Montague's intensional logic is a powerful tool for analyzing meaning in language. It combines types, intensions, and extensions to represent semantic objects and their relationships across possible worlds. This framework allows for precise modeling of complex linguistic phenomena.

Lambda calculus plays a crucial role in Montague's system, enabling compositional semantic representations. By using lambda abstraction and function application, we can build up complex meanings from simpler components, capturing the nuances of natural language semantics.

Montague's Intensional Logic

Components of Montague's intensional logic

• Types categorize semantic objects
  • Basic types include entities (e) and truth values (t)
  • Complex types are functions from one type to another (<a,b>)
    • Example: <e,t> is a function from entities to truth values (predicates)
• Intensions capture meaning across possible worlds
  • Denoted by the ^ operator applied to an expression
  • ^man represents the property of being a man in any possible world
  • Intensions are functions from possible worlds to extensions
• Extensions are the actual referents or truth values in a specific world
  • Denoted by the @ operator applied to an expression
  • @man is the set of individuals who are men in the current world
  • Extensions can vary across different possible worlds

Lambda calculus for semantic representation

• Lambda calculus is a formal system for representing functions
  • Used in Montague's intensional logic to model meaning
  • Allows for compositionality in semantic representations
• Lambda abstraction creates functions by abstracting over variables
  • λx[man(x)] represents the property of being a man
  • The variable x is bound by the lambda operator
• Function application combines a function with an argument
  • (λx[man(x)])(john) applies the property of being a man to the individual john
  • Results in the proposition that John is a man
• Compositionality principle states that the meaning of a complex expression is determined by:
  1. The meanings of its constituent parts
  2. The rules used to combine those parts

Possible Worlds and Semantic Interpretation

Possible worlds in semantic interpretation

• Possible worlds represent alternative ways the world could be
  • Used to model different scenarios or states of affairs
  • Montague's intensional logic evaluates expressions relative to possible worlds
• Accessibility relations connect possible worlds
  • Determine which worlds are relevant for evaluating the truth of a proposition
  • Different modal operators (necessity, possibility) rely on accessibility relations
• Semantic interpretation assigns meanings to expressions in context
  • In Montague's intensional logic, interpretation involves evaluating expressions across possible worlds
  • The intension of an expression determines its extension in each possible world
    • Example: The intension ^man maps each possible world to the set of individuals who are men in that world

De dicto vs de re readings

• De dicto (about the saying) readings apply modal operators to entire propositions
  • "Necessarily, the number of planets is eight" (□(^(λx[number-of-planets(x) = 8])))
  • The necessity operator □ scopes over the whole proposition
• De re (about the thing) readings apply modal operators to specific entities
  • "The number of planets is necessarily eight" (λxnumber-of-planets(x) = 8)
  • The necessity operator □ scopes over the entity (the number 8)
• Scope ambiguity arises when a sentence has both de dicto and de re readings
  • Montague's intensional logic disambiguates these readings using:
    1. The ^ and @ operators
    2. Lambda calculus to specify scope