©1992-2001 The MIT Press. Reproduction by any means strictly prohibited.


To the Student

The most important thing for you to know about this book is that it is designed to be used with a teacher. You should not expect to learn logic from this book alone (although it will be possible if you have had experience with formal systems or can make use of the website at http://mitpress.mit.edu/LogicPrimer/). We have deliberately reduced to a minimum the amount of explanatory material, relying upon your instructor to expand on the ideas. Our goal has been to produce a text in which all of the material is important, thus saving you the expense of a yellow marker pen. Consequently, you should never turn a page of this book until you understand it thoroughly.

The text consists of Definitions, Examples, Comments, and Exercises. (Exercises marked with asterisks are answered at the back of the book.) The comments are of two sorts. Those set in full-size type contain material we deem essential to the text. Those set in smaller type are relatively incidental--the ideas they contain are not essential to the flow of the book, but they provide perspective on the two logical systems you will learn.

In this age of large classes and diminished personal contact between students and their teachers, we hope this book promotes a rewarding learning experience.

To the Teacher

We wrote this book because we were dissatisfied with the logic texts now available. The authors of those texts talk too much. Students neither need nor want page after page of explanation that require them to turn back and forth among statements of rules, examples, and discussion. They prefer having their teachers explain things to them--after all, students take notes. Consequently, one of our goals has been to produce a text of minimal chattiness, leaving to the instructor the task of providing explanations. Only an instructor in a given classroom can be expected to know how best to explain the material to the students in that class, and we choose not to force upon the instructor any particular mode of explanation.

Another reason our for dissatisfaction was that most texts contain material that we are not interested in teaching in an introductory logic class. Some logic texts, and indeed some very popular ones, contain chapters on informal fallacies, theories of definition, or inductive logic, and some contain more than one deductive apparatus. Consequently, we found ourselves ordering texts for a single-semester course and covering no more than half of the material in them. This book is intended for a one-semester course in which propositional logic and predicate logic are introduced, but no metatheory. (Any student who has mastered the material in this book will be well prepared to take a second course on metatheory, using Lemmon's classic, Beginning Logic, or even Tennant's Natural Logic.)

We prefer systems of natural deduction to other ways of representing arguments, and we have adopted Lemmon's technique of explicitly tracking assumptions on each line of a proof. We find that this technique illuminates the relation between conclusions and premises better than other devices for managing assumptions. Besides that, it allows for shorter, more elegant proofs. A given assumption can be discharged more than once, so that it need not be assumed again in order to be discharged again. Thus, the following is possible, and there is no need to assume P twice:

  1	(1)	P → (Q & R)		assume
  2	(2)	P			assume
  1,2	(3)	Q & R			from 1,2
  1,2	(4)	Q			from 3
  1,2	(5)	R			from 3
  1	(6)	P → Q			from 4, discharge 2
  1	(7)	P → R			from 5, discharge 2
  1	(8)	(P → Q) & (P → R)	from 6,7

Clearly, the notion of subderivation has no application in such a system. The alternative approach involving subderivations allows a given assumption to be discharged only once, so the following is needed:

 	(1)	P → (Q & R)		assume
 	(2)	P			assume
 	(3)	Q & R			from 1,2
 	(4)	Q			from 3
 	(5)	P → Q			from 4, discharge 2
 	(6)	P			assume
 	(7)	Q & R			from 1,6
 					(same inference as at 3!)
 	(8)	R			from 7
 	(9)	P → R			from 8, discharge 6
 	(10)	(P → Q) & (P → R)	from 5,9

The redundancy of this proof is obvious. Nonetheless, an instructor who prefers subderivation-style proofs can use our system by changing the rules concerning assumption sets as follows: (i) Every line has the assumption set of the immediately preceding line, except when an assumption is discharged. (ii) The only assumption available for discharge at a given line is the highest-numbered assumption in the assumption set. (iii) After an assumption has been discharged, that line number can never again appear in a later assumption set. (In other words, the assumption-set device becomes a stack or a first-in-last-out memory device.)

There are a number of other differences between our system and Lemmon's, including a different set of primitive rules of proof. What follows is a listing of the more significant differences between our system and Lemmon's, together with reasons we prefer our system.

In many cases, we have deliberately not used quotation marks to indicate that an expression of the formal language is being mentioned. In general, we use single quotes to indicate mention only when confusion might result. (We hope no one is antagonized by this flaunting of convention. Trained philosophers may at first find the absence of quotes disconcerting, but we believe that we are making things easier without leading the student astray significantly.)

We have tried to present the material in a way that reveals clearly the systematic organization of the text. This manner of presentation makes it especially easy for students to review the material when studying, and to look up particular points when the need arises. Consequently, there is little discursive prose in the text, and what seemed unavoidable has been relegated to the Comments. We hope to have produced a small text that is truly student-oriented but that still allows the instructor a maximum of flexibility in presenting the material.