Sid's Paradox

September 1, 1997

Can you distinguish fact from opinion?

In our common discourse we often make a distinction, usually a casual one, between matters of fact and matters of opinion. For example, if I claim, “Boulder is a pleasant city to live in,” the claim would ordinarily be regarded as an opinion, by which we might mean that it reflected a subjective experience not extensible to others’ experiences of Boulder in an unambiguous way. On the other hand, if I make the claim, “Boulder is a city in the state of Colorado,” then this latter claim would ordinarily be considered a true fact, worthy of inclusion in an encyclopedia, say, as a guide to others of an actual condition pertaining in the world.

Our distinctions between fact and opinion are not always casual. In a judicial setting, much effort is made (in particular regarding the admissibility and weight of evidence) to separate fact from opinion. This point is greatly stressed in legal training, and making such distinctions is clearly an essential part of a judge’s duties.

In mathematics such distinctions are made so formal as to be obviated, that is, opinions are not called opinions at all, but rather “conjectures,” which are inspired surmises relevant to theory but unproved (and perhaps unprovable).

Even in daily life, in making many kinds of decisions, the distinction between fact and opinion can be an important one. For emphasis I briefly list a few more examples.

opinion It is warm in this room.
fact It is 74 degrees Fahrenheit in this room.
opinion Most mathematicians are Platonists on weekdays and formalists on Sundays.
fact? Computers have revolutionized communication.
opinion? Representative democracy is the best means of preserving individual liberty.

Given that we are at pains in many ways to distinguish fact from opinion, an obvious question arises:

Is the difference between matters of fact and matters of opinion…
a matter of fact or a matter of opinion?

What is your answer to this question? Consider it carefully before continuing.


Certainly one might debate whether the question above is well-formed or ill-conceived. Nonetheless, regardless of any philosophical, linguistic, or formalist objections to the question itself which might be raised, each of us acts in daily life as though we know the difference between fact and opinion, or at least as though the difference is knowable. Philosophical niceties aside, it is difficult to see how we could function otherwise. On that basis, I claim the question is worth trying to answer.

And so, have you answered it? Good. That brings us to a small but rather delicious difficulty. Perhaps you have guessed it already. Depending on how you answered the question, you have made one of the following assertions:

The difference between matters of fact and matters of opinion is a matter of opinion.
The difference between matters of fact and matters of opinion is a matter of fact.

Here is the difficulty: characterize your answer. Is it a fact or an opinion? Consider your answer to this second question carefully before continuing.


As you answered the second question, you very likely noticed that it is subject to the same challenge. That is, if you asserted that “the difference between fact and opinion” is an opinion, one may immediately question whether this assertion is a fact or an opinion. And the answer to this, of course, is likewise challengeable. And so on. We have an infinite regress.

So…envision each potentially infinite string of possible answers as a string of F’s and O’s (F for fact and O for opinion). One might, for instance, get something like,

F O F F F F F F F F…

This would represent the assertion that the distinction between matters of fact and matters of opinion is a matter of fact, that this answer is an opinion, and that the assertion that it is an opinion is factual, and that that is a fact, and that that is a fact, ad infinitum. In fact (if you'll forgive the expression!) this is a very common response among the many people to whom I have posed these questions. The other most common are these:

F F F F F F F F F F…
O O O O O O O O O…
O F O F O F O F O…
O O F F F F F F F F…

You may wish to pause for a few moments to work out what each of these strings asserts. At this point, do your earlier answers still hold, or do you wish to reconsider them? Try jotting down a string corresponding to your own answer to the infinite regress of questions.

Certain questions arise:

  1. Since the possible strings of answers are exactly the possible infinite binary strings, the set of possible answers is itself uncountably infinite. However, can we distinguish among these any subset that is (somehow or another) consistent and plausible? For example, it is hard to image how anyone could argue that
    F F F F F F O F F F…           
    is a sensible or self-consistent response. If such a distinction can be made, how many would there be? Finitely many? Countably many?
  2. The fact (there it is again!) that attempting to distinguish formally between facts and opinions leads to an infinite regress casts doubt on whether the notions of fact and opinion are susceptible of precise definition. Or is there, perhaps, some way of resolving a distinction between the two that avoids an infinite regress?
  3. If a formal distinction between fact and opinion is not resolvable, what does this imply about the status of knowledge, as such? Can a fact be a fact if it is not, in some sense, formally a fact? And, if not, how then can we have knowledge without certainty of at least a few facts? The flavor of this paradox is akin to certain classical problems in formal logic. Russell's Paradox, in particular, comes to mind, and suggests that there is a problem here in comprehension, where this word is used in the sense of comprehension axioms in set theory. Specifically, are we vainly attempting to define a “set of all facts,” when to do so would necessarily involve asserting a fact (to wit: the defined set of facts) which is a member of the set we are attempting to define?
  4. Is mathematics really immune from this difficulty? Define a “fact” for a given theory T to be a theorem of T which is provable in T, and an “opinion” of T to be a conjecture which is neither provable nor disprovable in T. Godel's Incompleteness Theorem then assures us that the distinction between fact and opinion (in mathematical theories) cannot be fully determined.

This problem occurred to me some years ago, and I have never seen it addressed elsewhere. I would be pleased to hear any thoughts about it that readers cared to relate. Feel free to write me using the contact form.