Ruth Barcan Marcus
Ruth Barcan Marcus was a philosopher of logic who restored the "modal" concepts of necessity and possibility to the "quantified" logic that analyzes truths in terms of set membership. She is said to have created "quantified modal logic."

The original predicate logic developed by Aristotle in his Prior Analytics contained the quantifiers "for some" and "for all" - e.g., all men are mortal. Aristotle's logic also contained the modal notions of necessity ("must") and possibility ("may"), but modality had disappeared from texts on symbolic logic since Gottlob Frege and Bertrand Russell had reduced philosophy to a "truth-functional" analysis of statements that are plain assertions - true or false.

For them and for Ludwig Wittgenstein and Rudolf Carnap, all of knowledge in general and science in particular is reducible to the collection of all true statements.

C. I. Lewis reinvented modal logic in the 1920's and tried to add it to the symbolic logic of the great Principia Mathematica of Russell and Alfred North Whitehead.

Willard Van Orman Quine mostly ignored Lewis's modal logic, and he reacted negatively to Marcus's suggestion in 1946 that modality operators (a box '◻' for "necessarily" and a diamond '◇' for "possibly") could be transposed or interchanged with quantification operators (an inverted A '∀' for "for all" and a reversed E '∃' for "for some"), while preserving the truth values of the statements or propositions.

Marcus asserted these transpositions in what are now called the "Barcan formulas."

∀x ◻Fx ⊃ ◻ ∀x Fx       ∀x ◇Fx ⊃ ◇ ∀x Fx

∃x ◻Fx ⊃ ◻ ∃x Fx       ∃x ◇Fx ⊃ ◇ ∃x Fx

Quine had generated a number of apparently paradoxical cases where truth value is not preserved when "quantifying into a modal context." But these can all be understood as a failure of substitutivity of putatively identical entities. Information philosophy shows that two distinct expressions that are claimed to be identical are often not identical in all respects, e.g., reference and sense. So a substitution of one expression for the other may not be identical in the relevant respect. Such a substitution can change the meaning, the intension of the expression.

Perhaps Quine's most famous paradox is this argument about the number of planets:

(1) 9 is necessarily greater than 7

for example, is equivalent to

'9 > 7' is analytic

and is therefore true (if we recognize the reducibility of mathematics to logic)...

Given, say that

(2) The number of planets is 9

we can substitute 'the number of planets' from the non-modal statement (2) for '9' in the modal statement (1) gives us the false modal statement

(3) The number of planets is necessarily greater than 7

But this is false, says Quine, since the statement

(2) The number of planets is 9

is true only because of circumstances outside of logic.

Marcus analyzes this problem in 1961, which she calls 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.

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 Kripke would call a "rigid designator" 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 now accepts. 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."

Names and Necessity
In his 1943 paper in the Journal of Philosophy, "Notes on Existence and Necessity," Quine wrote:
One of the fundamental principles governing identity is that of substitutivity – or, as it might well be called, that of indiscernibility of identicals. It provides that, given a true statement of identity, one of its two terms may be substituted for the other in any true statement and the result will be true. It is easy to find cases contrary to this principle...

The principle of substitutivity should not be extended to contexts in which the name to be supplanted occurs without referring simply to the object.

The relation of name to the object whose name it is, is called designation; the name 'Cicero' designates the man Cicero. An occurrence of the name in which the name refers simply to the object designated, I shall call "purely designative." Failure of substitutivity reveals merely that the occurrence to be supplanted is not purely designative, and that the statement depends not only upon the object but on the form of the name. For it is clear that whatever can be affirmed about the object remains true when we refer to the object by any other name.

To say that two names designate the same object is not to say that they are synonymous, that is, that they have the same meaning. To determine the synonymity of two names or other expressions it should be sufficient to understand the expressions; but to determine that two names designate the same object, it is commonly necessary to investigate the world. The names 'Evening Star' and 'Morning Star', for example, are not synonymous, having been applied each to a certain ball of matter according to a different criterion. But it appears from astronomical investigations that it is the same ball, the same planet, in both cases; that is, the names designate the same thing. The identity: Evening Star = Morning Star is a truth of astronomy, not following merely from the meanings of the words.

In 1947, Marcus wrote an article on "The Identity of Individuals " asserting the "necessity of identity." Her work was written in the dense expressions of symbolic logic, with little explanation. We present it for historical completeness,

2.33*. ⊢ (β1I2) ≡ (β1Im2).
((βI1m2) (β1I1) ) ⥽ (β112)    2.21, 2.3, subst, 14.26
(β1Im2) ⥽ (β1I2)                      2.6, 2.32*, subst, adj, 18.61, mod pon
(β1I2) ≡ (β1Im2)                       18.42, 2.23, subst, adj, def

A direct consequence of 2.33* is

2.34*. ⊢ (β1Im2) ⥽ (B1 ≡ B2)

Five years later, Marcus's thesis adviser, Frederick B. Fitch, published his book, Symbolic Logic, which contained the simplest proof ever of the necessity of identity, by the simple mathematical substitution of b for a in the necessity of self-identity statement.

23.4
(1) a = b,
(2) ◻[a = a],
then (3) ◻[a = b], by identity elimination. (p.164)

Clearly this is mathematically and logically sound. Fitch substitutes b from (1), for a in the modal context of (2). This would be fine if these are just equations. But as Barcan Marcus knew very well from Lewis's work on strict implication, substitutivity in statements also requires that the substitution is intensionally meaningful. In the sense that b is actually just a, substituting b is equivalent to keeping a there, a tautology, something with no new information. To be informative and prove the necessary truth of the new statement, we must know more about b, for example, that its intrinsic information in b is identical to that of a.

Fourteen years after her original identity article, Marcus presented her work at a 1961 colloquium at Boston University attended by Quine and Kripke.

Marcus called for disassociating directly referential names (including descriptions that are functioning as unambiguous names) from the kind of meaningful descriptions that lead to Quine's "referential opacity." This led years later to Kripke's "rigid desgnators."

It would also appear to be a precondition of language that the singling out of an entity as a thing is accompanied by many - and perhaps an indefinite or infinite number - of unique descriptions, for otherwise how would it be singled out? But to give a thing a proper name is different from giving a unique description. For suppose we took an inventory of all the entities countenanced as things by some particular culture through its own language, with its own set of names and equatable singular descriptions, and suppose that number were finite (this assumption is for the sake of simplifying the exposition). And suppose we randomized as many whole numbers as we needed for a one-to-one correspondence, and thereby tagged each thing. This identifying tag is a proper name of the thing...

This tag, a proper name, has no meaning. It simply tags. It is not strongly equatable with any of the singular descriptions of the thing...

The principle of indiscernibility may be thought of as equating a proper name of a thing with the totality of its descriptions.

Marcus also argued that not every singular description prevents it from being substituted in a logical context. Some descriptions can become proper names.
If we decide that 'the evening star' and 'the morning star' are names for the same thing,... then they must be intersubstitutable in every context. In fact it often happens, in a growing, changing language, that a descriptive phrase comes to be used as a proper name - an identifying tag - and the descriptive meaning is lost or ignored. Sometimes we use certain devices such as capitalization and dropping the definite article, to indicate the change in use. 'The evening star' becomes 'Evening Star', 'the morning star' becomes 'Morning Star', and they may come to be used as names for the same thing.
Marcus reprised the proof of her claim about the necessity of identity. She explicitly added Leibniz's Law relating identicals to indiscernibles to her argument.

(x)(y) (x = y) ⊃ ◻ (x = y)

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

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)

(3) x eq y = x I y

Statement (2) is Leibniz's Law, 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.

Arthur N. Prior's book Formal Logic appeared the following year with Marcus's latest argument, incorporating Leibniz's Law. Prior may have discussed the necessity of identity with Marcus?

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. Wiggins was the first to claim explicitly that the self-identity claim (x = x) is a property φx that must by (2) be a property of φy.

But the property "= x" is what information philosophy recognizes only as an intrinsic (internal) property of x. It names the property of being self-identical. It is linguistic nonsense to interpret this as (y = x). An identical property for y is the self-identity of y (y = y).

In the physical and logical worlds, no entity can fail to be identical to itself. So we can speak of the necessity of identity. But this is a tautology, empty of meaning, like A = A, if the only strict identity is self-identity.

Marcus was the first to prove the "necessity of identity" using Leibniz's Law – the "Identity of Indiscernibles." Like Frege, Wittgenstein, and others, she used it only to establish self-identity.

Ten years after Marcus, Saul Kripke published a similar argument in his 1971 article "Identity and Necessity." Unfortunately, it is Kriple's 1970 lectures (though not published until 1982), and not Marcus's 1961 work nor Wiggins 1965 treatment, that is best known for the idea of the "Necessity of Identity," as well as the need for directly referential names when quantifying into modal contexts.

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

Wiggins (1965)Kripke (1971)
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) ' and we shall get as consequence of (1)

(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 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

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

(5) (x)(y)((x = y). ⊃. ◻(x = y))
First, the law of the substitutivity of identity says that, for any objects x and y, if x is identical to y, then if x has a certain property F, so does y:

(1) (x)(y) [(x = y) ⊃ (Fx ⊃ Fy)]

[Note that Kripke omits the critically important universal quantifier (F), "for all F."]

On the other hand, every object surely is necessarily self-identical:

(2) (x) ◻(x = x)

But

(3) (x)(y) (x = y) ⊃ [◻(x = x) ⊃ ◻ (x = y)]

is a substitution instance of (1), the substitutivity law. From (2) and (3), we can conclude that, for every x and y, if x equals y, then, it is necessary that x equals y:

(4) (x)(y) ((x = y) ⊃ ◻ (x=y))

This is because the clause ◻(x = x) of the conditional drops out because it is known to be true.

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?

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.

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 information between them. Instead of claiming that y has x's property of being identical to 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.

In his 1980 book, Sameness and Substance, David Wiggins elaborates his 1965 argument, but this time he credits Marcus very nicely

This proof adapts a famous proof of the necessity of identity which was given by Ruth Barcan Marcus in 1947. Its merit when given in this form is that it makes evident that all substitutions within the Barcan proof can be made in manifestly extensional positions, lying outside the scope of 'necessarily.'

Miss Barcan's proof was long received with incredulity by those committed to the mutual assimilation (much criticized in more recent times by Kripke and others) of the categories of necessity and a priority, and rejected on the grounds that the identity of evening and morning star was an a posteriori discovery. But even if statement ascertainable a priori to be true and necessary true statement coincided perfectly in their extensions, Miss Barcan's theorem could still stand in our version. For the conclusion is not put forward here as a necessarily true statement. (On this we remain mute.)

References
Barcan, R. C. (1946). "A functional calculus of first order based on strict implication." The Journal of Symbolic Logic, 11(01), 1-16.
Barcan, R. C. (1946). "The deduction theorem in a functional calculus of first order based on strict implication." The Journal of Symbolic Logic, 11(04), 115-118.
Barcan, R. C. (1947). "The identity of individuals in a strict functional calculus of second order." The Journal of Symbolic Logic, 12(01), 12-15.
Kripke, Saul. 1971. "Identity and Necessity." In Munitz 1971, 135-164.
Kripke, Saul. 1981. "Naming and Necessity." Blackwell Publishing.
Marcus, R. B. (1961). Modalities and intensional languages. Synthése, 13(4), 303-322.
Munitz, Milton, ed. 1971. Identity and Individuation. New York: New York University Press.
Quine, W. V. 1943. "Notes on Existence and Necessity." The Journal of Philosophy, 40 (5) p.113
Quine, W. V. 1947. "The Problem of Interpreting Modal Logic." The Journal of Symbolic Logic 12 (2) p.43
Quine, W. V. 1953. From a Logical Point of View, Cambridge, MA: Harvard University Press.
Wiggins, David. 1965. "Identity Statements," in Analytical Philosophy, Second Series, Oxford: Blackwell.
Wiggins, David. 1980. Sameness and Substance. Cambridge University Press.
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