# Are There Finitary Logical Constraints on Converting Recursive Syntax to Semantics?

From the logical standpoint can we continue these series on and on and still be able to make sense of it?

```1. I know                                                      <--- no problem
2. I know that you know                                        <--- that's fine
3. I know that you know that I know                            <--- that's okay
4. I know that you know that I know that you know              <--- that's fine too!
5. I know that you know that I know that you know that I know  <--- em...
...
N. I know that you know that I know that you know ...
```

I am personally struggling mentally when I go beyond the fourth level (although I have some moments of "awakening" when it seems clear to me, and then it disappears again).

Is there any hard stop that makes it loose sense at some point or we can construct such sentences forever and it will still have meaning?

• How far we can still "grasp" iterative constructs "at once" has to do not with "logic" (or rather semantics, the theory of meaning) but with psychology, the size of operational short term memory and the like. The same way arithmetic is not concerned with how far we can physically count. But in principle there is no problem with interpreting any number of these iterations, just as there is no problem with interpreting arbitrarily large integers. It is simply done the same way they are constructed, piecemeal. Aug 12, 2020 at 4:25
• Interestingly, I find the iterations easier to understand (excluding the first) when the embedding is even, i.e. 2, 4, and more difficult when it's odd, i.e. 3, 5. In particular, I struggled with 5, but 6 wasn't as difficult to understand. Anyone else have a similar experience? Aug 12, 2020 at 13:59

The sentences continue to make sense in theory, but beyond some point it just becomes too much for our human working memory to track. From this article:

Now try the fifth sentence: The malt that the rat that the cat that the dog worried killed ate lay in the house that Jack built. Are you still following me? That last example is perfectly grammatical, but more than one level of center-embedded recursion is hard to follow, for psychological rather than linguistic reasons. Center-embedding requires a memory device, such as a stack of pointers, indicating where to pick up the procedure once an embedded constituent has been completed. This is not so bad if there is just one embedded structure, since a single pointer can be held in memory to show where to pick up the original procedure. With multiple embedding, you need to keep track of several pointers, which can overstretch working memory. Examples of sentences with more than one level of center-embedding are rare in natural discourse.

• very good read, thank you for the reference! Aug 12, 2020 at 20:10

There is no limit on this structure of a self-reflexive and highly compressible* sentence.

This sentence's function is iterative and each sentence that makes sense can have a child sentence that also makes sense (because each level introduces a new [and identical] layer of alternating parties knowing the contents of the prior sentence). Because the first sentence is logical, every subsequent sentence will also contain a decipherable meaning and no confounding language can ever be introduced.

*By compressible, I refer to the structure being carefully repetitive:

x(1,2,3) == ['I know', ' that you know', ' that I know']

n0 == x(1)

for 1:i

n1 == sum(n) + x(2)

n2 == sum(n) + x(3)

end

view(n)

This representation obviously isn't meaningful with respect to actual programming but it might help you appreciate that the system can't eventually lose meaning because it's just a loop — if it makes sense at first, it should make sense ad infinitum, ad nauseam.

• thanks! i'm puzzled why it feels so weird and almost indecipherable after to cross some threshold (4th level for me)... Aug 12, 2020 at 20:02

Excellent philosophical question!

As Conifold has stated, there is no limit to logical reference from a theoretical standpoint, but there is from a practical one.

While you are using identical subordinate clauses, one could also use varied prepositional phrases in a sentence:

1. There is the boy in the house.
2. There is the boy in the house on the sofa.
3. There is the boy in the house on the sofa of the owner.
4. There is the boy in the house on the sofa of the owner with the name Bill. ....

Do the sentences ever become meaningless? No. Remember, just because you don't comprehend the proof of the Laplace Transform doesn't make the proof meaningless. This is a lesson lost on poor critical thinkers who aren't aware of the Dunning-Kruger Effect. : ) Do they become incomprehensible, yes. And since they do, we just split them up into comprehensible pieces. So, we can rewrite 4 as follows:

There is the boy in the house on the sofa. He's sitting on the sofa which is owned by Bill.

One could also use mathematical operations to differentiate between meaning and comprehension. Consider that 1 can be added to itself, and we can write 1+1, 1+1+1, ..., 1+1+1+...+ 1. Would anyone ever claim that there is a logical limit to how many times we can add? No, but we can make the sentence more comprehensible:

Σ1 from terms 1 to n.

This helps to show a fundamental difference between syntax and semantics. The ability to process syntax to lead to semantics is what both the calculator and the brain do. Thus, as long as one follows the rules of syntax, one can have a meaningful, but incomprehensible sentence. In computation, the study is called formal language. In natural language, the study of such a topic falls under psycholinguistics under headers such as chunking.

And for the record, if you're interested in such topics, you can approach these sorts of philosophical thoughts in the philosophy of language. John Searle in his Speech Acts recognizes distinctions among linguistics which studies specific languages and their features, linguistic philosophy which is an approach to doing philosophy by examining the nature of the language, and the philosophy of language which he describes as "the attempt to give philosophically illuminating descriptions of certain general features of language, such as reference, truth, meaning, and necessity[.]" (pg. 4).

• appreciate your answer, however, I think you moved the focus from the central point that I was trying to make -- the principal "loopiness" of these kinds of statements. In your examples it's obvious that we simply adding some details each time, and it's clear that it will make sense forever Aug 12, 2020 at 19:58
• Thank you. I did shift the focus somewhat because I wasn't sure what your familiarity with the more abstruse nature of recursion is, and was worried about closure. Did you select "loopiness" in respect to GEB's use of the "strange loop" or just good intuition? I can add a more technical segment if you want.
– J D
Aug 12, 2020 at 20:02
• I revised the title, and upvoted.
– J D
Aug 12, 2020 at 20:03
• Yes! Exactly, Hofstadter's "strange loops" came into my mind while pondering this one. Aug 12, 2020 at 20:07
• Well, it's important to draw a distinction between natural and formal languages, first, because natural language has built into it biological constraints where as formal logic and language would approach it from the place of an axiomatic construct. If the question really is, despite that a person can't comprehend the recursion, is it still meaningful, then in a technical sense yes, because as long as the rules of the language can be used to reduce to something meaningful, then, the original statement has meaning.
– J D
Aug 12, 2020 at 20:17