Oxford University Press's
Academic Insights for the Thinking World

# The logic of unreliable narrators

In fiction, an unreliable narrator is a narrator whose credibility is in doubt – in other words, a proper reading of a narrative with an unreliable narrator requires that the audience question the accuracy of the narrator’s representation of the story, and take seriously the idea that what actually happens in the story – what is fictionally true in the narrative – is different from what is being said or shown to them. Unreliable narrators are common in fiction. Notable examples include Agatha Christie’s The Murder of Roger Ackroyd, Ken Kesey’s One Flew Over the Cuckoo’s Nest, Akira Kurosawa’s Rashômon, and Ron Howard’s A Beautiful Mind.

There are all sorts of interesting philosophical questions one might ask about unreliable narrators and how they function as a storytelling device. Here, however, I am going to point out some purely logical features of unreliable narrators.

Presumably, although the full account is no doubt more complex, one of the primary factors that determines whether a narrator is reliable, and to what extent, is the ratio between the number of (fictionally) true claims made by the narrator to the total number of claims made by the author. All else being equal, the higher this ratio, the more reliable the narrator is. Now, consider two stories. The first story – S1 – involves the narrator making n claims for some number n:

S1 = {C1, C2, C3Cn}

And let’s assume that, for some number mn, m of these claims are true. So the relevant ration is m/n. The second story – S2 – is exactly like the first except for the addition of one more claim by the narrator: the claim that he or she is unreliable, which we shall call U:

U = “I am an unreliable narrator”

Hence:

S2 = {C1, C2, C3Cn, U}

Now, there are two possibilities. Either U is true, or U is false. If U is true, then the ratio of truths to falsehoods is (m+1)/(n+1). But, for any positive finite numbers m and n where mn:

(m+1)/(n+1) > m/n

So, although the narrator of S2 might well be unreliable, he or she is more reliable than the narrator of S1 which fails to contain the admission of unreliability U. Note that this also implies that the narrator of S1 must have been unreliable as well.

If U is false, however, then then the ratio of truths to falsehoods is (m)/(n+1). But, for any positive finite numbers m and n where mn:

m/n > (m)/(n+1)

So, although the narrator of S2 might well be reliable, he or she is less reliable than the narrator of S1 which fails to contain the admission of unreliability U.

The latter fact – that a reliable narrator claiming to be unreliable in fact makes them less reliable – is perhaps unsurprising and uninteresting, the fact that an unreliable narrator admitting their unreliability makes them more reliable is more interesting. Before examining this fact further, however, it is worth noting that there is a Truth-teller like phenomenon in the vicinity as well.

Consider a story where the narrator, in addition to narrating the story, also claims to be a reliable narrator (perhaps, along time-honored traditions, by beginning the story with “Everything I am about to tell you is true”). Via computations similar to the above, if the narrator of a story not containing a claim to reliability of this sort is generally reliable then the narrator of a story otherwise identical but supplemented with such a claim to reliability is even more reliable, and if the narrator of a story not containing a claim to reliability of this sort is generally unreliable, then the narrator of a story otherwise identical but supplemented with such a claim is even less reliable.

Now, although it involves fictional truth (i.e. what claims we ought to make-believe to be true when consuming a fiction) rather than actual truth, at this point this puzzle looks like nothing more than a variant of the Liar paradox and the Truth Teller. But there is a secondary puzzle that arises once we have noted the Liar-like behavior of “I am an unreliable narrator.”

Whether or not a narrator is reliable, and more generally, the extent to which a narrator is reliable, is typically not something the author of a work announces at a press conference or prints on the cover of a book or DVD, but is instead something that the reader or viewer of a work has to decipher for him-or-herself from clues included in the story. On the face of it, one piece of evidence that we might think to be definitive in this regard is an admission of unreliability by the narrator him-or-herself. But, as we have seen, such an admission in fact makes the narrator more reliable, rather than less, if the narrator is in fact generally unreliable. In short, the sort of claim that we might, on the face of it, take to be good evidence of the presence of an unreliable narrator turns out to be much less useful than we might have first thought.

On the other hand, the results given above do suggest a sort of informal decision procedure for determining whether or not the narrator of a work is generally reliable or not. When confronted with a story where the evidence seems indeterminate with regard to whether, and to what extent, we should “believe” the narrator, we can just imagine a story that is similar except that the narrator claims to be reliable. If the narrator of the original story was generally reliable, the narrator of this new story will be even more reliable, and if the narrator of the original story was generally unreliable, then the narrator of this new story will be even more unreliable. Presumably, the more pronounced reliability, or unreliability, in the new story will be easier to detect than the original degree of reliability or unreliability in the original story was. If there still isn’t enough evidence to decide, then simply add another claim to reliability on the part of the narrator. And if this isn’t enough, add another one. Presumably at some point the reliability, or unreliability, of the narrator will become so extreme that it will be impossible not to spot, in which case the narrator of the original story will be generally reliable or generally unreliable if and only if the narrator of this new expanded story is (although not, of course, to the same degree).

Now, clearly the recipe just given is absurd – this algorithm for detecting whether or not a narrator is reliable or not just won’t work. But it strikes me as a little bit difficult to say exactly where it has gone wrong.

Featured image: Book by Kaboompics // Karolina, Public Domain via Pexels.

1. Paul Canniff

I don’t see why multiple copies of the same claim would necessarily have a cumulative effect.

2. Pragmatic guy

I realize that you are not interested in the pragmatics, but the Book of Mormon is potentially an example to discuss.
It proclaims its reliability ad nauseam, but that has the effect of undermining its credibility rather than reinforcing it. Presumably this is a version of the idea that someone can “protest too much”. Why insist on the logical redundancy that what you are saying is true if not to reassure doubters? and why be so interested in reassuring them unless there are good grounds for doubt?

Here’s the beginning of BoM:

Be it known unto all nations, kindreds, tongues, and people, unto whom this work shall come: That we, through the grace of God the Father, and our Lord Jesus Christ, have seen the plates which contain this record… And we also know that they have been translated by the gift and power of God…, wherefore we know of a surety that the work is true. And we also testify that we have seen the engravings which are upon the plates; and they have been shown unto us by the power of God, and not of man. And we declare with words of soberness, that an angel of God came down from heaven, and he brought and laid before our eyes, that we beheld and saw the plates, and the engravings thereon; and we know that it is by the grace of God the Father, and our Lord Jesus Christ, that we beheld and bear record that these things are true.

That’s a paradigm case of someone’s “protesting too much”.

Very interesting. The opening claim from “The DaVinci Code” immediately sprung into mind as the perfect example of a narrative voice making a claim for reliability in the realm of fictionally-true (although its presentation as factually true complicates affairs slightly).
I’m left wondering what sorts of stories could be crafted using a narrator which oscillates between claiming to be reliable and claiming to be unreliable – and being reliable while claiming to be unreliable, and vice versa. I think keeping this paradox in mind could allow for some interesting storytelling.

4. Eric Hiatt

Has logic taken into account the fact that it’s not “decoupled” from the Universe itself? That is, when a brain, or whatever, processes a logical process, the configuration of the Universe has literally changed. There are no sets of decoupled statements that apply to a given Universe because as soon as one statement has been processed we’re literally dealing with a new Universe (e.g entropy has increased). You could pretend that logical statements are being processed “simultaneously” in spacetime, but now you’re dealing with the variable simultaneity of relativity. Besides, it would always take time for all these statements to “communicate”.

What’s strange about paradoxes like these and the Yablo paradox is that statements are both altering the Universe – changing its configuration – but then also attempting to use information from this new configuration. May the paradox lies in a fundamental violation of physical reality. Maybe these paradoxes are similar to what happens when there’s no cosmic speed limit or energy isn’t conserved or you make the assumption that division by zero is possible (at which point you can prove anything). I’m almost certain the issue has something to do with the fact that we treat logic as if it’s decoupled from physical reality, which it can’t be.

5. David Williams

A calculus of narrative reliability would best be founded upon the axiom that any and all strings of factual claims contain a number of untruths ranging from none of them to all of them. This way the inclusion of a statement such as “I am an unreliable narrator” becomes redundant at no cost to completeness (consistency possibly being another matter).
Any steps we then take to further empiricise the concept of ‘unreliable narrator’ within this calculus would almost certainly require us to augment (taint?) our investigations with induction, abduction, and Bayesian reasoning. Still, the logical song should remain the same despite this.

6. tqk

I think where you write “But, for any positive finite numbers m and n where m ≤ n: (m+1)/(n+1) > m/n”, the “≤” should be a “<".

I don't see what justifies your assumption that "more pronounced" (un)reliability will be easier to detect (given the way you measure reliability). In fact, your example with the narrative can be considered an argument by contradiction to show that your assumption was false.

Of course, there are informal notions of reliability under which greater unreliability tends to be more obvious (e.g. the lie that I am 8 feet tall is more obviously unreliable than the lie that I am 6 feet tall). However, your measure of reliability does not capture this.