**Reviews:**

“*More Precisely* is
a very useful tool for philosophers with a limited formal background who
nevertheless aim to leave a lasting mark on the discipline.”

— **Nathan Salmon**,
Distinguished Professor of Philosophy, University of California, Santa Barbara

“Back in the day, philosophy
graduate students might have been expected to complete a course in metalogic.
It would be far better to complete a course that covers the topics presented
here. This is excellent coverage of a wide range of mathematical ideas and
techniques that are at the heart of many contemporary issues in metaphysics and
epistemology.”

—**F. Thomas Burke**,
University of South Carolina

__Comments
on First Edition__

“[*More Precisely*]
addresses a need by giving an elementary presentation of a number of technical
concepts used in philosophy, which previously were not collected together. It
should be especially useful for students preparing for graduate work whose
undergraduate training is likely to have skipped over at least some of the
concepts that the book covers. The material in each chapter is presented in a
very clear and engaging way, without presupposing any background beyond basic
high school mathematics.”

—**Susan Vineberg**,
Wayne State University

“This is the book I
wish I had when I was learning philosophy. It does an excellent job introducing
students to the formal tools that philosophers use in an accessible yet
rigorous way. From set theory to possible words semantics to probability theory
to the mathematics of infinity, all the key concepts are taught in a way that
will provide students the foundation for being solid philosophers. Moreover,
the concepts are taught in an engaging way, and frequent examples are given to
explain why these formal notions are philosophically relevant. I’m going to ask
all my incoming graduate students to read this book. And there’s enough
material in here that most philosophy professors could learn something from the
book as well.”

—**Bradley Monton**,
University of Colorado, Boulder

“This is a splendid
and innovative book. It explains accurately and accessibly a wide range of
technical topics frequently drawn upon in philosophy. Wonderfully informative
and clear, this book is a lifeline for students at undergraduate or graduate
level.”

—**Chris Daly**, University of Manchester

“This is a great
resource! Philosophers have always used the tools of mathematics to make their
claims clearer and more precise, even more so since the end of the nineteenth
century. Until now we haven’t had a systematic way to acquire those tools.
Steinhart’s book remedies the situation, presenting the fundamental ideas
thoroughly and comprehensively. Highly recommended for anyone getting into the
serious study of philosophy.”

—**Anthony Dardis**,
Hofstra University

Description:

*More
Precisely* is
a rigorous and engaging introduction to the mathematics necessary to do
philosophy. Eric Steinhart provides lucid explanations of many basic
mathematical concepts and sets out the most commonly used notational
conventions. He also demonstrates how mathematics applies to fundamental issues
in various branches of philosophy, including metaphysics, philosophy of
language, epistemology, and ethics. This second edition adds a substantial
section on decision and game theory, as well as a chapter on information theory
and the efficient coding of information.

**Key
Features:**

- A
primer on the mathematical and formal aspects of philosophy, including set
theory, relations, functions, probability, and more
- Suitable
as a course text or as supplemental reading for students who wish to fill gaps
in their technical knowledge
- This
edition adds chapters on decision theory and information theory
- Diagrams
and tables are used to illustrate and organize technical ideas
- Supplemental exercises and solutions are
provided on a companion website

Contents:

**Preface**

**Chapter 1. Sets** • Collections of Things •
Sets and Members • Set Builder Notation • Subsets • Small Sets • Unions of Sets
• Intersections of Sets • Difference of Sets • Set Algebra • Sets of Sets •
Union of a Set of Sets • Power Sets • Sets and Selections • Pure Sets • Sets
and Numbers • Sums of Sets of Numbers • Ordered Pairs • Ordered Tuples •
Cartesian Products

**Chapter 2. Relations** • Relations • Some
Features of Relations • Equivalence Relations and Classes • Closures of
Relations • Recursive Definitions and Ancestrals • Personal Persistence • The
Diachronic Sameness Relation • The Memory Relation • Symmetric then Transitive
Closure • The Fission Problem • Transitive then Symmetric Closure • Closure
under an Operation • Closure under Physical Relations • Order Relations •
Degrees of Perfection • Parts of Sets • Functions • Some Examples of Functions
• Isomorphisms • Functions and Sums • Sequences and Operations on Sequences •
Cardinality • Sets and Classes

**Chapter 3. Machines** • Machines • Finite
State Machines • Rules for Machines • The Careers of Machines • Utilities of
States and Careers • The Game of Life • A Universe Made from Machines • The
Causal Law in the Game of Life • Regularities in the Causal Flow • Constructing
the Game of Life from Pure Sets • Turing Machines • Lifelike Worlds

**Chapter 4. Semantics** • Extensional
Semantics • Words and Referents • A Sample Vocabulary and Model • Sentences and
Truth-Conditions • Simple Modal Semantics • Possible Worlds • A Sample Modal
Structure • Sentences and Truth at Possible Worlds • Modalities • Intensions •
Propositions • Modal Semantics with Counterparts • The Counterpart Relation • A
Sample Model for Counterpart Theoretic Semantics • Truth-Conditions for
Non-Modal Statements • Truth-Conditions for Modal Statements

**Chapter 5. Probability** • Sample Spaces •
Simple Probability • Combined Probabilities • Probability Distributions •
Conditional Probabilities • Restricting the Sample Space • The Definition of
Conditional Probability • An Example Involving Marbles • Independent Events •
Bayes Theorem • The First Form of Bayes Theorem • An Example Involving Medical
Diagnosis • The Second Form of Bayes Theorem • An Example Involving Envelopes
with Prizes • Degrees of Belief • Sets and Sentences • Subjective Probability
Functions • Bayesian Confirmation Theory • Confirmation and Disconfirmation •
Bayesian Conditionalization • Knowledge and the Flow of Information

**Chapter 6. Information Theory** •
Communication • Exponents and Logarithms • The Probabilities of Messages •
Efficient Codes for Communication • A Method for Making Binary Codes • The
Weight Moving across a Bridge • The Information Flowing through a Channel •
Messages with Variable Probabilities • Compression • Compression Using Huffman
Codes • Entropy • Probability and the Flow of Information • Shannon Entropy •
Entropy in Aesthetics • Joint Probability • Joint Entropy • Mutual Information
• From Joint Entropy to Mutual Information • From Joint to Conditional
Probabilities • Conditional Entropy • From Conditional Entropy to Mutual
Information • An Illustration of Entropies and Codes • Information and
Mentality • Mutual Information and Mental Representation • Integrated Information
Theory and Consciousness

**Chapter 7. Decisions and Games** •
Act Utilitarianism • Agents and Actions • Actions and Their Consequences •
Utility and Moral Quality • From Decisions to Games • Expected Utility • Game
Theory • Static Games • Multi-Player Games • The Prisoner’s Dilemma •
Philosophical Issues in the Prisoner’s Dilemma • Dominant Strategies • The Stag
Hunt • Nash Equilibria • The Evolution of Cooperation • The Iterated Prisoner’s
Dilemma • The Spatialized Iterated Prisoner’s Dilemma • Public Goods Games •
Games and Cooperation

**Chapter 8. From the Finite to the Infinite** •
Recursively Defined Series • Limits of Recursively Defined Series • Counting
through All the Numbers • Cantor’s Three Number Generating Rules • The Series
of Von Neumann Numbers • Some Examples of Series with Limits • Achilles Runs on
Zeno’s Racetrack • The Royce Map • The Hilbert Paper • An Endless Series of
Degrees of Perfection • Infinity • Infinity and Infinite Complexity • The
Hilbert Hotel • Operations on Infinite Sequences • Supertasks • Reading the
Borges Book • The Thomson Lamp • Zeus Performs a Super-Computation •
Accelerating Turing Machines

**Chapter 9. Bigger Infinities** •
Some Transfinite Ordinal Numbers • Comparing the Sizes of Sets • Ordinal and
Cardinal Numbers • Cantor’s Diagonal Argument • Cantor’s Power Set Argument •
Sketch of the Power Set Argument • The Power Set Argument in Detail • The Beth
Numbers • The Aleph Numbers • Transfinite Recursion • Rules for the Long Line •
The Sequence of Universes • Degrees of Divine Perfection

**Further Study**

**Glossary of Symbols**

**References**

**Index **

About the Author:

**Eric
Steinhart** is
a professor of philosophy at William Paterson University. He is the author
of *Your Digital Afterlives: Computational
Theories of Life after Death* (Palgrave Macmillan, 2014), *On Nietzsche* (Wadsworth, 1999),
and *The Logic of Metaphor: Analogous Parts
of Possible Worlds* (Kluwer Academic, 2001).

Target Audience:

The
book demonstrates
how mathematics applies to fundamental issues in various branches of philosophy,
including metaphysics, philosophy of language, epistemology, and ethics.