Möbius Logic: Puzzles that cannot reach binary resolutions

Have you ever seen a Möbius strip? It can be constructed fairly easily from a long, thin strip of paper. One end is taped to the other so that it would form a ring, except with one difference: there is a half-twist in the middle, so that "Side A" of the piece of paper at one end is attached to "Side B" of the piece of paper at the other end. It is now possible to go from one side of the paper to the opposite side of the paper without crossing the edge simply by following the circuit of the loop. One circuit around the loop will end you exactly opposite your starting point, and another circuit around will land you back where you began.

Building a Möbius strip with logic
This has real-life application to logic in that some conundrums have what you might call a "Möbius topography". The old logical puzzle

This sentence is false.

has a Möbius topography. Like the Möbius strip, it is self-referencing and self-reversing. Following forward without changing sides, you will find yourself on the opposite side from where you started after one circuit: If you assume "this sentence is false" is false, then it is proved to be true. If you follow the loop twice and assume it is true as shown, it then tells you it is false so that the second circuit around will land you back where you began. The joke shop version of the same is an index card which has a riddle printed on it:

How do you keep a fool occupied all day? (Over)

The same text is printed on the reverse of the card so as to complete the effect. That's the basic structure of a Möbius puzzle: self-referencing, self-reversing.

Examples of Möbius logic in popular culture
The time loop with a twist is the subject of much entertaining fiction, and whether the mechanism for creating the loop is a time machine or a prophecy. In ancient days, the Oedipus story has a man seeking to avoid his fate who thereby causes it; one of the themes is the logical conundrum of fate. The Harry Potter series has the villain of the piece trying to save himself from a prophecy of his destruction and setting in motion a chain of events that may very well cause his destruction.

Examples of Möbius logic in popular philosophy
The classification of certain logical puzzles as Möbius puzzles has practical applications. The old puzzle "Can God make a rock so large he cannot lift it?" (or the Simpsons version, "Can God make a burrito so hot he cannot eat it?") are both Möbius-style logic puzzles, variations of "Can an unstoppable force stop itself?" In religious/anti-religious polemic, atheist champion Michael Martin advances a number of Möbius-style arguments on the irrationality of the concept of God (see section 5); for all of his entertaining examples he has not actually proven whether the concept of God is inconsistent or incoherent, only that crafting an argument about God's attributes to be self-referencing and self-reversing creates a Möbius puzzle. This particular technique does not demonstrate whether the concept of God is incoherent any more than constructing a paper Möbius strip demonstrates the incoherence of paper; that particular technique only demonstrates the "unresolvable" property of a self-reversing puzzle in an endless loop.

Some of the internal debates in religious philosophy are precisely about the power of God and whether an unstoppable force can stop itself. Religious philosophers debate whether omnipotence is unstoppable and therefore an inescapable deterministic chain and, if so, exactly how omnipotent it is to be stuck in a deterministic prison caused by the fact of such power.

In other areas of logic, the debate about whether our brains are merely physical and if so irrational is a nicely tantalizing exercise in asking "Have you ever considered how irrational you are?" In this debate, rational people mount rational arguments conclusively proving their irrationality, QED ... Except that if they were really that irrational, it's debatable whether they would have ever successfully proved it based on evidence and logic. A sense of perspective is occasionally missing from these discussions.

The Point
All of the debates used as examples here are debates well worth having. I would simply point out that when the debate is framed as a Möbius loop, it has been framed in a way that is entertaining but renders progress impossible. We have the tools to recognize such a logical structure. Once a presentation has been identified as a Möbius conundrum, we can know from the outset that no resolution can come from that particular way of framing the question.

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