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This important course, sometimes disparagingly referred to as “baby logic,”
is often an undergraduate™s ¬rst and only exposure to the rigorous study of
reasoning. When we teach this course, we cover much of the ¬rst two parts
of the book, leaving out many of the sections indicated as optional in the
table of contents. Although some of the material in these two parts may seem
more advanced than is usually covered in a traditional introductory course,
we ¬nd that the software makes it completely accessible to even the relatively
unprepared student.
At the other end of the spectrum, we use LPL in an introductory graduate-
level course in metatheory, designed for students who have already had some
exposure to logic. In this course, we quickly move through the ¬rst two parts,
thereby giving the students both a review and a common framework for use
in the discussions of soundness and completeness. Using the Grade Grinder,
students can progress through much of the early material at their own pace,
doing only as many exercises as is needed to demonstrate competence.
There are no doubt many other courses for which the package would be
suitable. Though we have not had the opportunity to use it this way, it would
be ideally suited for a two-term course in logic and its metatheory.
Our courses are typically listed as philosophy courses, though many of the
students come from other majors. Since LPL is designed to satisfy the logical
needs of students from a wide variety of disciplines, it ¬ts naturally into logic
courses taught in other departments, most typically mathematics and com-
puter science. Instructors in di¬erent departments may select di¬erent parts
of the optional material. For example, computer science instructors may want
to cover the sections on resolution in Part III, though philosophy instructors
generally do not cover this material.
If you have not used software in your teaching before, you may be con-
cerned about how to incorporate it into your class. Again, there is a spectrum
of possibilities. At one end is to conduct your class exactly the way you always
do, letting the students use the software on their own to complete homework
assignments. This is a perfectly ¬ne way to use the package, and the students
will still bene¬t signi¬cantly from the suite of software tools. We ¬nd that
most students now have easy access to computers and the Internet, and so
no special provisions are necessary to allow them to complete and submit the
At the other end are courses given in computer labs or classrooms, where
the instructor is more a mentor o¬ering help to students as they proceed at
their own pace, a pace you can keep in step with periodic quizzes and exams.
Here the student becomes a more active participant in the learning, but such
a class requires a high computer:student ratio, at least one:three. For a class

To the instructor
12 / Introduction

of 30 or fewer students, this can be a very e¬ective way to teach a beginning
logic course.
In between, and the style we typically use, is to give reasonably traditional
presentations, but to bring a laptop to class from time to time to illustrate
important material using the programs. This requires some sort of projection
system, but also allows you to ask the students to do some of the computer
problems in class. We encourage you to get students to operate the computer
themselves in front of the class, since they thereby learn from one another,
both about strategies for solving problems and constructing proofs, and about
di¬erent ways to use the software. A variant of this is to schedule a weekly
lab session as part of the course.
The book contains an extremely wide variety of exercises, ranging from
solving puzzles expressed in fol to conducting Boolean searches on the World
Wide Web. There are far more exercises than you can expect your students
to do in a single quarter or semester. Beware that many exercises, especially
those using Tarski™s World, should be thought of as exercise sets. They may, for
example, involve translating ten or twenty sentences, or transforming several
sentences into conjunctive normal form. Students can ¬nd hints and solutions
to selected exercises on our web site. You can download a list of these exercises
from the same site.
Although there are more exercises than you can reasonably assign in a
semester, and so you will have to select those that best suit your course, we
do urge you to assign all of the You try it exercises. These are not di¬cult
and do not test students™ knowledge. Instead, they are designed to illustrate
important logical concepts, to introduce students to important features of the
programs, or both. The Grade Grinder will check any ¬les that the students
create in these sections.
We should say a few words about the Grade Grinder, since it is a truly
innovative feature of this package. Most important, the Grade Grinder will
free you from the most tedious aspect of teaching logic, namely, grading those
kinds of problems whose assessment can be mechanized. These include formal
proofs, translation into fol, truth tables, and various other kinds of exercises.
This will allow you to spend more time on the more rewarding parts of teaching
the material.
That said, it is important to emphasize two points. The ¬rst is that the
Grade Grinder is not limited in the way that most computerized grading
programs are. It uses sophisticated techniques, including a powerful ¬rst-order
theorem prover, in assessing student answers and providing intelligent reports
on those answers. Second, in designing this package, we have not fallen into
the trap of tailoring the material to what can be mechanically assessed. We

To the instructor / 13

¬rmly believe that computer-assisted learning has an important but limited
role to play in logic instruction. Much of what we teach goes beyond what
can be assessed automatically. This is why about half of the exercises in the
book still require human attention.
It is a bit misleading to say that the Grade Grinder “grades” the home-
work. The Grade Grinder simply reports to you any errors in the students™
solutions, leaving the decision to you what weight to give to individual prob-
lems and whether partial credit is appropriate for certain mistakes. A more
detailed explanation of what the Grade Grinder does and what grade reports
look like can be found at the web address given on page 15.
Before your students can request that their Grade Grinder results be sent
to you, you will have to register with the Grade Grinder as an instructor. This registering with
the Grade Grinder
can be done by going to the LPL web site and following the Instructor links.

Philosophical remarks
This book, and the supporting software that comes with it, grew out of our
own dissatisfaction with beginning logic courses. It seems to us that students
all too often come away from these courses with neither of the things we
want them to have. They do not understand the ¬rst-order language or the
rationale for it, and they are unable to explain why or even whether one claim
follows logically from another. Worse, they often come away with a complete
misconception about logic. They leave their ¬rst (and only) course in logic
having learned what seem like a bunch of useless formal rules. They gain little
if any understanding about why those rules, rather than some others, were
chosen, and they are unable to take any of what they have learned and apply
it in other ¬elds of rational inquiry or in their daily lives. Indeed, many come
away convinced that logic is both arbitrary and irrelevant. Nothing could be
further from the truth.
The real problem, as we see it, is a failure on the part of logicians to ¬nd a
simple way to explain the relationship between meaning and the laws of logic.
In particular, we do not succeed in conveying to students what sentences
in fol mean, or in conveying how the meanings of sentences govern which
methods of inference are valid and which are not. It is this problem we set
out to solve with LPL.
There are two ways to learn a second language. One is to learn how to
translate sentences of the language to and from sentences of your native lan-
guage. The other is to learn by using the language directly. In teaching fol,
the ¬rst way has always been the prevailing method of instruction. There are
serious problems with this approach. Some of the problems, oddly enough,

To the instructor
14 / Introduction

stem from the simplicity, precision, and elegance of fol. This results in a dis-
tracting mismatch between the student™s native language and fol. It forces
students trying to learn fol to be sensitive to subtleties of their native lan-
guage that normally go unnoticed. While this is useful, it often interferes with
the learning of fol. Students mistake complexities of their native tongue for
complexities of the new language they are learning.
In LPL, we adopt the second method for learning fol. Students are given
many tasks involving the language, tasks that help them understand the mean-
ings of sentences in fol. Only then, after learning the basics of the symbolic
language, are they asked to translate between English and fol. Correct trans-
lation involves ¬nding a sentence in the target language whose meaning ap-
proximates, as closely as possible, the meaning of the sentence being trans-
lated. To do this well, a translator must already be ¬‚uent in both languages.
We have been using this approach for several years. What allows it to
work is Tarski™s World, one of the computer programs in this package. Tarski™s
World provides a simple environment in which fol can be used in many of
the ways that we use our native language. We provide a large number of
problems and exercises that walk students through the use of the language in
this setting. We build on this in other problems where they learn how to put
the language to more sophisticated uses.
As we said earlier, besides teaching the language fol, we also discuss basic
methods of proof and how to use them. In this regard, too, our approach
is somewhat unusual. We emphasize both informal and formal methods of
proof. We ¬rst discuss and analyze informal reasoning methods, the kind
used in everyday life, and then formalize these using a Fitch-style natural
deduction system. The second piece of software that comes with the book,
which we call Fitch, makes it easy for students to learn this formal system
and to understand its relation to the crucial informal methods that will assist
them in other disciplines and in any walk of life.
A word is in order about why we chose a Fitch-style system of deduction,
rather than a more semantically based method like truth trees or semantic
tableau. In our experience, these semantic methods are easy to teach, but
are only really applicable to arguments in formal languages. In contrast, the
important rules in the Fitch system, those involving subproofs, correspond
closely to essential methods of reasoning and proof, methods that can be used
in virtually any context: formal or informal, deductive or inductive, practical
or theoretical. The point of teaching a formal system of deduction is not
so students will use the speci¬c system later in life, but rather to foster an
understanding of the most basic methods of reasoning”methods that they
will use”and to provide a precise model of reasoning for use in discussions of

Web address / 15

soundness and completeness.
Tarski™s World also plays a signi¬cant role in our discussion of proof, along
with Fitch, by providing an environment for showing that one claim does
not follow from another. With LPL, students learn not just how to prove
consequences of premises, but also the equally important technique of showing
that a given claim does not follow logically from its premises. To do this, they
learn how to give counterexamples, which are really proofs of nonconsequence.
These will often be given using Tarski™s World.
The approach we take in LPL is also unusual in two other respects. One
is our emphasis on languages in which all the basic symbols are assumed to
be meaningful. This is in contrast to the so-called “uninterpreted languages”
(surely an oxymoron) so often found in logic textbooks. Another is the inclu-
sion of various topics not usually covered in introductory logic books. These
include the theory of conversational implicature, material on generalized quan-
ti¬ers, and most of the material in Part III. We believe that even if these topics
are not covered, their presence in the book illustrates to the student the rich-
ness and open-endedness of the discipline of logic.

Web address

In addition to the book, software, and grading service, additional material can
be found on the Web at the following address:


Note the dash (-) rather than the more common period (.) after “www” in
this address.

Web address
Part I
Propositional Logic

Chapter 1

Atomic Sentences
In the Introduction, we talked about fol as though it were a single language.
Actually, it is more like a family of languages, all having a similar grammar
and sharing certain important vocabulary items, known as the connectives
and quanti¬ers. Languages in this family can di¬er, however, in the speci¬c
vocabulary used to form their most basic sentences, the so-called atomic sen-
Atomic sentences correspond to the most simple sentences of English, sen- atomic sentences
tences consisting of some names connected by a predicate. Examples are Max
ran, Max saw Claire, and Claire gave Scru¬y to Max. Similarly, in fol atomic
sentences are formed by combining names (or individual constants, as they
are often called) and predicates, though the way they are combined is a bit
di¬erent from English, as you will see.
Di¬erent versions of fol have available di¬erent names and predicates. We names and predicates
will frequently use a ¬rst-order language designed to describe blocks arranged
on a chessboard, arrangements that you will be able to create in the program
Tarski™s World. This language has names like b, e, and n2 , and predicates
like Cube, Larger, and Between. Some examples of atomic sentences in this
language are Cube(b), Larger(c, f), and Between(b, c, d). These sentences say,
respectively, that b is a cube, that c is larger than f , and that b is between c
and d.
Later in this chapter, we will look at the atomic sentences used in two
other versions of fol, the ¬rst-order languages of set theory and arithmetic.
In the next chapter, we begin our discussion of the connectives and quanti¬ers
common to all ¬rst-order languages.

Section 1.1
Individual constants
Individual constants are simply symbols that are used to refer to some ¬xed
individual object. They are the fol analogue of names, though in fol we


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