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Modal Logic and Metaphysics
Although the modes of necessity, possibility, and impossibility had been part of Aristotelian logic (indeed, even future contingency was analyzed), Gottlob Frege's logic of propositional functions included only one mode - simple affirmation and denial of statements and the universal and existential quantifiers. The Principia Mathematica of Alfred North Whitehead and Bertrand Russell followed Frege and ignored other modalities.
Although the Scholastics considered some questions of modality, it was the Harvard logician C.I. Lewis who advanced beyond Aristotle and developed the first modern version of modal logic. He wrote two textbooks,
Lewis was critical of the
Lewis's inclusion of intension (meaning) was criticized by Willard Van Orman Quine, who thought symbolic logic should be limited to "extensional" arguments, counting the number of members of classes in a set theory basis for logic. In Quine's 1943 article, "Notes on Existence and Necessity," (revised to appear ten years later as part of the chapter "Reference and Modality" in his landmark book,
These latter are Quine saw the need for serious restrictions on the significant use of modal operators (p.127). Just three years later, Ruth Barcan Marcus, publishing under her maiden name Ruth C. Barcan, added a modal axiom for possibility to the logical systems S2 and S4 of C.I. Lewis. Lewis was pleased, although by that time, he had given up any work on logic. Quine reacted negatively to Marcus's suggestion in 1946 that modal operators (Lewis's diamond '◇' for possibly, and a box '◻' for "necessarily" suggested by Barcan's thesis adviser, F.B. Fitch) could be transposed or interchanged with universal and existential 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 this commuting of quantification and modal operators in what A.N. Prior called the "Barcan formulas."
∀x ◻Fx ⊃ ◻ ∀x Fx ∀x ◇Fx ⊃ ◇ ∀x Fx
In his 1943 article, 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.
∃x ◻Fx ⊃ ◻ ∃x Fx ∃x ◇Fx ⊃ ◇ ∃x Fx Information philosophy has shown that two distinct expressions that are claimed to be identical are never identical in all respects. 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. Quine called this "referential opacity." This is a problem that can be solved with unambiguous references.
Frege had insisted that we must look past the reference or designator (his " Perhaps Quine's most famous paradox of referential opacity 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 analyzed this problem in 1961, which she called the "familiar example,"
(27) 9 eq the number of planets 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 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 puts it. Both expressions,
The Necessity of Identity
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
(x)(y) (x = y) ⊃ ◻ (x = y)
which reads "for all x and for all y, if "x = y," then necessarily "x = y."
Quine found this relationship in the 1952 Textbook, 23.4 (1) a = b, (2) ◻[a = a], then (3) ◻[a = b], by identity elimination. (p.164)
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 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 on Identity
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 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,
Saul Kripke on Identity
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
David Lewis on Identity
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 Except, says an information philosopher, "in some respects."
Modal Logic and Possible Worlds
In the "semantics of possible worlds," 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 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.
There are no genuine possibilities in Lewis's "possible worlds"!
It comes as a shock to learn that every "possible world" is just as actual, for its inhabitants, as our world is for us. There are no alternative possibilities, no contingency, that things might have been otherwise, in any of these possible worlds. Every world is as physically deterministic as our own.
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 possibilities" exist in minds and in quantum-mechanical "possibility functions"
It is what call "actual possibilism," the existence in our actual world of possibilities that may never become actualized, but that have a presence as abstract entities that have been embodied as ideas in minds. In addition, we include the many possibilities that occur at the microscopic level when the quantum-mechanical probability-amplitude wave function collapses, making one of its many possibilities actual.
Actual possibles can act as causes when an agent chooses one as a course of action.
Why Modal Logic Is
Modal logicians from Ruth Barcan Marcus to Saul Kripke, David Lewis, and the Not Metaphysics
necessicist Timothy Williamson are right to claim metaphysical necessity as the case in the purely abstract informational world of logic and mathematics. But when information is embodied in concrete matter, which is subject to the laws of quantum physics and ontological chance, the fundamental nature of material reality is possibilist.
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
Possible world semantics is a
Ludwig Wittgenstein’s core idea from the
4.11 The totality of true propositions is the total natural science (or the totality of the natural sciences). 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 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
Possible worlds semantics defines 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. Normal | Teacher | Scholar |