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Try googling 'mathematical gem'. I just got 465,000 results. Quite a lot. Indeed, the metaphor of mathematical ideas as precious little gems is an old one, and it is well known to anyone with a zest for mathematics. A diamond is a little, fully transparent structure all of whose parts can be observed with awe from any angle.

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One of the central tasks when reading a mystery novel (or sitting on a jury) is figuring out which of the characters are trustworthy. Someone guilty will of course say they aren’t guilty, just like the innocent – the real question in these situations is whether we believe them.

The guilty party – let’s call her Annette – can try to convince us of her trustworthiness by only saying things that are true, insofar as such truthfulness doesn’t incriminate her (the old adage of making one’s lies as close to the truth as possible applies here). But this is not the only strategy available. In addition, Annette can attempt to deflect suspicion away from herself by questioning the trustworthiness of others – in short, she can say something like:

“I’m not a liar, Betty is!”

However, accusations of untrustworthiness of this sort are peculiar. The point of Annette’s pronouncement is to affirm her innocence, but such protestations rarely increase our overall level of trust. Either we don’t believe Annette, in which case our trust in Annette is likely to drop (without affecting how much we trust Betty), or we do believe Annette, in which case our trust in Betty is likely to decrease (without necessarily increasing our overall trust in Annette).

Thus, accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved. But is this reflective of an actual increase in the number of lies told? In other words, does the logic of such accusations makes it the case that, the higher the number of accusations, the higher the number of characters that must be lying?

Consider a group of people G, and imagine that, simultaneously, each person in the group accuses one, some, or all of the other people in the group of lying right at this minute. For example, if our group consists of three people:

G = {Annette, Betty, Charlotte}

then Betty can make one of three distinct accusations:

“Annette is lying.”

“Charlotte is lying.”

“Both Annette and Charlotte are lying.”

Likewise, Annette and Charlotte each have three choices regarding their accusations. We can then ask which members of the group could be, or which must be, telling the truth, and which could be, or which must be, lying by examining the logical relations between the accusations made by each member of the group. For example, if Annette accuses both Betty and Charlotte of lying, then either (i) Annette is telling the truth, in which case both Betty and Charlotte’s accusations must be false, or (ii) Annette is lying, in which case either Betty is telling the truth or Charlotte is telling the truth (or both).

This set-up allows for cases that are paradoxical. If:

Annette says “Betty is lying.”

Betty says “Charlotte is lying.”

Charlotte says “Annette is lying.”

then there is no coherent way to assign the labels “liar” and “truth-teller” to the three in such a way as to make sense. Since we are here interested in investigating results regarding how many lies are told (rather than scenarios in which the notion of lying versus telling the truth breaks down), we shall restrict our attention to those groups, and their accusations, that are not paradoxical.

The following are two simple results that constraint the number of liars, and the number of truth-tellers, in any such group (I’ll provide proofs of these results in the comments after a few days).

Result 1: If, for some number m, each person in the group accuses at least m other people in the group of lying (and there is no paradox) then there are at least m liars in the group.

Result 2: If, for any two people in the group p₁ and p₂, either p₁ accuses p₂ of lying, or p₂ accuses p₁ of lying (and there is no paradox), then exactly one person in the group is telling the truth, and everyone else is lying.

These results support an affirmative answer to our question: Given a group of people, the more accusations of untrustworthiness (i.e., of lying) are made, the higher the minimum number of people in the group that must be lying. If there are enough accusations to guarantee that each person accuses at least n people, then there are at least n liars, and if there are enough to guarantee that there is an accusation between each pair of people, then all but one person is lying. (Exercise for the reader: show that there is no situation of this sort where everyone is lying).

Of course, the set-up just examined is extremely simple, and rather artificial. Conversations (or mystery novels, or court cases, etc.) in real life develop over time, involve all sorts of claims other than accusations, and can involve accusations of many different forms not included above, including:

“Everything Annette says is a lie!”

“Betty said something false yesterday!”

“What Charlotte is about to say is a lie!”

Nevertheless, with a bit more work (which I won’t do here) we can show that, the more accusations of untrustworthiness are made in a particular situation, the more of the claims made in that situation must be lies (of course, the details will depend both on the number of accusations and the kind of accusations). Thus, it’s as the title says: accusation breeds guilt!

Note: The inspiration for this blog post, as well as the phrase “Accusation breeds guilt” comes from a brief discussion of this phenomenon – in particular, of ‘Result 2′ above – in ‘Propositional Discourse Logic’, by S. Dyrkolbotn & M. Walicki, Synthese 191: 863 – 899.

The post Accusation breeds guilt appeared first on OUPblog.

        

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