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.

Where the material conditional gets its truth conditions

Oh, the material conditional. Some love it, some hate it. But can we all agree that explaining it to the uninitiated is a perennial headache? If you've taught baby-logic, you know how this goes. There you are, giving a just lucid shpeel on deductive systems, until you get to this part:

A	B		A → B
T	T		T
T	F		F
F	T		T
F	F		T

and the tires screech to a halt. Why are those bottom two values True? -- they demand. The first two rows don't bother them. But if A is false, why should it be that A → B is true, regardless of the truth of B?

You could just say it's a convention, get over it. But why is this the convention adopted in classical logic? My colleague Jonathan Livengood and I discussed this, and came up with a better answer:

Suppose we agree on the first two rows of the above truth table. If implication (→) is both non-trivial and asymmetric, then this its only possible truth table.

Here's why. Start by writing down all the possibilities for these bottom two rows. There are only four, and A → B has to be one of them.

A	B		1	2	3	4
T	T		T	T	T	T
T	F		F	F	F	F
F	T		T	F	F	T
F	F		F	T	F	T

Column 1 is trivial, because it has the same values as B. If this were the correct column, then saying A → B would mean the same thing as just saying B. So, assuming → is not trivial, we can throw this column out.

Column 2 has a symmetry property that implication doesn't. Namely, it stays the same if we reverse the A and B cells. If this were material implication, then A → B would be true if and only if B → A is true. So, assuming → is asymmetric, we can throw this column out too. (This column is actually the usual truth table for A ↔ B.)

Column 3 has exactly the same problem: it stays the same when we reverse the cells containing A and B. So we can throw it out for the same reason. (This column is actually the usual truth table for A&B. So, plausibly, we can also observe that "implies" should mean something different than "and.")

And that's it! If → is non-trivial and asymmetric, then Column 4 is the only option left: the standard, not-just-conventional truth table for material implication.

Get Started Playing Ehrenfeucht-Fraisse Games

Ehrenfeucht-Fraisse games are a very useful method in logic, when you're trying to figure out if two models are elementarily (logically) equivalent or not. This may be of special interest to philosophers of science, since the 'empirical equivalence' of two models is much better characterized by elementary equivalence than it is by isomorphism.

At any rate, I've written a friendly introduction to this method (PDF). Here's a little of what you'll find there.

Ehrenfeucht-Fraisse games involve two players: a copier (who tries to copy the other player's moves), and a spoiler (who tries to spoil the other player's copying). I call the players, Erin and Fred:
Erin (the spoiler) and Fred (the copier).
All that Erin and Fred need to play a game is a pair of models (L-structures) M and N. The ordinals are nice, simple structures that can be used to illustrate how this works, so let's make M and N two ordinals -- say 7 and 8. We begin by setting the number of moves in the game -- say 3.

Game play then consists of Erin first choosing an element of one ordinal, followed by Fred choosing an element of the other ordinal. The game is over when they've each chosen three elements. If the 3 elements from 7 are in the same order as the three elements of 8, the Fred (the copier) wins. If they're not, then Erin (the spoiler) wins. Below is an example game in which Fred has won.

Erin always goes first. In this game, the choices from the left ordinal (7) match the order of the choices from the right ordinal (8), and so Fred wins.
One of the most interesting applications of these games in model theory is established by a theorem due to Ehrenfeucht:

Theorem: Two L-structures M and N are elementarily equivalent iff for every Ehrenfeucht-Fraisse game on M and N, Fred has a winning strategy.

This fact has led to many interesting results, including a simple proof of which ordinals are elementarily equivalent. If this sounds interesting, it is! For more details, see my friendly introduction (PDF).

Comments and suggestions are always appreciated. Enjoy!

PHIL-∞: An Infinite Seminar?

Can every philosophy seminar be correctly completed in either finite time or infinite time? Here's a (gödelesque) answer in the negative, in the form of a philosophy seminar I'm calling PHIL-∞ (pronounced Phil-Infinity):

Course Description. PHIL-∞ is a (very high-level) philosophy research seminar. Students participate in term-paper research, with the goal of determining the truth or falsity of the following claim:

All courses in the philosophy department can be completed in finite time.

(Technical note: students can neglect relativistic corrections to the notion of absolute time, since all coursework is guaranteed to take place in roughly the same inertial, weak-gravitational regime.)

Course Requirements. In order to complete the course, students must correctly complete all the required coursework. The only required coursework in PHIL-∞ is the following term-paper assignment:

1. Write down the title of each course in the philosophy department. For example, if PHIL-101 "Intro to Philosophy" is listed as a philosophy department course, then students must write this down. Obviously, PHIL-∞ will also have to appear somewhere on their list.


2. For each course on the list formulated in 1, do one of the following:
   
   
   1. If the course requires infinite time to complete, then write down 'infinite' below the course title.
   
   
   2. Otherwise (if the course can be completed in finite time), then write down 'finite' below the course title. Then complete the term-paper assignment for that course by writing it out completely.

Remark. By enrolling in this course, students are making a very unfortunate mistake indeed. If you haven't taken a moment to imagine what it would take to complete PHIL-∞, try it now. (I go through it below.)

Imagine a hard-working student, who writes down each course on the department listing, and below each one either writes 'infinite,' or else writes 'finite' and does the term paper. But what does one write below PHIL-∞? It would be incorrect to write down 'infinite,' because then the student would have completed the course in finite time.

Therefore, the student must write down 'finite,' and then rewrite the term paper requirement for PHIL-∞. This entails writing down all the department courses, and so on.

But sooner or later, the student will get to PHIL-∞ again, and start the cycle all over. This cycle would continue indefnitely, so it would also be incorrect to write down 'finite' below PHIL-∞.

So it appears that there is no correct way to complete PHIL-∞, in finite or in infinite time?.

The question I leave to you then, is: why should that be the case?