Buy Narrow, Sell Wide: Anthology

Collecting the commentary and correcting the LaTeX

First time here? I'm dabchick—unequal parts economist, philosopher, and mathematician, with an incurable habit of connecting dots and a Quixotic fantasy of being a renaissance man. After years of vanishing into online rabbit holes, I've developed a lust for thought and a preference for first-principles reasoning. Each week, I challenge myself to write, hoping that someone, somewhere, may marginally improve their world model. It may even be me. That’s what the comment section is for!

To repent for the sad experience of LaTeX on Substack for dark-mode users, and to facilitate easy reading, I am collecting my essay series on Logic, Action, and Interpretation here in one place.

Introduction

Western philosophy's fascination with mathematics goes back to the Ancient Greeks; the entrance to Plato's Academy was said to be marked by the words ‘Let no one ignorant of geometry enter here.’ Quite a few major philosophers were mathematicians in their own right; Descartes, Pascal, and Leibniz immediately come to mind. This fascination probably has its roots in the sense of absolute certainty that mathematics brings.

This influence of mathematical thought naturally created the ‘transcendental realist’ epistemic perspective that Kant critiqued. Crediting David Hume for shaking him out of his “dogmatic slumber”, Kant proceeded to posit transcendental idealism as the appropriate epistemic position for man qua man, adopting an anthropocentric, transcendental idealist position that cut the speculative metaphysics of Leibnizian, Berkley, and the Scholastics down to size, while simultaneously answering the skeptical challenge presented by Hume.

This critical position, straddling the extremes of dogmatism and skepticism, was the essence of Kant’s “Copernican revolution”. Demonstrating that both the skeptics and dogmatists were holding human reason up to an impossible theocentric standard that only mathematics could conceivably achieve, Kant changed the game by flipping the standard to an anthropocentric, grounded one. In other words, he showed that and how synthetic a priori truths are possible.

Given our nature as finite cognisers, reason must work with the data from our senses, i.e. appearances, rather than direct intuitions of reality as it is in itself, and therefore, the appearances must be structured in some ways that make this action possible. Without going into the depths of the transcendental deduction of the categories here, the essential thrust of the critical position can be captured in the famous aphorism.

‘Thoughts without content are empty, intuitions without concepts are blind.’

The role of formal mathematical thinking in leading the ‘transcendental realists’ to error is clear - they expected too much from human knowledge and held it to the impossible standard of mathematical knowledge. In mathematics, certainty and precision are only obtained because the concepts are explicitly constructed by us.

Similarly, logic has been a central aspect of Western philosophy since the Greeks. It is still a thriving branch of study, despite Kant's proclamation that ‘from Aristotle's time on, logic has not gained much in content, by the way, nor can it by its nature do so.' The importance of Frege in this genealogy is undeniable; for one, he was the first to develop a comprehensive formal language, although it was so unnecessarily complicated that nobody used it. The notation we are familiar with today comes from Guiseppe Peano.

Praxeology is sometimes derided for not being sufficiently formal, which is a strange charge. I believe it is analogous to the mathematicians’ historical demands of philosophy, insisting upon a standard of knowledge that is to be found nowhere outside logic itself. Formalisations of particular aspects of praxeological thought, for instance, gedanken experiments such as the evenly-rotating economy, are of quite a different nature than the formalisms of mathematics. The use of such models can have, at most, a pragmatic justification for facilitating economic understanding, but it is thus only in contact with an immanent apprehension of action that we can ascertain their use.

In other words, thymology reveals the relevance of the models, whether formal or praxeological. Thus, Roderick Long rephrases Kant’s famous aphorism as

“Praxeology without thymology is empty, thymology without praxeology is blind.”

Formal languages ‘considered as they are in themselves’ do not intrinsically have any interpretations - rather, we endow certain models with certain interpretations. In this sense, they are said to be ‘empty’ without the interpretation. Similarly, the theorems of praxeology are ‘empty’ in the sense that they do not refer to the content of action, only its logical form.

The a priori of action, which we can grasp by being entities that act, is thus a synthetic a priori truth grounded in the possibility of action. The ‘action axiom’, so to speak, is the a priori cognition of those categories of action that are present universally in every particular action, regardless of its particular content. The categories of action - means, ends, etc. - are universally applicable to all possible actions; that is to say, they are the grounds of possibility for us to perceive action at all. Thus, they have only a praxeological use as opposed to what Kant would term a transcendental use, e.g. it is a category error to speak of “the purpose of the universe” since the categories of action simply do not apply to the universe considered as a whole.

Specific applications of the praxeological categories are something we intuitively learn as we attempt to model other minds in the course of our own lives, but this power can be sharpened and made precise. In this process, the whole discipline of economics can be reconstructed as a subfield of praxeology and becomes an invaluable tool with which to understand the world. This is precisely what Ludwig von Mises accomplished, as best laid out in his magnum opus, Human Action.

Thymology gives praxeology the material upon which to develop its theories in the way that interpretations of models give us some reason to work on particular models rather than others. If Peano arithmetic did not yield conventional arithmetic when interpreted, it would be discarded, not arithmetic.

Furthermore, unlike what you may infer from Russell’s 200-page proof of 1+1=2, we don’t use axioms to justify our knowledge, but we use our prior established mathematical knowledge to justify the fittedness of axiomatic systems.

Formal systems are just one tool in our kit, and we use many such tools to analyse the world and synthesise our knowledge of it into a working world model that informs our actions. I attempt to use the simplistic tool-application illustration with respect to formal models to analogise the use of praxeology applied to the analysis of actions. This analogy is highly informal!

What is a formal language exactly? Hopefully, the precise characterisation of formal languages, the fact that we can concisely define them at all, should suffice to demonstrate the gulf between formal languages and ordinary languages and, thus, between mathematics and philosophy.

The Begriffsschrift

Frege's 1879 publication, the Begriffsschrift, attempted to develop logical formalism as a language for pure thought. The logical calculus he designed for this purpose would be employed in his doomed effort to reduce arithmetic to logic. Gödel would later show that the reduction of arithmetic to pure logic is an impossible task, thus vindicating Kant on the synthetic a priori nature of mathematics. But even though Frege was ultimately wrong about the foundations of arithmetic, his work on logic still renders him a pivotal figure in the history of thought, and it is through this that Roderick T. Long finds in him a connection between the two Ludwigs (Wittgenstein and von Mises), which is what has inspired this piece.

Perhaps Frege's seminal contribution to logic was his introduction of quantifiers, which revolutionised logic. Before Frege, Aristotle dealt with quantifiers, as evidenced by his famous `All men are mortal' syllogism, but did not deal with complex propositions in detail. In particular, Aristotelian syllogisms were monadic rather than polyadic. Regardless, his position as the `father of logic' is disputed and, in some part, folklore. It is likely the Stoics who ought to be credited for it.

Many of their writings were lost, making it difficult to determine the first investigator in various areas of propositional logic. However, Sextus Empiricus tells us that Diodorus Cronus and his student Philo debated whether a conditional statement is true simply when its antecedent is false while its consequent is true or whether a more substantive connection is needed between the two. This debate remains relevant in modern discussions of conditionals.

Structure & Interpretation

Predicate logic is a cornerstone for every conceivable formal model; many axiomatic-deductive formal models can be constructed if we allow for (syntactically constructed) strings over some alphabet, together with a set of axioms and rules for inference. Whenever we have a structure corresponding to a model under some interpretation, we say that the formal syntactical strings ‘satisfy’ certain propositions under the given interpretation.

So far, however, no such formal model exists that adequately captures the praxeological method. This should hardly be surprising, given that the subject of praxeology is human action in all its complexity. Thus, the following discussion will serve as an extended analogy rather than a formal construction. Praxeology plays the same functional role for thymology as formal modelling for its interpretations.

Let us make these notions precise by beginning with specific examples.

We only need concern ourselves with first-order languages for this discussion. Formal languages, as opposed to natural languages, are obtained by adhering to a fixed, finite set of rules and are purely syntactic, i.e. constructed without any reference to their meaning. A formal language consists of an Alphabet that is common to all first order languages by convention, and a Symbol set that varies with the language. Before getting into the construction, let us provide a concrete example to motivate it. The mathematically adept reader may skip ahead.

A group is a mathematical object that consists of G: a set of elements, and ◦: a binary function (one that takes two values as input) on it with identity and inverse. The axioms of Group theory are as follows: A group consists of a set, a binary function, and an identity element such that the following hold:

1. Associativity — for all x, y, z ∈ G: (x◦y)◦z = x◦(y◦z)

2. Existence of Identity — for all x ∈ G: x◦e = e◦x = x

3. Existence of Inverse — for all x ∈ G, there exists y ∈ G: x◦y = e

One simple theorem you can prove for any group is that the left inverse exists for every element and is the same as the right inverse for that element. The proof is left as an exercise; the solution is in here.

To generalise from this (admittedly trivial) example, note that the group theory as a whole, which contains the theorem on left inverses in particular, is comprised of a set of axioms from which all theorems about groups “follow”. This naturally brings us to the notion of first-order languages.

First-Order Languages: Syntax

The extent to which first-order languages suffice for all of mathematics is out of the scope of this essay; how to apply it in the case of group theory as outlined above, and the notions of necessity, consequence, and satisfaction, will be clear enough to take back to our analysis of praxeology. Let us recall that a first-order language is made up by an Alphabet consisting of:

1. A (possibly infinite) set of variables v_0, v_1, v_2, …

2. Logical connectives ¬ (not), ∧ (and), ∨ (or), → (if-then), ↔ (if and only if)

3. Quantifiers ∀ (for all), ∃ (there exists)

4. A symbol for equality vis- ́a-vis the context ≡

5. Parentheses ),(

and Symbol Set denoted by S consisting of sets of n-ary functional and relational symbols, for every positive integer n, along with a set of constants. Any one of these sets could be empty, but at least one must be non-empty. For instance, in the group theory model above, there was only one binary function and one constant, namely, the identity element e.

While it is not strictly necessary to go through the inductive construction of terms and formulae, it is a good idea to familiarise yourself with the syntax and basic concepts of formal logic. Suffice it to say that there is a simple algorithm that outputs all the ‘sensible’ strings of symbols like ∀v∃vR^(3)v_1v_2v_3 (where R(3) is a 3-ary relation symbol), and never outputs ‘senseless’ strings such as ∃∀R(7)∀v_4v_7 by pure syntactic processes.

The task of formalisation is now to come up with axiomatic strings and rules to manipulate strings so that the resulting strings can be “read back” in theorems. A ruthless mechanisation of truth is achieved, tightening the epistemic screw, so to speak, and allowing for the possibility of pristine truth claims, such as the Sylow theorems.

However, we also need to extract some semantic meaning from these strings to employ the formal method to group theory and define this extraction procedure precisely. This is captured in the notion of interpretations.

First-Order Languages: Semantics

Consider the binary operation from our group theory example. In the following, however, we will use Polish notation rather than the standard one we have been using. We will write the operation applied to two elements as ◦xy instead of x◦y to prevent any ambiguities in reading. This indicates that the binary function takes x and y as inputs (in that order). Note that the string only has meaning because we choose to endow it with some. Hence, we can axiomatise group theory in a purely formal way. It is a good exercise to formally rewrite the strings in this manner and identify the alphabet and symbol set.

The logical connectives are always interpreted in the same way by convention, regardless of all the other particulars of the formal language. They are defined in terms of truth tables, and it can be shown that they are a “sufficiently powerful” system using truth tables.

Any structure that satisfies these axioms is said to be a group, and a whole host of group-theoretic results follow that we need not get into here. But what exactly is a structure, or more accurately, an S-structure, where S is a particular symbol set? Formally, a structure ש is determined by considering a domain A and specifying:

1. An n-ary function on A for each n-ary function symbol in S.

2. An n-ary relation on A for each n-ary relation symbol in S.

3. An element of A for every constant in S.

Thus, if S = {◦, e}, we can consider the set of integers to be the domain, associate the composition function with the addition operation on the integers and associate the symbol e with the integer zero. This is the additive group structure on the set of integers; this particular S-structure is denoted in ordinary group-theoretic terms as (Z, +, 0). It is trivial to verify that this structure conforms to the axioms of group theory as written above, but keep in mind that it does so purely syntactically once we have specified the structure. Even more precisely, we define an S-structure ש as a pair (A,α) such that

1. A is a non-empty set; the domain or universe of ש

2. α is a map from S to A satisfying the following:

   1. For each n-ary relation symbol R, α(R) is a n-ary relation on A

   2. For each n-ary function symbol f, α(f ) is a n-ary function on A

   3. For each constant c, α(c) is an element of A

The map α: S → A does not have to be one-to-one or invertible. Now, we have dealt with the symbol set S and conventionalised the use of logical connectives through truth tables. All that is left is to deal with the variables v_0, v_1, . . . which we do by considering assignments that let us interpret syntactic formulae semantically. Given an S-structure A = (A, α), an assignment is a map from the set of variables in the common alphabet to the domain β: A (the alphabet) → A.

For instance, if the variable set is denoted v_0, v_1, . . ., the map β(v_k) = (−1)^k is an assignment that assigns each variable to one of the integers ±1. Now we can give a precise definition of what an interpretation of a first order language is through the notions of structure and assignment, thus completing what we set out to accomplish; the task is delightfully simple.

Interpretations

An S-interpretation τ is defined as a pair (ש, β) consisting of an S-structure ש and an assignment β. We will drop the S- prefix for aesthetic presentation, but keep in mind that all these concepts are relative to the symbol set of some first-order language. Thus, we simply speak of structures and interpretations. Now, we can read formulas through the lens of interpretations and breathe meaning into these strings of symbols. For instance, under the group structure (Z, +, 0) that we saw above, consider the string ∀v_0(◦v_0v_0 ≡ v_1); if we define the assignment β(v_k) = 2, then the string is ‘interpreted’ as the false statement that every integer added to itself yields the value 2 (the term v_0 is not ‘translated’ under β because its occurrence in the formula is in the scope of the quantifier; only free variables are translated under β). Similarly, the formula ∀v_0∃v_1∃v_2(◦v_1v_2 ≡ v_0) is interpreted as the true statement that every integer is the sum of two integers.

Satisfaction

The satisfaction relation makes precise the notion of a formula being true under a given interpretation. Again, we need not get into the precise inductive definition of this relation, which involves a long list of truth-preserving transformations. The intuitive notion of a formula being satisfied by an interpretation will suffice (refer to the examples listed in the previous section), and the fact that τ satisfies ϕ is denoted by τ |= ϕ. The intuitive notion of course being that we say τ |= ϕ if the statement ϕ is true under the interpretation τ.

Now, generalise this notion of satisfaction from single formulae to sets of formulae (which is simple enough): an interpretation τ is said to satisfy a set of formulae φ if it satisfies each formula ϕ in φ. In this case, we say that τ is a model for φ and denote this by τ |= φ. Although the same symbol is used, note that in the first case, we are dealing with a single formula, and in the second, we are dealing with sets of formulae.

Consequence

Using the notion of satisfaction, we can now define the consequence relation. Again, we will use the same symbol |=, but the consequence relation is between a set of formulae and another formula, so there is no ambiguity involved. The notion of consequence is simple enough once we have the satisfaction relation; given a set of formulae φ, a formula ϕ is said to be a consequence of φ if, for any interpretation τ |= φ, we have that τ |= ϕ. In other words, a consequence of φ is any formula ϕ that holds under all interpretations satisfying φ.

For an example of the notion of consequence, let us go back to our theorem on the existence of a left inverse in groups. The formula for this theorem is ∀v_0∃v_1(◦v_1v_0 ≡ e), and this can be formally derived (rules of derivation are another inductively defined formal notion that we will deal with intuitively) from the strings that represent the axioms of group theory. Thus, under all interpretations of the group theoretic axioms, the theorem on the existence of left inverses is true. Hence, in any group, every element has a left inverse.

Summing Up

To get back to our more familiar notions of praxeology and thymology, I want to suggest a prism to view this framework, namely the prism of structure and interpretation as laid out above. Praxeology serves us as an analogue of first-order languages in the study of action; its theorems do not make any reference to the content of action.

The study of this content, the interpretation of human behaviour through the structure of purposeful action, is precisely what Mises termed as thymology. The theorems of praxeology are precisely the consequences of the action axiom, and thus, they hold for every interpretation of the structure, i.e the theorems of praxeology are valid for every particular action because the theorems of praxeology must be satisfied under every interpretation. This is the relationship between praxeology and thymology, between the logical form and the content of action. This is the reason for the universal applicability of praxeology to purposeful behaviour.

To sparsely summarise what we have laid out so far:

1. Formal systems are tools that reduce our thinking to machine-verifiable symbol manipulation.

2. First-order formal languages are constituted by an Alphabet, a Symbol Set, and manipulation rules for constructing and augmenting strings of symbols.

3. They only derive any meaning from what we imbue them with; more formally, they must be interpreted.

Notions of satisfaction and consequence were also developed, which the reader can look back on to peruse. We can now eat the pudding to see its proof.

Praxeology and Thymology

The ‘proof’, so to speak, of the action ‘axiom’ (that humans act — they employ scarce means to achieve valued ends) is often understood to be via negativa, i.e. by demonstrating that denying it would be self-defeating; the denial of the axiom is itself an action, aimed at some end, and employing some means. Only the radical sceptic does not face direct self-defeat, though it could be considered a mark against a position to have to retreat to radical scepticism to dispute it.

Since transcendental proofs in general do not constitute defeaters for sceptical scenarios, however, it is more fruitful to see the ‘axiom’ as an invitation of sorts to interlocutors to look and see for themselves that the praxeological categories are universal, i.e. constitutive of every possible action.

Every action is indeed aimed at some end using some means. This is a constitutive fact about action. It is not a contradiction to speak of an action aimed at no end in particular, it is simply false. More accurately, it is a category error. It is meaningless to speak of an action that isn’t aimed at any particular end. Actions conform to our categories in the sense that any phenomenon that didn’t would not be an action for us. Such phenomena as involuntary breathing or blinking are not actions but merely behaviours. Thus, the axiom is indeed synthetic a priori.

Much like Kant’s categories of experience, there are also the categories of action; every action is aimed at some end, employing some scarce means (at least the actor’s time and body), and subject to success or failure, with uncertainty. The explication of these categories and the consequences of their particular structure is the task of praxeology. If the categories are the grounds for any possible experience, the praxeological categories are the grounds for any possible action. The relevance of the consequences of such an identification of our experiential synthesis is a pragmatic question; the proof of the pudding is in the eating.

Just like the categories require the schemas to “latch on” to their subject, just like the strings of our group-theoretic language require an interpretation to be semantically sensible, praxeology requires thymology to apply itself to particular actions. The formal conditions of praxeology are the necessary structure of any phenomenon that is to be perceived by us as an action, but in identifying the content, the flesh that is held up by the formal skeleton, we need thymology. Without this content, we are merely left with an empty formalism. Thoughts without content are empty; intuitions without concepts are blind.

Transcendental Deduction and Performative Contradictions

As mentioned above, arguments for the validity of the action axiom are often patterned as “performative contradictions”. The thrust of the argument is that it is contradictory to deny the action axiom or argue against it because denying and/or arguing is itself an action. Thus, in making the argument, the interlocutor demonstrates precisely what they mean to deny - a performative contradiction. Since any attempt to argue for the negation of the action axiom must fail in this manner, the argument goes, one must accept the axiom. The power of Logic compels you.

The inference process, however, if it must command some normative capability, is subject to the same paradox as Carroll’s. The argument via negativa is simply insufficient for the task. A pragmatic appeal to the applicability of the praxeological theorems, and thus to the immanent application of the categories of action to particular actions, is simply unavoidable.

Transcendental arguments, in general, only demonstrate doxastic necessity; in other words, they demonstrate what we must believe to be true as opposed to what is. For pragmatic purposes, however, given that truth-discovery is an activity we engage in, revelation of doxastic necessity is sufficient. To borrow an analogy from Roderick Long, analysing human action without the teleological categories is like playing chess by hitting a ball across a net. That would not be chess. That would not be praxeology. Such proclamations simply do not count (for us) as statements about human actions.

In a broad sense, this is akin to Kant’s primacy of practical reason over pure reason. The argument via negativa is even structured like a transcendental deduction. Thus, the action axiom in praxeology is akin to the synthetic unity of apperception in the Kantian system: the lynchpin of the entire program.

There is not enough space here for a thoroughgoing defence of the discursive formulation of Transcendental Idealism from Stroudian attacks. The implications for Hoppe’s famous “argument from argument” will be the subject of a future post.

The Problem of Irrational Action

One consequence of the formalist paradigm, troubling to some, is that the notion of “rational action” is rendered oxymoronic, and thus that of an “irrational action”, contradictory. This seems to fly in the face of ordinary experience; don’t people act irrationally all the time? The resolution is not particularly satisfactory, but unfortunately, just correct - the term “rational” is being used in a technical sense here, much like the term “rational number”. Even seemingly “irrational” actions are, from the perspective of the praxeologist, either rational or not actions in the first place.

Thought can never be of anything illogical, since, if it were, we should have to think illogically... It used to be said that God could create anything except what would be contrary to the laws of logic. – The truth is that we could not say what an ‘illogical’ world would look like... It is as impossible to represent in language anything that ‘contradicts logic’ as it is in geometry to represent by its coordinates a figure that contradicts the laws of space or to give the coordinates of a point that does not exist.

-Wittgenstein.

“Buy narrow, sell wide”

In this consideration, Roderick Long brings up a Wittgensteinian example of an odd tribe that values wooden blocks not for their volume but rather for their width. Although it may seem idiosyncratic, the example is substantive in characterising, e.g. Dutch book arguments for the “irrationality” of certain beliefs. Let us state it in full and examine it for ourselves.

People pile up logs and sell them, the piles are measured with a ruler, the measurements of length, breadth, and height multiplied together, and what comes out is the number of pence which have to be asked and given. They do not know ‘why’ it happens like this; they simply do it like this: that is how it is done... Very well; but what if they piled the timber in heaps of arbitrary, varying height and then sold it at a price proportionate to the area covered by the piles? And what if they even justified this with the words: “Of course, if you buy more timber, you must pay more”?... How could I shew them that – as I should say – you don’t really buy more wood if you buy a pile covering a bigger area? – I should, for instance, take a pile which was small by their ideas and, by laying the logs around, change it into a ‘big’ one. This might convince them – but perhaps they would say: “Yes, now it’s a lot of wood and costs more” – and that would be the end of the matter. – We should presumably say in this case: they simply do not mean the same by “a lot of wood” and “a little wood” as we do; and they have a quite different system of payment from us.

The alleged irrationality, of course, is in the fact that an Alert Kirznerian Entrepreneur (AKE) could buy up piles of wood, spread them out, and sell them back to make a clean profit. The analogy with “money pump” arguments is perhaps clear now. Superficially, this seems embarrassing for the praxeologist. Must we defend the behaviour of this tribe as being “rational”?

YES.

From a wertfrei standpoint, buying up the wood piles and spreading them out is simply an act of production. The AKE is no more “exploiting” this tribe than his counterpart producing bread from wheat is. In a reductive sense, all production is mere rearrangement. The piled-up wood is simply a higher-order good that is converted to a consumer good (the pile spread out) with an application of human labour and time. The praxeologist qua praxeologist is agnostic to the normativity of consumer preferences, he simply takes them as given in his analysis.

The only sense in which such behaviour could be considered “irrational” would be if the tribe were sleepwalking, in which case we would not consider them to be acting at all. Thus, the “buying” and “selling” of wood piles would no longer be economic activity, since the teleology would be deflated out of the phenomenon. We would put on our psychological hats instead and investigate the cause of their peculiar symptoms, but we would not construe them to be acting.

Rationality is neither a psychological claim about empirical regularity nor a heteronomous commandment imposed from without. It is constitutive of all action insofar as it is action.

Defending Diminishing Marginal Utility

As a final application of our discursive thesis, let us tackle an objection to the Austrian formulation of the law of diminishing marginal utility, which can be found here. Our foil will be David Friedman’s critique.

Critique of a Version of Austrian Economics

David Friedman states the succinct case for DMU, quoting Rothbard.

Thus, if no units of a good (whatever the good may be) are available, the first unit will satisfy the most urgent wants that such a good is capable of satisfying. If to this supply of one unit is added a second unit, the latter will fulfill the most urgent wants remaining, but these will be less urgent than the ones the first fulfilled. Therefore, the value of the second unit to the actor will be less than the value of the first unit. Similarly, the value of the third unit of the supply (added to a stock of two units) will be less than the value of the second unit. … Thus, for all human actions, as the quantity of the supply (stock) of a good increases, the utility (value) of each additional unit decreases.

(Rothbard)

His objection to this demonstration takes the form of a counter-example. We will not stress the positive argument, but merely dissolve the counter-example through the discursivity thesis (“Praxeology without thymology is empty, thymology without praxeology is blind”).

This is Rothbard’s proof of the principle of declining marginal utility. To see why it is wrong, consider tires for my car. The marginal utility of the third tire, the benefit of having three tires instead of two, is less, not more, than the marginal utility of the fourth tire. Rothbard’s proof works as long as each unit is used for a different and unrelated purpose, since you rationally choose to achieve the most important purposes first. It breaks down any time having the earlier units makes it possible to use later units in ways they before could not have been used.

(Friedman)

The absence of praxeology here ‘blinds’ Friedman to the theory of marginal utility, so to speak. He conceives of units in terms of physical quantities rather than units of a homogeneous stock as seen by the acting person. This point is brought up in the next quotation from Rothbard in the essay (emphasis added).

It is possible that a man needs four eggs to bake a cake. In that case, the second egg may be used for a less urgent use than the first egg, and the third egg for a less urgent use than the second. However, since the fourth egg allows a cake to be produced that would not otherwise be available, the marginal utility of the fourth egg is greater than that of the third egg.

This argument neglects the fact that a “good” is not the physical material, but any material whatever of which the units will constitute an equally serviceable supply.Since the fourth egg is not equally serviceable and interchangeable with the first egg, the two eggs are not units of the same supply, and therefore the law of marginal utility does not apply to this case at all. To treat eggs in this case as homogeneous units of one good, it would be necessary to consider each set of four eggs as a unit.

(Rothbard)

Friedman summarises:

The fourth egg is not equally serviceable with the first because having the first three eggs increases the options for the fourth — it can be used to make a cake. The second glass of water is not equally serviceable with the first because it is not needed to quench my thirst. The fact that possession of earlier units changes what later units can be used for is what makes marginal utility decline — or, in the case of eggs and tires, increase.

(Friedman)

Physical homogeneity is not the criterion that characterises “units” of a supply of goods. Neither is the criterion of homogeneity purely psychological, however. Properly understood, it is praxeological. Machaj points out that the appropriate conception of homogeneity is rooted in the counterfactual conditions of any given action. Objects are homogeneous insofar as they serve the same end. This conception of homogeneity is rooted in the means-end framework of the acting person. Thus, in the tires/eggs example(s), the unit in question is properly understood as a set of four tires/eggs. The possession of earlier physical units only completes the praxeological unit when the fourth physical unit is added.

To speak of the ‘marginal utility’ of the fourth physical unit is a category error — the notion of marginal utility only properly applies to units of a homogeneous supply of goods in the praxeological sense, not the physical one. To understand what the appropriate ‘unit’ is in any given case, we must use our faculty of understanding, or verstehen. In other words, the praxeological law of diminishing marginal utility is emptywithout thymological understanding, i.e.

Friedman also brings up a secondary counter, however. Rothbard seems to contradict his careful analysis in a later section of the book.

Later in the book, Rothbard writes:

This individual has, of necessity, a diminishing marginal utility of money, so that each additional unit of money acquired ranks lower on his value scale. This is necessarily true.

If I have $999 in my possession, the addition of $1 may allow me to purchase a $1000 widget that I value more than anything else I could purchase with a budget of $999. Once again, however, the appropriate “unit” of money in question would be $1000, not $1. The appropriate application of the praxeological categories allows us to dissolve the apparent tensions in all such examples.

I do take issue, however, with Rothbard’s presentation in one aspect — it is preferable to present the law of diminishing marginal utility if one considers the diminution of supply rather than an increase. This is because the increased supply can allow for an expansion of the possibilities frontier, while a diminution can never do so. Thus, the law of diminishing marginal utility can be more cleanly conceived of as the fact that the end given up when supply is diminished must be the least valued end that can still be satisfied.