# Modal Systems and Possible Worlds

Philosophy I · Logic · https://tryals.app/learn/philosophy-i/modal-systems-and-possible-worlds

## A Semantics With Consequences

Possible-worlds semantics gave modal logic what it had lacked: a precise account of what modal claims *mean*, rather than a list of axioms. The payoff was immediate, questions about which axioms to adopt became questions about the structure of the accessibility relation, and those could be settled mathematically.

A **frame** is a set of worlds plus an accessibility relation; a **model** adds a valuation saying which atoms hold at each world. The **correspondence results** are exact: an axiom is valid on a frame precisely when the accessibility relation has the matching property. Reflexivity corresponds to $\Box P \rightarrow P$, transitivity to $\Box P \rightarrow \Box\Box P$, symmetry to $P \rightarrow \Box\Diamond P$.

**S5** is the strongest of the standard systems, with accessibility an equivalence relation. Its distinctive feature is that **iterated modalities collapse**: $\Box\Diamond\Box P$ reduces to $\Box P$, and any string of operators reduces to its last one. This makes S5 attractive for logical necessity, where whether something is necessary should not itself be a contingent matter.

The framework extends well beyond alethic modality. Temporal logic reads the operators over times, epistemic logic over epistemic alternatives, deontic logic over ideal worlds, and **counterfactuals** — "if the match had been struck, it would have lit" — are analysed as claims about the *closest* worlds where the antecedent holds, which is why they are not material conditionals.

The metaphysical question is what possible worlds **are**. **David Lewis** defended **modal realism**: they are concrete, as real as the actual world, and "actual" is merely indexical, meaning "this one", exactly as "here" means wherever the speaker is. The theory is elegant and almost universally rejected as extravagant. The usual alternative is **ersatzism**: worlds are abstract objects — maximal consistent sets of sentences, or properties the universe might have had — which keeps the semantics while avoiding an infinity of concrete universes.

The disagreement matters because the semantics only *explains* modality if worlds are something we understand better than modality itself. If a possible world just is "a way things could have been", the analysis is circular, and that objection has never been fully answered.

> **Common pitfall:** treating possible worlds as distant planets. They are not far away; they are not anywhere. Even on Lewis's realism, worlds are spatiotemporally isolated from one another, you cannot travel to one, and nothing in one can causally affect another.

## Practice questions

8 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. On Lewis’s modal realism, other possible worlds are spatiotemporally isolated from ours, which is why we cannot investigate them.

**Answer:** True

**Why:** True. Lewis held worlds are concrete but wholly isolated: no travel, no causal contact, no spatial relation. "Actual" is indexical, meaning simply "this one", exactly as "here" means wherever the speaker happens to be.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/on-lewiss-modal-realism-other-possible-worlds-are-spatiotemporally

### 2. Which are components of a modal MODEL, as opposed to a bare frame?

A. An accessibility relation between worlds
B. A valuation saying which atoms hold at each world
C. A similarity ordering ranking worlds by closeness
D. A set of worlds

**Answer:** A. An accessibility relation between worlds; B. A valuation saying which atoms hold at each world; D. A set of worlds

**Why:** A frame is worlds plus accessibility; a model adds the valuation. A similarity ordering is further structure again, required only once counterfactuals are being analysed, which is exactly why they resisted analysis for so long.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/which-are-components-of-a-modal-model-as-opposed-to-a-bare-frame

### 3. Which are genuine objections to using possible worlds to explain modality?

A. If a world just is a way things could have been, the analysis is circular
B. Worlds are causally isolated, so we cannot investigate them
C. The semantics gives the wrong verdicts for simple modal claims
D. Concrete realism postulates an extravagant infinity of universes

**Answer:** A. If a world just is a way things could have been, the analysis is circular; B. Worlds are causally isolated, so we cannot investigate them; D. Concrete realism postulates an extravagant infinity of universes

**Why:** The objections are metaphysical and epistemological, not technical: the semantics works beautifully. Circularity is the deepest, if worlds are explained *as* possibilities, they cannot explain possibility.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/which-are-genuine-objections-to-using-possible-worlds-to-explain

### 4. On modal realism, 'actual' is an indexical term rather than an absolute property. What follows from this distinction for the ontological status of our own world compared to others?

A. Our world alone serves as the causal anchor for neighbouring worlds
B. Our world is the only one that possesses concrete physical reality
C. Our world is metaphysically fundamental whilst others remain abstract
D. Our world has no privileged metaphysical reality over any other

**Answer:** D. Our world has no privileged metaphysical reality over any other

**Why:** Treating 'actual' as an indexical like 'here' means every world is actual to its own inhabitants. Ersatzists, not realists, treat other worlds as abstract, whilst Lewis's complete spatiotemporal isolation rules out causal relations between distinct universes.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/on-modal-realism-actual-is-an-indexical-term-rather-than-an

### 5. Sort each position by its account of what possible worlds are.

**Answer:**

- Modal realism: Worlds are concrete, as real as the actual world, "Actual" is indexical, meaning simply "this world"
- Ersatzism: Worlds are abstract objects such as maximal consistent sets of sentences, Only one concrete universe exists

**Why:** Lewis’s realism takes the semantics at face value and pays in ontology; ersatzism keeps one concrete universe and makes the alternatives abstract. Both preserve the formal semantics, they disagree about what it commits you to.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/sort-each-position-by-its-account-of-what-possible-worlds-are

### 6. Arrange these developments in the order they occurred in the history of modal logic.

**Answer:**

1. Modal axioms are proposed without a semantics to interpret them
2. Possible-worlds semantics is developed
3. Correspondence results link axioms to properties of accessibility
4. Metaphysical debate begins about what possible worlds are

**Why:** Axioms first, semantics second, correspondence results third, metaphysics last. The order matters: the metaphysical debate is a *consequence* of the semantics succeeding, not a preliminary to it.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/arrange-these-developments-in-the-order-they-occurred-in-the-history

### 7. In S5, any string of modal operators reduces to its final operator. Set the number of operators remaining after reducing a string of five.

**Answer:** 1 (within ±0.4)

**Why:** Just **one**, whatever the original length. This collapse is what makes S5 the natural choice for logical necessity, it encodes the thought that whether something is necessary is not itself a contingent matter.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/in-s5-any-string-of-modal-operators-reduces-to-its-final-operator

### 8. Match each branch of modal logic to what its operators range over.

**Answer:**

- Alethic → Ways the world could have been
- Temporal → Times, past and future
- Epistemic → Situations compatible with what is known
- Deontic → Situations in which all obligations are met

**Why:** The formal machinery is identical; only the interpretation of a "world" changes. Deontic worlds are *ideal* rather than possible, which is exactly why $\Box P \rightarrow P$ fails there.

Page: https://tryals.app/practice/philosophy-i/modal-systems-and-possible-worlds/match-each-branch-of-modal-logic-to-what-its-operators-range-over