Self-Reference without Paradox

How to Understand Self-Referential Sentences

Self-reference is the foundation of a new mystery religion. Adherents see paradoxes everywhere, even at the foundation of critical thinking—logic itself. “The Liar’s Paradox”, they say, “demonstrates that the law of non-contradiction isn’t absolute.”

“Logic can’t really give us the truth, because something something Godel’s incompleteness theorems.”

Nearly all the mystical paradoxes people bring up today invoke either 1) self-reference, 2) infinity, or 3) quantum mechanics. I’ve dealt with all three before, but I want to revisit the topic of self-reference.

Here’s a general technique for clearing up the paradoxes that are generated by self-referential sentences:

Pay attention.

Pay attention to your own mind and how it processes language. The magic and mystery of self-reference disappears when you take the time to observe your own mental processes. Take two examples:

Statement (1):    This sentence is false.

Statement (2):    This sentence has five words.

(1) Is the famous liar’s paradox, which superficially appears to be a contradiction—if it’s true, then it’s false, but if it’s false, then it’s true.

Contradictions cannot be made sense of, and yet, the liar’s paradox seems like it should make sense. Hence, this is the most famous paradox and has been around forever.

(2) Makes straightforward sense and is “true.”

So what’s going on here? How can one example of self-reference result in logical annihilation, while the other is trivially evaluated as true?

Pay attention.

Observe what your mind is doing when encountering the words.

Resolving the Liar

I have written about the resolution to the Liar’s paradox elsewhere, but let me summarize the argument here.

“This sentence is false” either 1) explodes into an infinity, or 2) collapses to zero.

It either generates an infinite regress, or it’s simply nonsense wearing a fancy suit.

To see why, ask the question, “What sentence exactly is false?” What do the words “this sentence” refer to?

In other words, is the claim:

Option A: “This sentence is false” is false.

Or simply,

Option B: This sentence is false.

If Option A, then it’s easy to see why it generates an infinite regress. The use of parentheses helps. The claim is:

(This sentence is false) is false.

How does our mind try to make sense of these words?

Well, outside the parentheses, we are told that something is false, which means that inside the parentheses, there must be some valid truth claim to evaluate. So we look inside the parentheses and see the words “this sentence is false.”

How do we evaluate such a claim? We again have to figure out what “this sentence” refers to. If it refers to the entire sentence—“this sentence is false”—then we are stuck generating the infinite regress:

“((This sentence is false) is false) is false…”, and so on. It’s like trying to walk to the end of a road that keeps elongating with every step you take. It won’t work.

The only other option is to evaluate “this sentence is false” by itself. We should first break it into two parts: (This sentence) + (is false).

The words (is false) tell us that we’re supposed to evaluate the truth value of a preceding proposition. But (this sentence) is not a proposition. It’s merely two words: “this” and “sentence.”

“This sentence” is not a valid truth claim. It’s essentially an undefined function; we cannot evaluate the words “this sentence” as true or false. That’s why I like to say the liar’s paradox either explodes to infinity or collapses to zero.

Just Make Sense

It sounds like a good resolution, but perhaps it proves too much? Are all examples of self-reference therefore invalid?

Of course not. We can clearly make sense of the following:

Statement (P): This sentence has five words.

Statement (Q): This sentence is in English.

Statement (R): This sentence does not contain the word “paradox.”

All three of these we can evaluate as true or false. The first two are true. The last is false. So what’s going on? Why do these not explode to infinity or collapse to zero? I propose the same answer:

Pay attention.

Observe what your mind is doing when encountering the words.

When you read (P), what does your mind do? It says, “Hey, check out this set of words. It’s supposed to contain five elements. The elements are: “This” “sentence” “has” “five” “words”, which totals five, and therefore (P) is true.”

No magic, no mystery, no explosion to infinity—the words “this sentence” refers to something definite. A proper use of self-reference.

Now consider (Q). How does your mind process that sentence?

It says, “Hey, check out these words. They are all supposed to be English. The words are “This” “sentence” “is” “in” “English”, which are all English words, therefore (Q) is true.”

Clear and simple. Now let’s do (R):

“Hey, check out this set of words. It’s not supposed to contain the word “paradox.” The words are “This” “sentence”… “paradox”. Since “paradox” is part of the set, (R) is false.”

Clear and simple. No magic, no infinity. Just good ol’ fashioned self-reference that does not destroy the fabric of reality.

Good Self, Bad Self

Every case I’ve encountered of self-reference works this way. By looking at the mind, all of the paradoxical examples are resolved, and all of the sensible examples are explained. It’s similar to computer programming. Most of the time, self-referential code works fine, but sometimes, it hangs the computer in a never-ending loop. When the latter happens, we conclude, “Huh, I guess the code/coder is bad” and never “Huh, I guess that means logic is broken.”

I suggest we approach all examples of self-reference with this common sense heuristic:

If you can’t make sense of it, or it’s impossible for a computer to execute, or it results in a contradiction, it’s bad. There’s a bug lurking somewhere.

If it’s possible to make sense of without contradiction, it’s good. That simple.

Paradoxes do not reveal anything fundamental about reality; they reveal our own confusion about things. When you take the time to carefully look at the processes in your own mind—or the processes in a computer—everything can be understood, the bugs can be discovered, and apparent contradictions can be resolved.