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## Praxeology and Mathematical Logic

04/24/2020

Sometimes critics of praxeology make this complaint about it. Praxeology is supposed to be logically deduced from the concept of action (Mises) or from the action axiom (Rothbard). If so, these deductions should be set down in rigorous form. We need to know what exactly follows from what. To do this, ordinary language isn’t adequate. Praxeology should be formalized, using mathematical logic. Then, we would be able to tell whether the deductions really worked.

In the time I’ve been lecturing on Human Action, this complaint has come up nearly every year. In this article, I’d like to respond to it. I should mention that it isn’t only critics of praxeology who think that the complaint has something to be said for it. Occasionally, supporters of praxeology have signaled an intention to formalize it, but these suggestions haven’t borne much fruit, so far as I am aware.

The economist George Schuller was one of the first to raise the objection. In a note that appeared in the American Economic Review in 1951, he said:

When a logical chain grows beyond the limits set by stated assumptions, it uses unstated assumptions. The number of unstated assumptions (axioms, postulates, or other) in Human Action is enormous. If Mises denies this, let him try to rewrite his book as a set of numbered axioms, postulates, and syllogistic inferences using, say Russell’s Principia, or, closer to home, Von Neumann’s Theory of Games as a model.

Murray Rothbard answered Schuller in this way:

The suggestion has been made that, since praxeology and economics are logical chains of reasoning based on a few universally known premises, to be really scientific it should be elaborated according to the symbolic notations of mathematical logic. This represents a curious misconception of the role of mathematical logic, or “logistics.” In the first place, it is the great quality of verbal propositions that each one is meaningful. On the other hand, algebraic and logical symbols, as used in logistics, are not in themselves meaningful. Praxeology asserts the action axiom as true, and from this (together with a few empirical axioms—such as the existence of a variety of resources and individuals) are deduced, by the rules of logical inference, all the propositions of economics, each one of which is verbal and meaningful. If the logistic array of symbols were used, each proposition would not be meaningful. Logistics, therefore, is far more suited to the physical sciences, where, in contrast to the science of human action, the conclusions rather than the axioms are known. In the physical sciences, the premises are only hypothetical, and logical deductions are made from them. In these cases, there is no purpose in having meaningful propositions at each step of the way, and therefore symbolic and mathematical language is more useful.

Simply to develop economics verbally, then to translate into logistic symbols, and finally to retranslate the propositions back into English, makes no sense and violates the fundamental scientific principle of Occam's razor, which calls for the greatest possible simplicity in science and the avoidance of unnecessary multiplication of entities or processes.

Contrary to what might be believed, the use of verbal logic is not inferior to logistics. On the contrary, the latter is merely an auxiliary device based on the former. For formal logic deals with the necessary and fundamental laws of thought, which must be verbally expressed, and logistics is only a symbolic system that uses this formal verbal logic as its foundation. Therefore, praxeology and economics need not be apologetic in the slightest for the use of verbal logic—the fundamental basis of symbolic logic, and meaningful at each step of the route. (Rothbard, Man, Economy, and State, p. 941)

I’d like to supplement Rothbard’s remarks with an additional point. But before doing so, a preliminary issue needs to be cleared out of the way. Both Human Action and Man, Economy, and State contain the authors’ remarks on all sorts of subjects in addition to the presentation of praxeology. Schuller seems not to realize this—no wonder he is puzzled by the “unstated assumptions” of Human Action. Does he think, e.g., that Mises’s discussion of the Industrial Revolution is supposed to be logically derived from axioms?

Now we at last come to the supplement to Rothbard. He stresses the fact that in praxeology we want to know the meaning of each step in a deduction. What I would like to add is that praxeological deduction is material rather than formal. Not only does the praxeologist want to understand the meaning of each step: he needs to understand the meaning of the step in order to see that the deduction of the step is valid.

Consider this example of a praxeological truth: if someone prefers one apple to one orange at a given time, it’s not the case that he prefers one orange to one apple at that time. To understand why this statement is true, you need to understand what “prefers” means. You can’t just identify the statement as a formal contradiction without knowing what the word means, as you could with “Someone prefers one apple to one orange at a given time, and it’s not the case that he prefers one apple to one orange at a given time.” That could immediately be seen to be an instance of the necessarily false (p and not p), even if you don’t know what a preference is. To be sure, you could state as an axiom that if at time t, A prefers x to y, then it’s not the case that at time t, A prefers y to x. But that wouldn’t accomplish much. You would still need to understand what “prefers” means to realize that the statement is true.

Oddly enough, Rudolf Carnap, a leading logical positivist, emphasizes exactly the point about implication that I’ve tried to make. (He of course isn’t defending praxeology when he does so.) The philosopher Bruce Aune summarizes Carnap in an easy-to-follow way: “As Carnap emphasized, not everything reasonably regarded as an implication is a formal relation: some implications are ‘material,’ based on the meaning of nonlogical words. An example of such a nonformal implication is that between ‘a is warmer than b’ and ‘∼ (b is warmer than a).’ The latter follows from the former, but it does so for nonlogical reasons” (Metaphysics: The Elements (Minneapolis: University of Minnesota Press, 1985), p. 132).

Thus, praxeology is not “inexact” and “nonrigorous” because it reasons in words, and people who say otherwise have a mistaken view of implication.

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David Gordon is Senior Fellow at the Mises Institute and editor of the Mises Review.