Cognitive Science > Introduction to Semantics > Meaning Derivations & Lambda Notation

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[edit] Intension & Extension

  • Names = individuals
  • sentences = truth/false
  • predicates = functions from individuals to truth values
  • Extension
    • truth value of sentence in certain situation/world
    • Frege: "reference" = Bedeutung
  • Intension
    • parameterized truth value, i.e. truth conditions
    • Frege: "sense" = Sinn
  • → individual vs individual concept

[edit] Fragments of English

Extensional framework from now on.
  • 1. Ontology
    • D = set of individuals
    • truth values {0,1}
    • functions from D to {0,1}
  • 2. Lexicon
    • [|Object language|] = Meta language
  • Rules

[edit] I & E of functions

  • Intension
    • function as rule
    • LaTex: [|limps|] = f: D -> {0,1} for all x \in D, f(x) = 1, iff x limps
  • Extension
    • as tables of mapping
    • [|limps|] = [Cassandra → 1]

[edit] Formal Concept of functions

  • right-unique relations
  • <x,y> \in R bzw. y (value) = R(x Argument)
  • domain (x) vs range (y)
    • Bsp.: father → D = set of persons, R = set of fathers

[edit] Lambda Notation

Gives us a way to name functions in a way that makes it immediately clear how they are defined.
  • LaTex: \lambda x[x+1] successor function LaTex: (= f(x) = x+1)
  • LaTex: \lambda variable [description of value)
  • Abstraction
    • form the LaTex: \lambda -notation
    • Bsp.: [|Cassandra sleeps|] = [|sleeps|] ([|Cassandra|]) = s(cass) =LaTex: \lambdax.s(x) (Cassandra) = s(Cassandra)
  • Conversion
    • replace LaTex: \lambda -notation, replace variable by argument
  • Complex arguments
    • Sets
    • LaTex: \lambda X[X \cap \{ 1,2,3 \}] -> X = set
    • Functions
    • LaTex: \lambda f[f(3)] - \lambda x[x^2] -> \lambda x[x^2] (3)
  • Function Composition
    • f°g = f(g(x))
    • LaTex: \lambda f[\lambda x[f(g(x))]]] (für fLaTex: ) (für gLaTex: ) =
    • LaTex: \lambda x[\lambda y[y+1](\lambda z[z+z](x)] = \lambda x[\lambda y[y+1](x+x)] = \lambda x[x+x+1]
Source: http://wiki.stuhlmueller.info/Cognitive_Science/Introduction_to_Semantics/Meaning_Derivations_%26_Lambda_Notation