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1. NEB Dissertation Grants
Abstract of Dissertation Project
APPLICAN'l"S NAME: Ochoa, Michael R.
ADDRESS: Department of Philosophy, 105 Newcomb Hall,
Tulane University, New Orleans, LA 70118
TELEPHONE NUMBERS: home (504) 865-1201
work (504) 865-5305
DEPARTMENT AND INSTITUTION: Dept. of Philosophy
Tulane University
DISSERTATION DIRECTOR: Graeme Forbes
REQUESTED GRANT PERIOD: June 1995 to May 1996
DESCRIPTIVE TITLE OF DISSERTATION: "A Philosophical
and Mathematical Theory of Truth"
ABSTRACT
The goal of this project is to provide an internal
defense of Mathematical Science against the post-modern
charge of "logo-centrism" by developing a philosophically and
mathematically satisfying theory of Truth. In the 1930's
Alfred Tarski showed that the principle obstacle to
developing such a theory was the occurrence of semantic
paradoxes (i.e. the Liar: "This sentence is false"). His
solution to this problem was to construct a hierarchy of
formal languages, each which contained a totally defined
truth predicate for the preceding language. Insofar as none
of these languages contained their own truth predicate, the
formulation of such sentences as the Liar was impossible.
The cost of this solution was that it resulted in an infinite
number of truth predicates, each indexed to the preceding
language of the hierarchy. More recent attempts, notably
Saul Kripke's fixed point approach and Jon Barwise and John
Etchemendy's employment of situation semantics and non-well-
founded set theory, avoid this problem, but imply a counter-
intuitive split between what is true and what can be said to
be true.
The current project is divided into four parts. The
first three contain rigorous analyses of the theories of
Tarski, Kripke, Barwise and Etchemendy in light of the
concerns raised by Post-Modernism. The goal of this analysis
is to separate the intuitions which motivate the mathematical
project from those which make it susceptible to the charge of
"logo-centrism". The fourth chapter will demonstrate that
the conflict arises out of an equivocation over the term
"universality". An account of Truth will be given which
extends the formal results and resolves this conflict.
To date, 'all of the formal research needed to carry out
the project has been completed. Drafts of the first three
chapters will be completed by the beginning of the grant
tenure. The fourth chapter will be written and the whole re-
written during the grant tenure.

2. - /
NEH Dissertation Grants
Resume
APPLICANT'S NAME: Ochoa, Michael R.
PRE­DOCTORAL EDUCATION: B.A. cum laude, Amherst
College, Department of
Philosophy, 5/1989;
M.A., Tulane University,
Department of Philosophy, 9/93
DOCTORAL PROGRAM
Department: Philosophy
Institution: Tulane University
Major field of Study: Logic
Minor Field of Study: History of Philosophy
Required Languages: N/A
Other Language Qualifications or Skill Relevant to the
Dissertation: Proficiency in the Major
Branches of Mathematical Logic
Name of Dissertation Director: Graeme Forbes
Fu II Title of Dissertation: A Philosophical and
Mathematical Theory
of Truth
Date Advanced to Candidacy: 11/94
Date Ph.D. is Expected: 5/96
INSTITUTIONS OTHER THAT THE DOCTORAL INSTITUTION
(LISTED ABOVE) WHERE THE APPLICANT IS CURRENTLY
TEACHING: N/A

3. OTHER INFORMATION
Relevant Employment History:
Tulane University: Elementary Symbolic Logic, Fall
1991­Spring 1994; Introduction to Philosophy, Fall
1994
Louisiana State University: Graduate Seminar on the
Philosophy of Language (Frege, Russell, Grice,
and Davidson), Spring 1994
Johns Hopkins University: Introduction to Logic at the
Center for Talented Youth, Summer 1994
University of New Orleans: Symbolic Logic, Spring 1995
Grants, Fellowships, or Honors Received:
Tulane University: Graduate Fellowship, 9/90­5/92;
Full Tuition Scholarship, 9/90­5/95; Teaching
Assistantship, 9/91­5/95; Graduate Student
Representative to the Graduate Student Association
9/90­5/91; Co­Founder/Chair Philosophy Graduate
Student Colloquium 9/90­5/91; Graduate Student
Representative to the Faculty of the Department of
Philosophy, 9/92­5/93
Amherst College: Amherst College Scholarship, 9/85-
5/89: National Merit Scholarship, 9/85­5/89
Scholarly Papers Presented:
Tulane Department of Philosophy Graduate Students'
Colloquium: "Wittgenstein on Rule Following", 11/92;
"Frege and Russell on Reference", 3/93; "Necessity and History
in Hegel's phenomenology of Geist", 11/94

4. Graduate Courses Taken at Tulane University
Course JJllil
Mathematical Logic
Philosophy of Language
The Metaphysics of Identity
Contemporary Issues in
Mathematical Logic
Non­Well­Founded­Set­Theory
A. N. Whitehead's Process and
Reality
Contemporary Epistemology
Ethical Realism
Topics in the Philosophy of
Mind: Perception
Anti­Consequentialism
Plato's Statesman
Aristotle's Nichomachean
Ethics
Aristotle's Qn the Soul
Aristotle's po Ijtics
Aristotle's Metaphysics
Hegel's phenomenology of Geist
Locke
Heidegger's Being and Time
Kierkegaard
Environmental Ethics
L2a1a
Fall 90
Spring 92
Spring 91
Fall 91
Spring 93
Spring 91
Fall 90
Fall 91
Spring 91
Spring 93
Fall 90
Spring 92
Spring 93
Fall 92
Fall 94
Spring 92
Fall 92
Fall 91
Spring 91
Fall 90
Jnstructor Grade
Graeme Forbes A
Graeme Forbes A
Graeme Forbes A
Graeme Forbes A­
Michael Mislove audit
Andrew Reck A -
Bruce Brower A
Bruce Brower I
Radu Bogdan A-
Eric Mack A -
Ronna Burger A -
Ronna Burger A -
Ronna Burger A-
Ronna Burger audit
Ronna Burger audit
Robert Berman A-
Bruce Thomas A
Frank Schalow audit
John Glenn 8+
Michael 8+
Zimmerman

5. Project Description
That Mathematics is the most precise of inquiries cannot be
doubted. Mathematics also provides our paradigm of certainty.
Mathematics does not provide knowledge of individual entities or of
particular sorts of entities, but of all things to the extent that they
conform to universal principles. Nor does Mathematics distinguish
actual entities from theoretical or fictional entities. Thus,
Mathematics stands removed from knowledge of Reality, the realm
of actual individuals of various kinds.
To establish Mathematical Science as an epistemic project, as
j
opposed to a merely technical project, the precision and certainty of
Mathematics must be wed to inquiries concerning actual individuals
of various sorts. The first step in this program is to provide a
bridge between the ephemeral realm of Mathematics and Reality.
This is the subject of the present inquiry and will take the form of a
mathematical definition of Truth. Subsequently, it must be
demonstrated what kinds of things there are and what particulars
there are of these kinds.
In the last century there have been a number of attempts to
express in the form of a mathematical definition the ancient idea
(dating at least to Aristotle's Metaphysjcs book Gamma) that Truth
is correspondence to Reality. To be more precise, correspondence is
a definable two-place relation which holds between Reality and the
possible bearers of Truth (whether these are sentences, thoughts,
theories, or something else need not yet concern us). That Truth is
mathematically definable implies that the precision and certainty of

6. Mathematics can be extended to Science. That the terms of the
relation are distinct implies that Idealism is false. That the
conditions of a sentence, thought or theory being true are
independent of what sentence, thought or theory is being considered
implies that Relativism is false.
Though appealing to mathematicians and scientists, the
Correspondence Theory of Truth has been one of the principle targets
of the post-modern critique of Rationality. Such diverse projects as
Pragmatism, Deconstructionism, and even various forms of
Feminism and Environmental Ethics, find a common enemy in the
"Iogocentric" project which motivates the attempt to define Truth
as correspondence. Some even cite it as one of the primary sources
of contemporary social ills. If the attempt to "ground" Mathematics
to Reality fails, then Science will lose its claim to having unique
privileged access· to Reality as a whole. Science will be confined to
the project of technical mastery or become one of many competing
fictions about Reality. Mathematics will be confined to an
ephemeral realm unconnected to Reality. On the other hand, were the
attempt an unquali'fied success, it would establish a "totalizing
discourse" which would make impossible any other sorts of inquiry.
The tyranny of Mathematics would undermine the claims of Justice
and Beauty.
This project will provide an internal defense of Mathematical
Science aga.inst the criticisms of Post-Modernism. In examining the
three major attempts to give a mathematically precise formulation
of the Correspondence Theory we shall make explicit the set of
intuitions which motivate the naive version of the project sketched
-

7. above. These intuitions concern the concepts of objectivity,
universality, truth, uniqueness, primacy, and precision. We shall see
that the premises which make the naive model vulnerable to the
post-modern critique can be separated from those which are
necessary to preserve the epistemic claims of Mathematical
Science. In particular, it will be demonstrated that "universality" is
equivocal and Science need not maintain it in its strongest sense.
The result of this analysis will be a new, dynamic understanding of
the relationship between Theory and Reality. The project is divided
into four chapters as follows.
In the first chapter it will be shown how Model Theory was
developed in the 1930's by Alfred Tarski for the specific purpose of
mathematically defining Truth as correspondence. From this
standpoint Tarski described two ordered structures, a language and
that universe which the language was about. He then tried to reduce
the notion of Truth to a relationship of structural similarity
between that language and the universe. Tarski showed that the
attempt to do so from within the language being considered raises
the problem of semantic paradoxes (e.g. the ancient Liar paradox:
"This sentence is false"). His solution to this problem was to
construct a second language, a meta-language, from which to define
the semantic relationship between the universe and the first,
"object", language. This procedure led to a hierarchy of formal
languages, each which contained a totally defined truth predicate for
the preceding language. Insofar as none of these languages contained
their own truth predicate, the Liar could be formulated in none of
them. However, the cost of this solution was that it resulted in not

8. one, but an infinite number of truth predicates, each indexed to the
preceding language in the hierarchy.
In the second chapter we will examine the fixed-point
approach to the Liar problem. This basis for an alternative to
Tarski's hierarchy was investigated in Martin & Woodruff's "'True in
L' in L" (1974). However, it was not until Saul Kripke's "Outline of a
Theory of Truth" (1975) that a complete mathematical theory was
proposed which succeeded in defining a single truth predicate for a
given language from within that language. Formally, Kripke defined
the set of true (false) sentences of a given language to be one of two
ordered members of the (least) fixed point of a monotonic evaluation
scheme which allows for truth-value "gaps" (e.g. Kleene's strong
three-valued logic or van Fraassen's supervaluation schema). This
procedure allowed Kripke to give a purely structural
characterization of problematic "ungrounded" sentences such as the
Liar and the Truth-Teller (Le. "This sentence is True"). Intuitively,
some method (e.g. empirical evaluation) classifies as either true or
false all sentences which do not contain semantic terms (e.g. "true"
and "false"). The iteration of some evaluation procedure then
determines the truth-values of increasingly complex sentences
which contain semantic terms (e.g. "It is true that it is snowing"
and "It is false that it is true that it is snowing"). From the fixed
point on, iterations of the evaluation procedure classify no new
sentences, but "ungrounded" sentences such as the Liar and the
Truth-Teller remain unclassified. The two principle problems with
this approach are, 'first, that it may be susceptible to strengthened
versions of the Liar (e.g. "This sentence is either false or

9. ungrounded"). Second, there seem to remain sentences of the object
language which are true, but which remain unclassified (e.g. "The
Liar sentence is not true").
In the third chapter we shall examine the approach to Truth
and the Liar developed by Jon Barwise and John Etchemendy in I.h.e
Liar: An Essay on Truth and Circularity (1987). There they provide
an account of the Liar which employs two technical innovations. The
Situation Semantics developed by Barwise in The Situation in Logic
enable them to distinguish sentences, statements, and propositions,
thereby offering a more "fine-grained" account than the previous
theories which dealt only with sentences. Abandoning standard
Zermelo-Fraenkel Set Theory for Peter Aczel's Non-Well-Founded
Set Theory, they are also able to model circular phenomenon such as
Kripke's "ungrounded" sentences, rather than simply excluding them
from the evaluation procedure. In addition to these technical
innovations they change the focus of the discussion by abandoning
the attempt to define truth in favor of diagnosing the difficulty of
which the Liar is symptomatic. Ultimately, their project leads to
problems similar Kripke's in that it implies a problematic rift
between what is true and what can be said to be true.
In the fourth chapter it will be shown that the seeming failure
of these theories satisfactorily to resolve the Liar paradox does not
stand in the way of the larger project. In particular it will be shown
that there are several senses in which the definition of Truth may be
universal: 1) It may specify the relationship between Natural
Language and Reality (the actual universe); 2) For each individual
sentence, statement or thought of a given language it may provide
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10. - - - - - - - - - - - - - -
the necessary and sufficient conditions for its being true; 3) It may
provide the necessary and sufficient conditions for the truth of all
sentences, statements or thoughts of a given language. The final
definition which we settle on will be universal in the second sense,
and it will be shown that this is sufficient for the claims of
Mathematical Science. The criticisms of Post-Modernism will be
deflected to interpretations of the program which are commited to
the third sense.
To date all of the formal aspects of the project have been
researched. This partially entailed taking a course on the principle
theorems of Kurt Gadel and Alfred Tarski as well as a course on the
Paris-Harrington independence results in First-Order Arithmetic
with Graeme Forbes in the Department of Philosophy of Tulane
University. Further results were studied in a course on Peter
Aczel's Non-Well-Founded Set Theory taught by Michael Mislove in
the Department of Mathematics at Tulane University. Drafts of the
first three chapters will be completed by the beginning of the grant
tenure. The grant will allow me time to formulate the original
results of the fourth chapter and put the whole into its final form.

11. Selected Bibliography
Aczel, Peter, Non-Well-Founded Sets, Center for the Study of
Language and Information Lecture Notes No. 14; 1988.
Barwise, Jon, Handbook of Mathematical Logic; North-Holland
Publishing Company, 1977.
The Situation in Logic, Center for the Study of Language and
Information Lecture Notes No. 17; 1989.
Barwise, Jon and Etchemendy, John, The Liar: An Essay OIJ Truth and
Circularity; Oxford University Press, 1987.
Boolos, George and Jeffery, Richard, Computabjlity and Logic, third
edition; Cambridge University Press, 1980.
Forbes, Graeme, "Aspects of the Liar Paradox" (unpublished).
Mathematical Logic and the Limits of Computation
(unpublished) .
Independence Results in First-Order Logic (unpublished).
Gupta, Anil, The Revision Theory of Truth, The MIT Press; 1993.
Martin, Robert, The ParadQx Qf the Liar; Yale University Press, 1970.
Recent Essays Qn Truth and the Liar Paradox; Oxford
University Press, 1984. Includes: Kripke, Saul, "Outline of a
Theory Qf Truth", Martin, Robert and Woodruff, Peter,
"'True in L' in L".
Tarski, Alfred, Logic, Semantics and Metamathematics: Papers from
1923-1938; ClarendQn Press, 1956. Includes: "The Concept of
Truth in FQrmal Languages", "SQme ObservatiQns Qn the
Concepts of Q-Consistency and Q-Completeness", "The
Establishment of Scientific Semantics", "On the Concept of
Logical Consequence".
Yablo, Steven, "Truth and Reflection", Journal of Philosophical Logic
13: 213-232; 1984.