Necessity is often opposed to chance and contingency. In a necessary world there is no chance. Everything that happens is necessitated, determined by the laws of nature. There is only one possible (necessary!) future.

The great atomist Leucippus stated the first dogma of determinism, an absolute necessity.

"Nothing occurs at random, but everything for a reason and by necessity."

Information philosophy claims that there is no physical necessity. The world is contingent. Necessity is a logical concept, an idea that is an important part of a formal logical or mathematical system that is a human invention.

Like certainty, analyticity, and the a priori, necessity and necessary truths are useful concepts for logicians and mathematicians, but not for a metaphysicist exploring the fundamental nature of reality, which includes irreducible contingency.

Most philosophers cannot imagine denying these true statements. But information philosophy now puts them in the correct historical perspective of new information creation and human knowledge acquisition. Both these facts became known long before humans developed the logical and mathematic apparatus to declare them a priori and analytic.

Willard Van Orman Quine's claim that all knowledge is synthetic is correct from this perspective. And since nothing in the world was pre-determined to happen, this acquisition of knowledge was ultimately contingent.

We can loosely call some knowledge synthetic a priori (Kant) or even necessary a posteriori (Kripke) if we find these descriptions useful, but neither is metaphysically true.

Of course truth itself is another human invention. So we should probably say metaphysically valid, where validity is defined as a procedure within our axiomatic metaphysical apparatus.

Information metaphysics begins by establishing the meaning of intrinsic information identicals, so we can provide an axiomatic ground for "A is A" and "1 = 1," which are usually considered fundamental laws of thought. See Identity.

An affirmative truth is one whose predicate is in the subject; and so in every true affirmative proposition, necessary or contingent, universal or particular, the notion of the predicate is in some way contained in the notion of the subject

An absolutely necessary proposition is one which can be resolved into identical propositions, or, whose opposite implies a contradiction... This type of necessity, therefore, I call metaphysical or geometrical. That which lacks such necessity I call contingent, but that which implies a contradiction, or whose opposite is necessary, is called impossible. The rest are called possible.

In the case of a contingent truth, even though the predicate is really in the subject, yet one never arrives at a demonstration or an identity, even though the resolution of each term is continued indefinitely...

(Necessary and Contingent Truths)

First, we should note that Leibniz's definitions refer to propositions and predicates. In this respect, he is the original logical and analytic language philosopher. He shared the dream of Bertrand Russell, Ludwig Wittgenstein, and Rudolf Carnap, that all our knowledge of the world could be represented in propositions, "logical atoms," as Russell and Wittgenstein called them, "atomic sentences."

Secondly, Leibniz's truths are always tautological, as Wittgenstein emphasized. They are of the form, "A is A," propositions "which can be resolved into identical propositions." Their truth ultimately lies in the identity of the subject with the predicate.

Note that Leibniz's "absolutely necessary" compares to modern modal logic axioms that define not only necessity, but the necessity of necessity, like the axiom that extends the model system M to become Lewis's S4, necessarily A implies necessarily necessarily A!

The analytic philosopher Arthur Pap gave a clear account of the "necessity of necessity" argument in 1958. He asked the fundamental question "Are Necessary Propositions Necessarily Necessary?" Any contingency of truth must be denied. Necessary truths are independent of the physical world, outside space and time.

The question whether "it is necessary that p" is, if true, itself a necessary proposition is of fundamental importance for the problem of explicating the concept of necessary truth, since it is likely that any philosopher who answers it affirmatively will adopt the necessity of the necessity of p as a criterion of adequacy for proposed explications of necessary truth. He will, in other words, reject any explication which entails the contingency of such modal propositions as failing to explicate the explicandum he has in mind. The same holds, of course, for the concept of logical truth: since all logical truths are necessary truths (whether or not the converse of this proposition be true also), any criterion of adequacy for explications of "necessary truth" is at the same time a criterion of adequacy for explications of "logical truth." This question cannot be decided by formal reasoning within an uninterpreted system of modal logic, containing the usual explicit definition of "necessary" in terms of "possible": p is necessary = not-p is not possible. Indeed, an uninterpreted system of modal logic can be constructed without even raising the question of the necessity of the necessity of p; thus there is no postulate or theorem in Lewis' system S2 that bears on the question, nor is the question informally discussed in the metalanguage. In Appendix II to Lewis and Langford's Symbolic Logic (New York and London, 1932) it is pointed out that Lewis' system of strict implication "leaves undetermined certain properties of the modal functions, ◇ p, — ◇ p, ◇ ~ p, and ~ ◇ ~ p." Accordingly "Np ⥽ NNp," as well as "Np ⊃ NNp" (N . . . = it is necessary that . . .). is both independent of and consistent with the axioms of the system, and whether an axiom of modal iteration, e.g. "what is possibly possible, is possible" (which can be shown to be equivalent to "what is necessary, is necessarily necessary") should be adopted must be decided by extrasystematic considerations based on interpretation of the modal functions. Now, let us refer to the thesis that necessary propositions are necessarily necessary henceforth as the "NN thesis." What appears to be the strongest argument in favor of the NN thesis is based on the semantic assumption that "necessary" as predicated of propositions is a time-independent predicate, where a "time-independent" predicate is defined as a predicate P such that sentences of the form "x is P at time t" are meaningless.

("The Linguistic Theory of Logical Necessity,"Semantics and Necessary Truth, p.120)

1. P. (for example, P is the proposition that A = A. A is A, A is identical to A.)
2. It is true that P.
3. It is necessarily true that P.
4. P is true in all possible worlds.
5. P is necessarily true in all possible worlds.

In recent years, modal logicians claim to prove the "necessity of identity" using the converse of Leibniz's Law – the "Identity of Indiscernibles."

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 one of the properties of x is that x = x, so if y shares that property "= x" of x, we can say y = x. Necessarily, x = y. QED.

Our rule that the only identity is self-identity becomes in information philosophy that 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..

The necessity of identity in symbolic logic is

Despite many such arguments in the philosophical literature over the past forty or fifty years, this is a flawed argument. Numerically distinct objects can only be identical "in some respect," if they share qualities which we can selectively "pick out". We can say that a red house and a blue house are identical qua house. They are different qua color.

Here is Saul Kripke's argument against the possibility of contingent identity statements:

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)]

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.

This is an argument which has been stated many times in recent philosophy. Its conclusion, however, has often been regarded as highly paradoxical. For example, David Wiggins, in his paper, "Identity-Statements," says,

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?

(Identity and Necessity, p. 136)

Where are Kripke's errors? First we must unpack his "indiscernibility of identicals." Instead of (x)(y) [(x = y) ⊃ (Fx ⊃ Fy)], we must say that we can clearly discern differences between x and y, their names and their numerical distinctness, unless we are merely talking about a single object using two different names. For example, Hesperus = Phosphorus qua names referring to the planet Venus.

"A sea battle must either take place tomorrow or not,
but it is not necessary that it should take place tomorrow,
neither is it necessary that it should not take place,
yet it is necessary that it either should or should not
take place to-morrow."
(De Interpretatione IX, 19 a 30)

The major founder of Stoicism, Chrysippus, took the edge off strict necessity. Like Democritus, Aristotle, and Epicurus before him, Chrysippus wanted to strengthen the argument for moral responsibility, in particular defending it from Aristotle's and Epicurus's indeterminate chance causes. Whereas the past is unchangeable, Chrysippus argued that some future events that are possible do not occur by necessity from past external factors alone, but might depend on us. We have a choice to assent or not to assent to an action.

Later, Gottfried Leibniz distinguished two forms of necessity, necessary necessity and contingent necessity. This basically distinguished logical necessity from physical (or empirical) necessity.

In two-stage models of free will, chance or indeterminism in the generation of alternative possibilities for action breaks the causal chain of determinism. Actions are not directly determined by reasons or motives, but by an agent evaluating those possibilities in the light of reasons and motives.

The thinking agent generates new ideas and chooses to act on one of them. Thoughts are free. Actions are willed. Free and Will are two temporal stages stages in the process of free will.

Chance is regarded as inconsistent with logical determinism and with any limits on causal, physical or mechanical determinism.

Despite abundant evidence to the contrary, many philosophers deny that chance exists. If a single event is determined by chance, then indeterminism would be true, they say, and undermine the very possibility of certain knowledge. Some go to the extreme of saying that chance would make the state of the world totally independent of any earlier states, which is nonsense, but it shows how anxious they are about chance.

Bertrand Russell said "The law of causation, according to which later events can theoretically be predicted by means of earlier events, has often been held to be a priori, a necessity of thought, a category without which science would not be possible." (Nature of the External World, p.179)

The core idea of indeterminism is closely related to the idea of causality. Indeterminism for some is simply an event without a cause. But we can have an adequate causality without strict determinism, which implies complete predictability of events and only one possible future.

An example of an event that is not strictly caused is one that depends on chance, like the flip of a coin. If the outcome is only probable, not certain, then the event can be said to have been caused by the coin flip, but the head or tails result was not predictable. So this causality, which recognizes prior events as causes, is undetermined and the result of chance alone.

Events are caused by a combination of caused and uncaused prior events, but not determined by events earlier in the causal chain, which has been broken by the uncaused causes.

Despite David Hume's critical attack on the logical necessity of causes, many philosophers embrace causality strongly. Some even connect it to the very possibility of logic and reason. And Hume himself strongly, if inconsistently, believed in necessity while denying causality. He said "'tis impossible to admit any medium betwixt chance and necessity."

Even in a world with chance, macroscopic objects are determined to an extraordinary degree. This is the basis for an adequate physical causality.

We call this kind of determinism (determined but not pre-determined) "adequate determinism." This determinism is adequate enough for us to predict eclipses for the next thousand years or more with extraordinary precision. Newton's laws of motion are deterministic enough to send men to the moon and back.

The presence of quantum uncertainty leads some philosophers to call the world indetermined. But indeterminism is misleading, with strong negative connotations, when most events are overwhelmingly "adequately determined." The neural system is robust enough to insure that mental decisions are reliably transmitted to our limbs. Our actions are determined by our thoughts and our choices. But our thoughts themselves are free. This means that our actions were not pre-determined from before we began thinking.