Let me define "unanswerable question" as one that must use at least one term whose definition depends on the answer to the question.

Let me define "a ranking" as a sequence by which priority is assigned.

True or false: Any ranking of internally consistent answers to an unanswerable question is necessarily arbitrary.

Edit: for christo183...a mathematical idea from a comment I made to Mr. Ross' answer: Regarding fixed point theories...their truth/provability is conditional...try arriving at a fixed point proof (for all continuous functions f:[0,1]-->[0,1] there exists x such that f(x)=x) on a Mobius strip where complements are identified. Are we in Oregon, USA up and the Australians down?)

It is obvious that a fundamental property of a finite array...its cardinality...is independent of the point of view...a 3x5 array has the same number of elements as when turned sideways: a 5x3 array. It is a fundamental implicit assumption in accepted theories of mathematics that this property extends to infinite arrays...an assumption accepted without proof or evidence. I would contend that this assumption is unjustified, and in fact by all evidence outside the hermetically sealed world of mathematics, is wrong.

In as much as our own consciousness is an inescapable part of any analysis/experiment we do, and given the undecidable nature of the question, "what is consciousness?", there is a non-orientable aspect to any thought. "any surface is nonorientable if and only if it contains a Möbius band as a subspace"--> we ourselves make everything we think about non-orientable in that sense. Any thought of fixed points forgets about the thinker.

• I assume you are referring to self-referential questions or is that not the case? By ranking do you mean whether we say the sentence is first true or first false? Can we use "This sentence is false" as an unanswerable question? – Frank Hubeny Oct 19 '18 at 23:59
• If there was an example that brought this question to mind, can you provide it? This interesting question needs specifics. – Mark Andrews Oct 20 '18 at 0:09
• I don't know if this question should be considered a paradox. It just doesn't provide enough information to be answered without a great amount of assumption on the part of the one who answers. Like what does it mean for something to be arbitrary, and if something is necessarily arbitrary -- its arbitrariness follows from a principle -- what is said principle? – Ethan NOPE Oct 20 '18 at 6:43
• In my answer, I assumed that what was arbitrary was that which didn't follow from a principle, but that is by no means the only possible definition. – Ethan NOPE Oct 20 '18 at 6:44
• Mark Andrews-what brought this to mind is the question of whether there is a boundary between order and disorder...As I see it there can be none because if one exists then by complementation across the border disorder would be bounded (excluded from that which is bounded) - contradiction of the very definition of disorder: to be disordered is to be unbounded in any way...to lack any and all boundaries. Boundaries only appear to the bounded...the unbounded permeates everything and cannot be excluded. – 21stCenturyParadox Oct 20 '18 at 13:53

At its core, this is a question about Self Reference. I want to start by reducing this to propositional or sentential self-reference, rather than dealing specifically with questions and answers, then I want to point you in the direction of how you might usefully start to address this within a framework of implicit hierarchies of sets or predicates.

Curiously enough, I've suggested an analysis of the propositional form of asking questions on this site in the past. The idea is that a well-formed question is a particular act querying a proposition that may or may not be true - we divide the scope of possible assertions into those that imply the truth of some proposition and those that imply its falsehood, and we are desiring someone to present us with a proposition in one of those categories that we say suffices to have answered it.

It would seem impossible to give a consistent answer to this question. To answer in the affirmative is to assert that one's answer is false, thereby satisfying the requirements needed to make the underlying proposition true, but this runs contrary to what we said (similarly with the negative).

But this is not particularly difficult to schematically break down into a propositional form. One could use a different form of indexical reference here to break down exactly what "your answer to this question" is - for example, "the statement you make within some space of time subsequent to my having made this utterance made in the intention of providing me with an answer given the intention you take me to have had in asking it".

So the "Question" part of this is a meandering but ultimately tractable way of hooking into a deeper phenomenon, which is propositional or sentential self-reference. Really, all I'm doing in asking the Liar Interrogation is prompting you to state a conventional liar sentence (or its negation, which is in fact the same proposition), and that is a well known topic of interest in philosophical analysis.

Now, let us try to formulate the "meta" question - can one non-arbitrarily assign a sequence of priorities to formulations of a self-referential proposition?

And, as has been well demonstrated, yes, you can. The key here is to understand the application of a Fixed Point theorem (relevant SEP section) for an evaluation operation. If, for example, you want to try to cash out this idea of reference in terms of Truth conditions, and by introducing a predicate for affirming the truth of sentences into our language, you can construct incremental refinements of languages including Truth, and respecting our definitions for what it means for a given sentence to be true (e.g. `Tr('x') iff x`).

These refinements result in a hierarchy of languages, each of which has a larger body of admissable statements, which we can index using an ordinal number. And, eventually, by considering these sets of sentences and the operation of incremental augmentations of truths, we reach a Fixed point, where our operation of adding new truths has stabilized and we're not adding anything new, such that `Tr('x') = x`, and this is a language exhibiting self-referential properties.

The existence of such fixed points falls out as a mathematical theorem from how we've gone about incrementally refining our language. What's particularly interesting is that as part of this proof, we don't only show that fixed points exist, but we also prove that there is a least such fixed point. That is, sentences of a relevant self-referential nature can be connected ordinally to a sequence of operations of incremental refinements of language, and it is possible to identify the first ordinal level at which our required self-referential behaviour occurs.

Our process is ordinally indexed from the get go, so all we need to do to rank our "answers" in terms of this ordinal sequence, et voila. Non-trivial, non-arbitrary orderings of self-referential statements in terms of the languages that sufficiently interpret them.

I've been a bit fuzzy about this because I don't really intend to go through the nuts and bolts of exactly what kind of self-referential behaviour you're looking to exploring further. However, I would be confident that the most correct answer to your question is "no, it is not necessarily arbitrary, and indeed it is quite important to understand why there is an appropriate ordering involved in current theories of propositional self-reference".

• Just as a note, I mentioned that it is "possible to identify the first ordinal level...". That's slightly stretching it, given the possibility of dipping into transfinite induction - it's more like we can invoke the axiom of choice to show that we can do so, even if we might practically struggle to say "oh, it's exactly this ordinal many steps up the chain". – Paul Ross Oct 20 '18 at 21:18
• It is interesting that given the display of certainty in math and by mathematicians, that mathematics has not solved all the problems of the world. It is my understanding that the focus of math for the early Greeks was "that which is known". Euclid's parallel postulate not withstanding, that changed in a big way in the late 19th century to "that which is assumed." While the investigations of Morely and Shelah have lead to a deeper understanding of the stability of certain theories, the range of theories to which their analysis applies is of zero measure in the set of all possible theories. – 21stCenturyParadox Oct 23 '18 at 12:54
• One of my favorite readings is this piece by Paul Cohen... math.upenn.edu/~kazdan/proof/notes/… – 21stCenturyParadox Oct 23 '18 at 12:58
• Regarding fixed point theories...their truth/provability is conditional... try arriving at a fixed point proof (for all continuous functions f:[0,1]-->[0,1] there exists x such that f(x)=x) on a Mobius strip where complements are identified. – 21stCenturyParadox Oct 23 '18 at 13:11
• I spelled Morley incorrectly – 21stCenturyParadox Oct 23 '18 at 13:19

Necessarily arbitrary?