Accuracy, Applicability, and Tarskian Semantics

by Erik Curiel (Guest Post)

First is a concise statement of my problem with contemporary accounts of those semantics, as based on the idea of truth as, in some sense, prior to that of meaning. Second is my problem with Tarskian semantics in particular, which seems to be far and away the most popular formal theory of semantics used to construct particular accounts of the semantics of scientific theories, no matter what else the philosopher using it thinks about semantics

I. Scientific Semantics as Based on Notions Like Truth

Carnap, in the Introduction to Semantics (ch.B, §7, p.22) concisely expresses the seductive intuition that grounds essentially all contemporary thought on the semantics of scientific theories:

… to understand a sentence, to know what is asserted by it, is the same as to know under what conditions it would be true.

As appealing as this idea is, its naive application leads to severe problems. This is so no matter the details of the architectonic form of one’s account of a theory and its semantics, whether it falls, e.g., under the purview of either the syntactical or the semantical account of scientific theories and their semantics, or some other view entirely, so long as the foundation of that view takes as ineliminable a concept such as truth that must be grounded on accuracy of prediction. My gripe is not with any particular conception of truth, nor with the idea of truth itself. Truth is just the notion that specific instances of the generic form of semantics I oppose most commonly employ in their respective foundations—that genus of semantics that attributes semantic content to a theoretical representation based on the accuracy of the fit of its predictions to the results of the empirical, quantitative measurements made on the system it purports to model. In other words, my argument is with accounts of semantics that make semantic content devolve in the end upon the accuracy of a theory’s models, irrespective of how exactly it is that the accuracy comes into play in fleshing out the theory’s semantic relations and content (as justifying referential relations, as characterizing adequacy, as being required for truth, or what have you).

The heart of the problem is that such accounts cannot differentiate inaccuracy from inapplicability as a defect in a theoretical representation of a physical system: a semantics grounded on a notion like truth can rule a model of a system inadmissible only on the grounds that it does not model the behavior of the system accurately enough. That, however, is too coarse-grained a measure of the way models can fail to provide semantically sound representations of physical systems.

Consider the example of a model of a body of liquid as provided by the classical theory of fluid mechanics. When the liquid is not too viscous, is in a state near hydrodynamical and thermodynamical equilibrium and the level of precision and accuracy one demands of the model is not at too fine a spatiotemporal scale, then the classical theory yields excellent models of the liquid’s behavior over a wide range of states and environments. When the state of the liquid, say, begins to approach turbulence, the representation the theory provides begins to break down. It does so, however, in a subtle way, one that cannot be wholly accounted for by adverting merely to the fact that the theory’s model becomes inaccurate. In particular, there is a regime in which the dynamical equations of motion of the theory no longer provide accurate predictions by any reasonable measure, and yet all the quantities the theory attributes to the liquid, and all the kinematical constraints the theory jointly imposes on those quantities (e.g., the continuity of mass-density, the conservation of energy, etc.), will still be satisfied. In a strong sense, then, the theory can still provide a meaningful — and appropriate — model of the liquid even though the model is not adequately accurate.

A semantics whose fundamental terms require, by way of relation to empirical phenomena, no more than accuracy in prediction (as do all those grounded on truth, referential relations, and so on), however, cannot admit such models as part of the theory, period, for the models are not accurate. This view is inadequate for (at least) two reasons. First, it does not allow us, within the scope of the theory itself, to understand why such models are not sound even though all the quantities the theory attributes to the system are well defined and the values of those quantities jointly satisfy all kinematical
constraints the theory requires. Second, we miss something fundamental about the meaning of various theoretical terms by rejecting such models out of hand merely on the grounds of their inaccuracy. It is surely part of the semantics of the term `hydrostatic pressure’, e.g., that its definition as a physical quantity treated by classical fluid mechanics breaks down when the fluid approaches turbulence; because, however, the theory’s equations of motion stop being accurate long before, in a precise sense, the quantity loses definition in the theory, any semantics that rejects the inaccurate models in which the term still is well defined will not be able to account for that part of the term’s meaning. Thus, an adequate semantics for physical theory must be grounded on notions of meaning derived from relations in some sense prior to the accuracy of the theory’s representations of the dynamical behavior of the physical systems it treats, relations that govern the applicability of the theory’s representational resources to the system at issue.

II. Tarskian Semantics

Let’s take, at a minimum, Tarskian semantics as applied to scientific theories to require the following:

  1. a theory is (characterized by) the collection of its (Tarskian) models
  2. the semantic content of the theory is completely exhausted by the association of each model to the (possible) systems it adequately represents

In particular, no semantic content of intrinsic physical significance can accrue to the theory in virtue of relations among its models.

It is usual to take a model to be fully characterized by a solution to the theory’s equations of motion, and, indeed, I see no other reasonable way to go. Tarskian semantics then has the consequence that no structure intrinsic to the family of all solutions to the equations of motion can have semantic content of intrinsic physical significance. This seems prima facie wrong. Families of models (classes of solutions to the equations of motion) may have on their own semantic content that forms part of the semantic content of the theory but that is not formulable in a traditional Tarskian semantics. For example, the claim that the equations of motion have a well set initial-value formulation in the sense of Hadamard indubitably informs part of a theory’s semantic content, but it is one that, in its essence, consists of relations among models and cannot be reduced to the interpretation of a single model. Thus, the simple aggregation of the meaning of all individual models does not exhaust the semantic content of a theory.

Erik Curiel is a philosopher at London School of Economics, specializing in philosophy of physics, philosophy of science, and ancient philosophy. For more on Curiel’s work, visit his homepage.


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