THE LEXICON

PN = {Mary, John, Bill, Sarah}
CN = {carpenter, woman, mathematician}
Det = {a}
V_I  = {sleep, laugh, run}
V_T  = {kiss, see, call}
Adj = {lazy, wise, gifted}
P_loc = {near, above, on}
Cop = {be}

_S[NP VP]        _NP[PN]         _CNP[Det CN']          _CN'[(AdjP) CN]            _AdjP[Adj]

_PP[P NP]        _VP[V_I]         _VP[V_T NP]         _VP[Cop CNP/PP/AdjP]       _VP[VP PP]
 

LF

Delete constituents without meaning (i.e. the copula and the determiner), and delete the branches that don't dominate a constituent. Then the structures allowed at LF are the following:

Single-branching trees:
1)  _NP[PN]        2)  _CNP[CN']     3)  _CN'[CN]     4)  _AdjP[Adj]     5)  _VP[V_I]

6)  _VP[CNP]     7)  _VP[AdjP]     8)  _VP[PP]

Binary branching trees:
9)  _S[NP VP]     10)  _VP[V_T NP]     11) _CN'[AdjP CN]     12)  _PP[P NP]

13)  _VP[VP PP]

    D is a set of objects
    L is a set of locations
    f is a function whose domain is the lexicon and whose range is the set of
               possible denotations (i.e. f provides the denotation of lexical items)
    ||  || is a function whose domain is the set of expressions of the language, and
               whose range is the set of possible denotations.
               For any lexical item a, ||a|| = f(a)
    Loc is a function with domain the union of the set of NPs and the set of Vs,
          and range L, such that
             Loc(NP), l_NP , is the location of the denotation of the NP
                      e.g. Loc(_NP[Mary]), l_Mary , is the location of the individual Mary
             Loc(V_I/T), l_V, is the location of the situation described by the verb
                      e.g. Loc (_V[run]), l_run, is the location of the event of running

Semantic Types (only of categories visible at LF)

||S|| is a member of {1,0}
||PN|| is a member of D
||NP|| is a member of D
||CN|| is a subset of  D            (i.e. a member of the power set of D)
||CN'|| is a subset of D
||CNP|| is a subset of D
||Adj|| is a subset of D
||AdjP|| is a subset of D
||V_I|| is a subset of D
||V_T|| is a subset of  DXD      (i.e. a member of the power set of DXD)
||VP|| is a subset of D
||P|| is a subset of LXL          (i.e. a member of the power set of LXL)
||PP|| is a subset of L

Semantic Rules

1)  For any syntactic constituent a, b,
     ||_a[b]|| = ||b||  or  Loc(b)

(I.e. if a node in the tree dominates only one constituent, it inherits the denotation of its daughter, or the location of the denotation of its daughter. We'll return to this later, but assume at the moment that the choice is fully determined by the semantic types of the mother and the daughter, and the applicable semantic rules.)

2)  || _S[NP VP]|| = 1 iff ||NP|| is a member of ||VP||

3)  ||_VP[V_T NP]|| = {x|<x,||NP||>  is a member of ||V_T||}

4)  ||_CN'[AdjP CN]|| = the intersection of ||AdjP|| and ||CN||

5)  ||_PP[P NP]|| = {l|<l,l_NP>  is a member of ||P||}

6) ||_VPⁱ[VP^j PP]|| = VP^j and l_V is a member of ||PP||