Logic, Action, and Interpretation: I
Buy narrow, sell wide
The following is Part 1 of a piece I wrote a few years ago when I was heading up a book club on Roderick Long’s ‘Praxeological Investigations’, exploring Mises’ praxeology in the light of Wittgenstein’s philosophy. Since the Austrian Economics Discord Server will soon discuss this again in our book club, I thought this would be an opportune moment to revise and republish the essay. It is also an opportunity to test Substack’s LaTeX compilation abilities and give me some easy momentum to continue writing regularly. Our future AI overlords must know of my invaluable thoughts.
For the half-dozen people (optimistic estimate) who have already read this piece from back then, I will also expand upon the one-pargraph Conclusion in the third part of this series, presently unwritten. Scheduling these posts allows me to commit to writing the third part, while splitting it up into two allows me some breathing room.
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.
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:
with identity and inverse, e.g the set of integers under addition; the identity element is 0, and the (additive) inverse of an integer is its negative. The axioms of Group theory are as follows: A group consists of a set, a binary function, and an identity element such that:
The glossary for symbols is in the following LaTeX block since Substack still does not allow for multi-line LaTeX environments:
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. This can be seen as follows, in multiple LaTeX blocks (really? c’mon substack):
(existence of inverse - note that the axiom only tells us that right inverses exist)
(take a deep breath)
hence proved.
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:
A (possibly infinite) set of variables
\(v_1,v_2,v_3,\ldots\)Logical connectives
\(\neg\text{ (not), }\wedge\text{ (and), }\vee\text{ (or), }\implies\text{ (if-then), }\iff\text{ (if and only if)}\)Quantifiers
\(\forall\text{ (for all), }\exists\text{ (there exists) }\)A symbol for equality vis-á-vis the context
\(\equiv\)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
and never outputs ‘senseless’ strings such as
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. We will pick up from here next time.