A necessary truth is a true statement whose negation must imply a contradiction in reality, such that the negation would be impossible.
So, if “One plus one equals two,” is a necessary truth, then the statement “One plus one does not equal two” will imply a contradiction. Given the meanings of “one” and “two,” we can immediately see that the addition of two “ones” (units) always does yield “two,” yet the statement “One plus one does not equal two,” contradicts this. It’s incomprehensible that one plus one should ever add to anything but two. So “One plus one equals two,” is commonly held to be a necessary truth, with its negation being impossible.
A contingent truth is a true statement whose negation does not imply a contradiction in reality, such that the negation could have been the case.
So, if “John married Jessica last Sunday,” is a contingent truth, then the statement “John did not marry Jessica last Sunday,” could have been true, without implying a contradiction in reality. Since John could have chosen not to marry Jessica, or to have married her on a different day, we can see that this is indeed a contingent truth.
The Objectivist View on the Necessary/Contingent Distinction
Causality (the Law of Cause and Effect) is the Law of Identity applied to action. This means that an entity’s actions follow from its nature. That is, the nature of the entity (its attributes, properties, etc.) causes the action it will take in any given situation. In any given context, there is only one action open to it: the one in accordance with its nature. Any other action would contradict its nature.
For an inanimate object, this means that there is one specific course of action it will take in response to a specific situation. Any other action would contradict its nature. (1)
Now, human beings have a very distinctive attribute: a consciousness with a specific kind of free will: the continuous choice to think or not. During the normal operation of consciousness, human beings, by their nature, must each choose to focus their minds on the process of thinking, or release their focus and let their consciousness drift. That this choice must be made is inherent in human nature, but which of the two alternatives an individual will choose is not determined by prior factors. (For more on this, see: The Formal Refutation of Determinism and The Validation of Free Will (Libertarian Volition))
Since an individual’s actions will be different depending on whether he chooses to think or not in various contexts, his particular actions in a given context do not follow inexorably from his nature. (Within his capacities as a human being. His nature does set limits on what he can do.)
Thus, Objectivism distinguishes between two types of facts: those that proceed from human choice, and those that do not. Those facts that proceed from human choice, Ayn Rand referred to as “man-made.” Those facts that do not proceed from human choice, she called “metaphysically given.” (Note here that “metaphysically-given facts” include those that result from the physical cause/effect chains of inanimate objects, without human involvement.)
Because the facts of the physical interactions of things in the universe–outside of human control–follow inexorably from the natures of the entities involved, and because any imagined alternative to these interactions, as they really happened, would imply that something, in some way, was not what it was, Objectivism regards all such facts as necessary. Any proposed alternative to the way things are, outside the control of humans, implies a contradiction somewhere in reality, and thus violates the Law of Identity. For metaphysically-given facts, it is literally impossible for things to have been otherwise than they are.
To clarify further how such alternatives involve a contradiction: In order for any fact to have been different, something in the history of the universe would have had to have acted differently. Without human intervention, (or the intervention of a volitional entity) to act differently would mean that the entity violates its nature.
The contradiction is: you have this entity that has a specific nature, and you are saying it is that entity. Yet it does something different than its actual nature allows in that circumstance, so it isn’t really that entity. You are saying that this thing is A and not A, at the same time and in the same respect.
In the case of man-made facts–those facts under the control of human choices, through actions–we can propose alternatives without implying a contradiction in reality. Because particular human choices could have been different without contradicting human nature, we can legitimately say that the facts produced by human choices (insofar as they are thus produced) could have been different. Thus, Objectivism regards all man-made facts as contingent.
So Objectivism holds that the necessary/contingent distinction matches the metaphysically-given/man-made distinction. The current fact that, say, “The vast majority of tigers have stripes,” is necessary. If human beings one day gain the ability to genetically engineer tigers to not have stripes, then the fact that certain tigers have stripes could become contingent at that time. But the fact that the vast majority of tigers have stripes in May of 2015 will remain necessary, regardless of human choices.
For more information on Objectivism’s metaphysical views, I recommend Objectivism: The Philosophy of Ayn Rand, by Leonard Peikoff.
The Mainstream Academic View on the Necessary/Contingent Distinction
It is quite common, both today and historically, for philosophers to draw a distinction between necessary and contingent truth that does not match the distinction between the metaphysically given and the man-made. Philosophers frequently hold that things Objectivism calls “metaphysically given” are contingent. For example, many would say that the truth of the statement, “The planet Jupiter exists,” is a contingent truth. They would say that things “could have been different,” such that Jupiter did not coalesce as a planet–or, in modern philosophical jargon, that “there are some possible worlds” that do not include Jupiter as a planet.
But Objectivism holds that this view–that some facts outside of human control can be contingent–is in error. Specifically, this view implies an element of what Objectivism refers to as a “primacy of consciousness” approach to metaphysics.
The primacy of consciousness approach holds that some consciousness, (or more than one) through its internal processes, is active in making reality what it is. This could be held to be God, society, the self, or the group speaking a given language. (The most obvious and explicit forms of the primacy of consciousness are the various forms of philosophical idealism. But there are many more subtle ways in which this view manifests, and the common view of the necessary/contingent distinction is one of them.)
The primacy of consciousness approach contrasts with the “primacy of existence” approach. This view holds that the function of all consciousness, as such, is to perceive reality, not create it. Consciousness can affect reality in certain, delimited ways by its control over the body, but it does not, by its mere processing, project itself out into its primary object–i.e. extramental reality–nor does it create an internal reality as its primary object. In the primacy of existence approach, the facts of reality are what they are, independent of any of the processes of consciousness, including wishes, hopes, fantasies, emotions, classifications, and linguistic conventions. Objectivism holds firmly to the primacy of existence view.
In the case of the necessary/contingent distinction, the primacy of consciousness approach implicitly informs the mainstream view. This is because the idea that a state of affairs “could have been different” implies that some choice was involved somewhere in producing it. Yet choice is a feature of consciousness–specifically, human consciousness–not inanimate objects reacting in the only way they can to their circumstances.
So when someone regards a fact of reality (not produced by human action) as contingent, he is implying that some consciousness controls the nature of reality. Again, this could be God, society, the self, or the group speaking a given language.
The intuitive pull of calling “empirical”/”synthetic” truths contingent derives from the fact that someone can, at least superficially, imagine an alternative to them. (2) And the feeling is that, if one’s own consciousness can imagine things differently, then the consciousness that has shaped reality could have shaped it differently, as well. If the imagined alternative to a fact presents an obvious contradiction, such that it boggles the imagination and/or reason, then the feeling is that no consciousness could have made reality that way, making the alternative impossible.
Objectivism restricts the idea of contingent facts to those facts that result from human choices, through human actions. These are facts one can legitimately say could have been otherwise. The alteration of any facts outside of those that result from human choices (the metaphysically given) would result in a contradiction somewhere in reality, even if it is not immediately apparent to one’s imagination. These metaphysically-given facts include many facts that most philosophers would call “empirical” or “synthetic.” These facts could not have been otherwise than they are.
(1) Some interpretations of quantum mechanics hold that the randomness of the observations we make of particles in experiments is fundamental and metaphysical–that is, they hold that certain properties of a particle have no definite value until the particle is observed. So these interpretations would hold that the interactions of inanimate entities is not deterministic, and thus could be considered contingent. But Objectivism supports the rejection of these interpretations on the basis of the axiomatic Law of Identity: particles must have some definite nature, including a definite set of properties, before they are observed. Any randomness in the world is epistemic: it represents a lack of knowledge of the relevant determining factors on our part.
(2) Note that Objectivism rejects the dichotomies between “analytic” and “synthetic” truths and between a priori and a posteriori, as commonly understood. But these are essays for another time.
[Edited 5-25-15 to add paragraphs 7 and 8 of the second section.]