1.9. The circle
Two derivations of one law have now been rebuilt, and they give way at the same place. The pieces are under The finding and Mises's route; this part sets them side by side and says plainly what the audit does and does not establish about them. It is short because most of the work is already done.
Both deny needing what both use. Rothbard's man has "no need for him to formulate hypothetical value scales" (p. 33); Mises's scales have "no independent existence apart from the actual behavior of individuals" (ch. IV, §2). Two sentences, and they are the same rule: a man's ranking shows up in what he does, and nowhere else.
Then each derives the law from a premise that rule forbids. Rothbard's premise ranks the wants the man would serve with every stable he could cut from his own. Mises's asks less of that plan and more of the ranking, and it is written over exactly the same stables — the two claims quantify over the same region, every subset of the stock in hand, and neither reaches an inch past it. Reach was never the trouble, on either route. What both premises are is a settled answer, held in advance, for every one of those stables at once — and that is the scale in advance both authors say a man does not need.
What each then helps himself to. Rothbard sees the difficulty and exempts the law from the rule by name, in a subordinate clause, with the derivation as the exemption's only warrant. Mises does not see it, or does not say so: he sets down his counterfactual definition and announces that in doing so "We do not transcend the sphere of praxeological reasoning" (ch. VII, §1). One reasons in a circle; the other declares. Neither gives the premise a warrant that stands apart from the law it is used to prove.
The weaker premise does not help. Mises's premise is strictly the weaker on the allocation, and a reader might hope that weakening further reaches something about actual choices alone. Here is the reason to expect not. The law compares two supplies, and the man holds one of them. Whatever premise delivers the law at one moment has to say something about the supply he does not hold, and the only thing that can be said about it is what he would do with it. Weakening the premise changes how much it says in that tense; it cannot change the tense. That is a reason, not a result: nothing here proves that no premise about actual choices can reach the law, and no one has tried to prove it. What the audit has is two routes, each resting on a standing counterfactual its author disowns. A third reading, where the man held the smaller supply at an earlier moment, speaks only of what he did; it is set out under The ladder as a history, and it pays in a different coin.
One shape, if you like. This last is a reading and not a finding, and it is offered as one. The two defects the audit turned up in Rothbard — the circle here, and the inference about units set out under The other findings — have a common form. Each time, what is offered as a warrant does not reach what it is offered for. The exemption is offered for the plan and reaches only the law; the definition of a supply is offered for the plan's indifference to which units and reaches only the units. Mises's declaration fits the same form trivially, since it reaches nothing. Whether that is one habit or two coincidences is not something a proof assistant can say.
What is and is not established. Established: that each derivation follows from its stated premise, that each premise is a claim about what the man would do, that each author restricts value scales to what he does do, and that neither warrants the exception. Not established: that the law is false, or that praxeology cannot have it. A route from actual action does exist, and is a theorem; what it reaches and what it costs is under The ladder as a history. The question the audit began with — whether the certainty Mises claimed for this law is the certainty of a theorem — has an answer of the same width. The derivation is as certain as its premise. The premise is the part nobody has derived.