Q&A: The Distinction Between Paradox and Recursion
The Distinction Between Paradox and Recursion
Question
Hi
I’d like the advice of someone who can formulate, in intuitive language (without examples from mathematics and the like), the difference between paradox and recursion.
From what I understand, in both cases we’re dealing with an umbrella definition (there are many different meanings for both concepts), but in the past I came across an attempt to generalize each of the concepts. From those rough generalizations came the distinction between the two:
A paradox is circular (for example, the liar paradox) and therefore has no "vector," unlike recursion (for example, the statement "A Jew is someone whose mother is Jewish").
Thanks to those who answer.
Answer
To my mind, this is like asking: what is the difference between a paradox and a chair? What connection is there at all between recursion and paradox? There is no connection at all.
The illusion that there is a connection is probably created because sometimes people let recursion continue infinitely (they don’t set a stopping condition), and then a mechanism can arise that does not halt. For example, the following recursive definition: a Jew is someone whose mother is Jewish, without defining that Sarah was Jewish (that is the stopping condition). But even when you do set a stopping condition (like Sarah), it is still recursion, because there is a function here that, during computation, calls itself. So there is no basis whatsoever for identifying recursion with paradox.
But even infinite recursion is not really a paradox, and that is for two reasons: 1. Paradoxes are not necessarily loops (contrary to what was said to you). For example, the Swedish army paradox (the surprise drill) is not created by a loop. Self-reference paradoxes are only one type of paradox. 2. A mechanism that does not halt is not a paradox. A paradox is an argument that shows that some claim has two contradictory truth values (true and false). A mechanism that does not halt gives no truth value to a claim at all (you never reach an answer). True, one could quibble here and say that having truth and falsehood together is like having no truth value.
And finally, let us look at a recursion with no stopping condition that nevertheless does not create a paradox (and also not really a loop): a Jew is someone whose mother is Jewish, and as for Sarah, she is Jewish if her daughter is Jewish. This is at most an anti-paradox (a claim that can receive either of the two truth values—not both together as in a paradox, but separately). See columns 195-6.
Discussion on Answer
Yes. I gave as an example the Swedish army paradox, which has no loop in it. The full definition, in my view, is when a claim cannot be assigned a truth value or a false value. That can be because of circularity or for some other reason.
Something still doesn’t sit right for me in your last answer ("when a claim cannot be assigned a truth value or a false value"). But let’s leave that aside.
Would you at least agree with me that there is a broad common denominator between loop-type paradoxes and recursions?
Absolutely not. I explained.
A paradox contains two matters that stand in contradiction to one another.
Sometimes, when trying to examine the state of those matters, one falls into infinite recursion.
For example, in the liar paradox: "This sentence is false"—then you check: it’s false, so it’s true, so it’s false… infinite recursion that creates a situation where at every stage of the examination the sentence changes its status.
Whereas in everyday language we are used to a sentence having a static status, in processes this is not so.
Therefore, what is considered a paradox in static language may be nothing more than a non-terminating loop in the world of processes.
Also in the surprise exam paradox, after concluding that it cannot take place on any day, you return to the beginning and conclude that it will indeed be a surprise and the exam will take place on one of the days, and then the whole analysis starts over again and you get an infinite loop.
Decisor,
I don’t always get the chance to agree with you, but maybe this time it happened.
Michi wanted to show that there are paradoxes that are not circular, but the example he brought in order to illustrate this (the surprise exam) seems to me דווקא to say the opposite. So his explanation is not clear to me.
In general, I assume that the concept of paradox is an umbrella definition, and by its nature it is somewhat loose.
Nevertheless, see the definition that appears on Wikipedia, which in my opinion also does not fit with Michi’s explanation:
"In general, a paradox can be defined as a concept or claim whose reason for being true ultimately becomes the reason for refuting it, and so on repeatedly."
Thank you.
By my sins, I’m not managing to understand the main point of your explanation.
Isn’t it true that your own definition of paradox assumes that self-reference—or circularity—is an essential part of all paradoxes (and not just a certain type of them, as you say)? Because according to you, the two truth values feed each other (give each other validity).
And if so, then there is in any case a broad common denominator between the two phenomena: both are logical phenomena in which, when implemented as an argument or explanation, a problem of meaning is created; both are circular (some form of self-reference); and both continue infinitely.
Am I mistaken?