The Dodecahedron of Opposition


Back in my seminary days, a bunch of us Aquinas fans (dubbed “the moldy thomists”) started a blog called Tu Quoque. For over a decade now it has lain dormant – and today, at the request of a couple contributors, it has been officially deleted.

Requiescat in pace, tu quoque.

On its way to blog heaven (or internet archive purgatory), I managed to nab pretty much my favorite blog post of all time. Writing as “SoulDevice” it was my a satirical response to Gregory Boyd, Thomas Belt, and Alan Rhoda’s paper The Hexagon of Opposition: Thinking Outside the Aristotelian Box. I figured if they could make Aristotle’s square of opposition into a hexagon, I could double down and go full Dodecahedron on their open theist butts.

Although I wrote it on my lunch break as a joke, what followed was an amazing exchange over at the now defunct wherein some of the article’s authors and many commenters attacked “Beaumont’s view” as allegedly represented by this silly article. I found the whole thing quite entertaining, but unfortunately all that is left of the dialogue is in the few comments made on the original TuQuoque blog which I present along with the original, unedited article below.


The Dodecahedron of Opposition (original article with comments)

TUESDAY, JULY 12, 2005

The Dodecahedron of Opposition:
Thinking Outside the Boydian Hexagon
Douglas Beaumont


Gregory Boyd, Thomas Belt, and Alan Rhoda have proposed a new solution to the problem of God’s omniscience and the fact that it does not fit with their desired view of God.[1] They report that “one common line of reasoning supporting this traditional belief is the following:

P1: All propositions are either true or false (bivalence).
P2: God knows the truth value of all propositions (omniscience).
P3: The future can be exhaustively described in terms of what either will or will not come to pass.
C: Therefore, God knows the future exclusively as that which either will or will not come to pass.

They admit that the argument is formally valid, however, they argue, “the third premise (P3) can be plausibly denied. This premise, we maintain, is arbitrarily restrictive. There are three, not two, distinct modes in terms of which future events may be described. . . . What P3 overlooks, however, is that may also be the case (3) that S might and might not obtain.” The authors claim that “S’s obtaining is indeterminate— neither inevitable nor impossible,” and go on to propose that western philosophy’s failure to recognize this possibility is that the Aristotelian Square of Opposition “fails to make the logical possibility of genuine indeterminacy sufficiently explicit.” When the authors add these extra possible states of affairs to the square they derive a “Hexagon of Opposition”.

I argue that in fact, even this hexagon is too restrictive, for it only allows for future indeterminacy. Why not the past? It is logically possible that I might or might not have written this article. Thus, to really cover our bases we need to add past indeterminacy statements. As is obvious from reality the past affects the future. For example if five minutes ago I had made the statement S1 that: “In two minutes event (E) might or might occur” these two possibilities would exist as subcontraries that are both possibly true. However – the same event could also be referred to from a future vantage point , viz. S2: “Two minutes ago event (E) might or might not have occurred.” These are also subcontraries that could have both been possibly true depending on the truth value of S1 which is dependant on E. To use the author’s example: S = “Hilary will be president in 2008” might or might not be true in 2004. However, the truth value of S might or might not be true. Therefore my statement about S (Ss) must be assigned values as well. When added to the possible future possibilities which are contingent on the possibly past possibilities it looks something like Figure 1 below. I like to think of this as the “Dodecahedron of Opposition.” Note that the traditional square(s) and hexagon(s) are still present, but now their restrictive nature has been replaced by possible future / past possibilities.

Figure 1: The Dodecahedron of Opposition

But does this really exhaust the possibilities? Suppose that statement (Ss1) is made: “Event (E2) [such that statement S1 might or might not be true] might, or might not, occur.” In this case E2 not only has both possibly true (in the case that S1 is true in that it might be both true or false) and possibly false possible possibilities (in the case that S1 is true in that it might be both true or false), but E2’s truth value is dependant on the truth value of the possibility of S1 being true – which itself is a possibility!


In the interest of cool titles further refinements to this model (which is obviously true despite the fact that it has been missed by the greatest thinkers of the past 5,000 years up until Greg Boyd and myself) will have to wait until I can find out what a 144 sided figure is called.

[1] All quotes from Gregory Boyd, Thomas Belt, and Alan Rhoda, The Hexagon of Opposition: Thinking Outside the Aristotelian Box



Johnny-Dee said…
I was at an EPS meeting where Boyd and Rhoda presented papers that (seem to be) proto-types for the paper you mention in this post. I think their idea is interesting, but ultimately, this confuses ontology with epistemology (one of the main flaws of this brand of openness theology).
I hadn’t considered your point about their schema not being broad enough to cover all their bases. I’d have to think through it before I give my full endorsement, but I think you have a good objection (and the graphic really sells it too).
As far as the Aristotelian square of opposition goes, I say don’t try to fix what ain’t broken.
Matthew Graham said…
Soul Device,
You are my hero man. That was an insightful critique aided by a helpful visual. Thanks for your contributions!
6:04 PM
Alan Rhoda said…
Very interesting, though it ought to be clear that as a logical structure the hexagon is just an extension of standard modal logic where necessity and contingent are taken as the base operators, rather than necessity and possibility.
Applied to the possibility space of the future (which is defined relative to a specified ‘present’ moment), the extra complications are unnecessary for an openness metaphysic. When the question is what is NOW true of the future, or what is true of the future as of a given point in time, then future indeterminacy is the only relevant indeterminacy since the past and present are already settled by that time. For that purpose, the hexagon is sufficient and certainly correct as a depiction of the metaphysics.
An Ockhamist or a Molinist would, however, dispute the hexagon’s correctness as a depiction of how truth values ought to be distributed among propositions about the future (cf. Johnny-Dee’s charge above). For more details and a response, see our more recent paper “Open Theism, Omniscience, and the Nature of the Future”.
The dodecahedron (or any more intricate developments along the same lines) is illuminating with respect to historical counterfactuals or to propositions about cross-time states of affairs that require temporal indexing at multiple moments.
Good work.
11:09 AM
Anonymous said…
My thoughts can be found here:;=7&p;=1#m57100
Alan is too nice sometimes. ;o)
Tom Belt
12:15 PM
Tom Belt said…
How can I enlarge that graphic, Doug, to make it out better?
1:22 PM
Soul Device said…
To all involved on this thread – specifically Tom and Alan,
I am honored by your recognition of my now apparently infamous Dodecahedron of Opposition. I am also rather embarrassed and somewhat baffled at the amount of dialogue that the post has drawn.
To be perfectly honest, my “article” was written because I thought it would be funny to use the word “dodecahedron” in a title (’tis a humorous word no one can deny). Beyond that it was pretty much just some B.S. I hammered out during lunch.
Anyway, thank you all for the entertaining commentary and criticisms, but please do not try to evaluate my position based on anything in that post. I’m not sure what the article meant, if it meant anything, so I doubt anyone else will figure it out!
Alan – thank you for your kind responses both here and on Open Theism’s Board, and Tom – well, there are a lot of non-caffeinated brands on the market today that are just as good as the real thing! 😉
8:02 PM
Tom Belt said…
Thanks for dropping in on the OV site, Doug. Appreciated it. Like the site here very much. Wish you all the best.
I do decaf sometimes too. You may want to consider changing your lunch diet as well when it means your “hammering out B.S.” ;o)
Juuuuuust having fun.
10:21 PM
Soul Device said…
Hahaha. That’s a line from one of my favorite movies “Real Genius”. Like you said, just having some fun. 😉