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Now in a new edition --the classic presentation of the theory of
computable functions in the context of the foundations of
mathematics. Part I motivates the study of computability with
discussions and readings about the crisis in the foundations of
mathematics in the early 20th century, while presenting the basic
ideas of whole number, function, proof, and real number. Part II
starts with readings from Turing and Post leading to the formal
theory of recursive functions. Part III presents sufficient formal
logic to give a full development of G del's incompleteness
theorems. Part IV considers the significance of the technical work
with a discussion of Church's Thesis and readings on the
foundations of mathematics. This new edition contains the timeline
"Computability and Undecidability" as well as the essay "On
mathematics."
Propositional Logics presents the history, philosophy, and
mathematics of the major systems of propositional logic. Classical
logic, modal logics, many-valued logics, intuitionism,
paraconsistent logics, and dependent implication are examined in
separate chapters. Each begins with a motivation in the
originators' own terms, followed by the standard formal semantics,
syntax, and completeness theorem. The chapters on the various
logics are largely self-contained so that the book can be used as a
reference. An appendix summarizes the formal semantics and
axiomatizations of the logics. The view that unifies the exposition
is that propositional logics comprise a spectrum. As the aspect of
propositions under consideration varies, the logic varies. Each
logic is shown to fall naturally within a general framework for
semantics. A theory of translations between logics is presented
that allows for further comparisons, and necessary conditions are
given for a translation to preserve meaning. For this third edition
the material has been re-organized to make the text easier to
study, and a new section on paraconsistent logics with simple
semantics has been added which challenges standard views on the
nature of consequence relations. The text includes worked examples
and hundreds of exercises, from routine to open problems, making
the book with its clear and careful exposition ideal for courses or
individual study.
This series of books presents the fundamentals of logic in a style
accessible to both students and scholars. The text of each essay
presents a story, the main line of development of the ideas, while
the notes and appendices place the research within a larger
scholarly context. The essays overlap, forming a unified analysis
of logic as the art of reasoning well, yet each essay is designed
so that it may be read independently. The topic of this volume is
prescriptive reasoning. Descriptive claims say how the world is,
was, or will be; prescriptive claims say how the world should be.
We have fairly clear rules for reasoning with descriptive claims.
The goal of the first essay, "Reasoning with Prescriptive Claims,"
is to clarify how to reason with prescriptive ones. The first step
in doing so is to justify our viewing prescriptions as true or
false. That justification is part of a general approach to
reasoning in which many kinds of personal evaluations are taken to
be true-false divisions. That view has been implicit if not
explicit in analyses of reasoning from formal logic through
argument analysis. in "Truth and Reasoning" I set out reasons for
adopting that methodology. Theories, too, seem to be descriptive or
prescriptive. Some say how the world is, others how the world
should be. Yet, as shown in "Prescriptive Theories?," on close
examination the distinction evaporates. Unless, that is, one says
that certain theories about values use an entirely different notion
of truth than is used in science and is codified in our usual
methods of reasoning. Absent that, there seems to be no
justification for constructing and evaluating what are typically
thought of as prescriptive theories differently from descriptive
ones. Many discussions of how to evaluate prescriptive claims are
given in terms of what is rational or irrational to do. In the
final essay, "Rationality," what we mean by the idea of someone
being rational is investigated and the limitations of that label in
evaluating reasoning or actions is shown.
In "Classical Mathematical Logic," Richard L. Epstein relates
the systems of mathematical logic to their original motivations to
formalize reasoning in mathematics. The book also shows how
mathematical logic can be used to formalize particular systems of
mathematics. It sets out the formalization not only of arithmetic,
but also of group theory, field theory, and linear orderings. These
lead to the formalization of the real numbers and Euclidean plane
geometry. The scope and limitations of modern logic are made clear
in these formalizations.
The book provides detailed explanations of all proofs and the
insights behind the proofs, as well as detailed and nontrivial
examples and problems. The book has more than 550 exercises. It can
be used in advanced undergraduate or graduate courses and for
self-study and reference.
"Classical Mathematical Logic" presents a unified treatment of
material that until now has been available only by consulting many
different books and research articles, written with various
notation systems and axiomatizations.
This series of books is meant to present the fundamentals of
reasoning well in a clear manner accessible to both scholars and
students. The body of each essay gives the main development of the
subject, while the footnotes and appendices place the research
within a larger scholarly context. The topic of this volume is the
nature and evaluation of reasoning in science and mathematics.
Science and mathematics can both be understood as proceeding by a
method of abstraction from experience. Mathematics is distinguished
from other sciences only in its greater abstraction and its demand
for necessity in its inferences. That methodology of abstraction is
the main focus here. The study of these subjects is not just of
academic interest. First comes clear thinking, then comes clear
research and clear writing. The essays: Background Models and
Theories Experiments Mathematics as the Art of Abstraction
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