Reading Field 2008: §15.1

The first two parts of Hartry Field’s new book on truth and paradox are very much expository, although well-written and insightful.
Now, turning to his own proposal, he concludes that the most satisfactory approach so far would have been Lukasiewicz’s continuum valued semantics.
It would allow for ‘[...] the naive theory of truth [..] in sentential logic based on continuum-valued semantics [...]‘ (2008:231).
In other words, it would allow for a truth predicate satisfying the T equivalences and the Intersubstitutivity Principle
However, continuum valued semantics is restricted to sentential logic and does not apply to a full first order language.
Therefore, Field now attempts its generalisation to quantificational languages.
His approach is algebraic and his starting point a space where V is a set of semantic values and partially ordered by \leq.
As a first approximation the model aimed for is specified as deMorgan lattice such that it has the following properties:
# with a |^| b = inf{a,b} and a |_| b = sup{a,b}:
v(A & B) = v(A) |^| v(B) and v(A v B) = v(A) |_| v(B)=sup{v(A), v(B)}.
# greatest lower bound – operation and least upper bound operation are distributive
# There is an element 1 such that for any x \in V x \leq 1 and an element 0 sucht that for any x, 0 \leq x.
# 1 is join-irreducible
# An operator `*’ is defined as orthocomplementation
The quantified sentences can also be interpreted in such semantics, universal quantification as the supremum of all instances and existential quantification

For the interpretation of conditionals he states, for the time being, four necessary conditions:
(I) v(A -> B)=1 iff A \leq B
(IIa) if B \leq C then [v(A -> B) \leq v(A->C)]
(IIb) if A \leq B then [v(A->C) \leq v(B -> C)]
(III) if v(A)=1 & v(B) = 0 then [v(A -> B) =0]

With these requirements, the semantics would already exceed Kleene’s, because the latter’s conditional would not meet (I).
I suppose Field here thinks of the following.
The Kleene value space is {0,u,1}, and his strong table for his conditional (`->k’) is best be represented such that v(A ->k B)=1 iff sup{v(A)*, v(B)}=1.
Assume that v(A)=0 and v(B)=u.
Then, v(A) \leq v(B). However, v(A ->k B) = sup{1,u} = u 0.

For V= {0,1} the definitions above and these necessary conditions coincide with classical semantics.

So far, however, the outline is much too inclusive.
The description applies to a multitude of semantics many of which do not suffice yet for a language with truth predicate.
For instance, it still applies to bivalent classical semantics which would model an inconsistent truth theory.
Continuum valued semantics, too, must be excluded, although I do not immediately see why. Is it because of what Field gives in §4.5?

The further restriction required is conservativeness, which Field here gives as follows.
A model M+ fulfils the conservativeness requirement iff
# M V-valued model
# there is a classical \omega-model of the base language
# M is a reduct of M+
# in M+, the value of T is identical to that of A itself. M+ satisfies what Field has introduced as the model-theoretic intersubstitutivity condition.
# Whichever objects are assigned to the variables, if any variable v the sentence ^Sent(v)^ has the value 0 then v(^True(v)^)=0, too. In other words, no object which is not a sentence is in the extension of the truth predicate.

Thus, the semantics aimed for must not only have all the algebraic properties from above but also have a classical reduct, validate intersubstitutivity and restrict the truth predicate to sentences.
This is the result of section 15.1 and this is what the reader expects from the following sections and chapters.