Kripke Semantics: Modal Logic and Possible Worlds

1. A Formal Semantics for Modality

Quine's skepticism was partly fueled by the fact that, in his time, modal logic lacked a rigorous semantic theory comparable to Tarski's semantics for classical logic. Kripke's great achievement was to provide one.

Your background in model theory for first-order logic is the perfect starting point. In a Tarskian model, you have a single domain of objects, and the interpretation function assigns extensions (sets of objects or tuples) to predicates within that single domain. This works perfectly for evaluating truth in one world—the actual one. But how do we evaluate a statement like "It is possible that Socrates was not a philosopher"? This requires us to consider a different scenario, a different "way things could have been."

Kripke's central idea was to generalize the Tarskian model. Instead of a single world, a Kripke model includes a whole set of possible worlds. The modal operators for possibility ($◊$) and necessity ($\square$) are then treated as quantifiers over these worlds.

The Kripke Model

A Kripke model is a structure that formally captures this idea. For your mathematical background, the definition will look familiar—it's a labeled directed graph.

A Kripke model $M$ consists of:

1. A non-empty set of possible worlds, $W$.
2. An accessibility relation $R$, which is a binary relation on $W$. We write $w_{1}R{w}_2$ to mean that world $w_2$ is "possible relative to" or "accessible from" world $w_1$. The intuitive idea is that $R$ specifies which worlds are considered live possibilities from the standpoint of a given world.
3. A valuation function $V$, which for each atomic proposition $p$ and each world $w\in W$, assigns a truth value, $V(p,w)\in \{\text{True, False}\}$. This tells us what is true at a given world.

With this structure, we can define what it means for any formula $\varphi$ to be true at a world $w$ in a model $M$, written as $(M,w)\models \varphi$. The rules for standard connectives are relativized to worlds (e.g., $(M,w)\models \varphi \wedge \psi$ iff $(M,w)\models \varphi$ and $(M,w)\models \psi$). The crucial new rules are for the modal operators:

• $(M,w)\models \square \varphi$ if and only if for all worlds $w^{\prime }$ such that $wR{w}^{\prime }$, we have $(M,w^{\prime })\models \varphi$.

  In words: A statement is necessarily true in a world $w$ if it is true in all worlds accessible from $w$.

• $(M,w)\models ◊\varphi$ if and only if there exists some world $w^{\prime }$ such that $wR{w}^{\prime }$ and $(M,w^{\prime })\models \varphi$.

  In words: A statement is possibly true in a world $w$ if it is true in at least one world accessible from $w$.

This framework is incredibly powerful. It provides a formal, rigorous, and extensional semantic theory for an intensional logic, built from familiar set-theoretic components.

2. The Metaphysical Status of Possible Worlds

Kripke's formalism was a triumph, but it immediately raises a profound philosophical question. The semantics is defined in terms of a set $W$. But what are the elements of this set? What exactly is a "possible world"?

It's crucial here to distinguish between the mathematical formalism and its philosophical interpretation. Kripke gave us the former, but doing so sparked a new debate about the latter.

The rest of our lesson concerns the "impure semantics." What are the leading philosophical theories about the content of $W$? Broadly, they fall into two camps: realism and ersatzism (a form of anti-realism).

A. Modal Realism (Concretism)

The most famous, and most radical, answer comes from the philosopher David Lewis. For Lewis, "possible worlds" are not just a formal tool. They are as real and as concrete as the universe we inhabit.

• What is a possible world? A possible world is a maximal, spatiotemporally connected universe. Our world is one such universe. Other possible worlds are other universes, causally and spatiotemporally isolated from ours. There is no physical travel between them.
• What is 'actuality'? Lewis argues that "actual" is an indexical term, like "I", "here", or "now". When we say our world is the "actual world," we are just saying it's this world, the one we are in. Inhabitants of other worlds correctly call their worlds "actual."
• Ontology: This view, known as modal realism or concretism, commits us to the existence of a vast plurality of concrete worlds. For any way things could be, there is a world where they are that way. There is a world where you pursued pure maths instead of philosophy, a world with talking donkeys, and so on. They are all equally real; ours is simply the one that is actual for us.

B. Ersatzism (Abstractionism)

Most philosophers find Lewis's "ontological slum" of infinite concrete worlds too much to accept. They prefer an ersatz theory of possible worlds. "Ersatz" means "substitute," and the idea is that possible worlds are abstract substitutes for concrete ones. They are representations of ways the world could be, rather than being those ways themselves.

There are several varieties, but they share a core idea:

• What is a possible world? A possible world is an abstract entity. For example:
  • It could be a maximal consistent set of propositions. The actual world is the set of all true propositions; other possible worlds are different maximal sets that could have been true.
  • It could be a maximal possible state of affairs, as advocated by Alvin Plantinga. The actual world is the one maximal state of affairs that actually obtains.
• Ontology: Ersatz theories are more ontologically conservative. They posit the existence of abstract entities (like propositions or states of affairs), but not a plurality of concrete universes. Only one world, the actual one, is concrete.

You can now read a more detailed breakdown of these competing views in the Stanford Encyclopedia of Philosophy.

C. Kripke's Agnosticism

Finally, what did Kripke himself think? Famously, he was largely non-committal. He tended to treat possible worlds as a formal tool for clarifying our modal talk, without necessarily endorsing a heavy-duty metaphysics like Lewis's. He suggested that when we consider a possibility, we don't 'discover' a pre-existing world through a telescope; rather, we stipulate a scenario. The 'worlds' are part of the model, a tool for analysis, and a philosophical theory of their ultimate nature is a separate, and perhaps unnecessary, question.

Conclusion

In this lesson, we have seen how Kripke semantics provided a powerful and elegant formal framework that revitalized modal logic. By treating modal operators as quantifiers over a set of possible worlds, it offered a rigorous way to model necessity and possibility.

However, this formal success did not end the philosophical discussion; it transformed it. The debate shifted from whether modal logic was coherent to what its ontological commitments were. The central question became: "What is the metaphysical status of these 'possible worlds'?"

Key Takeaways:

• Kripke semantics provides a model theory for modal languages using a structure $⟨W,R,V⟩$, where $W$ is a set of possible worlds, $R$ is an accessibility relation, and $V$ is a valuation function.
• Necessity ($\square$) is treated as a universal quantifier over accessible worlds, and possibility ($◊$) as an existential quantifier.
• This formalism raises the metaphysical question of what possible worlds are. This is a question about the 'impure' or 'applied' semantics.
• Modal Realism (Lewis): Possible worlds are concrete universes, just like ours, which are spatiotemporally and causally isolated. 'Actual' is an indexical term.
• Ersatzism (e.g., Plantinga): Possible worlds are abstract entities (e.g., maximal sets of propositions or states of affairs) that represent ways the world could be. Only one world is concrete.
• Kripke's View: Kripke himself remained largely agnostic, suggesting worlds can be seen as a useful stipulative device for analyzing modal claims, without a specific metaphysical commitment.

Preview of the Next Lesson:
We have now mapped out the main positions in the debate over the nature of possible worlds. In our next lesson, we will delve deeper into this conflict. We will compare David Lewis's modal realism with its ersatzist alternatives, evaluating the arguments for and against each position to understand the philosophical price and the explanatory payoff that each theory of modality offers.