Tarski and the Philosophical Foundations of Logical Consequence

The Common Concept of Logical Consequence

Before constructing a formal definition, Tarski first undertook a careful analysis of what he called the "common concept" of logical consequence—the intuitive notion we employ in everyday reasoning. He identified three essential features that any successful definition must capture.

As you've just read, Tarski distilled our intuitive notion of consequence into three core properties:

1. A Modal Element: Logical consequence involves necessity. It’s not just that the premises happen to be true and the conclusion true; it must be that it is impossible for the premises to be true while the conclusion is false.
2. A Formal Element: The relationship of consequence holds in virtue of the logical form of the sentences, not their specific subject matter. If an argument is valid, any other argument with the same form will also be valid, regardless of what the non-logical terms (like names and predicates) refer to.
3. An A Priori Element: We should be able to know that a conclusion follows from its premises without needing to conduct empirical investigations of the world. Our knowledge of logic seems to be independent of experience.

The central challenge for Tarski was to find a single, mathematically precise definition that could capture all three of these features simultaneously.

The Model-Theoretic Definition

Tarski's ingenious solution was to replace the vague, metaphysical notion of "possibility" with the precise, mathematical concept of a model. As you know from your study of model theory, a model for a language consists of a set of objects (the domain) and an interpretation function that assigns extensions to all the non-logical constants of the language (names, predicates, function symbols).

Building on his work on truth, Tarski defined "truth-in-a-model." With this tool in hand, the definition of logical consequence becomes remarkably elegant.

Let's read two short texts that lay out this definition. The first, from the Stanford Encyclopedia of Philosophy, is a direct and formal statement. The second, from the Internet Encyclopedia of Philosophy, provides a helpful conceptual bridge using the "locked room metaphor."

Let's consolidate these ideas. Tarski's model-theoretic definition of logical consequence can be stated as follows:

A sentence $X$ is a logical consequence of a set of sentences $K$ if and only if every model that makes all sentences in $K$ true also makes $X$ true.

In other words, there is no possible interpretation of the non-logical terms (i.e., no model) in which the premises are true and the conclusion is false. This is often written as:

$K\models X$

This single definition brilliantly captures Tarski's three desiderata:

• It captures the modal element by quantifying over all possible models. The "must be true" of consequence becomes "true in all models."
• It captures the formal element because the logical constants (like and, not, for all) have their meaning held fixed by the definition of truth in a model, while the non-logical terms are allowed to vary from model to model. Validity depends only on the fixed-meaning logical terms.
• It aims to capture the a priori element because determining whether $K\models X$ is a mathematical question about a class of abstract structures (models), not an empirical question about the world.

Philosophical Presuppositions and Challenges

Tarski's definition was a monumental achievement, and it remains the standard definition of logical consequence in modern logic. However, from a philosophical perspective, it is not without its own deep assumptions and challenges. Analysing these is key to understanding the philosophy of logic.

1. The Demarcation of Logical Constants

The entire Tarskian apparatus rests on a prior division of vocabulary into logical and non-logical terms. The meanings of logical terms are held fixed, while non-logical terms are re-interpreted in each model. But what is the principled basis for this division?

• Why are and and not logical, but is taller than and is a planet are not?
• What about is identical to (=)? Or is a member of (∈)?

Tarski himself was aware of this problem and suggested that the line might be somewhat conventional. But if the line is arbitrary, then what counts as a logical consequence also seems arbitrary. This is known as the problem of logical constants, a topic we will explore in more detail later in the course.

2. The Status of Models

The definition relies on the notion of a "model," but what exactly are models, philosophically speaking? And how does quantifying over them capture genuine logical necessity? The philosopher John Etchemendy famously argued that there are two ways to think about models, neither of which is fully satisfactory.

To summarise Etchemendy's dilemma:

• Representational Approach: If models represent genuinely possible worlds, then the definition seems to capture necessity. However, it fails to account for formality. It collapses logical consequence into general metaphysical necessity. For example, "Water is H₂O" is metaphysically necessary, but we don't think "Water exists" logically entails "H₂O exists." The link isn't formal.
• Interpretational Approach: If models are just different ways of interpreting non-logical language about the actual world, then the definition captures formality. However, it fails to capture necessity. It's possible that a statement is true under every re-interpretation of its terms in the actual world, but is still not a necessary truth. For example, if there happen to be fewer than five objects in the universe, the sentence "There are at least five objects" would be false in every model (since no model's domain would have five objects), making its negation a Tarskian 'logical truth', which seems wrong.

This critique suggests that Tarski's definition might not have perfectly captured the intuitive concept of consequence after all.

3. Reliance on Set Theory

Finally, the entire machinery of model theory is built within set theory (usually ZFC). This raises the question of whether logic is truly fundamental, or if it depends on the truths of mathematics. If the collection of all things in the actual world is "too big" to be a set (a proper class), then there is no model that corresponds to the real world. In that case, how can we be sure that "truth in all set-theoretic models" guarantees truth in reality? This suggests that logical knowledge, on the Tarskian account, may not be entirely free of substantive assumptions about the world (or at least, about the universe of sets).

Conclusion

Today we have seen how Tarski leveraged his theory of truth to create a powerful and precise definition of logical consequence. This model-theoretic account has become the cornerstone of modern logic.

Key Takeaways:

• Tarski's Goal: To formalise the intuitive concept of logical consequence by capturing its modal, formal, and a priori features.
• The Definition: A sentence $X$ is a logical consequence of a set of premises $K$ ($K\models X$) if and only if there is no model in which all sentences in $K$ are true and $X$ is false.
• Philosophical Presuppositions: This elegant definition rests on several significant philosophical assumptions:
  1. A clear distinction can be made between logical and non-logical vocabulary.
  2. The collection of "models" adequately captures the modal notion of possibility required for logical necessity (a claim challenged by Etchemendy).
  3. Set theory provides a neutral and adequate framework for defining logic.

Preview of the Next Lesson:

We have briefly introduced John Etchemendy's influential critique of Tarski's definition. In our next lesson, we will dive deeper into this argument. We will evaluate Etchemendy's critique that Tarski's account conflates logical and material consequence, examining whether the model-theoretic definition truly captures the necessity of logic or merely a form of accidental, universal truth.