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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 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 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
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.
∃x ◻Fx ⊃ ◻ ∃x Fx ∃x ◇Fx ⊃ ◇ ∃x Fx Perhaps Quine's most famous paradox is this argument about the number of planets:
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
The failure of 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 Saul 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 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
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,
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 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,
Five years later, Marcus's thesis adviser, Frederick B. Fitch, published his book,
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 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...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 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 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 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:
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
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
In his 1980 book, 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.'
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.113Quine, W. V. 1947. "The Problem of Interpreting Modal Logic." The Journal of Symbolic Logic 12 (2) p.43Quine, 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.Normal | Teacher | Scholar |