > 7)

for it leads to the falsehood

(29) ◻(the number of planets > 7).

Since the argument holds (27) to be contingent (~ ◻(9 eq the number of planets)), 'eq' of (27) is the appropriate analogue of material equivalence and consequently the step from (28) to (29) is not valid for the reason that the substitution would have to be made in the scope of the square.

("Modalities and Intensional Languages," in Synthése, Vol. 13, No. 4 (Dec., 1961), p. 313)

The failure of substitutivity can be understood by unpacking the use of "the number of planets" as a purely designative reference, as Quine calls it.

In (27), "the number of planets" is the empirical answer to the question "how many planets are there in the solar system?" It is not what Marcus would call a "tag" of the number 9. The intension of this expression, its reference, is the "extra-linguistic" fact about the current quantity of planets (which Quine appreciated).

The expression '9' is an unambiguous mathematical (logical) reference to the number 9. It refers to the number 9, which is its meaning (intension).

We can conclude that (27) is not a true identity, unless before "the number of planets" is quantified, it is qualified as "the number of planets qua its numerosity, as a pure number." Otherwise, the reference is "opaque," as Quine describes it. But this is a problem of his own making.

As Marcus says, when we recognize (27') as contingent, ~◻(9 eq the number of planets), it is not necessary that 9 is equal to the number of planets, its reference to the number 9 becomes opaque.

The substitution of a possible or contingent empirical fact that is not "true in all possible worlds" for a logical-mathematical concept that is necessarily true is what causes the substitution failure.

When all three statements are "in the scope of the square" (◻), when all have the same modality, we can "quantify into modal contexts," as Quine puts it. Both expressions,
'9' and 'the number of planets, qua its numerosity,' will be references to the same thing,
They will be identical in one respect, qua number. They will be "referentially transparent."

In her third article back in 1947, "The Identity of Individuals," Barcan had first proved the necessity of identity. This result became a foundational principle in the modern incarnation of Leibniz's "possible worlds" by Saul Kripke and David Lewis

Her proof combined a simple substitution of equals for equals and Leibniz's Law.

Quine described in his 1953 Reference and Modality (p.153) as in the form

which reads "for all x and for all y, if "x = y," then necessarily "x = y."

Quine found this relationship in the 1952 Textbook, Symbolic Logic, by F. B. Fitch, who was Ruth Barcan's thesis adviser. Although Fitch mentions her work in his foreword, he does not attribute this specific result to her where he presents it. His proof is based on the trivial assumption of substitutability, which he calls "identity elimination."

Then in 1961, Marcus published a three-step proof of her claim, using Leibniz's Law relating identicals to indiscernibles.

In a formalized language, those symbols which name things will be those for which it is meaningful to assert that I holds between them, where 'I ' names the identity relation... If 'x' and 'y' are individual names then

(1) x I y

Where identity is defined rather than taken as primitive, it is customary to define it in terms of indiscernibility, one form of which is

(2) x Ind y =_df (φ)(φx eq φy)

("Modalities and Intensional Languages," in Synthése, Vol. 13, No. 4 (Dec., 1961), p. 305)

Statement (2) says that the indiscernibility of x from y, by definition means that for every property φ, both x and y have that same property, φx eq φy.

A few years after Marcus' 1962 presentation, David Wiggins developed a five-step proof of the necessity of identity, using Leibniz' Law, as had Marcus. He did not mention her.

David Wiggins and Peter Geach debated back and forth about the idea of "relative identity" for many years after Geach suggested it in 1962.

Ruth Barcan Marcus pubilshed her original proof of the necessity of identity in 1947 and repeated her argument at a 1961 Boston University colloquium.

Whether Wiggins knew of Marcus 1961 is not clear. He should have known of her 1947, and there is similarity to her 1961 derivation (which uses Leibniz's Law). Wiggins gives no credit to Marcus, a pattern in the literature for the next few decades and still seen today.

Saul Kripke clearly modeled much of his derivation after Wiggins, especially his criticism of the derivation as "paradoxical". Kripke gives no credit to either Marcus or Wiggins for the steps in the argument, but his quote from Wiggins, that such a claim makes contingent identity statements impossible, when they clearly are possible, at least tells us he has read Wiggins. And we know Kripke heard Marcus present at the 1961 colloquium.

Here is Wiggins (1965),

I WANT to try to show (i) that there are insuperable difficulties any term + relation + term or subject + predicate analysis of statements of identity, (ii) that, however important and helpful the sense-reference distinction is,¹ this distinction does not make it possible to retain the relational or predicative analysis of identity statements, and (iii) that a realistic and radically new account is needed both of ' = ' and of the manner in which noun-phrases occur in identity-statements.

Till we have such an account many questions about identity and individuation will be partly unclear, and modal logics will continue without the single compelling interpretation one might wish.

The connexion of what I am going to say with modal calculi can be indicated in the following way. It would seem to be a necessary truth that if a = b then whatever is truly ascribable to a is truly ascribable to b and vice versa (Leibniz's Law). This amounts to the principle

(1) (x)(y) ((x = y) ⊃ (φ)(φx ≡ φy))

Suppose that identity-statements are ascriptions or predications. Then the predicate variable in (1) will apparently range over properties like that expressed by ' ( =a) ' ² and we shall get as consequence of (1)

Note that Wiggins predicates the property "= x" of y. Kripke writes this as "x = y," logically equivalent, but intensionally predicating "= y" of x!

(2) (x)(y) ((x = y) ⊃ (x = x. ⊃ . y = x))

There is nothing puzzling about this. But if (as many modal logicians believe), there exist de re modalities of the form

◻ (φa) (i.e., necessarily (φa)),

then something less innocent follows. If '( = a) ' expresses a property, then '◻ (a = a)', if this too is about the object a, also ascribes something to a, namely the property ◻ ( = a). For on a naive and pre-theoretical view of properties, you will reach an expression for a property whenever you subtract a noun-expression with material occurrence (something like ' a ' in this case) from a simple declarative sentence. The property ◻ ( = a) then falls within the range of the predicate variable in Leibniz's Law (understood in this intuitive way) and we get

Note (3) is almost Kripke's (3), but with intensional "y = x." Wiggins needs one more step. His (4) is Kripke's (3)

(3) (x) (y) (x = y ⊃ (◻ (x = x). ⊃ .◻ (y = x)))

Hence, reversing the antecedents,

(4) (x) (y) (◻(x = x ). ⊃ (x = y) ⊃ ◻ (x = y)))

But '(◻ (x = x)) ' is a necessary truth, so we can drop this antecedent and reach

(5) (x) (y) ((x = y). ⊃ .◻ (x = y)))

Now there undoubtedly exist contingent identity-statements. Let 'a = b' be one of them. From its simple truth and (5) we can derive '◻ (a = b)'. But how then can there be any contingent identity-statements?...

4. The derivation of (2) itself, via x's predicate ' ( = x)', might be blocked by insisting that when expressions for properties are formed by subtraction of a constant or free variable, then every occurrence of that constant or free variable must be subtracted. '( a = a )' would then yield ' ( = )', and (2) could not be derived by using ' ( = x ) ' . One would only get the impotent

(2') (x = y) ⊃ (x = x. ⊃ . y = y)

The paradox could still be derived however. Suppose that a is contingently b. Then <> ~{a=b); i.e., it is possible that not a=^H This gives the predicate '◇ ~ (a = ) ' . This is true of b. Then by (1), if a = b, this predicate is true also of a. This yields '◇ ~ (a = a) '. But ' (x) ◻(x = x)' is a logical truth and implies ' ~ ◇ ~ (a = a)'.

1 G. Frege, 'On Sense and Reference ', Translations from the Philosophic Writings of Gottlob Frege, ed. P. T . Geach and M. Black (Oxford, 1952), pp. 56-4]
2 Quotation marks are used under the convention that they serve to form a designation of whatever expression would result in a particular case from rewriting the expression within the quotation-marks with genuine constants in the place of free variables and dummy-expressions.

("Identity Statements," in Analytical Philosophy, Second Series, Oxford: Blackwell. pp.40-41
)

Kripke simplifies Wiggins (1965). We can compare the two expositions:

Kripke does not cite Wiggins as the source of the argument, but just after his exposition above, Kripke quotes David Wiggins as saying in his 1965 "Identity-Statements"

Now there undoubtedly exist contingent identity-statements. Let a = b be one of them. From its simple truth and (5) [= (4) above] we can derive '◻ ( a = b)'. But how then can there be any contingent identity statements?

(Identity and Necessity, p. 136)

Kripke goes on to describe the argument about b sharing the property " = a" of being identical to a, which we read as merely self-identity, and so may Kripke.

If x and y are the same things and we can talk about modal properties of an object at all, that is, in the usual parlance, we can speak of modality de re and an object necessarily having certain properties as such, then formula (1), I think, has to hold. Where x is any property at all, including a property involving modal operators, and if x and y are the same object and x had a certain property F, then y has to have the same property F. And this is so even if the property F is itself of the form of necessarily having some other property G, in particular that of necessarily being identical to a certain object. [viz., = x]

Well, I will not discuss the formula (4) itself because by itself it does not assert, of any particular true statement of identity, that it is necessary. It does not say anything about statements at all. It says for every object x and object y, if x and y are the same object, then it is necessary that x and y are the same object. And this, I think, if we think about it (anyway, if someone does not think so, I will not argue for it here), really amounts to something very little different from the statement (2). Since x, by definition of identity, is the only object identical with x, "(y)(y = x ⊃ Fy)" seems to me to be little more than a garrulous way of saying 'Fx' and thus (x) (y)(y = x ⊃ Fx) says the same as (x)Fx no matter what 'F' is — in particular, even if 'F' stands for the property of necessary identity with x. So if x has this property (of necessary identity with x), trivially everything identical with x has it, as (4) asserts. But, from statement (4) one may apparently be able to deduce various particular statements of identity must be necessary and this is then supposed to be a very paradoxical consequence.

(Identity and Necessity, p. 137-138)

The indiscernibility of identicals claims that if x = y, then x and y must share all their properties, otherwise there would be a discernible difference. Now Kripke argues that one of the properties of x is that x = x, so if y shares the property of '= x," we can say that y = x. Then, necessarily, x = y.

However, two distinct things, x and y, cannot be identical, because there is some difference in extrinsic external information between them. Instead of claiming that y has x's property of being identical to x ("= x") , we can say only that y has x's property of being self-identical, thus y = y. Then x and y remain distinct in at least this intrinsic property as well as in extrinsic properties like their distinct positions in space.

David Lewis, the modern metaphysician who built on Leibniz' possible worlds to give us his theory of "modal realism," is just as clear as Leibniz on the problem of identity.

[W]e should not suppose that we have here any problem about identity. We never have. Identity is utterly simple and unproblematic. Everything is identical to itself; nothing is ever identical to anything else except itself. There is never any problem about what makes something identical to itself, nothing can ever fail to be. And there is never any problem about what makes two things identical; two things never can be identical.

("Counterparts or Double Lives,"in Rea, M.C., ed., Material Constitution, p.126)

Except, says an information philosopher, "in some respects."

In the "semantics of possible worlds," necessity and possibility in modal logic are variations of the universal and existential quantifiers of non-modal logic. Necessary truth is defined as "truth in all possible worlds." Possible truth is defined as "truth in some possible worlds." These abstract notions about "worlds" – sets of propositions in universes of discourse – have nothing to do with physical possibility, which depends on the existence of real contingency.

Propositions in modal logic are required to be true or false. Contingent statements that are neither true or false are not allowed. So much for real possibilities, which cannot be based on truths in some possible worlds.

Historically, the opposition to metaphysical possibility has come from those who claim that the only possible things that can happen are the actual things that do happen. To say that things could have been otherwise is a mistake, say eliminative materialists and determinists. Those other possibilities simply never existed in the past. The only possible past is the past we have actually had.

Similarly, there is only one possible future. Whatever will happen, will happen. The idea that many different things can happen, the reality of modality and words like "may" or "might" are used in everyday conversation, but they have no place in metaphysical reality. The only "actual" events or things are what exists. For "presentists," even the past does not exist. Everything we remember about past events is just a set of "Ideas." And philosophers have always been troubled about the ontological status of Plato's abstract "Forms," entities like the numbers, geometric figures, mythical beasts, and other fictions.

Traditionally, those who deny possibilities in this way have been called "Actualists."

In the last half-century, one might think that metaphysical possibilities have been restored with the development of modal logic. So-called modal operators like "necessarily" and "possibly" have been added to the structurally similar quantification operators "for all" and "for some." The metaphysical literature is full of talk about "possible worlds."

The most popular theory of "possible worlds" is David Lewis's "modal realism," an infinite number of worlds , each of which is just as actual (eliminative materialist and determinist) for its inhabitants as our world.

Modal logicians now speak of a "rule of necessitation" at work in possible world semantics.The necessarily operator ' ◻ ' and the possibly operator ' ◇ ' are said to be "duals" - either one can be defined in terms of the other (◻ = ~◇~, and ◇ = ~◻~), so either can be primitive. But most axiomatic systems of modal logic appear to privilege necessity and de-emphasize possibility. They rarely mention contingency, except to say that the necessity of identity appears to rule out contingent identity statements.

The rule of necessitation is that "if p, then necessarily p," or p ⊃ ◻p. It gives rise to the idea that if anything exists, it exists necessarily. This is called "necessitism." The idea that if two things are identical, they are necessarily identical, was "proved" by Ruth Barcan Marcus in 1947, by her thesis adviser F.B.Fitch in 1952, and by Willard Van Orman Quine in 1953. David Wiggins in 1965 and Saul Kripke in 1971 repeated the arguments, with little or no reference to the earlier work.

This emphasis on necessitation in possible-world semantics leads to a flawed definition of possibility that has no connection with the ordinary and technical meanings of possibility.

Modal logicians know little if anything about real possibilities and nothing at all about possible physical worlds. Their possible worlds are abstract universes of discourses, sets of propositions that are true or false. Contingent statements, that may be true or false, like statements about the future, are simply not allowed.

They define necessary propositions as those that are "true in all possible worlds." Possible propositions are those that are only "true in some possible worlds." This is the result of forcing the modal operators ◻ and ◇ to correspond to the universal and existential quantification operators for all ∀ and for some ∃. But the essential nature of possibility is the conjunction of contingency and necessity. Contingency is not impossible and not necessary (~~◇ ∧ ~◻).

We propose the existence of a metaphysical possibilism alongside the notion necessitism.

Actual possibles can act as causes when an agent chooses one as a course of action.

There are two reasons for the failure of modal logic to represent metaphysical reality. The first is that information is vastly superior to language as a representation of reality. The second is that truths and necessity cannot be the basis for metaphysical possibility.

Possible world semantics is a way of talking about universes of discourse - sets of true propositions - that considers them “worlds.” It may be the last gasp of the attempt by logical positivism and analytic language philosophy to represent all knowledge of objects in terms of words.

Ludwig Wittgenstein’s core idea from the Tractatus had the same goal as Gottfried Leibniz’s ambiguity-free universal language,

4.11 The totality of true propositions is the total natural science (or the totality of the natural sciences).

(Tractatus Logico-Philosophicus)

Information philosophy has shown that the meaning of words depends on the experiences recalled in minds by the Experience Recorder and Reproducer. Since every human being has a different set of experiences, there will always be variations in meaning about words between different persons, as Gottlob Frege pointed out.

The goal of intersubjective agreement in an open community of inquirers hopes to eliminate those differences, but representation of knowledge in words will always remain a barrier and source of philosophical confusion. The physical sciences use analytic differential equations to describe the deterministic and continuous time evolution of simple material objects, which is a great advance over ambiguous words. But these equations fail at the quantum level and where discrete digital messages are being exchanged between biological interactors. Moreover, while mathematical methods are precise, their significance is not easily grasped.

The very best representation of knowledge is with a dynamic and interactive model of an information structure, what Wittgenstein imagined as a “picture of reality.” Today that is a three-dimensional model implemented in a digital computer with a high-resolution display, even a virtual reality display. While computer models are only “simulations” of reality, they can incorporate the best “laws” of physics, chemistry, and biology.

Sadly, modal logicians have never proposed more than a handful of specific propositions for their possible worlds, and many of these generated controversies, even paradoxes, about substitutivity of presumed identicals in modal contexts. Word and object have degenerated to words and objections. By comparison, molecular models of the extraordinary biological machines that have evolved to keep us alive and let us think can be “shown,” not said, as Wittgenstein imagined.

His later work can be summed up as the failure of language to be a picture of reality. Information philosophy gives us that picture, not just a two-dimensional snapshot, but a lifelike animation and visualization of the fundamental nature of metaphysical reality.

Our information model incorporates the irreducible ontological chance and future contingency of quantum physics. The claimed “necessity of identity,” and the “necessary a posteriori” of natural and artificial digital “kinds” with identical intrinsic information content are just more “ways of talking.” There is no necessity in the physical world.

Truths and necessity are ideal concepts “true in all possible worlds,” because they are independent of the physical world. They have great appeal as eternal ideas and “outside space and time.”

Possible worlds semantics defines necessity as “propositions true in all possible worlds” and possibility as “propositions true in some possible worlds.” There is no contingency here, as the only allowed propositions are either true or false. Modal logicians have little knowledge of our actual physical world and zero factual knowledge, by definition, of other possible worlds. The possible worlds of “modal realism” are all actual worlds, deterministic and eliminatively materialist. There are no possibilities in possible worlds, even the “many worlds” of physics.

A necessicist metaphysics is only a half-truth. Without metaphysical possibility, we cannot account for the information in the universe today, nor can we explain the cosmic, biological, and human creation of new information in our free and open future.

Necessitism and possibilism can be considered as another variation of the great duals of idealism and materialism or the One and the Many.