Consider the sentence: 'Necessarily, the number of planets is greater than seven.' Is this true or false in standard possible worlds semantics (S5), and why?
ATrue, because 'the number of planets' picks out the number 8, which necessarily exceeds 7
BFalse, because 'the number of planets' is a non-rigid description — in some possible worlds there are fewer than eight planets
CTrue, because the sentence is about actual astronomy, which doesn't change across worlds
DFalse, because mathematical claims cannot be expressed in natural language modal semantics
This is a de dicto reading: necessity attaches to the proposition 'the number of planets is greater than seven.' 'The number of planets' is a non-rigid description — it picks out whatever number happens to be the count of planets in each world. In worlds with six planets, this description picks out 6, and 6 > 7 is false. So the sentence is false: there exist accessible worlds where it fails. Compare: 'Necessarily, 8 is greater than 7' is true, because '8' is a rigid numeral that picks out 8 in every world. The distinction between rigid (names, numerals) and non-rigid (descriptions) is precisely what Kripke's insight resolves.
Question 2 Multiple Choice
Two modal logics differ only in their accessibility relations: Logic A has a reflexive, transitive relation; Logic B has an equivalence relation (reflexive, symmetric, transitive). What is the key difference in what they count as necessary?
ALogic A allows more sentences to be necessary because reflexivity adds more worlds
BIn Logic B (S5), if something is possible, it is necessarily possible — the accessibility relation links all worlds to all worlds, making necessity absolute
CLogic A is stronger because transitive accessibility means necessity propagates further
DThere is no meaningful difference between these logics for natural language
Logic B corresponds to S5 (reflexive + symmetric + transitive = equivalence relation). In S5, accessibility is universal: every world can access every other. This makes necessity 'absolute' — if p is necessarily true at any world, it's necessary at all worlds. The S5 axiom '◇p → □◇p' (if possibly p, then necessarily possibly p) holds. Logic A (reflexive + transitive, corresponding to S4) lacks symmetry: some worlds may not access back to the actual world, so necessity in distant worlds doesn't necessarily propagate back. S5 is typically used for metaphysical modality; S4 is used for epistemic contexts where knowledge doesn't propagate symmetrically.
Question 3 True / False
In possible worlds semantics, a sentence is necessarily true if and mainly if it is true in the actual world.
TTrue
FFalse
Answer: False
This conflates truth with necessary truth. A sentence is true if it holds at the actual world; it is *necessarily* true if it holds at every accessible possible world. Many sentences are true at the actual world without being necessarily true — 'Barack Obama was the 44th U.S. president' is actually true but not necessary, since there are possible worlds where someone else held that office. Necessary truth is a far stronger claim than actual truth. The whole point of modal semantics is to capture this distinction formally, using possible worlds to define the 'at every world' quantifier.
Question 4 True / False
According to Kripke's possible worlds semantics, a proper name like 'Aristotle' rigidly designates the same individual across all possible worlds, whereas a definite description like 'the teacher of Alexander' may pick out different individuals in different worlds.
TTrue
FFalse
Answer: True
This is Kripke's doctrine of rigid designation. Names are rigid: 'Aristotle' refers to that specific person — Aristotle of Stagira — in every world where he exists, even worlds where he never taught Alexander. Descriptions are typically non-rigid: 'the teacher of Alexander' picks out whoever taught Alexander in each world, which could be a different person in different worlds. This explains why 'Aristotle was necessarily Aristotle' is true (a rigid designator refers to the same object in all worlds) but 'Aristotle was necessarily the teacher of Alexander' is false (in some worlds, Aristotle might have pursued a different career).
Question 5 Short Answer
Why does the accessibility relation matter in possible worlds semantics, and how do different properties of this relation correspond to different modal logics?
Think about your answer, then reveal below.
Model answer: The accessibility relation determines which worlds count when evaluating modal claims. 'Necessarily p' means p is true at all worlds accessible from the current world — so whether a sentence is necessary depends entirely on which worlds are considered accessible. Different properties of accessibility correspond to different modal axioms: reflexivity (every world accesses itself) corresponds to the T axiom (what's necessary is true); transitivity corresponds to the S4 axiom (what's necessarily necessary is necessary); symmetry combined with transitivity and reflexivity gives S5, where accessibility is universal and necessity is absolute.
This is why modal logic is not one logic but a family: K, T, S4, S5 each impose different constraints on accessibility and thus make different sentences come out as valid. For metaphysical modality, S5 is usually assumed — metaphysical necessity holds in all worlds without restriction. For epistemic modality (what is known), symmetry fails because knowledge is not symmetric between agents. The power of possible worlds semantics is that it gives a uniform semantic framework that models all these different notions of modality by varying one parameter: the accessibility relation.