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 http://www.opentheismboard.org 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
Thinking Outside the Boydian Hexagon
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. 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.
 All quotes from Gregory Boyd, Thomas Belt, and Alan Rhoda, The Hexagon of Opposition: Thinking Outside the Aristotelian Box