Thursday, July 19, 2007

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.

(Graphics courtesy of the the shareables at Wikipedia.)


Anastasia Theodoridis said...

Wonderful analogy and clear analysis; thank you.

Yup, one hears lots of theological issues framed as a Möbius loop. Can God create evil if evil is the transgression of His will? Could Christ, in His perfect, incorruptible freedom, have sinned? Does God punish God to pay back God? All perfectly meaningless questions, yet it isn't at all unusual to hear people giving answers to 'em.


Anastasia Theodoridis said...

I've thought of another one:

Can God force a person to be free?

Mark said...

There is an interesting thing you can do with a Möbius strip. Take that tracing of the line down the center that you drew in the first paragraph. Poke a hole there with a scissors and cut along the line.

Did it "fall out" as you expect.

Does "cutting" a theological Möbius logical puzzle sometimes have the same effect? Perhaps these puzzles aren't as useless as you might think. :)

Weekend Fisher said...

I love Möbius strips, truth be told. I know, geek toys, but there it is. A logician's merry-go-round.

I had to restrain myself writing this, had more examples that I cut out on the last edit so I didn't bore everyone to tears overdoing the point.

My main point is that creating a Möbius-style logical conundrum is fun but doesn't actually prove much except that such loops are both endless and entertaining.

"Can God force a person to be free" is a fun one. Or "Is God able to limit his power?" is another version of "stopping the unstoppable force".

Gotta love Möbius strips.

Anastasia Theodoridis said...


If you then take the results of your experiment, and cut THOSE in the very same manner as you described cutting the original strip, you get another very interesting result.


Anastasia Theodoridis said...

"Is God able to limit his power?"

Well, now, that one may not be a Möbius strip, actually, depending on what's meant by it. (It's a bit ambiguous as stated.)

We EOs say God's powers are such that He can freely decide which ones to exert and when, and how and to what degree. He can restrain Himself in the use of His power, or turn it on full blast, according to what's best in any given situation.


Weekend Fisher said...

And that construction of God and power is the natural conclusion of the fact that God is personal (loves, has a will) and has wisdom.

It's when you get up to the Calvinist/hyperCalvinist end of the scale where God is in danger of being perceived as The Unstoppable Force where that question takes on a Mobius construction. The point I'd make is that reasoning from power to determinism is self-refuting.

Colin said...

Hello "Weekend Fisher" this is my first time to your blog, but I was tempted by your comments over at the Philosopher's Carnival.

There, you mention (what is not in the original post) that the looping nature of the Liar indicates that it is not a counterexample to LNC. As far as I can tell, your argument is something like this:

1) the Liar is only a counterexample to LNC if we cannot explain away its paradoxical behavior

2) we can explain away its paradoxical behavior by citing its 'looping' effect on its own truth-value (granting a classical exhaustive/exclusive conception of the relation of truth and falsity)

3) thus, the Liar and other paradoxes that satisfy such an explanation can be set aside; if we restrict ourselves to 'non-looping' discourse there are no counterexamples to LNC

I don't contest this. I think it is quite right. But there are various angles that might still motive the dialetheic argument.

One such angle, the most moderate I think, is to simply ask: but what if we want an account of truth and falsity and logic pertaining to all meaningful constructions of a language, rather than just the fragment which includes only 'non-looping' discourse? If we include the Liar, then we have to revise our conception of truth, hence logic, to accomodate its paradoxical behavior.

Another angle on this matter is to cite 'revenge' problems associated with this talk of 'non-looping' discourse. If you can show how to formally construct a meaningful predicate that is extensionally correct -- it picks out all and only the looping, paradoxical sentences -- then I can just construct a 'strengthened Liar':

"this sentence is either false or loopy"

If true, it is untrue (false or loopy). If false, it is true (so not false). If loopy, it is true (so not loopy). If those are the only three options, you have problems even advancing this account of the paradoxes. It just raises new paradoxes.

Weekend Fisher said...

Hi Colin

I'd stopped checking comments here back in July of 2007, and just happened across your comment tonight. The odds of you finding this are small, but who knows? You might have asked for an email on follow-up comments.

Anyhow, the very last part of your comment is where I'd want to pick up, where you said: "If loopy, it is true (so not loopy)". It doesn't actually follow that something true is not loopy. Something can be loopy like a circle instead of loopy like a Mobius strip. A self-referential, non-reversing statement would be like that.

Btw when it comes to dialetheism, my suspicion is that "This sentence is false" and its cousins are a specific species of logic that really is an exception to LNC. We can explain its paradoxical behavior, and this means that LNC is still valid in the general case -- for arguments that don't have a Mobius topology.

So LNC is valid for the general case, and we can recognize one category of exceptions by the fact that they're self-referencing AND self-reversing.

Kind of like Euclidean geometry works for most cases -- and is still hugely useful and covers almost all normal purposes -- but there is such a thing as non-Euclidean geometry. Similar to that, I think there'll ultimately be LNC logic which works for most cases -- still hugely useful and covering almost all normal purposes -- but here we can recognize that there is such a thing as non-LNC logic. I'm not sure if Mobius Logic will be the only alternative logic besides LNC-logic.

Thanks for commenting.