Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 1 Expanding Our Formalism, Part 2 1 1. Lambda Notation for Defining Functions A Practical Concern: • Most expressions of natural language will have some kind of function as their extension... • These more complex functions are very awkward to write in our current notation… (1) Example: The Extension of “Or” [[ or ]] = q: Dt  D<t,t> for every x ∈ Dt , q(x) = px : Dt  Dt for every y ∈ Dt, px(y) = T iff y=T or x=T (2) Lambda Notation, Part 1 a. Syntax: [ λx : x ∈ D . ϕ(x) ] b. Semantics: The function whose domain is D (i.e., which takes as argument anything in the set D), and for all x ∈ D, maps x to ϕ(x) (3) Examples a. [ λx : x ∈ { 0, 1, 2, 3 } . x + 3 ] = (i) { <0,3> , <1,4>, <2,5>, <3,6> } (ii) f : { 0, 1, 2, 3 }  { 3, 4, 5, 6 } for all x ∈ { 0, 1, 2, 3 }, f(x) = x + 3 b. [ λx : x ∈ { Beatles , Metallica } . the drummer for x ] = (i) { <Beatles, Ringo Starr> , <Metallica, Lars Ulrich> } (ii) g : { Beatles, Metallica }  { y: y is a drummer } for all x ∈ { Beatles, Metallica }, g(x) = the drummer for x 1 These notes are based on the material in Heim & Kratzer (1998: 34-53).Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 2 (4) Lambda Notation: Functions Taking Arguments [ λx : x ∈ D . ϕ(x) ] (a) = the unique y such that <a,y> ∈ [ λx : x ∈ D . ϕ(x) ] = the function ‘[ λx : x ∈ D . ϕ(x) ]’ taking a as argument (5) Examples a. [ λx : x ∈ { 0, 1, 2, 3 } . x + 3 ](2) = 5 b. [ λx : x ∈ { Beatles , Metallica } . the drummer for x ](Metallica) = Lars Ulirch (6) The Rule of ‘Lambda Conversion’ (LC) The following equation is a consequence of how our notation is defined… Since we’ll be using it quite a bit, it’s nice to have a name for it: ‘Lambda Conversion’ [ λx : x ∈ D . ϕ(x) ](a) = ϕ(a) (7) Examples a. [ λx : x ∈ { 0, 1, 2, 3 } . x + 3 ](2) = 2 + 3 = 5 b. [ λx : x ∈ { Beatles , Metallica } . the drummer for x ](Metallica) = the drummer for Metallica = Lars Ulrich (8) The True Power of This Notation • The real advantage of lambda notation is that it offers a very handy and simple way of defining functions that yield other functions as values • The way to represent such functions is incredibly simple: You just embed one lambda formula inside another one!Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 3 (9) Example 2 [ λx : x ∈ N . [ λz : z ∈ N . x + z ] ] This is the function which takes a number x as argument and returns the function which takes a number z as argument and returns x + z (10) Example [ λx : x ∈ N . [ λz : z ∈ N . x + z ] ](3) = (by LC) [ λz : z ∈ N . 3 + z ] (11) Convention for Sequences of Arguments Now that we can embed ‘lambdas inside of lambdas’, we can also write out formulae that look like the following: ‘[ λx : x ∈ N . [ λz : z ∈ N . x – z ] ](3)(4)’ b. How You Read The Formula Above: • The function ‘[ λx : x ∈ N . [ λz : z ∈ N . x – z ] ](3)’ taking (4) as argument. c. Equation That Follows From (11b) [ λx : x ∈ N . [ λz : z ∈ N . x – z ] ](3)(4) = [ λz : z ∈ N . 3 – z ](4) (12) Simplification of Lambda Expressions: Examples a. (i) [ λx : x ∈ N . [ λz : z ∈ N . x – z ] ](3)(4) = (by LC) (ii) [ λz : z ∈ N . 3 – z ](4) = (by LC) (iii) 3 – 4 = (iv) -1 b. (i) [ λx : x ∈ N . [ λy : y ∈ N . [ λz : z ∈ N . (x+y) – z ] ] ](1)(5)(6) = (ii) [ λy : y ∈ N . [ λz : z ∈ N . (1+y) – z ] ](5)(6) = (iii) [ λz : z ∈ N . (1+5) – z ](6) = (iv) (1+5) – 6 = (v) 6 – 6 = (vi) 0 2 To save space, I will write ‘N’ for the set { x : x is a whole number greater than 0 }Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 4 (13) Crucial Question How do we use lambda notation to represent a function like the following? f: De  Dt for all x ∈ De, f(x) = T iff x smokes Answer: It involves a new, special part of the lambda notation.3 (14) Lambda Notation, Part 2 4 a. Syntax: [ λx : x ∈ D . IF ϕ(x) THEN y, ELSE z ] b. Semantics: The function whose domain is D (i.e., which takes as argument anything in the set D), and for all x ∈ D, maps x to y if ϕ(x), and maps x to z otherwise (15) Example [ λx : x ∈ De . IF x smokes THEN T, ELSE F ] = a. The function whose domain is De, and for all x ∈ De, maps x to T iff x smokes b. [[ smokes ]] (16) Another Example [ λx : x ∈ De . IF x dances THEN T, ELSE F ] = a. The function whose domain is De, and for all x ∈ De, maps x to T iff x dances b. [[ dances ]] (17) Still Another Example [ λx : x ∈ Dt . IF x = F THEN T, ELSE F ] = a. The function whose domain is Dt, and for all x ∈ Dt, maps x to T iff x = F b. [[ it is not the case that ]] 3 The notation we will use for representing functions like [[smokes]] will (for a little while) differ from what you find in Heim & Kratzer (1998). We will eventually move to the system found in Heim & Kratzer (1998), and point out how it relates to the system we’re using here. 4 Again, this notation is not found in Heim & Kratzer (1998). A highly technical discussion of it can be found at the following: http://en.wikipedia.org/wiki/Lambda_calculus#Logic_and_predicatesSeth Cable Semantics and Generative …