Denotation, truth, meaning

Constants, variables, negation as before
Quantifiers: ,
Forms
n-place predicates w/ proper number of arguments
connectors and forms
negation of a form
quantified forms
equality of constants, variables
2

Two main components (as before):
Universe of discourse: U
Extension (valuation) of U: V
An assignment function g( ) for variables
Vocabulary, syntax: predicate calculus
Denotation: as before, except a denotation
involving g( )
The denotation of a variable is its interpretation
with respect to g( )
3

The valuation!
For constants: the extension (i.e.
individual)
For predicates: the extension (i.e. set)
For variables: the value assigned by g()
4

Predicates and constants:
[[A]] M, g = V(A)
Variables:
[[A]] M, g = g(A)
As usual: [[A&B]], [[AVB]], [[AB]],
etc.
[[R(x1,x2,x3...xn)]] M,g = 1 iff
< [[x1]] M,g ,[[x2]] M,g ... [[xn]] M,g > [[R]] M, g
5

Quantified expressions:
[[xA]] = 1 iff for all uU, [[A]] M,g[u/x] = 1
[[xA]] = 1 iff for some uU, [[A]] M,g[u/x] =
1
6

Universal: need to show T for all
relevant elements
Existential: need to show T for one
When LF is involved: T means any one
LF alternative is T
7

Logical Form (LF): syntax/semantics
interface
Derived from Surface Structure (SS)
WH- movement
From DS to SS
(FYI: also happens from SS to LF; not this
semester, though…)
Quantifiers: as before
Quantifier Raising
Quantified NP’s at SS raise, adjoin to S node in LF
8

Like wh- movement in overt syntax
Is an instance of Move-
Quantified NP’s at SS raise, adjoin to S
node in LF (QR)
Extension of overt syntax
Ambiguity results whenever multiple
orderings arise among QR possibilities
9

Logical Form (LF): syntax/semantics
interface
Derived from (i.e. after) Surface Structure (SS)
Quantifiers: allow counting, subsets
Propositional representations don’t allow this
Predicate representations do
10