This article has been accepted for publication in Analysis Published by Oxford University Press.
Punny Logic
Noah Greenstein

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 premiss and conclusion. Given the premiss:

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 premiss 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 premiss 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 most common definitions of terms’.
	In the first meaning, by definition, kleptomaniacs steal, that is, they take things.
	So then it is true that they ‘take things literally’.
	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. P can’t be defined as Q and then not defined as Q: (P=Q & P≠Q) is a contradiction and to be avoided classically. But let’s start with a classical formulation. Let:

•	TTL1 to be Take Things Literally in the 1st sense: to steal
•	TTL2 to be 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 → Hy ]
For anyone who takes things literally (definitionally), Then it is hard to explain puns to them

What we want, however, is closer to:

(3) ∀z [[ Kz → TTLz ] → Hz ]
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 classically 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 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:

(4) ∀z [ Kz → { TTLz ]1 → Hz }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 (Hintikka 1989). IF logic allows for fine-grained scope allocation. So let:

•	S mean to steal.
•	D mean to take things definitionally.

Then:

(5) ∀x ∀y ∃u/∀x ∃v/∀y [ Kx → ( x=u & y=v & Su & Dv → TTLvu ) → Hy ]
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: Stealing pairs with Kleptomania and taking things Definitionally pairs with being Hard to explain puns to.
	The two pairs are also linked together by the way they must be mutually, self-referentially, instantiated. x can’t be instantiated independently of y because x is identical to u, which depends on y. y can’t be instantiated independently of x because y is identical to v, which depends on x. This makes the pairs, and hence the meanings, chain-linked together, just like in the scope configuration of Equation (4): [ - { - ]S - }D. Given this linked instantiation and each pair meaning to Take Things Literally, Equation (5) therefore provides a formalisation of the original pun.

	While this article may have forever ruined the above pun for you (and me), it does potentially give insight into what people find funny. Classically, Porphyry associated risibility -- the ability to laugh -- with humans and rationality (Owen 1853: 612). However, humour has generally been a philosophically neglected topic (Morreall 2012) and formal-logically non-existent. The ability to formally represent meaning ambiguity shows humour to be an untapped resource of novel logical structure.
	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 analyse theories’ contradictions and incommeasureabilities. Experiments are then the punchlines that help determine scientific interpretations.
	On the other hand, opposed to what we scientifically learn about the world, what we find funny, or don’t find funny, often reveals something about ourselves. Given that the above analysis cuts in both directions, the methods of logic and science can be brought to bear on humour. Perhaps, then, there’s something to be learned about the human condition as well, and some of our philosophical oversight can be rectified.

References:

Hintikka, J. and G. Sandu. 1989. Informational Independence as a semantical phenomenon. in Logic, Methodology and Philosophy of Science, Vol. 8, J. E. Fenstad, I. T. Frolov, and R. Hilpinen (eds.), Amsterdam: Elsevier, pp. 571–589; reprinted in Hintikka, J., 1998, Paradigms for Language Theory and Other Essays. Jaakko Hintikka: Selected Papers. Volume 4, Dordrecht: Kluwer, 52–70.

Morreall, J. 2012. Philosophy of humor. The Stanford Encyclopedia of Philosophy (Spring 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2013/entries/humor/>.

Owen, O.F. 1853. The Organon, or Logical treatises, of Aristotle: With the introduction of Porphyry. London: H.G. Bohn.

Abstract:
Logic and humour tend to be mutually exclusive topics. Humour plays off ambiguity, while classical logic falters over it. Formalising puns is therefore impossible, since puns have ambiguous meanings for their components. However, I will use Independence-Friendly logic to formally encode the multiple meanings within a pun. This will show a general strategy of how to logically represent ambiguity and reveals humour as an untapped source of novel logical structure.

Keywords:
ambiguity, formalisation, humour, independence friendly logic, philosophical logic