Questions: Intensionality and Possible Worlds Semantics

Question 1

John believes that Superman can fly. It is a fact that Superman is Clark Kent. Does it follow that John believes Clark Kent can fly?

• A. Yes — 'Superman' and 'Clark Kent' denote the same individual, so they can be substituted in any sentence without changing its truth value
• B. No — belief contexts are intensional; 'Superman' and 'Clark Kent' have the same extension but different intensions, and John may not know they refer to the same person
• C. Yes — belief is about facts in the actual world, and it is a fact that Superman = Clark Kent, so any belief about one applies to the other
• D. No — 'Clark Kent' does not have the same extension as 'Superman' because one is a public identity and one is private

Answer: (B)

This is the classic intensionality puzzle. In purely extensional semantics, co-referring terms can always be substituted salva veritate. But in intensional contexts — especially propositional attitude reports like 'John believes...' — substitution can fail. 'Superman' and 'Clark Kent' have the same extension (they pick out the same individual in the actual world) but different intensions (they pick him out via different descriptions, and could pick out different individuals in other possible worlds). John's belief is indexed to his mental states, which track descriptions and modes of presentation, not just individuals. Option A is the classic mistake; option C explicitly commits it.

Question 2

What is an intension, as distinguished from an extension, in possible worlds semantics?

• A. An intension is the emotional or connotative meaning of a word; an extension is its literal, denotative meaning
• B. An intension is a function from possible worlds to extensions; an extension is the set of individuals (or truth value) a term denotes in the actual world
• C. An intension is the full set of properties associated with a concept; an extension is just the concept's referent
• D. An intension applies only to modal operators like 'necessarily'; extensions apply to all non-modal expressions

Answer: (B)

In possible worlds semantics, an extension is a term's denotation at a particular world — for a noun phrase, the set of individuals it picks out; for a sentence, its truth value. An intension is the rule that assigns extensions across all possible worlds — a function from worlds to extensions. 'The morning star' and 'the evening star' have the same extension in the actual world (both denote Venus) but different intensions, because in a world where Venus didn't exist or the two objects were distinct, they would pick out different things. The intension is what distinguishes genuinely co-referential terms from terms that merely happen to co-refer.

Question 3

Because 'the morning star' and 'the evening star' both refer to Venus in the actual world, the sentence 'The morning star is the evening star' is necessarily true — true in most possible worlds.

• T. True
• F. False

Answer: False

This is a contingent identity, not a necessary one. The statement is true in the actual world as an a posteriori astronomical discovery — it was not known until observation revealed it. In possible worlds where Venus doesn't exist, or where two distinct objects occupy those orbital positions, the sentence would be false. Necessary truth ('true in all possible worlds') is a much stronger claim. The morning star/evening star example is precisely Kripke's argument that some identity statements are contingent even though co-referential expressions are involved.

Question 4

In possible worlds semantics, 'Possibly P' is true at world w if and only if there is at least one world accessible from w in which P is true.

• T. True
• F. False

Answer: True

This is the standard Kripkean semantics for the possibility operator. The accessibility relation ◇ between worlds determines what counts as 'possible' relative to a given world — different kinds of modality (epistemic, deontic, metaphysical) correspond to different accessibility relations. For epistemic modality ('It might be raining'), accessible worlds are those compatible with what the agent knows. For metaphysical possibility, accessible worlds are those where the laws of nature or logic could obtain. 'Necessarily P' is then the dual: P is true in all accessible worlds.

Question 5

Why can't co-referring expressions like 'the morning star' and 'the evening star' always be substituted for each other in intensional contexts like belief reports? What does this reveal about the difference between extensions and intensions?

Model answer: Co-referring expressions have the same extension (they pick out the same individual in the actual world) but different intensions (they are associated with different descriptions or modes of presentation, and could pick out different things in other possible worlds). In intensional contexts like belief reports, truth does not depend only on what is actually true — it depends on the possible worlds compatible with the agent's mental states. John's belief worlds may include 'the morning star is a planet' without including 'the evening star is a planet' if John has never connected the two descriptions. Substituting one for the other in a belief report changes which set of possible worlds is being quantified over, potentially changing the truth value. This shows that extensions alone are insufficient for intensional semantics: we need intensions — functions from possible worlds to extensions — to capture meaning in opacity-creating contexts.

The substitution failure is the core diagnostic of intensionality. An extensional context is one where substituting co-referring terms preserves truth; an intensional context is one where it may not. Possible worlds semantics explains this by distinguishing what a term denotes now (extension) from how it picks out things across all possible circumstances (intension). Belief verbs, modals, and conditionals all create intensional contexts for the same underlying reason: their truth depends on quantification over possible worlds, not just evaluation at the actual world.