Type Theory
Kareem Khalifa
Philosophy Department
Middlebury College
Overview
I. The Basics
II. Functions
III. Composition &
functional application
IV. Types of functions
I. The Basics
A. Compositionality
B. Syntax & semantics
C. Two ways to represent
semantic rules
II. Functions
A. Two kinds of
denotation
B. Functions
C. Saturation
D. General structure of functions
Type C =
F(argument)
Type <B, C>
Type B
Function F
F’s argument
E. Example
t
This refers to
BLUE(Cookie
Monster)
<e, t>
e
BLUE
Cookie Monster
A. Binary
B. BIGGER
III. Semantic composition = functional
application
t
e
Cookie Monster
BIGGER (Cookie Monster, Elmo)
<e, t>
BIGGER (x, Elmo)
<e, <e, t>>
e
is bigger
than
Elmo
BIGGER (x, y)
IV. Types of functions
A. How to determine
types
B. The List
General structure of &-statements
t
= P&Q
t
P
<t, t>
= ___&Q
<t, <t, t>>
t
&
Q
= ___&___
NOTE: This means that the topmost nodes of any statement of the form “P&Q” is going
to have this structure. I think this will also apply for all other truth-functional
connectives (save negation ~, which I look at in the next few slides.)
Example
t
= BRUNETTE(Emma) &
BRUNETTE (Elisabeth)
t
e
Emma
<e, t>
BRUNETTE
= ___& BRUNETTE
(Elisabeth)
<t, t>
t
<t, <t, t>>
&
e
<e, t>
Elisabeth
BRUNETTE
= ___&___
General structure of ~-statements
t
= ~P
<t, t>
t
~
P
= ~___
NOTE: ALL truth-functional connectives have as their second highest nodes t and <t,t>.
This is, in effect, what makes them truth-functional AND binary.