Title

Beyond First Order Analysis

First order predicate logic with quantification is first order because all the terms are individual terms.

Type theory to track generalized semantic types.

Examples which suggest that there should be predicates and relations which involve argument terms which are themselves predicates.

Predications on apparent predicates of individuals.

Predicates and formulas as relata

Adjectives

Not enough quantifiers.

First order type theory

Let e be the type of Individuals (Entities)
t be the type of truth-values.

For any two types s and t , then á s ,t ñ is the type of functions from things of type s to things of type t .

The first order logic relates its syntactic categories to semantic types as follows:

Individual Terms	e
Formulas	t
1-place predicates of individuals	á e, t ñ
2-place relations between individuals	á e,á e, t ñ ñ etc.

We can generalize the notion of predicates and relations as follows:

1-place predicates of things of type s	á s , t ñ
2-place relations between things of type s and things of type t .	á s ,á t , t ñ ñ etc.

Predicates (of Individuals) as Argument Terms

Margo is swimming.

Margo is healthy.

Swimming is healthy.

Swimming (Margo)

Healthy (Margo)

Healthy (Swimming)

Everything which is painful is good for you
Jogging is painful
Jogging is good for you.

Consider consider:

I consider Margo healthy.
I consider Margo happy.
She regards me as happy.
She strikes me as happy.

Formulas as arguments

Verbs of Propositional Attitude

I believe that it will rain tonight.
I doubt that it will snow tonight.
I could be wrong.
It is possible that I could be wrong.
It is likely that it will rain tomorrow.
It's unlikely that it will snow before October.

Adjectives

That is a red book
That is a book
That is red..

red(x) & book(x)
book(x)
red(x)

A male linguist walked in.
A linguist walked in.
A male (person) walked in.

That is a big book
That is a book
That is big.

That is a very small elephant
That is an elephant.
That is very small.

That is a big flea.
That is a flea.
That is big.

Adjectives which are absolute can be treated as one-place predicates of individuals, but relative adjectives create predicates which are relative to the set denoted by the modified noun. They are treated as restrictive functions from predicates to predicates and would have functions of type á á e, t ñ ,á e, t ñ ñ as semantic values.

Such a function f is restrictive iff for any set P to which it applies, V(f( P)) Í V(P).

Consider also:

An alleged murder walked in.
A murderer walked in.
An alleged person walked in.

That is a fake flea.
That is a flea.
That is fake.

Not Enough Quantifiers

Messy, non isomorphic renderings of other quantifiers:

Consider how to formalize

At least two ducks quacked.

Ex (duck(x) & Ey ( (duck(y) & x ‚ y) & (quack(x) & quack(y)) ))

Impossible cases:

Most (of the) ducks quacked.
More than half of the ducks quacked.

?x ( duck'(x) ???? quack'(x) )

There is no defineable operator to put in for the initial ? which will capture this meaning by filling in truth functions to relate the restriction to the scope part of the translation.

Moral: Quantifier determiners are relations between sets of individuals.