1.4. The claims🔗

Swap dominance — the agent's plan beats every one-swap alternative to it.

Take any sub-stock of the units on hand, and the ends the agent would serve with it. Now make one swap: drop a served end, and put in its place an unserved one that the good could have served but the plan left out. The claim is that the agent prefers the bundle he would have served to the swapped bundle.

It is asserted of one given plan, and it is subjunctive throughout — it speaks of sub-stocks the agent may not hold and swaps he does not make.

Source: Rothbard, MES, ch. 1, §5.B, pp. 24–27 (Mises Institute ed.): "action uses scarce means to satisfy the most urgent of the not yet satisfied wants" (p. 24); the counterfactual framing is Rothbard's own ("suppose ... faced with the necessity of giving up one horse"; "he gives up the least urgent of the wants which the larger stock would have satisfied", p. 25), backed by the reallocation argument (p. 27: "follows from the defined interchangeability of units and from disregard of past events"). Mises, Human Action, ch. VII.1.

Status: explicit-in-tradition as doctrine; the one-swap form, and the pulling apart of the two halves below, are our-reconstruction. Rothbard's premise fuses two claims in one sentence: that ACTION employs the means, and that the wants served are "the most urgent of the not yet satisfied wants" (MES p. 24). This claim keeps the ordering half and states it subjunctively, over what the agent WOULD serve. The subjunctive is Rothbard's own — his derivation compares a stock of six with a stock of five and asks which want "the larger stock would have satisfied" (p. 25), a comparison no single actual allocation delivers. The action half is dropped rather than answered: nothing here bridges an act to a preference.

A tension remains, and it is Rothbard's own. It is not that this claim reaches stocks the agent does not hold; that reach is his (see Anything about units not on hand, below). It is that the claim attributes a STANDING disposition. The plan is one object, at one time, assigning served ends to every sub-stock at once — a value scale formulated in advance of the choices it covers. That is the thing Rothbard says is not needed. He holds that praxeology "may deal with utilities only as deduced from the concrete actions of human beings" (MES p. 882 n. 8), and inside this very chapter he writes, in words he introduces with "It must be reiterated", that "value scales do not exist in a void apart from the concrete choices of action", adding that "there is no need for him to formulate hypothetical value scales" (MES p. 33). His actor needs no scale in advance: it is revealed choice by concrete choice as he goes. This claim gives him one up front.

Yet it is on that same page that he exempts this law from the restriction, in a subordinate clause and without argument: choices cannot be predicted "except that they will follow the law of marginal utility, which was deduced from the axiom of action". So the one thing that may be said in advance about the choices is that they follow the law — which is what licenses attributing the standing plan this claim is asserted of, and that plan is the premise the law is derived from. The warrant for the premise is the conclusion.

The circle is in the WARRANT, not in the proof. Nothing in the Lean is circular: marginal_utility derives the law from this claim, and this claim is asserted, never proved. What is circular is Rothbard's licence for asserting it.

Asserted of ONE plan, not of every plan of that shape — and that is not a hedge. Given one plan, a rival can always be defined, so a claim about all of them could be refuted just by building one. Asserted of the single plan a theorem is handed, there is no such rival to build. Whether a rival one swap away could also be swap-dominant then becomes a theorem (no_rival_swap_dominant, Apodictic.Urgency) instead of a premise.

Does not say:

1. Anything about actual action. No bridge from action to preference is part of this claim.

2. Anything about alternatives that differ by more than one swap.

3. Anything about independence of uses — that is the theorems' hypothesis IndependentUses.

4. Anything about units not on hand (subStock ⊆ stock.units), which is all the plan speaks about. The reach over stocks the agent does not hold is Rothbard's own, in both directions. He builds the law upward over stocks the actor lacks — the second unit, then the third "added to a stock of two units" (MES pp. 23–24) — and downward, six horses to five (p. 25). Holding none of a good is no obstacle either: "If the actor has no units of some goods in his possession, this does not affect the principle" (p. 32). The hypothetical scales he refuses on the next page are a different thing: scales for a wholly different multi-good endowment, (3X, 4Y, 2Z) against (6X, 8Y, 5Z), where what he permits inside the actual stock is "adding and subtracting" units of it (p. 33). That line is anchored to the stock the agent has, not drawn at some distance from it, and it is closed under the single-unit steps he iterates himself — first horse, then second, then third. Taken downward from a stock, those steps reach every sub-stock of it. So quantifying over all of subStock ⊆ stock.units is that region exactly, and not a reach beyond his text.

   Where this claim does depart from him is in INDEXING, not in reach: it speaks of arbitrary sub-stocks where he speaks of sizes. The gap between the two is interchangeability, which he assumes throughout and this claim does not: it is the hypothesis Homogeneous. Stated without that assumption, the claim distinguishes cases he never had to tell apart.

5. Anything about interchangeability of units. That two sub-stocks of the same size would serve the same ends is the hypothesis Homogeneous, and no part of this claim.

6. That such a plan exists, or that it is unique without further properties of Prefers.

Apodictic.SwapDominant.mk

swap : ∀ subStock ⊆ stock.units,
  ∀ served ∈ plan.wouldServe subStock,
    ∀ unserved ∈ stock.serves,
      unserved ∉ plan.wouldServe subStock →
        praxis.Prefers agent time (↑(plan.wouldServe subStock))
          (insert unserved (↑(plan.wouldServe subStock) \ {served}))

Served in order of urgency — nothing serviceable and more urgent is passed over for something less urgent.

If the good can serve an end, and that end is more urgent than something the agent WOULD serve, then he would serve it too. Like SwapDominant it is subjunctive throughout, and asserted of one given plan.

This is the premise of MISES's route to the law (Apodictic.Mises.ladder_from_order), where SwapDominant is the premise of Rothbard's. The two are not equivalent: Rothbard's premises entail this one given asymmetry (Apodictic.Contrast.rothbard_entails_servedInOrder), and this one does not entail his (Apodictic.Model.servedInOrder_not_entail_swapDominant). Mises asks less of the agent's allocation and more of his value scale.

Source: Mises, Human Action, ch. VII, §1: "If the supply available increases from n–1 units to n units, the increment can be employed only for the removal of a want which is less urgent or less painful than the least urgent or least painful among all those wants which could be removed by means of the supply n–1"; and the bridge sentence the derivation turns on, "It is nothing else than the reverse of the statement that what satisfies more is preferred to what gives smaller satisfaction." Compare Mises ch. IV, §2, where the agreement is interpretive rather than a constraint: "Every action is always in perfect agreement with the scale of values or wants because these scales are nothing but an instrument for the interpretation of a man's acting." Rothbard states the doctrine in the same family: "action uses scarce means to satisfy the most urgent of the not yet satisfied wants" (MES p. 24).

Status: our-reconstruction. Mises argues for the law by a dilemma about whether action terminates, not from a stated ordering premise; this claim is what that argument needs and does not supply. The dilemma is machine-checked NOT to deliver it (Apodictic.Model.recurs_does_not_deliver_ladder), and his refutation of his own second horn does not go through on the action axiom he names (Apodictic.Model.existence_does_not_refute_second_horn).

The warrant problem is Mises's too, and not Rothbard's bookkeeping. Mises restricts value scales exactly as Rothbard does — "These scales have no independent existence apart from the actual behavior of individuals" (ch. IV, §2) — and then, in ch. VII, §1, declares the openly counterfactual definition licit without argument: "We do not transcend the sphere of praxeological reasoning in establishing the following definition". Rothbard exempts the law circularly; Mises asserts the exemption. Neither warrants it.

Does not say:

1. Anything about bundles. That the served bundle beats each one-swap rival is SwapDominant, a different claim of a different shape.

2. That the ends a good can serve are comparable. That is the situational condition Stock.ComparableServiceable, and the route that uses this claim needs it separately.

3. Anything about actual action; no bridge from an act to a preference appears here or anywhere in the library.

4. Anything about units not on hand: subStock ⊆ stock.units, the same region SwapDominant speaks of.

Apodictic.ServedInOrder.mk

inOrder : ∀ subStock ⊆ stock.units,
  ∀ served ∈ plan.wouldServe subStock,
    ∀ better ∈ stock.serves, praxis.PrefersEnd agent time better served → better ∈ plan.wouldServe subStock

Asymmetry of preference — if one bundle is preferred to another, the other is not preferred to the first.

The "strict" reading of Prefers. Apodictic.Contrast.rothbard_entails_servedInOrder argues by contradiction from a preference running both ways, and without asymmetry there is no contradiction to draw; Apodictic.no_rival_swap_dominant spends it the same way.

Source: tacit. No sentence in Mises or Rothbard asserts it. Rothbard's value scales are built from acts of choice, and one act cannot demonstrate a preference in both directions — but that is an argument from the bridge (demonstrated preference), not a statement of his; and the bridge as stated here (DemonstratedPreference, of one trade) yields a preference in one direction only. The nearest explicit treatment is Nozick's reconstruction (1977), which is a critic's formalization and not the tradition's own words.

Status: our-reconstruction. Recorded as ours rather than attributed, precisely because the search for a source came back empty.

Does not say: not transitivity, not totality, not comparability of any particular pair. It is asserted of one agent at one time, and of the bundle-level relation, from which the end-level case follows since PrefersEnd is Prefers on singletons.

Apodictic.AsymmetricPreference.mk

asym : ∀ (X Y : Set praxis.End), praxis.Prefers agent time X Y → ¬praxis.Prefers agent time Y X

Acts in order of urgency — nothing serviceable and more urgent is passed over for something less urgent, in what the agent ACTUALLY does with the stock he holds.

If the agent serves an end, and prefers to it another end he believes the good can serve, he serves that end too. The actual-action twin of ServedInOrder: that claim is subjunctive, asserted of a plan over sub-stocks the agent may not hold; this one is asserted of one allocation of one stock at one time. It is what the TEMPORAL reading of the law (Apodictic.Temporal) spends in place of a standing plan.

Source: Rothbard, MES, ch. 1, §5.B, p. 24, in the passage that narrates the acquisition of horses one by one: "The first horse will fulfill the most urgent wants that a horse can serve; this follows from the universal fact that action uses scarce means to satisfy the most urgent of the not yet satisfied wants". Mises, Human Action, ch. IV, §2, states the same agreement and gives its ground: "Every action is always in perfect agreement with the scale of values or wants because these scales are nothing but an instrument for the interpretation of a man's acting."

Status: our-reconstruction. The words are Rothbard's; the claim is not quite his. Both authors deny that a value scale exists apart from the action it is read from. Mises: "The scale of value manifests itself only in real acting; it can be discerned only from the observation of real acting. It is therefore impermissible to contrast it with real acting and to use it as a yardstick for the appraisal of real actions" (ch. V, §4). Rothbard: "praxeology may deal with utilities only as deduced from the concrete actions of human beings" (p. 882). On their account the sentence holds of every act by interpretation, and no act could contradict it. Here PrefersEnd is a ranking kept distinct from the allocation, so the claim constrains the act: an act that passes over a preferred end he believes the good can serve, while serving a lesser one, refutes it. Only the ordering half of Rothbard's sentence is taken, not its "scarce means" half.

Does not say:

1. Anything about sub-stocks the agent does not hold, or about any other time. In particular nothing about how this allocation relates to one made at another time with a different stock — that relation, if any, is a theorem's to state and pay for.

2. That action reveals preference. This claim runs from the ranking to the act. The bridge from an act to a preference (DemonstratedPreference) is stated of trades only, and allocations are not trades; a bridge that read the ranking off the act would make this claim hold by construction, which is the authors' reading above.

3. That the ends a good can serve are comparable (Stock.ComparableServiceable), nor anything about the order between two ends both served.

4. That the scale at this time bears on the scale at any other (constancy). Mises denies it: "value judgments are not immutable" (ch. V, §4). The library does not assert it as a claim. As a condition it appears once, for one pair of holdings in one trade (Trade.RankingHolds); where the temporal reading would need it, the library exhibits the failure instead (Apodictic.Model).

5. How much is served: not that every unit is used, nor that as many ends are served as the stock allows. An agent who serves nothing satisfies it.

6. Anything about ends the agent does not believe the good can serve. Passing those over is a matter of belief, not of this claim.

Apodictic.ActsInOrder.mk

inOrder : ∀ served ∈ allocation.served,
  ∀ better ∈ stock.serves, praxis.PrefersEnd agent time better served → better ∈ allocation.served

Demonstrated preference, in exchange — the agent of a trade prefers, at the time of trading, the holding he chose to what refusing would have left him with, both counted by his beliefs at that time.

The bridge from what an agent DOES to the ranking Prefers, which the vocabulary otherwise keeps apart. Asserted of the trade handed to a theorem, not of every trade.

Source: Rothbard, MES, p. 307: "Any action demonstrates choice based on preference: preference for one alternative over others." p. 85, of exchange: "both people make it because they expect that it will benefit them; otherwise they would not have agreed to the exchange." p. 260: "We deduce the existence of a specific value scale on the basis of the real act". The doctrine is named "demonstrated preference" at p. 882, n. 8, citing Rothbard, Toward a Reconstruction of Utility and Welfare Economics (1956), pp. 224 ff.

Status: explicit-in-tradition as doctrine; our-reconstruction as a claim. For Rothbard the scale is DEDUCED from the act (p. 260), so the sentence cannot fail — it defines what the scale says. Here Prefers exists apart from the act, so the claim constrains it: a trade made against the agent's own ranking at that moment — a slip, a compulsion he would disown — refutes it.

Does not say:

1. That the agent is better off afterwards. The comparison is made with his beliefs and his scale AT THE TRADE; what the holding does for him later, and how he ranks it later, are not touched (Trade.BeliefsHold, Trade.RankingHolds).

2. Anything about the two goods alone. It ranks whole holdings; the descent to "Y over X" is a theorem's to pay for (Trade.SeparableFromRest, Trade.KeptServesOtherEnds).

3. That the trade was voluntary. Under a threat the claim still holds — the victim prefers handing over to what refusing would cost — and demonstrates nothing about the goods. Trade.Voluntary is what makes refusal the old holding.

4. Anything about other acts, other agents, or other times: not that every act demonstrates preference.

Apodictic.DemonstratedPreference.mk

demonstrates : praxis.Prefers trade.agent trade.time (trade.after trade.time) trade.refusal

Time preference — of two ends that are the same satisfaction, the agent at now prefers the one attained sooner, provided the sooner one is not already past.

Source: Rothbard, MES, ch. 1, p. 15: "A fundamental and constant truth about human action is that man prefers his end to be achieved in the shortest possible time. Given the specific satisfaction, the sooner it arrives, the better." / "This is the universal fact of time preference." Mises, Human Action, ch. XVIII, §2: "Satisfaction of a want in the nearer future is, other things being equal, preferred to that in the farther distant future." / "Time preference is a categorial requisite of human action."

Status: explicit-in-tradition as doctrine; our-reconstruction as a claim. The authors fix "the same satisfaction" by the satisfaction it gives: ice in summer and ice in winter "are not the same, but different goods" (MES pp. 15–16, n. 15), and Mises's ice is "for all practical purposes different commodities". Read that way the sentence cannot fail — whatever is preferred later was a different good. Here SameSatisfaction is supplied apart from Prefers, so the claim constrains the ranking: an agent who prefers the same satisfaction later refutes it. The reading on which it cannot fail is built as the rejected encoding (Apodictic.TimePreference.valuation_reading_is_free). Carried by Apodictic.TimePreference.less_durable_preferred.

Does not say:

1. Anything about ends that are not the same satisfaction. "Other things being equal" is carried whole by SameSatisfaction: a perishable good, two enjoyments that cannot be had together (Mises's Carmen and Hamlet), unequal certainty of getting it — each is a pair that is not the same satisfaction, and the claim is silent there. So is anticipation. A pleasure put off brings the looking forward with it, from now until it comes; a pain got over with spares the waiting for it. Either way the later option holds a satisfaction that begins now and the earlier does not. That is the literature's own account: savouring is "positive utility derived from anticipation of future consumption", dread "negative utility resulting from contemplation of the future" (Loewenstein, Economic Journal 1987, p. 667). It counts as a diagnosis only where the anticipation is shown apart from the choice it explains; inferred from the choice to wait, it is the valuation reading again.

2. Anything about bundles. Preferring a sooner bundle needs DatedFrame.LiftsOverRest.

3. Anything about ends already past at now.

4. That Before is transitive, irreflexive or total. The claim uses "sooner" only as a label on the pair.

5. How MUCH sooner is better — no rate, no discounting.

6. That the preference shows in any act. That bridge is DemonstratedTimePreference, of one act.

Apodictic.TimePreference.mk

sooner : ∀ (soon late : praxis.End),
  praxis.SameSatisfaction agent soon late →
    praxis.Before (praxis.attained soon) (praxis.attained late) →
      ¬praxis.Before (praxis.attained soon) now → praxis.PrefersEnd agent now soon late

Demonstrated time preference — an act that takes a satisfaction and forgoes the same satisfaction later shows that the agent, at the time of acting, prefers the sooner.

The bridge from an act to the ranking, for acts across dates, as DemonstratedPreference is for trades. Asserted of the act handed to a theorem, not of every act.

Source: Mises, Human Action, ch. XVIII, §2: "He who consumes a nonperishable good instead of postponing consumption for an indefinite later moment thereby reveals a higher valuation of present satisfaction". Rothbard, MES, p. 307: "Any action demonstrates choice based on preference: preference for one alternative over others."

Status: explicit-in-tradition. It is also the premise Mises's regress runs on without stating it — that an agent with no preference for the sooner postpones — read the other way round (Apodictic.TimePreference.consumption_reveals). Carried by that theorem and by Mises's regress (Apodictic.TimePreference.regress).

Does not say:

1. Anything about acts that forgo something other than the same satisfaction later. "Nonperishable" is carried by SameSatisfaction.

2. That the chosen satisfaction is present, rather than merely sooner.

3. Anything about other acts or other times.

Apodictic.DemonstratedTimePreference.mk

demonstrates : ∀ later ∈ act.forgone,
  praxis.SameSatisfaction act.agent act.chosen later →
    praxis.Before (praxis.attained act.chosen) (praxis.attained later) →
      praxis.PrefersEnd act.agent act.time act.chosen later

Appraisal from the past, weak form — nobody appraises a good as a medium of exchange on a day unless that good changed hands, against anything, on some earlier day.

Asserted of one record of exchanges and one good. "Appraises as a medium" is a belief that the good serves an onward end (MediumFrame.AppraisesAsMedium), so the claim is about what any prospective acquirer or holder can believe, not only about acquirers.

Source: Mises, Human Action, ch. XVII, §4: "he who considers acquiring or giving away money is, of course, first of all interested in its future purchasing power and the future structure of prices. But he cannot form a judgment about the future purchasing power of money otherwise than by looking at its configuration in the immediate past." Ch. XVII, §8: "A medium of exchange without a past is unthinkable." Ch. XVII, §1: "For all that is to be predicated of money is valid for every medium of exchange." Rothbard, MES p. 275: "Demand for a good as a medium of exchange must be predicated on a previously existing array of prices in terms of other goods." Rothbard, Power and Market (MES ed. p. 1021): a money must be one that "can be exchanged for a previously existing money".

Status: explicit-in-tradition as doctrine; the weak form is our-reconstruction. Mises states the premise exclusively — the judgment rests on the good's own price, from the immediate past, and on nothing else. This form keeps only that an exchange of the good came first, on any earlier day and against any good, money included.

Does not say:

1. That the earlier exchange was on the immediately preceding day.

2. That the appraisal rests on that exchange, or on that exchange alone — not that promises, edicts or cross rates play no part. Without quantities the vocabulary cannot say "rests on"; see History.OwnPastPrice.

3. That the earlier exchange was barter (History.FormedInBarter).

4. That the earlier acquirer wanted the good for itself; the regress derives that, from further conditions.

5. Anything about day 0. That nobody appraises the good as a medium on the first day of the record is the separate condition History.BeginsBeforeMedium, the termination Mises asserts.

6. That the record is complete. The claim is asserted of the record it is handed: an earlier exchange that happened but is missing from that record does not satisfy it.

Apodictic.AppraisalFromPast.mk

fromPast : ∀ (n : ℕ) (agent : praxis.Agent),
  praxis.AppraisesAsMedium agent (history.date (n + 1)) m → ∃ k ≤ n, history.ExchangedOn k m

Two partial demands — every end a good can serve is either one reached by passing the good on, or not: the demand for a medium of exchange divides without remainder into the two demands Mises names.

Asserted of one frame and one good. Its content is excluded middle for the one predicate Onward m: the vocabulary cannot decide it, and nothing in logic supplies it constructively, so where a theorem needs the split, the split is this claim.

Source: Mises, Human Action, ch. XVII, §4: "Thus the demand for a medium of exchange is the composite of two partial demands: the demand displayed by the intention to use it in consumption and production and that displayed by the intention to use it as a medium of exchange."

Status: explicit-in-tradition as doctrine; the reading of "composite" as an exhaustive sort of ends is our-reconstruction.

Does not say:

1. That the non-onward part is consumption or production. Mises names them; this claim leaves the sort "not onward", and the narrow reading is the separate condition History.OtherEmploymentsNarrow.

2. That the two demands add up, or have sizes at all. No quantities.

3. That an agent can tell which kind an end is, or that the sort can be computed. It says each end IS one or the other.

4. That excluded middle holds for any other predicate.

Apodictic.TwoPartialDemands.mk

sorted : ∀ (e : praxis.End), praxis.Onward m e ∨ ¬praxis.Onward m e