Etchemendy's Critique of Tarski

1. Formal vs. Material Consequence

To understand Etchemendy's critique, we first need a clearer grasp of the distinction between logical (or formal) consequence and material consequence. In a sense, logical consequence is what we're after: a conclusion that follows from premises due to its logical structure, irrespective of the specific facts of the world. Material consequence, by contrast, is a weaker notion that often depends on such facts.

Let's begin by reading a short text that introduces this distinction.

The article you just read illustrates material consequence with an inference that depends on the meanings of non-logical terms (like 'mother' and 'cousin'). Etchemendy's critique uses a slightly different, though related, sense of "material consequence." For Etchemendy, the issue is not about holding the meaning of terms like 'cousin' fixed, but about how Tarski's definition can be swayed by accidental, universal facts about the world.

Tarski himself was aware of a related concept. He noted that if we were to treat all terms in the language as logical constants (i.e., fix their interpretation across all models), his definition would yield a notion he called 'material consequence'.

As you've read, Tarski notes: "Even if all terms of the language are considered logical, the definition results in a characterization of a special concept of consequence, that of material consequence."

Let's put these pieces together. Etchemendy's central charge is that Tarski's definition doesn't just produce material consequence in this extreme case; he argues that it is always at risk of conflating logical necessity with accidental generality, thereby failing to properly distinguish logical from material consequence.

2. Etchemendy's Argument: Reduction and its Perils

Etchemendy's 1990 book, The Concept of Logical Consequence, frames Tarski's project as a reductive analysis. Tarski attempts to reduce the intuitive, modal concept of logical consequence (involving necessity and possibility) to a precise, non-modal, mathematical property: the property of truth preservation across a class of set-theoretic models.

The philosophical question is: does this reduction succeed? Does the mathematical property perfectly track the logical one? Etchemendy argues that it does not. A successful reduction must be both sound and complete; that is, the proposed property must not be too broad or too narrow. Etchemendy argues that Tarski's analysis fails on both counts.

1. Overgeneration: The analysis is too broad. It classifies certain arguments as logically valid which are, intuitively, not. This happens when an accidental feature of the world (or of the available models) makes a conclusion true in all models of the premises, but not out of logical necessity.
2. Undergeneration: The analysis is too narrow. It fails to classify certain arguments as logically valid. This could happen if the class of available models is too restricted to provide a counterexample to an argument that is, intuitively, invalid.

The core of the problem, for Etchemendy, lies in the "interpretational" reading of Tarski's models, which we touched upon in the last lesson. If models are just re-interpretations of the non-logical terms over a fixed, actual domain of objects, then the analysis is fatally tied to the contingent features of that domain.

3. An Example of the Conflation

Let's make this concrete by focusing on the overgeneration problem. This is the clearest way to see the alleged conflation between logical and material consequence.

Consider the sentence:
$S_1:\exists x\exists y(x\ne y)$
This sentence states, "There exist at least two distinct objects." Is this a logical truth? Most logicians would say no. It is a contingent claim about the size of the universe. A world with only one object is perfectly conceivable.

Now, let's see what Tarski's definition might say.
Suppose, as a matter of contingent fact, our universe contained exactly three objects: $a$, $b$, and $c$. According to the purely interpretational approach, a "model" is just a re-interpretation of the non-logical terms over this fixed domain, $\{a,b,c\}$. In every such model, the sentence $S_1$ is true. But this doesn't make it a logical truth.

The problem becomes sharper if we consider a different world. Imagine a world that contains only one object, let's call it 'Tweedledee'. Let this be our entire domain of discourse. Now consider the sentence:
$S_2:\forall x\forall y(x=y)$
This sentence states, "There is at most one object." In our hypothetical world containing only Tweedledee, this sentence is true. Furthermore, under the interpretational view, every model we can construct is just a re-assignment of predicates over this single-element domain. In all such models, $S_2$ remains true.

Therefore, according to Tarski's definition applied in this context, $S_2$ would be a logical truth.
$\models S_2$
This is the conflation Etchemendy warns about. A purely accidental fact about the domain (that it has only one member) has been converted into a supposed logical necessity. Tarski's account, on this reading, has mistaken a material truth (a truth about how the world happens to be) for a logical truth. The analysis overgenerates.

4. What Are Models, Anyway?

Etchemendy's critique is powerful, but it rests on a particular—and perhaps uncharitable—reading of Tarski. It assumes Tarski's models are meant to be understood purely interpretationally, as substitutions within a fixed actual domain.

Let's revisit the dilemma from the previous lesson to see if there is a way out for Tarski.

As you've just read, the two views are:

• Models as re-interpretations: This is the view Etchemendy attacks. It captures formality but fails on necessity, leading to the conflation with material consequence.
• Models as possible worlds: This view aims to capture necessity by having models represent all genuinely possible states of affairs.

Most defenders of Tarski argue that Etchemendy attacks a straw man. They claim Tarski never intended his models to be restricted to the actual world. Instead, the power of Tarski's definition comes from its use of set theory. When Tarski says "all models," he is generally understood to mean "all set-theoretic structures of the appropriate type."

Since set theory allows for the construction of sets of any cardinality (finite or infinite), the class of Tarskian models is not restricted by the contingent size or properties of the actual world. There will be a one-element model, a two-element model, and so on.

• In a one-element model, $S_2:\forall x\forall y(x=y)$ is true.
• In a two-element model, $S_2:\forall x\forall y(x=y)$ is false.

Because there exists a model where $S_2$ is false (any model with more than one object), $S_2$ is not a Tarskian logical truth. The problem of overgeneration is avoided. On this more charitable reading, Tarski's quantification over all set-theoretic models is what allows his definition to escape the confines of the actual world and capture a robust notion of necessity. The critique is deflected because the models are not just re-interpretations in our world, but are abstract mathematical structures that implicitly represent a vast space of possibilities.

Conclusion

Today we have evaluated John Etchemendy's influential critique that Tarski's model-theoretic definition of consequence conflates logical and material consequence.

Key Takeaways:

• The Critique: Etchemendy argues that Tarski's analysis reduces logic to a mathematical property of models. He claims this reduction fails because, on an "interpretational" view of models, it is sensitive to accidental facts about the world.
• The Conflation: This sensitivity can lead the Tarskian account to "overgenerate" logical truths, mistaking statements that are merely accidentally true in all available interpretations for genuinely necessary logical truths (e.g., "There is at most one object" in a single-object universe).
• The Defense: The standard defense of Tarski is that Etchemendy's critique relies on an uncharitable, purely interpretational reading. By defining models within a rich set theory, Tarski quantifies over a class of structures that is not limited by the contingent features of the actual world. This allows the definition to correctly classify statements about domain size as contingent, thus avoiding the conflation.
• The Upshot: Etchemendy's critique forces a crucial clarification. It shows that the philosophical adequacy of Tarski's definition hinges on what we take 'models' to be and on the richness of the background set theory used to construct them. The debate highlights the deep connections between logic, modality, and the foundations of mathematics.

Preview of the Next Lesson:

Both Tarski's definition and Etchemendy's critique of it depend crucially on the initial step of dividing the language into logical and non-logical terms. The meaning of logical terms is held fixed, while the meaning of non-logical terms is varied across models. But on what principled basis do we make this division? In our next lesson, we will tackle this question directly by analysing the demarcation problem: what makes a term a logical constant?