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*The Opinions Expressed In This Blog Post Are Solely Those of the Author And Not Necessarily Those Of DebateDrills*

The Banach-Tarski paradox is famous in mathematics – take a solid, 3-dimensional sphere of volume 1, split it into 5 parts, and reassemble it – into two disjoint spheres each of volume 1. The result seems ludicrous—after all, where did the volume for the second sphere come from? You can’t just get something out of nothing–its ridiculous. Yet, the paradox is a valid theorem–the statement is true, and a proof exists. Here, the key trick in the proof is a particularly thorny statement, the axiom of choice. While the statement seems innocuous enough, (for any collection of objects, I can pick one), it has wide-reaching implications and demonstrates an important principle of logic – should you accept an axiom, you must also accept its logical consequences.

A similar argument could be made for the fact that conditionality is infinite. In the proceeding discussion, I will present two independent justifications for conditionality, one axiomatic, the second inductive. The end goal of both is to demonstrate that conditionality, much like counterplans and negative fiat, is a logical consequence of the structure of debate, both independent of and assuming constraints on education and fairness.

The first, and perhaps more intuitive of the two arguments is axiomatic. The argument here proceeds in two steps. First, we consider what the best model for the negative is in a debate. Generally speaking, the endgoal of this is to develop a sufficiently general and intuitive role of the negative consistent with an offense-defense view of debate. To consider this, imagine a world, ceteris paribus, where the actor specified in the resolution is endowed with a set S of all possible actions and a set T of topical actions. It is then useful to model debate in the following way: Player 1, (the aff), selects some element t of T, and Player 2 must somehow prove that t is undesirable. I contend that the way to prove that t is undesirable is by proving there is a move s in S such that the aggregate societal utility1 (calculated by some u: S → R) of playing s is larger than any combination of s and t. Note this should jive very well with your intuition of what it means to vote neg. Since the status quo, sq is in S, a disad that “outweighs the aff” simply says u(sq)> u(t). Similarly, a counterplan simply says that there exists a strategy s that cannot be desirably combined with t and for which u(s) > u(t). Really, up to this point, we haven’t done anything special, apart for formalizing the definitions of disads and counterplans through the opportunity-cost model of debate.2

What is perhaps most surprising about this rather simple formalization is that its immediate consequence is arbitrarily large conditionality. After all, inspection intuitively yields that S is very large; further, the advent of the permutation equips the negative with the ability to read any s in S (though they should not win if it is not competitive). Perhaps the key insight of this model, then, is the fact that negative conditionality doesn’t actually occur; rather, it is the byproduct of the axiom that the role of the negative is to test the opportunity cost of the affirmative. As such, it seems illogical that the negative should be constrained by an arbitrary number of conditionality advocacies. To see this, without loss of generality, let the negative advocate for n conditional advocacies {si}, i = {1, 2, … , n} and go for one in the 2nr. We will assume that this negative is rational such that it has gone for the best possible advocacy. Further, assume the negative goes for some conditional advocacy s such that u(sq) > u(t) > u(s) (where the counterplan is worse than the aff but there is a disad to the aff that is larger than the advantage). In this world, then, the aff has not met its burden that the affirmative is a good idea; e.g. there is no opportunity cost to doing it, since it would be more beneficial to do nothing. As a result, the judge in this case should vote negative for sq, an option never explicitly forwarded by the negative. As a result, the negative never advocated for sq, so sq is not in {si}, but the judge voted for sq by voting for the negative, so sq was in {si}, a contradiction, which completes the argument. Perhaps the most simple corollary to this, then, is the explanation for why “the status quo is always an option for the negative” is synonymous to “infinite conditionality.” The more important point, however, is that it shows that conditionality is definitionally infinite; as such, “n conditionality” is almost a misnomer, because the act of sticking a 2nr with an advocacy is necessarily not conditional in the most logical definition of the term.

A common criticism of the above arguments are that they pay no attention to the oft-lauded ideals of fairness and education in debates. While these issues are much murkier to formalize, and the approach to an axiomatic system which includes these has thus far eluded me, it is possible to assume their existence while simultaneous abstracting away from their quantification. The following model relies on the following three assumptions. (That is, these things must be won by the negative prior to this argument applying). First, I assume that the default method for evaluating the legitimacy of a theory interpretation is reasonability. Formally put, I will assume that a theory practice is reasonable if the total “abuse3” of a theory practice is smaller than some constant M. Second, I assume at least one conditional advocacy is okay.4

Finally, I will assume that conditional advocacies experience diminishing marginal returns; that is, the differential in abuse between 500 and 501 conditional advocacies is significantly smaller than the differential in abuse between 1 and 2 advocacies, and that this trend holds under finer division. This should make intuitive sense when one considers that changes in strat skew, time skew, etc. should be measured by the percentage change in total new advocacies one has to deal with instead of just the net number. With these three assumptions, then, we can begin the argument. The base case relies in a rather crucial way on assumption (2). Such a reliance on these axioms is mostly to deal with the thornier problem of including extra variables (education, fairness) and how they are measured (time, strat, research depth, etc.) into the discussion. Once the base case has been established, however, we can proceed to the inductive step. Particularly, assume that (n-1) conditional advocacies is justified. Two possible interpretations are then possible: first, we consider the differential in abuse between (n-1) and (0) advocacies, which we can say is ∆0. Second, let the differential in abuse between (n-1) and (n-2) advocacies be ∆1 and the differential between (n-1) and (n) advocacies be ∆2. Then diminishing marginal returns (3), yields that ∆0 > ∆1 > ∆2. Finally, combining this inequality with assumption (1), we get that if (n-1) advocacies were allowed, then M > ∆0 > ∆1 > ∆2, implying n advocacies are allowed. This finishes the argument, and proves that an arbitrarily large amount of conditional advocacies are allowed. In practice, due to the unfortunately finite nature of human beings, this is indistinguishable from infinite conditionality, and proves the argument we set out to show.

If I had to pick an argument, I believe the axiomatic interpretation is better; I included the inductive argument, however, both because it was how I first conceived of the problem of conditionality and because it is valid under more stringent assumptions. However, in both cases, we have results that always work because of the definition of debate that is independent of the specific content of those debates. This is an important argument, as it shows the developed theories are consistent regardless of how a debate goes down. This perhaps highlights one of the most important maxims in logic – if you accept something as an axiom, you must accept its logical consequences. And while infinite conditionality may seem at first counterintuitive, I posit that it is at least more intuitive than the Banach-Tarski paradox.

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1. Note explicitly that the shape of u is independent of the shape of the utility function. How exactly to best aggregate these utility functions is often a subject of discussion in impact calculus or framework debates.

2. This model is useful in two ways; one, it is consistent with the view that the role of the aff is to prove t is a good idea, where a good idea is defined in the sense it is the best possible option, and two, it provides a reasonable restriction on arguments while being relaxing enough for a wide variety of educational arguments.

3. Quantitatively, I expect this to look like some aµf(t) + bµe(t) where t is the theory interpretation and µf and µe are measures quantifying fairness and education respectively, and a and b are weights.

4. This should be relatively easy to justify, considering the impracticalities in evaluating dispositional or unconditional advocacies.

*All views in the article are the authors*