# Punny Logic

Update 12 Feb: This post had been expanded upon and, after submission, accepted for publication in Analysis published by Oxford University Press. View the final version here.

[draft]

It is hard to explain puns to kleptomaniacs because they take things literally.

On the surface, this statement is a statement of logic, with a premise and conclusion.

Given the premise:

Kleptomaniacs take things literally.

We may deduce the conclusion:

It is hard to explain puns to kleptomaniacs.

Now, whether the conclusion strictly follows from the premise is beside the point: it is a pun, and meant to be funny. However, as a pun, it still has to make some logical sense. If it didn’t make any sense, it wouldn’t, and couldn’t, be funny either. While nonsense can be amusing, it isn’t punny.

What is the sense in which the conclusion logically follows from the premise then, and how does this relate to the pun?

Puns play off ambiguity in the meaning of a word or phrase. In this case the ambiguity has to do with the meaning of to take things literally. It can mean `to steal`, or it can mean `to only use the simplest, most common definitions of terms`.

In the first meaning, by definition, kleptomaniacs steal, i.e. they literally take things.

So then “take things literally” is true.

In the second meaning, by deduction, since puns play off multiple meanings of things, it is hard to explain a pun to someone who only uses the single, most common definition of a term. That is, if they take things literally, they won’t recognize the multiple meanings required to understand a pun.

So if someone “takes things literally” it is true that it is hard to explain puns to them.

Therefore, between the two meanings, we can informally derive the statement: it is hard to explain puns to kleptomaniacs because they take things literally.

However, if we wanted to write this out in a formal logical language, then we would need a formal way to represent the two meanings of the single phrase.

Classically, there is no way to give a proposition multiple meanings. Whatever a proposition is defined as, it stays that way. `A` can’t be defined as `B` and then not defined as `B`: `(A=B & A≠B)` is a contradiction and to be avoided classically. But let’s start with a classical formulation:

Let:

`TTL1` mean to Take Things Literally, in the 1st sense: to steal

`TTL2` mean to Take Things Literally, in the 2nd sense: to use the most common definitions of terms.

Then

1. ∀x [ Kx → TTL1x ]
For anyone who is a Kleptomaniac, Then they take things literally (steal)
2. ∀y[ TTL2y → Py ]
For anyone who takes things literally (definitionally), Then it is hard to explain puns to them

What we want, however, is closer to:

1. ∀z [[ Kz → TTLz] → Pz ]
For anyone who is a Kleptomaniac, Then they take things literally, Then it is hard to explain puns to them

with only one sense of TTL, but two meanings.

Since TTL1 ≠ TTL2, we can’t derive (3) from (1) and (2), as is. And if TTL1 = TTL2, then we would have (1) A→B, and (2) B→C, while trying to prove (3) A→B→C, which logically follows. However, there would no longer be a pun if there was only one meaning of TTL.

What is needed is to be able to recompose our understanding of ‘to take things literally’ in a situation aware way. We need to be able to have the right meaning of TTL apply at the right time, specifically Meaning 1 in the first part, and the Meaning 2 in the latter.

Intuitively, we want something like this, with the scope corresponding to the situation:

1. ∀z [ Kz → { TTLz ]1 → Pz }2

In this formula, let the square brackets [] have the first meaning of TTL apply, while the curly braces {} use the second meaning. Only the middle — TTL — does double duty with both meanings.

Achieving this customized scope can be done by using Independence Friendly logic. IF logic allows for fine-grained scope allocation.

So let:

`S` mean to steal.

`D` mean to take things definitionally.

Then:

1. ∀x ∀y ∃u/∀x ∃v/∀y [ Kx → ( x=u & y=v & Su & Dv → TTLvu ) → Py ]
If anyone is a kleptomaniac then there is someone who is identical to them who steals… and if there is someone who takes things definitionally then there is someone identical to them for whom it is hard to explain puns to… and the person who steals and the person who takes things definitionally then both Take Things Literally.

The scope gymnastics are being performed by the slash operators at the start and the equality symbols in the middle part of the equation. What they are doing is specifying the correct meanings — the correct dependencies — to go with the correct senses: `S`tealing pairs with `K`leptomania and taking things `D`efinitionally pairs with being bad at `P`uns, while both pairs also meaning `Taking Things Literally`. With both pairs meaning `TTL`, and each pair being composed independently, Equation (5) therefore provides a formalization of the original pun.

#### Discussion

Finding new applications for existing logical systems provides a foundation for further research. As we expand the range of topics subject to logical analysis, cross-pollination between these subjects becomes possible.

For instance, using custom dependencies to associate multiple meanings to a single term is not only useful in describing puns. Scientific entities are often the subjects of competing hypotheses. The different hypotheses give different meanings — different properties, relations and dependencies — to the scientific objects under study. Logically parsing how the different hypotheses explain the world using the same terms can help us analyze the contradictions and incommeasureabilities between theories.

On the other hand, while this article may have forever ruined the above pun for you (and me), it does potentially give insight into what humans find funny. Classically, risibility, having the ability to laugh, has been associated with humans and rationality. Analyzing this philosophical tradition with the new logical techniques will hopefully provide existential insight into the human condition.

## 5 thoughts on “Punny Logic”

1. Quentin says:

There is one thing that makes the joke funny that you did not addressin your analysis, which is that the pun is self-referential:
– it is a pun
– it involves taking the expression “to take litterally” litteraly (or not).

Fortunately, you didn’t completely ruin the pun.

2. nogre says:

Hi Quentin, thanks for your comment.

I agree that there is something with the ” ‘to take literally’ literally” issue. The question is about how meta-literal can you get — TTL(TTL(TTL(x))) — similar to Knowing that you Know x, and then Knowing that you Know that you Know x, etc. This literal regress doesn’t look too complicated because the meanings at each level are clear. As for the analysis of the pun, it shouldn’t be a problem since the pun doesn’t require any more than 1 level of TTL.

However, I do believe I addressed the self referentiality with the complex scope operations. The instantiation scheme in Equation (5) chains together the variables: you can’t instantiate X without also instantiating Y, and vice versa, because of the way U and V depend on X and Y. This makes the different parts of the formula refer to the other parts, and hence back on themselves. One of the reasons I wrote this article is to demonstrate an instance of a self-referential instantiation pattern.

1. Quentin says:

I don’t agree. I think the original pun gives rise to questions such as: would a kleptomaniac understand this very pun? Your formulation gives no way to formulate such questions, and it does not suggest that kind of questions in any way.

The self-reference in (5) does not address the point, because you could have the same kind of pun with the exact same formulation, except that the predicates would make no reference to explaining puns or taking things litteraly.

For example: It’s rewarding to listen to the stars, because they’re bright.
Admit this pun is much less funny than the original, but it can be formalized the same way as (5).

3. nogre says:

Ah, you’re right… I used a pun about puns, which is self-referential in a way different than other puns, and not what I was specifically describing in (5). My goal was to describe a generic kind of pun, like the one that you brought up.

Now, I don’t think there is anything wrong about what I said above. It is a new question, after the pun has been formalized, to talk about it as a pun about puns. There isn’t anything about this sort of self-reference that is different than other self-referential language, at least that I have spotted, so I’d just say: treat it as you would any other sort of self-referential language.

I disagree that this:

>would a kleptomaniac understand this very pun?

is tough to answer. (5) is a pun and, as such, would be subject to “It is hard to explain puns to kleptomaniacs.” So (5) would not be easily understood by a kleptomaniac.

edit 1am feb 2:

If you wanted to get specific about it, then I could add a clause to (5):

let:

N mean to be a pun

Q means doesn’t understand

then:

∀x ∀y ∃u/∀x ∃v/∀y [[ Kx → ( x=u & y=v & Su & Dv → TTLvu ) → Py ] → ∀g (Ng → Qxg)]

So, if (5) then if g is a pun the kleptomaniac x doesn’t understand g.

1. Quentin says:

Ok thank you for your reply. I agree that your analysis is still correct, as an account of a generic kind of pun (which might display self-referential properties, or not).