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 must be 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.

Since facts about the world are empirical and a posteriori, and thus contingent, 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 "x is contingently y" Prepending the necessity operator, we have "Necessarily, x is contingently y." It changes nothing.

Consider Q(x = y), where "'Q' (for questionable) is a contingency operator parallel to ' ◻' and ' ◇ ' Can we interchange the operators in ◻Q(x = y) to get Q◻(x = y)? Would it change the meaning from x is possibly y and possibly not y?

Diodorus was known as "The Dialectician," testimony to his sophistry with words, or for his ability to create paradoxes. Epictetus wrote a diatribe "Against those who embrace philosophical opinions only in words," in Book 2, Chapter 1, of his Discourses. It is our major reference to Diodorus and his famous Master Argument (the κυριεύων or κύριος λόγος).

Diodorus' Master Argument is a set of propositions designed to show that the actual is the only possible and that some true statements about the future imply that the future is already determined. This follows logically from his observation that if something in the future is not going to happen, it must have been that statements in the past that it would not happen must have been true.

The Master Argument was central in the Hellenistic debates about determinism, as shown by Cicero's descriptions in On Fate.

It is closely related to the problem of future contingency, also discussed by Diodorus, but made famous in Aristotle's example of a Sea-Battle in De Interpretatione 9. Aristotle thought that statements about the past and present must be either true or false. But statements about the future are only potentials, possibilities, so they lack any truth value until their potential becomes actual at some time in the future.

Note that there are in fact some things in the past that can be changed in the future. It is the truth value of a statement made in the past. The truth value of Aristotle's statement "there will be a sea-battle next week," can "actually" be changed if the event does not happen, showing that the concept of a "fixed past," so important in analytic language philosophy debates about free will, has some changeability. In analytical language philosophy, the "fixed past" is far from fixed.

Diodorus was a great logician and word-juggler. Like Socrates, he wrote little or nothing and preferred verbal debates. The Dialectician was a precursor of the later language game players, Ludwig Wittgenstein, Jacques Derrida, and Daniel Dennett.

Diodorus applied Democritus' great insight that much knowledge is pure convention (νόμος), but "in reality" there is only atoms and a void. For Diodorus, language definitions were conventional and quite arbitrary. Perhaps his most famous example (also attributed to Eubulides) was the linguistic puzzle of how to define a "heap" (metaphysicians call this the Sorites puzzle, from Greek σωρείτης so-ri'-tes, meaning "heaped up"). When does a number of grains become a heap? One? No. Two? No. Three? Etc. Or, given a heap of grains, as you take grains away, at which point does it stop being a heap?

That Aristotle believes in an open and ambiguous future with alternative possibilities is also shown by his denial of the logical Master Argument for determinism of Diodorus Cronus, in the form of Aristotle's famous "sea battle."

Diodorus argued from an assumed necessity of past truths (which is understandable, if a misapplication of logic to physical reality) that something is impossible that neither is or ever will be true.

Aristotle reframed the argument as the truth or falsity of the statement that a sea battle will occur tomorrow. Despite the law of the excluded middle (or principle of bivalence), which allows no third case (or tertium quid), Aristotle concluded that the statement is neither true nor false, supporting an ambiguous future.

What is, necessarily is, when it is; and what is not, necessarily is not, when it is not. But not everything that is, necessarily is; and not everything that is not, necessarily is not. For to say that everything that is, is of necessity, when it is, is not the same as saying unconditionally that it is of necessity. Similarly with what is not. And the same account holds for contradictories: everything necessarily is or is not, and will be or will not be; but one cannot divide and say that one or the other is necessary.

I mean, for example: it is necessary for there to be or not to be a sea-battle tomorrow; but it is not necessary for a sea-battle to take place tomorrow, nor for one not to take place — though it is necessary for one to take place or not to take place. So, since statements are true according to how the actual things are, it is clear that wherever these are such as to allow of contraries as chance has it, the same necessarily holds for the contradictories also. This happens with things that are not always so or are not always not so. With these it is necessary for one or the other of the contradictories to be true or false — not, however, this one or that one, but as chance has it; or for one to be true rather than the other, yet not already true or false.

(De Interpretatione, IX, 19a23-39 ) ⁶

Aristotle never denied the law of the excluded middle, merely that the truth or falsity of statements about future events does not exist yet. Note that this implies some things in the past may be changed in the future, i.e., the truth values of past statements about the future.