Logic is an abstract human invention, a formal system of ideas much like mathematics, or purely theoretical physics. It is a kind of language, another human invention. Although we can see language as arguably the latest natural evolution of a biological communication system that uses arbitrary symbols for messages in and between all organisms, logic and mathematics are purely the products of rational human minds.

There is nothing material about logic. It is purely abstract and immaterial information. Its application to the material world is fraught with danger. A purely materialistic metaphysics cannot understand the fundamental nature of physical reality, cannot understand metaphysics, without including immaterial forms, the "Ideas" of Plato.

Where symbols in ordinary language are notoriously ambiguous, logic is an attempt to formalize the allowed terms, the rules by which they are assembled into statements, and the principles for deductively reasoning from some statements (premises) to others (conclusions), such that true premises lead to true (valid) conclusions.

It was the vision of the great rationalist philosopher Gottfried Leibniz that we could develop an ambiguity-free language for logic and mathematics. That dream was pursued by Bertrand Russell, Ludwig Wittgenstein, Rudolf Carnap, and others.

Their logical truth-functional analyses are severely limited by the principle of bivalence, the excluded middle, that the only possible values are true and false. But the world is not limited to truth and falsity. Attempts to develop three-valued or many-valued logics have largely failed.

Modal logic is the analysis and qualification of statements or propositions as asserting or denying necessity, possibility, impossibility, and, most problematic, contingency.

The use of "necessity" and "impossibility" to describe the physical world should be guarded and understood to describe events or "states of affairs" that have extremely high or low probability. The term certainty, when used about knowledge of the physical world, normally represents only extremely high probability.

Possibility and contingency are not easily constrained to the binary values of true and false. To begin with, possibility is normally understood to include necessity. If something is necessary, it is a fortiori possible. Contingency is then defined as the subset of possibility that excludes necessity.

The modal operators are a box '◻' for necessity and a diamond ' ◇ ' for possibility. Impossibility is the negation of possibility, ¬◇, and contingency must negate necessity and also negate impossibility, so it is the logical conjunction of "not necessity" and "possibility" (¬◻ ∧ ◇).

Mathematically, contingency is a continuum of values between impossibility and necessity, the open interval between 0 and 1 that represents all the probabilities (excluding the certainties. It is the negation of the logical disjunction of necessity and impossibility, neither necessary nor impossible. (¬ (◻ ∨ ¬◇)).

But physically, contingency is the closed interval, including the endpoints of necessity (1) and impossibility (0). Theoretical physics today is often described as probabilistic and statistical, which is sometimes misunderstood to exclude perfect certainties like 0 and 1, but this is not the case. Even quantum physics, the basis of ontological chance in the universe, sometimes predicts certain outcomes, as explained by Paul Dirac.

With its four modes, necessity, possibility, impossibility, and contingency, modal analyses simply contain more than can be confined to two-valued truth-functions, whether in logic, usually called a priori truths, or language analysis, usually called analytic truths, nor in supposed metaphysical truths.

Truth is a binary relation of ideas, true or false. Facts of the matter have a continuous value somewhere between 0 and 1, with plus or minus estimates of the standard deviation of probable errors around that value.

In analytic language philosophy, we need more than the "truth" of statements and propositions with their apparent claims about "necessary" facts in the world. The logical empiricists equate necessity in the first-order logic of their "object language" with analyticity in their higher-order "metalanguage" of propositional functions.

Although we distinguish the a priori truths of logic from the analytic truths of language philosophy, many such "truths" were discovered long before modern methods were invented to demonstrate their "proofs.' In that sense, knowledge is usually discovered a posteriori and ultimately all knowledge is synthetic in the Kantian sense.

All facts about the world are (necessarily?) empirical and a posteriori, and thus contingent, so it is best to restrict the use of the concept "truth" to logic and to analytic discourse about statements and propositions. Truth is an appropriate concept in "ideal" formal systems like philosophical logic and mathematics where the extremes of necessity and impossibility are defined parts of the system. But the world itself cannot be confined to a Procrustean bed of true and false.

We therefore conclude that the logical empiricist's idea that the laws of nature can be described with linguistic statements or propositions is simply wrong. This is particularly the case for the laws of modern physics, which are now irreducibly probabilistic in view of the indeterministic nature of quantum mechanics, the uncertainty principle, etc.

The "evidence" that "verifies" or validates a physical theory is gathered from a very large number of experiments. No single measurement can establish a fact in the way that a single valid argument can assert the "truth" of an analytic statement. The large number of measurements means that evidence is statistical. Indeed, physical theories make predictions that are probabilities. Theories are confirmed when the a priori probabilities match the a posteriori statistics.

Probability is a theory, statistics are the results of experiments.

Information philosophy considers claims such as "If P, then P is true" to be redundant, adding no information to the (true) assertion of the statement or proposition "P." Further redundancies are equally vacuous, such as "If P is true, then P is necessarily true" and "If P is true, then P is necessarily true in all possible worlds."

Logically necessary and analytic statements are tautological and carry no new information. This is the paradox of analyticity. The statement "A is A" tells us nothing. The statement "A is B" is informative.

Adding "is true" and the like also add no new information. They cannot change the fundamental nature of a statement. For example, they cannot change a contingent statement into a necessary one. Consider the statement "A is contingently B." Prepending the necessity operator, we have "Necessarily, A is contingently B." It changes nothing.

In fact, that is to say in the empirical world, any fact F is at best probably "the case," with the probability approaching certainty in cases that are adequately determined. And, in any case, any past F was contingent and could possibly have been otherwise. The idea of a "possible world" is best understood as a way this actual world could have been.

There is, "in fact," only one actual world, the one that is the case. The original purpose of the invention of the idea that there are "possible worlds" – abstract entities – was to provide metaphysicians with other ways of talking about possibilities unrealized in our actual world.

The "sample space" of modern probability theory and the "phase space" of statistical physics are spaces for possible worlds. The 36 ways that two dice can be thrown, the 64 squares where a pawn can be located on a chessboard, the coarse-grained cells for gas particles in position-momentum space, and the minimum uncertainty volumes Δp Δx = ℏ of quantum physics, all can be used to describe possible worlds, how worlds can be, and thus how our world might be.

Information philosophy maintains that ontologically real possibilities "exist" or subsist as ideas, as pure abstract information, at the present time, alongside "actual" material objects. The ontological or existential status of ideas has always been a controversial question in metaphysics. The exact status of their "existence" in the past and future is equally controversial, and likely asymmetrical in the past and the future.

Possibilities in the past may be described as having been "actual possibles." Possibilities in the future are merely "possible possibles."

Possibilities in the past, for example the past alternatives for human actions or the past outcomes of experiments in probabilistic quantum physics, were mostly "roads not taken" and were condemned to "non-being," as the existentialists described it. But they were actual as possibilities in the past that were never "really" actualized. Thus, we can say they were at one time, that they "existed" as, "actual possibles."

The existence of alternative possibilities in the future raises the famous problem of future contingency, which, since Aristotle's De Interpretatione, has called into question the principle of bivalence (either P or not-P), since statements about the future may be (now) neither true nor false. P and not-P are (now) possible possibles about future actuals.

But what can be said about the existential status of these future alternative possibilities in the present time? What can "actual possibles" mean metaphysically? We shall show that possibilities are ideas, abstract entities, which from the time they are embodied in a physical system or in a human mind become "actual possibles." At later times, we are justified in describing them as past "actual possibles" that were never actualized.

Whenever one of many actual possibles is actualized, it does not mean that alternatives that existed as abstract entities at that moment are no longer possibly actualizable in the future. Unless they are forgotten, they remain as "actual possibles"for future use.

We can now describe the many possible worlds that exist within our actual world. They are ways our actual world may be.

Information philosophy provides two examples of future "possible possibles" that are transformed when one is actualized into past "actual possibles." One comes from the world of quantum physics (the source of ontological chance), the other from the human mind when evaluating alternatives and making a decision.

We distinguish three kinds of information structures and processes in our world, the physical, the biological, and the particularly human. All such processing systems can have multiple possibilities for the next step in their processes. These possibilities are abstract bits of information ("ideas") that must be embodied physically to be available as "actual possibles."

At the physical level, quantum events that are amplified to the macroscopic world start new causal chains in "adequately determined" physical processes.

Biological possibilities include sexual selection, where chromosomes for the zygote are randomly selected from the sperm and egg, as a genuinely new individual is created and novel information enters the universe.

For human beings, possibilities are ideas in minds about what to do next. Many of these ideas are constantly available in the normal repertoire of behaviors. That one is chosen over others does not remove the others from future actualization. They remain as "actual possibles" unless they are forgotten. Human minds also create genuinely new information, like that created in biological evolution, when they mentally consider an idea never before thought as an "actual possible."

Although our metaphysically actual possibles are not as numerous as the plurality of worlds of David Lewis or the many worlds of Hugh Everett III, they are plentiful enough. With ten billion humans, millions of other species, some with trillions of individuals that have behavioral repertoires, the numbers of possibilities being actualized in the world each second is vast.

There are many ways that our world may be. It is thus very strange that modal logicians, especially those who are necessitists, assume our actual world has only one way to be and all possibilities are found in worlds that are physically inaccessible, though modally accessible.

David Wiggins and Saul Kripke claimed that the proof of the necessity of identity appeared to make contingent identity impossible. Wiggins also argued against Peter Geach's idea of relative identity.

An information analysis of identity limits perfect and total identity to cases of self-identity, which includes an object's intrinsic internal information and the extrinsic information in dispositional relations of one object to others and to space and time. We can say that any object is absolutely identical to itself. We can also say that some objects are relatively identical if their "identity" is limited to their intrinsic internal information. We then discover a large number of relatively identical objects, both concrete and abstract, including some of those claimed as "natural kinds" by Kripke and Hilary Putnam, for example, atoms of gold and molecules of water (H₂O)

Kripke claims that such things, which we describe as informationally identical, are metaphysically necessary a posteriori. The domain of things that are intrinsic information identicals is much larger, including both natural and artifactual "digital clones," whether embodied or so-called "non-existent" abstract entities.

It was the claim for the necessity of identity that led to the leading modal systems including a "rule of necessitation," that if P, then necessarily P. (P ⊃ ◻ P) We should examine this claim carefully. If correct, it may only be a tautological or analytical statement about a universe of discourse, with no significance for the physical world. By contrast, our claim for intrinsic information identicals is a metaphysical and ontological claim about the fundamental nature of reality as including digital clones.

The first proof of the necessity of identity, by Ruth Barcan Marcus, was little more than the substitutivity of identicals, which may be seen as begging the question of that identity! It is best seen in a simple proof by her thesis adviser, Frederick B. Fitch,

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 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 is identical to that of a.

Most earlier identity claims showed only that a and b were references (names) for the same thing, Frege's Morning Star and Evening Star for example. But this is a new claim, that numerically distinct things are identical – in some respect.

Those earlier claims often referred to Leibniz's Law, the Identity of Indiscernibles, and Marcus in 1961, Wiggins in 1965, and Kripke in 1971 all added Leibniz's Law. But none of these changed the fact that contingent identities are merely possible, that substitution of b for a is valid if and only if you already know that a and b are intrinsic information identicals, and that such knowledge, gained a posteriori, is in no way made metaphysically necessary by substituting into the modal context of necessity. Wiggins offered the most complete argument,

If a and b refer to the same object, it is already a perfect and absolute identity. Calling it necessary adds nothing more than "is true" or "necessarily true in all possible worlds."

But it is for claims about "transworld identity" of individuals that modal realists following David Lewis require the "necessity of identity."

It is critical to note that the metaphysicians proposing possible worlds are for the most part materialists and determinists who do not believe in the existence of ontological possibilities in our world.

They are mostly actualists who say that the only 'possibilities' have always been whatever it was that has actually happened. This is Dan Dennett's position, for example, not far from the original actualist, Diodorus Cronus.

Moreover, all of their infinite number of worlds, e.g., David Lewis's possible worlds, are governed by deterministic laws of nature. This means that there are also no real possibilities in any of their possible worlds, only actualities there as well.

Now this is quite ironic, since the invention of possible worlds was proposed as a superior way of talking about counterfactual possibilities in our world.

Since information philosophy defends the existence of alternative possibilities leading to different futures, we can adopt a form of modal discourse to describe these possibilities as possible future worlds for our to-be-actualized world.

Saul Kripke recommended that his "possible worlds" are best regarded as "possible states (or histories) of the world," or just "counterfactual situations," or simply "ways the world might have been."

Kripke appears to endorse the idea of alternative possibilities, that things could have been otherwise.

David Lewis appears to have been a materialist and determinist.

The infinity is not as large as the absurdly extravagant number in David Lewis's possible worlds, which have counterparts for each and every living person with every imaginable difference in each of our counterparts, each counterpart in its own unique world.

Thus there are Lewisian worlds in which your counterpart is a butcher, baker, candlestick maker, and every other known occupation. There are possible worlds in which your counterpart eats every possible breakfast food, drives every possible car, and lives in every block on every street in every city or town in the entire word.

This extravagance is of course part of Lewis's appeal. It makes Hugh Everett's "many worlds" of quantum mechanics (which split the universe in two when a physicist makes a quantum measurement) minuscule, indeed quite parsimonious, by comparison.

Specifically, when an Everett universe splits into two, it doubles the matter and energy in the new universe(s) – an extreme violation of the principle of the conservation of matter/energy – and it also doubles the information. Apart from that absurdity, the two universes differ by only one bit of information, for example, whether the electron spin measured up or down in the quantum measurement.

Similarly, for every Lewisian universe, the change of one bit of information implies one other possible universe in which all the infinite number of other bits stay exactly the same. But Lewis imagines that every single bit in the universe may be changed at any time, an order of physical infinities that rivals the greatest number that Georg Cantor ever imagined. Is David Lewis ontologically committed to such a number?

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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. 1981. Naming and Necessity." Blackwell Publishing.
Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell.
Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell.
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