Montague Semantics (1973) › Intensional Logic

The most technically demanding aspect of Montague's framework involved handling intensionality.contexts where meaning couldn't be reduced to truth values or extensions (the sets of entities having properties). Consider the sentence "John seeks a unicorn." If we interpreted "a unicorn" extensionally, as denoting some particular unicorn, we'd face a problem: unicorns don't exist, so there's no unicorn for "a unicorn" to denote. Yet the sentence is perfectly meaningful and can be true.John might indeed be seeking a unicorn, futilely though his quest may be.

Montague solved this through intensions: functions from possible worlds to extensions. The intension of a property like "unicorn" was a function mapping each possible world to the set of unicorns in that world. In the actual world, this set is empty, but in other possible worlds.mythical worlds where unicorns exist.the set is non-empty. An intensional verb like "seek" operated on intensions rather than extensions: "John seeks a unicorn" meant that John stood in the seeking relation to the intension of unicorn, not to any particular unicorn. This explained the lack of existential commitment: seeking relates an individual to an intension, which can be meaningful even when its extension in the actual world is empty.

The same mechanism handled other intensional contexts. "John believes that Mary left" involved a belief relation between John and a proposition.the intension of "Mary left," understood as the set of possible worlds where Mary left. John's belief did not depend on Mary leaving in the real world. Modal operators like "necessarily" and "possibly" similarly operated on intensions: "necessarily P" was true if P's intension included all possible worlds; "possibly P" was true if P's intension included at least one possible world.

Temporal expressions required intensionality as well. The sentence "the President is tall" has different truth values at different times, depending on who holds office. Montague treated temporal expressions as involving implicit world-time indices: expressions were evaluated at world-time pairs rather than possible worlds alone. The present tense "is" evaluated at the current world-time, while past and future tenses shifted the time index. This allowed systematic treatment of temporal reasoning: "John was a student" was true at time t if "John is a student" was true at some earlier time.

Implementing intensional logic required a sophisticated type system. Montague distinguished:

• Type e: entities (individuals like John, books, cities)
• Type t: truth values (true/false)
• Type $\langle s,\alpha \rangle$: functions from possible worlds (type $s$) to expressions of type $\alpha$ (intensions)
• Type $\langle \alpha,\beta \rangle$: functions from type $\alpha$ to type $\beta$

Common nouns like "student" had type $\langle s,\langle e,t \rangle \rangle$: functions from possible worlds to functions from entities to truth values (i.e., intensions of properties). Intransitive verbs had the same type. Transitive verbs had type $\langle s,\langle e,\langle e,t \rangle \rangle \rangle$: functions from worlds to two-place relations. Noun phrases had type $\langle s,\langle \langle s,\langle e,t \rangle \rangle,t \rangle \rangle$: functions from worlds to functions from properties to truth values (intensions of generalized quantifiers). These complex types ensured compositional coherence: only compatible types could combine, guaranteeing semantic well-formedness.