Here's the idea: since people can have different attitudes towards
different propositions that happen to have the same truth value, we
-can't have sentences denoting simple truth values. Then if John
+can't have sentences denoting simple truth values. If we did, then if John
believed that the earth was round, it would force him to believe
Fermat's last theorem holds, since both propositions are equally true.
The traditional solution is to allow sentences to denote a function
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This system is modeled on the way Montague arranged his grammar.
-(There are significant simplifications: for instance, determiner
+There are significant simplifications: for instance, determiner
phrases are thought of as corresponding to individuals rather than to
-generalized quantifiers.) If you're curious about the initial `s`'s
+generalized quantifiers. If you're curious about the initial `s`'s
in the extensional types, they're there because the behavior of these
expressions depends on which world they're evaluated at. If you are
in a situation in which you can hold the evaluation world constant,
-you can further simplify the extensional types. (Usually, the
+you can further simplify the extensional types. Usually, the
dependence of the extension of an expression on the evaluation world
is hidden in a superscript, or built into the lexical interpretation
-function.)
+function.
The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
those attitude verbs that need to worry about intensions, and keep the
rest of the grammar as extensional as possible? This desire is
parallel to our earlier desire to limit the concern about division by
-zero to the division function, and let the other functions ignore
-division-by-zero problems as much as possible.
+zero to the division function, and let the other functions, like
+addition or multiplication, ignore division-by-zero problems as much
+as possible.
So here's what we do: