**Hartry****
Field.**** ***Saving Truth From Paradox. *Oxford: Oxford
University Press 2008. Pp. 406. (paperback ISBN-13: 978-0-19-923074-7).

**Anita Burdman Feferman & Solomon Feferman.** *Alfred Tarski*. Life and Logic.
Cambridge: Cambridge University Press 2008. Pp. 425. (paperback ISBN-13: 978-0-521-71401-3).

Alfred Tarksi considered himself to be ‘the
greatest living sane logician’, expressing simultaneously ‘supreme confidence in
his talent’ (1) as well as a criticism of life-long challenge Gödel (who
developed strange habits like wearing a mask). Tarski
is mostly known by philosophers for his work in semantics (justifying the study
of truth in a formal fashion) and *Tarski’s**
Theorem* (that languages with basic means of self-reference cannot contain
their own truth predicate). His work, besides these classics, comprehended wide
areas of logic. With others he invented meta-mathematics as the study of
properties of formal systems themselves (e.g. decidability, soundness,
independence of axioms…). He invented decision procedures, number theories,
types of algebras – and a lot more. After the Second World War he made the
University of California into the world’s centre for logic. Famous logicians
(like Dana Scott or Richard Montague) studied logic with Tarski.

Tarki’s biography by Anita and
Solomon Feferman is now available in a paperback
edition. They write from the perspective of former Tarski
students. Students knew Tarski from late night
session of carefully re-working single phrases of publications, Tarski, himself staying awake on coffee and amphetamines,
urging them on. The Fefermans euphemistically
describe a person others may consider an egocentric megalomaniac. Some of the
behaviors they describe as ‘a life-long need for women’ (cf. pp. 158, 178, 196,
200…) nowadays would be filed under ‘sexual harassment’ (of students). What is
more interesting about *Alfred Tarski* is less
the admiration for the person of the great logician one may share or not, but
the insightful view into the early days of analytic semantic theory (before the
Second World War) and Tarski’s empire building in
logic (after the Second World War). Even the story of a logic genius shows
itself to depend on many chance events. Most dramatically Tarski
only left on the eve of the Second World War to tour the USA. Had he stayed he,
converted to Catholicism, but being of Jewish descent (originally named ‘Teitelbaum’), most certainly would have been killed like
many other Polish logicians, famous nowadays for single theorems (like Lindenbaum or Presburger) as they
were murdered by the German occupants. For the whole war he had to fear for his
family, his wife and children surviving, other family members and colleagues
being killed.

The Fefermans
not only picture the biography of Tarksi, but also
set out, in six ‘Interludes’ beside the biographic narration, some of Tarki’s major achievements and areas of work. Thus students
and readers interested in the history of analytic philosophy and logic, and
being vaguely familiar with the areas Tarki’s name is
associated with certainly benefit from *Alfred Tarki:
Life and Logic*.

Tarski’s treatment of the
notion of truth and its paradoxes superseded the syntax centrism and hostility
to semantic concepts that prevailed in the Vienna Circle up to Tarski’s “The Concept of Truth in Formalized Languages” in
1935, including Carnap’s just published *Logical
Syntax of Language* (1933). Carnap devoted himself
to semantics, and Tarski’s work became classical. Tarski aimed at formalized languages only, as he took
natural languages to be universal (i.e. including their own semantics) and thus
inconsistent. His approach works by distinguishing the definition of truth in a
meta-language *L ^{+}* from the object-language

Hartry Field has devoted much
of his work in the last years to the study of the antinomies of truth and
property theory. *Saving Truth from Paradox *provides both an overview on
ways of dealing with the paradoxes of truth as well as an introduction to
Field’s own approach to save truth from paradox.

Field’s approach is a version of a
gap-approach (i.e. he denies *tertium** non datur *[TND] for the problematic sentences like the
Liar). Field uses several building blocks from other theories. Therefore the
first part of the book introduces *inter alia* Kripke’s fixed point construction for a theory of truth and
Lukasiewicz’s continuum valued logic. Field shares
some of the criticism of Tarski-style stratified
truth theories. He works, like Kripke’s construction,
with iteration instead of stratification: Starting with a ground-level of
sentences not involving ‘true’ more and more sentences (i.e. now sentences
involving ‘true’, speaking of other sentences involving ‘true’…) are assigned
to the positive extension of ‘true sentence’. As there are only countable many
sentences, somewhere (i.e. somewhere in the non-finite ordinals) the
construction has to settle into a fixed point, delivering the ultimate
extension of ‘true sentence’. Kripke’s own
three-valued construction contains no conditional and ultimately has to fall
back to stratification. Field therefore uses a three-valued or a continuum-valued
logic in the fashion of Lukasiewicz. To avoid some
pitfalls of Lukasiewicz’s construction Field
introduces a special conditional beside material implication. The conditional
is true at a stage if there is an ordinal (in the preceding iteration process)
starting from which the antecedent always has a lower semantic value than the
consequent; false is starting from some ordinal it always has a higher semantic
value; neither true nor false otherwise. This conditional has to be used where
TND fails (i.e. in the critical semantic sentences); where TND holds it is
identical to material implication. The logic of this conditional is, of course,
weaker than standard propositional reasoning. Field overall theory makes heavy
use of limit ordinal constructions. Field finally is able to derive his central
result: His construction can conservatively extend a model of the semantics
free ground language by evaluating all the truths evaluations, *and* do
this by having *both* the Truth Scheme [True(A)«A] and intersubstitutivity of ‘True(A)’ and ‘A’. ‘It is only
insofar as the unsubscripted predicate “True”
transcends the Tarskian hierarchy that it is nonclassical.’ (275)

The justification of Field’s
approach depends crucially on a comparison to other approaches to the
paradoxes. Field thus compares his approach to ‘classical solutions’ (one part
of the book) and paraconsistent solutions (the final part of the book).

Classical solutions keep classical
logic, and so have to give up at least one direction of the Truth Scheme.
Either way they have to endorse bizarre claims: Giving up the left-to-right
direction means having theorems saying that some sentence is true *without*
having that sentence itself *or* even having its negation! Giving up the
right-to-left direction means having some sentence as theorem *without*
being able to say that the sentence is true *or* even saying that it is
not true! *Saving Truth from Paradox* works meticulously through many
filiations of such theories and provides a veritable field guide in that area.
Such theories seem worse than giving up TND for some sentences.

Paraconsistent solutions keep the
Truth Scheme, but change the underlying logic, just like Field’s solution. In
distinction to Field’s ‘paracomplete solution’ which
has some sentences being neither true nor false, a
paraconsistent solution, at least dialetheism, may have some sentences being
both true and false. Field tries to argue that paraconsistent solutions face
worries worse than paracomplete solutions (*inter alia* problems of expressing determinate truth or
falsity, extending the ubiquity of true contradictions to simple arithmetic…).
It is not at all clear that these criticisms apply to paraconsistent solutions
in general, as Field focuses more or the less exclusively on Graham Priest’s
dialetheism and Priest’s criticism of Field. There are several paradigms of
paraconsistent logics (e.g. adaptive logics, which have interesting
conditionals), which may be better positioned to answer Field’s challenges and
have a better net balance of virtues and vices than Field’s solution. Whereas
the part dealing with the classical solutions in itself recommends *Saving
Truth from Paradox* the comparison with paraconsistent solutions is far from
settled. Sometimes intuitions clash: Dialetheism denies intersubstitutivity
of ‘True(A)’ and ‘A’ in the scope of negation, which
Field challenges as counterintuitive, whereas Field subscribes to *verum** ex quodlibet
sequitur *[e.g. A®(B®B)], counterintuitive to Relevant Logics
(one of the areas of paraconsistent logic).

Field himself considers some of
the typical challenges to gap-theories. Beside his theory of truth he considers
determinacy operators at length, constructing an additional theory of being
determinately true (once again involving fixed points somewhere beyond some
limit ordinal, where on pains of regaining the paradoxes the determinacy
iteration must not collapse). Field believes this theory to be immune to
revenge and almost free of counterintuitive drawbacks.

Notwithstanding the technical
sophistication of his overall treatment of matters, this positive
self-assessment needs further elaboration. For instance: Field makes short work
of the problem that one might introduce exclusion negation again by a postulate
'ØA is true iff A is not
true’. If that worked one would have a negation with TND and thus regain
paradox. Many gap-approaches have the problem that their meta-theory allows –
on pains of losing the power to express some semantic fact – the
re-introduction of exclusion negation, and thus of Strengthened Liars. Field
rejects such a postulate as it works ‘only if we assume Boolean laws for the
“not” used in making the stipulation’ (310). Nicely put, but unconvincing.
Compare: You have three collections of items and operations of moving one item
from one to the other; now the three collections are placed/distributed over a
border; there are three ways to do this, in all cases one collection is
opposite to the others (making now the across the border region); there is an
operation of moving an item from one of the opposing two collection to this
collection. This is perfectly structurally isomorphic to having three
collections of sentences, divided in the true sentences, the false sentences
and the gappy sentences. As one can introduce a
border with collections of marbles (the green vs. the non-green) it is possible
to have a border between the only true sentences and the other two collections.
The operation across the border is exclusive negation. Thus either a
Strengthened Liar is re-introduced (bad for Field) or although the semantics is
isomorphic to the marble model the semantic fact of a (possible) border cannot
be expressed: expressive limitations (also bad for Field). *Except*, the
structural analogy between marbles and sentences gives in – but this needs some
heavy duty metaphysical work, not yet delivered by Field.

Further on, Field proves a lot of
theorems about fixed points and limit ordinals, i.e. levels of iteration that
we finite beings certainly do not ‘reason up to in stepwise fashion’. We can,
of course, prove theorems about these infinite ordinals. What about the
reasoning about these limit stages and fixed points – where does it take place?
Field often distinguishes truth from truth in a model and validity (for some
semantics in some model) from genuine validity, for which then proving
soundness seemed bared by *Gödel’s Second Incompleteness Theorem* (cf.
45-49). If such reasoning is not feasible according to Field’s theory we
express something inexpressible, thus mystery. If answering these concerns one
supposes to talk at the ultimate fixed point stage it doesn’t sound that way:
What about the usage of ‘true’ at this stage? It seems we are at a level very like
the first semantic level above the ground language, but there is no where to
iterate to anymore to avoid paradox (as we have, by assumption, exhausted all
countable ordinals)! If this is a classical meta-language, we are back to Tarskian stratification and nothing is gained! Field answer
to that challenge (which could be put as asking for the truth theory for the
set theory ZF used in the model theory) that ‘we have an adequate truth theory
for ZF_{true} within ZF_{true}’
(356). He explicitly promises in the introduction ‘that there are languages
that are sufficiently powerful to serve as their own meta-languages’ (18). But
his construction contains its theory only in that sense of ‘theory’ that the
set of theorems containing ‘true’ is included within it. The meta-theorems he
proves are of another kind. They speak about the whole hierarchy. Field has to
be more explicit about the status of his meta-theory and its resources. This is
especially pressing in his treatment of determinate truth. He sees the problem
that with the determinacy operators (i.e. the operators ‘it is determinately
true that…’ for any amount of iteration) we do not have the same construction
as with ‘true’, where we have only iteration. We have the idea of a ‘super-determincacy’ operator claiming something to be true *tout
court*. Finally Field seems to yield to expressive limitations: ‘the claim
that I dispute is that the model theory *ought* to allow for
super-determinateness operator meeting intuitive preconceptions’ (357). The
meta-theorems he puts forward, I gather, are meant to be super-determinately
true, but he denies that they can be so.

*Saving Truth from Paradox* is a challenging book.
The reader has to have advanced background knowledge and understanding in
meta-logic and semantics. The treatment is at times Byzantine but most times
exiting.

**Manuel Bremer**

Philosophisches Institut, Universität Düsseldorf