Possible Worlds

Possible worlds are complete ways things could have been, including the way things actually are. In philosophy, they serve both as objects of metaphysical inquiry and as tools for analyzing modality: possibility, necessity, and contingency. Their use in logic and semantics does not by itself settle whether alternative worlds exist as concrete realities, abstract representations, or theoretical constructions. (plato.stanford.edu)

Historical development

Gottfried Wilhelm Leibniz gave possible worlds a prominent role in his philosophy. He understood alternative worlds through combinations of mutually compatible, or compossible, individuals. His theological account held that God actualizes the best possible world, while other worlds remain possible without being actual. This conception distinguished what happens from what could happen, rather than treating everything that does not happen as impossible. (plato.stanford.edu)

The modern framework emerged through twentieth-century developments in modal logic. Rudolf Carnap developed accounts using state-descriptions, while subsequent relational semantics, associated especially with Saul Kripke, made truth relative to worlds and relations between them. This transformed possible-world discourse into a systematic method for studying modal inference, independently of Leibniz’s theology. (plato.stanford.edu)

Possibility, necessity, and formal semantics

The basic interpretation identifies possibility with truth at some possible world and necessity with truth at every possible world. A contingent proposition is true at some worlds and false at others; an impossible proposition is true at none. These classifications depend on which possibilities are under discussion and which worlds the interpretation includes. (plato.stanford.edu)

In standard Kripke semantics, a propositional modal model contains a nonempty set of worlds $W$, an accessibility relation $R$, and a valuation $V$ assigning truth values to atomic propositions at worlds. An accessibility relation is a binary relation specifying which worlds count as relevant alternatives from a given world. The principal truth conditions are:

• $\Box p$ is true at $w$ exactly when $p$ is true at every world accessible from $w$.
• $\Diamond p$ is true at $w$ exactly when $p$ is true at some world accessible from $w$.

Thus, actual truth alone does not establish necessity. (plato.stanford.edu)

Different restrictions on accessibility yield different modal systems. Reflexivity validates $\Box p \rightarrow p$; transitivity validates $\Box p \rightarrow \Box\Box p$. An equivalence relation yields the familiar S5 framework. Within this approach, logical validity concerns truth across the relevant class of models, not merely truth at the actual world. (plato.stanford.edu)

What kind of entities are worlds?

The ontology of possible worlds is disputed. David Lewis defended modal realism, according to which other worlds are concrete realities, like ours, but spatiotemporally isolated from it. On this account, “actual” is indexical: inhabitants of any world can correctly call their own world actual. Actuality does not mark a privileged kind of existence. (plato.stanford.edu)

Actualist representational approaches instead understand worlds as representations available within actuality, rather than additional concrete universes. Proposed representations include maximal states of affairs and suitably organized propositions or descriptions. Actualism nevertheless requires an account of how such representations concern individuals that do not actually exist. Treating worlds as useful semantic objects therefore leaves substantive questions about their nature unresolved. (plato.stanford.edu)

Worlds are ordinarily intended to be complete alternatives, not isolated hypothetical episodes. A description such as “the bridge was never built” specifies only part of a world. Representational theories must distinguish a complete possibility from a partial description and from an inconsistent collection of claims. They must also explain what makes a representation genuinely possible. (plato.stanford.edu)

Individuals across worlds

Modal claims about particular objects raise the problem of transworld identity. Saying that a person could have pursued another occupation appears to attribute different circumstances to the same individual in another world. Identity-based accounts permit one individual to exist at multiple worlds; identifying that individual need not involve locating someone with identical observable characteristics. (plato.stanford.edu)

Lewis’s counterpart theory offers an alternative. Individuals belong to one world only, but have counterparts elsewhere related by relevant similarities. A claim about what someone could have done is analyzed through what an appropriate counterpart does. The counterpart relation is not identity, and its application may depend on context. This difference affects the interpretation of essentialism and claims about properties an individual could not lack. (plato.stanford.edu)

Counterfactuals and epistemic applications

Possible worlds provide influential analyses of counterfactuals, such as “If the switch had been pressed, the lamp would have lit.” Rather than considering every antecedent-world equally, Lewis–Stalnaker approaches examine the closest or most similar relevant worlds where the antecedent holds. Their accounts differ over whether a unique closest world is required. Similarity must be specified carefully, particularly when counterfactuals are used to analyze causation. (plato.stanford.edu)

In epistemology, worlds can represent alternatives compatible with an agent’s information. Standard epistemic logic interprets knowledge as truth throughout the accessible alternatives. Related semantics model belief, obligation, and temporal reasoning, with accessibility interpreted differently for each application. These uses need not treat every formal point as an independently existing universe. (plato.stanford.edu)

Limitations and extensions

Ordinary possible-world representations identify propositions that hold at exactly the same worlds. Consequently, all necessary truths receive the same world-set representation, although agents may understand or believe one without understanding another. This creates difficulties for representing finely differentiated content and for avoiding idealized assumptions of logical omniscience. (plato.stanford.edu)

Impossible worlds extend the framework with points at which logical or metaphysical impossibilities can hold. They have been used to distinguish necessarily equivalent contents, represent inconsistent beliefs, and analyze conditionals with impossible antecedents. Such extensions require additional semantic choices: an impossible world need not violate every logical principle, and different applications employ different kinds of impossibility. (plato.stanford.edu)