20. Kinds of Certainty
Summary
This short episode of Allan Di Donato's Critical Thinking course examines what we mean by certainty and distinguishes six different kinds. It sets the stage for the broader discussion of induction and probability that follows.
Certainty Is a Property of the Knower, Not the Proposition
We have said that deductive arguments end in certainty and inductive arguments end in probability. Strictly speaking, however, certainty is a characteristic of a person: it refers to one's confidence in a judgment. Some thinkers distinguish certainty from certitude, where certitude is the merely subjective feeling of confidence, and certainty is confidence proportioned to objective reasons for assenting to a secure truth.
A crucial point follows: I can be certain of the truth of a proposition, but the proposition itself is not "certain." It is simply true or false. Truth value has nothing to do with how we feel about a belief. We often hold beliefs we feel certain about that later turn out to be false. Since certainty means being in a state free from doubt, only beings that can doubt can be certain; we apply the term to propositions only in an analogous way, meaning that a claim is free from doubt as an object of doubt.
Strictly, there is no such thing as "almost certain," because certainty implies no doubt. But because we use the word for various levels of confidence, Di Donato adopts a colloquial understanding and ranks six kinds of certainty so we can see where inductive certainty fits among them.
The Highest Level: Mathematical, Logical, and Existential Certainty
Mathematical certainty sits at the very top. It is apodictic, the highest standard, requiring the necessary truth of its object: a truth impervious to doubt. This was the certainty René Descartes sought. Mathematical truths are analytically true (true by definition); numbers are universal, abstract entities, and arithmetic and geometry work the same way for everyone, everywhere. Within a self-contained, consistent system (like Euclidean or non-Euclidean geometry) there is always a single necessary answer, even when we cannot readily see it. The simplest example: one plus one is always, self-evidently, two by definition.
Logical certainty is likewise apodictic, since mathematics reduces to logic. We have it when it is logically impossible for a proposition to be false, because its denial would yield a contradiction. This rests on the law of non-contradiction: no contradiction can be true, so if denying a proposition produces a contradiction, the proposition is necessarily true. Example: "No square is a circle." Asserting the opposite is self-contradictory. Another: "There is such a thing as absolute truth." To deny it ("there are no absolute truths") is itself an absolute claim, and therefore self-defeating. Tautologies also carry logical certainty: statements whose predicate is already contained in the subject ("a bachelor is an unmarried man," "a triangle has three sides," "red is a color") are necessarily true.
Existential certainty is the third kind placed at the highest level. Here the truth is undeniable but not grounded in logic or definition alone; rather, actual existence provides evidence that cannot be contradicted. These are self-evident truths: once understood, they cannot be denied. The classic example is "I exist" (or "I am alive"). Unlike "two plus two is four," this is not necessarily true; I was not always here, and one day I will be gone. Yet at the moment I utter the words, denying them would require me to exist, so the denial proves the claim. Another example: "I feel pain now," which I can be certain of even if the feeling has no physical cause.
The Lower End: Virtual, Inductive, and Moral Certainty
Virtual certainty is a lower, arguable kind, though it sits at the top of the lower band. It is a psychological certainty grounded not in the thing known but in the knower. The confidence is something I could doubt but have no reason to: all known evidence points to the proposition's truth, yet new evidence might conceivably emerge. Examples include being virtually certain I did not eat breakfast this morning, or that I am speaking into a microphone right now. I could be dreaming or hallucinating, but nothing fits that explanation, so denying the obvious would be unreasonable.
Inductive certainty belongs at the lower end, because induction yields only probable conclusions, and probability is not certainty at all. Still, under the right conditions it can rise to the level of virtual certainty. We speak of inductive certainty when all the evidence is present and evaluated and a general conclusion is drawn: this is a perfect induction. The catch is that perfect inductions are usually unattainable. A perfect induction is theoretically possible whenever a situation has limits, and practically possible only when those limits are small and manageable. Example: "All the pens in my right hand are black" allows inductive certainty, because all the evidence is available in a tightly limited setting. By contrast, "All pencils contain graphite" is theoretically perfectible (the Earth and its pencils are finite) but practically impossible: I could never check every pencil, and a single old lead pencil could overturn the generalization. Without a perfect induction, there is no certainty.
Moral certainty is an inner certainty in the form of a conviction: a very high degree of probability that is sufficient for action though short of absolute certainty. The word "certainty" is used because the level of conviction is enough to determine how we act. This is a practical necessity, since everyone must act on limited information. Abolitionists, for instance, were morally certain that slavery was wrong, and many risked their lives acting on that conviction even though they could in principle have been mistaken. Di Donato adds that he is more certain that torturing innocent babies is wrong than that water is H2O.
Where This Leads
The episode offers a map of the spectrum of certainty, from apodictic mathematical and logical truths, through self-evident existential truths, down to the psychological and probabilistic certainties of virtual, inductive, and moral conviction. Since the course is focused on induction, the next step is to return to probability, beginning with a priori probability in the following episode.