Logic, Action, and Interpretation: II

Buy narrow, sell wide

Here is Part 2 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. Click here for Part 1. My apologies for the atrocious LaTeX formatting. I am trying to avoid it for the most part now until Substack fixes it.

Recall that we require a notion of interpretation to make sense of the semantic meanings of formal strings. This notion will allow us to bridge the gap between symbol-shunting and synthetic a priori truths and is in direct analogy to Mises’ notion of Verstehen.

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.

Conclusion

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.

We will wrap up next week by expanding upon this theme.