Cover Page



Title page

Copyright page


Preface to Revised Second Edition

Preface to Second Edition

Preface to Revised Edition

Introduction to Teacher

Chapter 1: A Brief Introduction to Key Terms

1.1 Arguments

1.2 Putting Arguments into a Standard Format

1.3 Multiple Conclusions

1.4 Deductive Validity

1.5 Soundness

1.6 Missing Premises and Conclusions


Chapter 2: Argument Forms and Propositional Logic

2.1 Formal Validity

2.2 Quotation Marks

2.3 Metalinguistic Variables

2.4 Non-formal Validity

2.5 The Need for Propositional Logic

2.6 The Type/Token Distinction



Chapter 3: Conjunction

3.1 Logical Conjunction

3.2 Distinguishing Deductive from Non-deductive Aspects of Conjunction

3.3 Phrasal Logical Conjunctions

3.4 Series Decompounding

3.5 Using ‘Respectively’

3.6 Symbolizing Logical Conjunctions



Chapter 4: Negation

4.1 Logical Negation

4.2 Some Other Negative Expressions

4.3 A Point about Methodology

4.4 A Point on Ambiguity

4.5 Symbolizing Logical Negations

4.6 Ambiguity and the Need for Groupers

4.7 Review of Symbols

4.8 Using ‘Without’

4.9 Argument Forms Continued

4.10 Symbolizing Logical Negations Continued



Chapter 5: Truth Tables

5.1 Well-formed Formulas

5.2 Scope

5.3 Main Connective

5.4 Truth Tables



Chapter 6: Disjunction

6.1 Logical Disjunction

6.2 Disjunction and Negation

6.3 Iterations and Groupers

6.4 Inclusive versus Exclusive ‘Or’

6.5 Symbolizing Logical Disjunctions Continued



Chapter 7: Conditionals

7.1 Conditionals with Constituent Statements

7.2 Conditionals without Constituent Statements

7.3 Logical Conditionals

7.4 Symbolizing Conditionals in PL

7.5 Necessary and Sufficient Conditions

7.6 Only If

7.7 Unless

7.8 Since, Because

7.9 Conditionals and Groupers

7.10 If and Only If

7.11 A Revised Grammar for Well-formedness in PL

7.12 Summarizing Truth Tables



Chapter 8: Truth Trees

8.1 Reviewing Validity

8.2 Tree Trunks and Compound and Atomic Statements

8.3 Truth Tree Rules

8.4 Strategies

8.5 Truth Trees and Invalidity

8.6 Propositional Logic and Counter-examples (Counter-models)

8.7 Logical Properties and Relations Revisited



Chapter 9: Property Predicate Logic

9.1 Limits of Propositional Logic

9.2 Singular Terms

9.3 Property Predicates

9.4 Quantifiers

9.5 Complex Predicates

9.6 Well-formedness in PPL

9.7 Quantifiers Modifying General Terms



Chapter 10: Evaluating Arguments in Property Predicate Logic

10.1 Quantifiers and Scope

10.2 The Truth Tree Method Extended

10.3 Super Strategy

10.4 Property Predicate Logic and Counter-examples (Counter-models)

10.5 PPL Logical Equivalences and Non-equivalences

10.6 Other Logical Properties and Relations



Chapter 11: Property Predicate Logic Refinements

11.1 Literal Meaning

11.2 ‘Any’ as an Existential

11.3 Restrictive Relative Clauses

11.4 Pronouns Revisited

11.5 Only

11.6 Restrictive Words in English

11.7 Evaluating Symbolizations of English in Logical Notation



Chapter 12: Relational Predicate Logic

12.1 Limits of Property Predicate Logic

12.2 Convention 1: Number

12.3 Convention 2: Order

12.4 Convention 3: Active/Passive Voice

12.5 Convention 4: Single Quantifiers

12.6 Variables



Chapter 13: Relational Predicate Logic with Nested Quantifiers

13.1 Multiply General Statements

13.2 Universal Quantifier Procedure

13.3 Existential Quantifier Procedure

13.4 Double Binding Variables

13.5 Systematic and Analytic Procedures

13.6 A Grammar for Well-formedness in RPL

13.7 Nested Quantifiers, Variables, and Scope

13.8 Order and Scope Refinements

13.9 Summary of the Overall Procedure for Symbolizing English Statements with Nested Quantifiers into RPL



Chapter 14: Extending the Truth Tree Method to RPL

14.1 RPL Arguments without Quantifiers

14.2 RPL Arguments without Nested Quantifiers

14.3 RPL Arguments with Nested Quantifiers

14.4 Choosing Singular Terms to Instantiate

14.5 Infinite Truth Trees for RPL Arguments

14.6 Summary of Truth Tree Strategies

14.7 Relational Predicate Logic and Counter-examples (Counter-models)


Chapter 15: Negation, Only, and Restrictive Relative Clauses

15.1 Negation

15.2 ‘Only’ as a Quantifier

15.3 Restrictive Relative Clauses

15.4 Quantifiers and Anaphora

15.5 Anaphora and Restrictive Relative Clauses

15.6 Anaphora Across Sentences

15.7 Quantification in English



Chapter 16: Relational Predicate Logic with Identity

16.1 Limits of Relational Predicate Logic

16.2 Extending the Truth Tree Method to RPL=

16.3 Sameness and Distinctness in English

16.4 Numerical Adjectives

16.5 Definite Descriptions



Chapter 17: Verbs and their Modifiers

17.1 Prepositional Phrases

17.2 The Event Approach

17.3 Indirect Support of the Event Approach

17.4 Adverbial Modification

17.5 Problems with the Event Approach




A1 Conjunction

A2 Negation and Disjunction

A3 Conditionals

A4 Property Predicate Logic

A5 Relational Predicate Logic

A6 Relational Predicate Logic with Identity

A7 Verbs and their Modifiers


Answers for Selected Exercises

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17


Logical Symbols


Praise for Meaning and Argument

“With care, imagination, and infectious enthusiasm, Lepore develops a novel and effective general technique of formalization which complete beginners should be able to grasp and use to deal with virtually any example in a first logic course.”

Bob Hale, University of Glasgow

Meaning and Argument is especially strong on the subtleties of translating natural language into formal language, as a necessary step in the clarification of expression and the evaluation of arguments. The range of natural language constructions surveyed is broader and richer than in any competing introductory logic text that I am aware of. As such, the book provides a solid and attractive introduction to logic not only for philosophy students, but for linguists as well.”

Richard Larson, University Stony Brook

“I can thoroughly recommend Ernest Lepore’s Meaning and Argument, particularly for those seeking to teach or learn how to paraphrase into formal symbolism, a much neglected aspect of logic. It contains a wealth of examples and is informed throughout by a deep theoretical knowledge of contemporary linguistics and philosophy of language.”

Alan Weir, Queen’s University Belfast

“Lepore’s book is unusual for a beginning logic text in that it contains no natural deduction proof system but rather concentrates on finding models and countermodels by means of a semantic tableaux method. It is also unusual in containing many translation examples that exemplify constructions that linguists have found interesting in the last decades. In both of these ways the book is well suited for use in educating philosophy students in the importance of logic even when these students do not intend to go further in the study of formal logic as a discipline.”

Francis Jeffry Pelletier, University of Alberta

Meaning and Argument is a beautiful display of both the power of first-order logic and the complexity of natural language. The book focuses on the use of logic to expose and remedy many difficulties with understanding a sentence’s exact meaning. Lepore’s user-friendly style makes the book enjoyable for beginning logic students, and his coverage of the details makes it useful for advanced students and professionals. There is no logic textbook that comes even remotely close to accomplishing what Meaning and Argument does.”

Kent Johnson, University of California at Irvine

Title Page


I (and others) have taught various drafts of this book; so first I thank the literally dozens of classes that have had to put up with my working out these materials in front of them. I enjoyed it immensely (I still do), but I also realize how frustrating it must have been for many students. Many other logic books have influenced me in one way or another, so many that I cannot even recall them all. In addition, so many discussions with colleagues and students have caused me to rethink much of what I had said that it’s impossible for me to thank each and every person who has contributed to the final product. I don’t even like the sound of that – the final product. The only way I was able to convince myself to let this book go into print was by thinking of it as but one more draft, to be revised as soon as I distribute the first published copy and begin a new semester.

Logic is not the sort of subject that jumps to mind in pondering friendship, but in reminiscing about who among my colleagues contributed to the development of this book I was struck with a rather strong (but not perfect) link between colleagues and long-term, recent, past, or current friends.

I began thinking about reasoning in high school. My then and still close friend, my first friend, is Brian McLaughlin. We spent years trying to solve various logical conundra like how on earth the Virgin Mary could have been a virgin and still have given birth to Jesus. In college I had the good fortune to find myself in the classroom of several terrific teachers, but three that stand out vis-à-vis logic are Ed – the Jet – Gettier, Terry Parsons, and Steve Herman. From Steve I learned about the joy of philosophy, and I’ll say more about him below. Ed Gettier continues to make me wonder what went wrong with my generation. I was under 20, barely articulate, a terrible Jersey public high school education holding me back, yet Ed never failed to figure out and answer what I was trying to ask. Like many others who went through UMass, I still treasure my collection of napkins chock full of Ed’s ideas and notes and passion for logic and philosophy. Terry Parsons was the first philosopher I got to know who knew a lot of logic, and, boy, could he teach it. I don’t have napkins from Terry, but I have precious notes. I continue to learn from him. It’s rare to hook up with someone as smart as Terry who also takes joy in teaching esoteric subjects and does it so well.

When I got to graduate school I thought I knew a fair amount of logic. I suppose I did, but not as much as I thought. John Wallace spotted that right away. He was the first philosopher who taught me how powerful a tool logic can be without having to ram it down everyone’s throat. Michael Root, the other chief influence on me in graduate school, taught me how to teach logic. I assisted him in several introductory logic courses. Lots of the examples in this book are cribbed from his excellent notes.

Sometime toward the end of my graduate education I came into personal contact with Donald Davidson. Under his influence I came to see that logic was an important device for thinking about natural language. I’m not sure why it took so long for me to figure that out. This book has ultimately been shaped by Davidson’s wisdom about the importance of semantics for thinking about natural language. Just one more debt I owe Donald.

In 1979, I spent a summer living with Barry Loewer while we were both attending a National Endowment of Humanities seminar run by Dick Grandy. I got a lot out of that seminar, but most memorable from that summer is the beginning of a long and continuing intellectual and loving friendship. Anyone who knows Barry knows that he knows his way around hard philosophy as well as anyone in the profession. After having been set straight by him, we merrily proceeded over the next decade to collaborate on well over a dozen articles, some of which we both still believe.

When I took a job at Rutgers University, I hooked up with Bas van Frassen – the Bas. We were both single and new to New Jersey. I still smile thinking about our hopping back and forth between my place on St Mark’s Place in the East Village and his home in Princeton. Bas did a lot for my self-confidence. If anyone as smart as he is didn’t think what I was saying was stupid, I figured I had to be on to something. More importantly, Bas, even when critical, knew how to dish it out gently and affectionately – we can all learn a lot from his example.

Around that same time I joined a trio, Dr E. Saarinen and Colin – the Ace – McGinn being the other two members. Esa, Colin, and I bounced around Helsinki, London, and New York City. Although we were all intensely interested in philosophical logic, I’m not sure how much influence we exerted over each other on that front. What I do recall quite fondly about the two of them, though, is how they pressed me to work things out for myself, and to rely less on others. I hope some of that comes through in this book. To the extent that it does, it’s intentional.

As the years passed, I came to know Willard Van Orman Quine. It took me about five years not to be intimidated just by being in his presence. Once I got over that initial star shock, I began to talk with him about logical form. It’s no accident that he has come to be known as the father of American philosophy. No introductory logic book can replace his Methods of Logic, and it would be folly to try to do so. My route was to try to amplify on his superior lead.

More recently I came under the spell of Jerry Fodor. We wrote a book together and a number of critical essays. No matter how strange one finds what Jerry sometimes says in and out of print, there is no denying how damn good he is at philosophy. No one else I know sees the dialectic as quickly and as well as he. I’m more than appreciative for the good number of times he’s challenged me on a piece of faulty reasoning before I put it into print.

Three years ago I decided I should try to publish my logic notes. That decision brought me back full circle to Steve Herman, my first philosophy teacher. I had a draft of the book ready to hand to Steve Smith, my friend and editor at Blackwell Publishing. I visited Steve Herman in Maryland one weekend and showed him that ‘final’ draft. After a few hours he told me he thought it could be turned into a good book with another year’s hard work. He also agreed to help. It never occurred to me to move any faster. Steve was off by at least two years; but it’s been great fun helping him finish my book.

There are many others. I’ve been blessed to have so many clear-headed, smart friends, young and old. During the past year or so a number of graduate students have become involved in helping me rewrite: in particular, Jeff Buechner, Kent Johnson, Jonathan Cohen, and Phillip Robbins. In the final stages of putting together this book, Herman Cappelen graciously taught it at Vassar College. He and his students offered keen criticism. Stephen Neale has taught us all much about descriptions, from which chapter 16 has benefited. On the penultimate draft, Bernie Linsky, Barry Schein, and especially Bob Hale showered me with criticism, all of which was very helpful. There are others I’m sure I’ve neglected to thank. Next time. The book is dedicated to two old friends, for whom it would be nothing close to exaggeration to say that this book is as much theirs as mine in ways only they know.

For Steve Herman and Dick Foley.
I’m a lucky person to be their friend.

Preface to Revised Second Edition

This is the revised second edition of Meaning and Argument. There was also a revised first edition. In this version, the revisions concern mostly added exercises and their solutions. We also corrected errors throughout – we hope the book is as close to error-free now as possible. We added some discussion of conversational inferences (e.g., “or” cases) and subjunctive conditionals.

We would like to thank the supportive criticism of Paul Gomberg as well as the invaluable assistance of Alex Anthony and Una Stojnic. We dedicate this new edition to all three of them.

Preface to Second Edition

This is the second edition of Meaning and Argument. There was also a revised first edition. In this latest version the revisions concern mostly issues surrounding anaphora. The account of “donkey” anaphora (anaphora between quantifiers and pronouns that are not in their syntactic scope) from the previous editions has been systematically reworked and extended to all of the connectives (via a “repair” algorithm). Furthermore, we have been able to treat anaphora between the premises, or from a premise to the conclusion, of an argument. We have also extended the account to plural pronouns. This is very much an active area of research, and as a result we introduce several competing accounts (for instance Geach’s “pronoun of laziness” account and Evans’ “E-type” account) with different areas of coverage. We have also added techniques for building countermodels, and revised each chapter and appendix in light of new research in order to improve reader comprehension.

Preface to Revised Edition

Some typographical errors and other infelicities were found in the first edition of this book. I want to thank Blackwell Publishing and in particular Jeff Dean for encouraging me to put together a new revised edition. Most of the corrections I have already put up on the home page for the book, It also seemed to me (and my students) that some of the exercises were less helpful than they should be. So I have revised many of the exercises – cutting here, adding there, re-ordering – throughout. In addition, I’ve elaborated on issues I thought should be elaborated upon – e.g., anaphora, predicative adjectives, ‘exactly’, negation of existentials, and numerical quantifiers.

I had wanted to add new chapters on natural deduction – in addition to the tableaux (‘truth tree’) method already in the book – but discovered that this addition would require too many chapters for merely a revised edition. So I have put up a natural deduction system with strategies and exercises on the book’s home page.

A number of people have sent me useful comments on this book since its publication. I’d like to thank them here. Herman Cappelen, Walter Dean, Tobyn DeMarco, Georg Dorn, Ray Elugardo, Edmund Gettier, Kent Johnson, Matt Philips, T. W. C. Stoneham, and Alan Weir.

A very special thanks goes to Sarah-Jane Leslie. Sarah-Jane has been a participant in every single aspect of this book and its uses since its publication. She was in the first class I taught using the book after it was published. She found several errors and has made numerous recommendations for how to improve this new edition, including writing up some of the new sections. She put together the home page for the book and continues to maintain it. I cannot thank her enough. I dedicate this new revised edition to her.

Introduction to Teacher

Often misunderstanding or conflict results from people not getting clear with each other. We take for granted that our language is an adequate vehicle of expression, but it isn’t. Although formal logic texts typically take on the task of helping students clarify ambiguous language, they tend to focus more on proof making, determining when a conclusion follows from its premises. Students should be taught to test the validity of formalized arguments, but a reasonable expectation is that they express themselves clearly before drawing conclusions. A primary task of logical skill development should be to give students tools for capturing adequately in a notation arguments they express in a natural language. To this end, language should play about one half of the project in a logic course, though it rarely does, at least not explicitly. Whether logic teachers are aware of it or not, it’s impossible to teach how to evaluate arguments from natural language without dabbling a bit in philosophy of language. (Of course, introductory logic can be taught exclusively on formal proofs in formal languages. Many try to do so. Why such courses should be taught in philosophy programs and not in mathematics or computer science programs is a mystery to me; indeed, why they should be taught at all eludes me as well.) Meaning and Argument shifts away from traditional emphasis on proofs (manipulation of a formal system) to symbolization of arguments of English in a formal system. It does so while still introducing students to what they ordinarily learn in an introductory logic course: truth tables, validity, propositional logic, predicate logic without and with identity, formal proofs, consistency, and so on. Meaning and Argument is not so bold as to claim that symbolization techniques are mechanical, but nor is it so cavalier as to suggest that manipulating natural language arguments into formal languages is an unteachable skill that one either has or doesn’t have.

Another distinctive feature of Meaning and Argument is that it shows how the need for expressive power and for drawing distinctions forces formal language development. In this sense the book is Quinean in spirit: symbolization (or, to use Quine’s term, regimentation) should be ruled by the maxim of ‘shallow analysis’. As Quine counsels us, ‘expose no more logical structure than seems useful for deduction or other inquiry at hand’ (Word and Object, p. 160). Symbolization is a tool useful for sorting out those linguistic constructions which can be rendered within the framework of a given logic from those that cannot be so regimented.

Some logic teachers might be tempted to dismiss rudimentary systems like the Propositional Calculus because they lack the complexity of logical apparatus needed to prove the validity of arguments we know intuitively to be valid. Meaning and Argument starts with these more elemental systems, however, using them to guide students through an idealized journey of thought. The purpose is to retrace what one might imagine were the paths which great logicians traveled as they elaborated simpler formal systems. In this way the book tries to get at why things might have happened as they did. The process is not necessary. Given modern tools and diagnostic devices, a mechanic can be trained to identify a malfunction and correctly replace the malfunctioning part without ever having torn apart an engine and rebuilt it. Intuitively there is something to be said on behalf of the latter approach, at least insofar as getting back on the road is one’s primary objective. But in the case of formal logic, we should want more than mechanical translation and testing of validity. Understanding formal systems and how they work and develop can be a valuable part of a curriculum that has as one of its goals to train systematic and scientific thinkers.

At the introductory level much that has passed as logic teaching is how-to, feel-good education, which has as its main purpose to help students get the right answers. Often this endeavor leads to well-intentioned efforts to give students recipes, step-by-step techniques, that render the processes of symbolization and proof making mechanical. Teaching students to treat conjunctions, conditionals, universal statements, or any other sort of logically complex expression as a purely grammatical matter is too facile, giving a temporary illusion of success at the early stages. This book stresses understanding; it sensitizes students to the contextual nature of language, and thus to the importance of not being lulled into automatic symbolizations based on grammatical structures alone. At each stage of system elaboration and development, the book seeks to answer the meta-logical questions:

Why is a particular formalism needed?
What form should it take?

These questions engage students in an inquiry, which allows them to see logical studies as a human enterprise aimed at achieving well-understood goals – clarity and good reasoning. When students realize that systems are elaborated and developed so that increasingly complex and ambiguous statements can be clearly stated and so that valid arguments can and invalid arguments cannot be proved, the entire subject matter begins to make sense. Instead of symbolizing statements and devising proofs without knowing why, students come to see the forest for the trees.

This book answers its main questions by proceeding from a simple formal language to increasingly more complex formal languages and by explaining the reason for each complication. The moves from propositional to property predicate logic, to relational predicate logic, and finally to relational predicate logic with identity are made clear to the student at each step. In this sense the book is progressive; yet, from the first chapter to the last, the book should be accessible to a novice.

In addition, this book introduces the student to the differences between logical and conversational implications. Throughout, the case is made that many logical fallacies may best be understood as failures to distinguish reasonable conversational inferences from strict logical deductions.

Instructional method: Meaning and Argument proceeds inductively in a pedagogical sense. An example either of a statement or an argument is given. The student’s intuitions are tested: What does the statement mean? Is the argument intuitively valid? Does the logical system presented, at its current level of development, enable us to capture this sense, clear up ambiguities, and prove validity? If not, how might the system be elaborated?

In the case of symbolization, Meaning and Argument contributes to the literature of logic pedagogy. It provides the student with procedures for symbolizing complicated locutions from English into the appropriate formal notation, a systematic and manageable process that leads almost invariably to correct symbolizations.

Relation of book to current linguistic theory: In the last 50 years linguists, logicians, philosophers, and computer scientists have assembled a substantial and highly sophisticated body of work on the structure of natural language. Inasmuch as Meaning and Argument aspires to be an introduction to the tools and techniques for studying natural language, one might expect the book to incorporate some of this burgeoning literature. However, anything less than a comprehensive, even if not a critical, review of that literature would be superficial and, I expect, confusing for most introductory students. Furthermore, an introductory book of this nature is better kept independent of current technical trends. For example, some very important recent contributions by formal semanticists are not acknowledged here. The bibliographies at the end of most chapters, however, list this burgeoning literature.

Chapter 1

A Brief Introduction to Key Terms

1.1 Arguments

Arguments crop up in conversations, political debates, lectures, editorials, comic strips, novels, television programs, scriptures, films, graffiti, posters, on the net, and so on. Ordinarily when we argue with others, we try to persuade them of some point. The forms of persuasion are many. We can persuade others by hitting them, by screaming at them, by drugging them, and so on. These kinds of persuasion are, unfortunately, prevalent. However, in this book we will use the term ‘argument’ exclusively to pick out sets of statements of the following sort:

Provided the fetus is a person, a fetus has a right to life. Should a fetus have a right to life, it is false that someone has the right to take its life. However, if abortions are moral, someone does have the right to take the life of a fetus. Consequently, if a fetus is a person, abortions are not moral.

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if caused by this, it is not caused by smoking.

Anyone who deliberates about alternative courses of action believes he is free. Since everybody deliberates about alternative courses of action, it follows that we all believe ourselves to be free.

What do these passages have in common in virtue of which each is an argument in our sense? In this book an argument is any set of statements – explicit or implicit – one of which is the conclusion (the statement allegedly being defended) and the others are the premises (statements allegedly providing the defense).1 The relationship between the conclusion and the premises is such that the conclusion purportedly follows from the premises. This description of an argument leads us to ask what statements are, and what it means to say that one statement follows from others.

1.1.1 What is a Statement?

A statement is any declarative sentence that is either true or false.2 The following are statements:

Galileo was an astronomer.
Provided the fetus is a person, a fetus has a right to life.
No one but Nixon knew the truth.
Lung cancer is not caused by smoking.
Everybody deliberates about alternative courses of action.
Martin Van Buren was not the ninth president of the United States.

Interrogatives, imperatives, and exclamations are sentences that are not statements; for example:

Is George Washington president?
Shave yourself!

1.1.2 Premises and Conclusion

A main aim in learning logic is to enhance skills in assessing arguments as we find them. But arguments cannot be assessed unless they are first identified as arguments. Although there are no sure signs of whether an argument is present, fairly reliable indicators exist.

Premise indicators: Premise indicators are terms that indicate that a premise will immediately follow. In the second argument in §1.1 above, the first sentence ends with ‘this is so for the following reasons’. This clause indicates that the statements which follow are the premises of this argument. The third argument has a second sentence that begins with the word ‘Since’, indicating that a premise is about to be introduced. Other such expressions include ‘because’, ‘for’, ‘after all’, ‘given’, ‘whereas’, ‘although’, ‘suppose’, ‘assume’, ‘let us presume’, ‘granted’, ‘here are the facts’. When a premise indicator starts a clause, then what follows the premise indicator is usually a premise.

Conclusion indicators: Likewise, when a conclusion indicator starts a clause, then what follows is usually a conclusion. In the first argument in §1.1 above, the last sentence begins with the word ‘Consequently’, indicating that it is the conclusion. Other conclusion indicators include ‘therefore’, ‘so’, ‘hence’, ‘it follows’, ‘…proves’, ‘…shows’, ‘we can now infer’, ‘it cannot fail to be’, ‘let us conclude’, ‘this implies’, ‘these facts indicate’, ‘this supports the view or claim’, ‘let us infer’, ‘as a consequence we can deduce’.

Many passages contain arguments but none of these indicators, and some passages contain one or more of them without stating a premise or a conclusion. Indeed, the indicators seen most frequently are often used as neither conclusion nor premise indicators, as in the following sentences.

How long has it been since you last saw him?
He is so good at what he does.
For two years he has been away.
After all these days, you come home.
Let us go hence.

None of these sentences is being offered as a premise for some conclusion or as a conclusion from some premises. Even though each contains familiar indicator terms, obviously the terms are not being used as indicators in these sentences.

Alternatively, one may simply announce that an argument is forthcoming, and then go on to affirm several statements. In this case, the context makes clear that the last statement is the conclusion and the others are premises. Still, some terms are almost always used as conclusion or premise indicators. It’s hard to imagine a context in which ‘on the assumption that’ is not a premise indicator or ‘as a consequence it follows’ is not a conclusion indicator.

1.2 Putting Arguments into a Standard Format

Having determined that some piece of discourse contains an argument, the next task is to put it into a standard format. This task may involve all of the following:

i. Identifying the premises and the conclusion.
ii. Placing the premises first. (Order does not matter.)
iii. Placing the conclusion last.
iv. Making explicit any premise or even the conclusion, which may be only implicit in the original but essential to the argument.

So standard forms for the above three arguments are:

Provided the fetus is a person, a fetus has a right to life.
Should a fetus have a right to life, it is false that someone has the right to take its life.
If abortions are moral, someone does have the right to take the life of a fetus.
If a fetus is a person, abortions are not moral.

The first three statements are premises, and the fourth is the conclusion.

Lung cancer is more common among male smokers than among female smokers.
If smoking were the cause of lung cancer, this would not be true.
The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup.
If it is caused by this, it is not caused by smoking.
Lung cancer is not caused by smoking.

The first four statements are premises, and the fifth is the conclusion.

Anyone who deliberates about alternative courses of action believes he is free.
Everybody deliberates about alternative courses of action.
We all believe ourselves to be free.

The first two statements are the premises, and the third is the conclusion.

In none of the arguments is a premise or a conclusion missing. Nothing said so far explains exactly how we were able to devise these standard forms based on what we were presented with. So far our process has been rather loose, and it cannot be tightened until we say something substantive about when one statement follows from some others.

1.3 Multiple Conclusions

As we have characterized arguments, no argument can have more than one conclusion. Of course, sometimes we do find passages with more than one conclusion. There are two types of cases. The first occurs when more than one conclusion is drawn from the same set of premises. For such a case we adopt the convention that distinct arguments can have the same premises, but different conclusions. So, for example, (1)–(4) below include two distinct arguments. (1)–(2) are premises, and (3)–(4) are conclusions. So (1)–(3) constitute one argument; and (1)–(2) and (4) constitute another distinct argument.

1. All women are mortal and rational.
2. Andrea is a woman.
3. So, Andrea is rational.
4. So, Andrea is mortal.

The second case occurs when we chain arguments together so that a single statement serves as both a premise and a conclusion. In this case the conclusion of one argument functions as a premise of another. (5)–(9) include two arguments.

5. Killing children is evil.
6. Children were being killed in Bosnia.
7. Therefore, someone was doing something evil in Bosnia.
8. When someone does something evil, he should be punished.
9. So, whoever killed children in Bosnia should be punished.

(5)–(6) are premises of an argument with (7) as its conclusion. However, (7) is also the premise of an argument, which along with (5), (6), and (8), has (9) as its conclusion.

Exercise 1.1 Standard form

1. If we are going to avoid a nuclear war in the next few years, we will have to adopt strong punitive measures now. But if we adopt such measures, many nations will be very unhappy. Thus, we are going to avoid a nuclear war in the next few years only if many nations are going to be unhappy.
2. The state will increase its financial support of our university only if the priorities of the legislature shift in favor of higher education. But if such a shift were to occur, the people who benefit from other state projects would complain bitterly. If the state does not increase financial support for the university, tuition will have to be raised. So, tuition will be raised.
3. If a man is to play some role in society, that role must be determined by nature or by society. However, if his role is determined by nature, that role will be the role of the selfish hunter on the make. Hence, either society determines a role for man, or man will play the role of the selfish hunter always on the make.
4. If it is true that 30 out of every 50 college coeds have sexual intercourse outside marriage, then it is very important to have birth control information available from the Student Health Service. It is very important to have birth control information available from the Student Health Service. Thus, we know that 30 out of every 50 college coeds have sexual intercourse outside marriage.

1.4 Deductive Validity

What is it for one statement to follow from others? The principal sense of ‘follows from’ in this book derives from the notion of a deductively valid argument.

A deductively valid argument is an argument such that it is not possible both for its premises to be true and its conclusion to be false.

So, consider argument (10)–(12).

10. The current Vice-President will win the next election.
11. If the current Vice-President wins the next election, then the country will prosper.
12. So, the country will prosper.

This is an argument in which the conclusion follows intuitively from its premises. But what is it about (10)–(12) that makes us think that (12) can be concluded on the basis of (10)–(11)? We know it’s not the truth of the premises, because we are in no position to know whether they are true. The relevant events haven’t even occurred yet. So what is it? One salient feature is that, were the premises true, the conclusion would have to be true as well. These considerations should help the reader to appreciate the difference between the deductive validity of an argument and the truth of its premises.

Another argument will make this point even more evident.

13. All fish fly.
14. Anything which flies talks.
15. So, all fish talk.

Even though each statement in (13)–(15) is obviously false, the argument is deductively valid, because it is not possible for its premises to be true and its conclusion to be false. If it so happened (even though we know otherwise) that the premises were true, would the conclusion have to be true as well? Say that, due to some strange release of radiation, all fish flew and anything which flew talked, it would then be, and have to be, true that all fish talk. In this sense the conclusion (15) follows from its premises (13)–(14), and the argument is deductively valid.

What about an invalid argument? Consider argument (16)–(18).

16. If God exists, then the creation is perfect.
17. God doesn’t exist.
18. So, the creation is imperfect.

Argument (16)–(18) is invalid. (16) merely tells us what the state of the creation would be if God exists. It doesn’t assert anything about the state of the creation if God does not exist. The creation could be perfect whether or not God exists. Consider an analogous argument. If you prepare for a chess game, you win. But suppose you don’t prepare for it. It doesn’t follow that you lose. Plainly you might win even if you don’t prepare. Your opponent might forfeit, for example.

In ordinary English the terms ‘valid’ and ‘true’, on the one hand, and ‘invalid’ and ‘false’, on the other, are often used interchangeably. People will say that statements are valid or invalid, and that arguments are true or false. But logicians use these terms in a much more restricted way, such that ‘valid’ and ‘invalid’ apply only to arguments, and ‘true’ and ‘false’ only to statements. We will adopt this restrictive practice in this book.

Exercise 1.2 Deductive validity

1. Mary either plays basketball or baseball.
    She does not play basketball.
    So, she plays baseball.
2. Al will play tennis or he will play baseball.
    Al will play tennis.
    So, Al will not play baseball.
3. You play in the National Basketball Association only if you are over three feet tall.
    Bill Clinton is over three feet tall.
    So, Clinton plays in the National Basketball Association.
4. Any creature with a kidney has a heart.
    Not every creature has a heart.
    So, not every creature has a kidney.
5. Bill ate steak for dinner this evening.
    So, Bill ate dinner this evening.
6. Everybody loves someone.
    So, someone is loved by everyone.
7. Some birds do not fly.
    Therefore, not every bird flies.
8. There is evil in the world.
    If there were a God, there would be no evil.
    Therefore, there is no God.

Exercise 1.3 Truth, falsity, and deductive validity

Devising a deductively valid argument for box 7 demonstrates that there is no connection between actual truth and validity. Someone may assert only false sentences, and yet his argument may be deductively valid.

Devising an argument for box 2, on the other hand, demonstrates that someone may assert only truths, and yet his argument may be deductively invalid.

Together these show that we might criticize in two distinct ways those who offer an argument in order to persuade us of some point: they might go wrong because they are ignorant (box 7), in which case what they say is false; or they might go wrong because they are illogical (box 2), in which case what they conclude does not follow from what they believe or assert as premises.

1.5 Soundness

A good argument is not simply deductively valid; (13)–(15) is deductively valid but unlikely to persuade anyone.

13. All fish fly.
14. Anything which flies talks.
15. So, all fish talk.

Though it is not possible for both (13) and (14) to be true and (15) false, the argument is unlikely to persuade any knowledgeable person, because (13)–(14) are patently false. Normally, good arguments are not only deductively valid. They also have true premises. Such arguments are called sound.3

Suppose an argument is valid, yet its conclusion is false. Then at least one of its premises must be false, and the argument, though valid, is unsound. Once we decide that an argument is deductively valid, we may direct attention to the question of its soundness.

1.6 Missing Premises and Conclusions

Intuitively, argument (19)–(20) is deductively valid, because an obviously true premise (21) is missing from it.

19. New York City is in New York State.
20. So, Manhattan is in New York State.
21. Manhattan is in New York City.

Almost anyone familiar with the facts about New York City, when presented with (19)–(20) as an argument, would effortlessly add (21) as a premise. But we must be careful about adding premises to arguments, since any argument can be turned into a deductively valid argument if the right ‘missing’ premise(s) are added. On its face, argument (22)–(23) appears to be invalid; but if we treat this argument like we did argument (19)–(20) by adding the ‘missing’ premise (24), we too easily turn a bad argument into a deductively valid argument.

22. The Yankees are losing.
23. So, we should elect a new president.
24. If the Yankees are losing, we should elect a new president.

Clearly, until some independent reason is provided, why should we add (24)? Why should we suppose a connection between the Yankees losing, on the one hand, and electing a new president, on the other?

The question we need to take a stab at is why many people would refrain from postulating (24) as a missing premise from (22)–(23), even though they intuit that (21) is implicit in argument (19)–(20)? According to one hypothesis, the difference has something to do with successful communication. When we communicate, we normally share a background of beliefs or assumptions and recognize that we share them. When I discuss politics with a colleague, we each take elementary facts about our government for granted, and we each assume that the other does too. We assume that Washington DC is the capital of the United States, that the White House is located there, that the President lives there, and so forth. The more we share, the easier it is to communicate. If we did not share any beliefs, we probably could not communicate at all.

Typically, a conversation depends in part on what we want to convey and in part on what we assume about the listener’s beliefs. We do not usually say what is already common knowledge, since that would be banal. Nor do we ordinarily say what we believe is inconsistent with our shared beliefs, since that would sabotage our conversation.

Although many facts and opinions are assumed in conversation, we frequently need to bring them to the fore, in order to evaluate properly the validity of an argument. Which premises to make explicit is not always easy to decide. The interpreter of an argument has a choice: to declare an explicitly invalid argument invalid or to make that argument valid by adding premises he thinks implicit but unstated. No matter how unpromising an argument may seem, you can always conjure up premises sufficient to make the conclusion follow. When should you add premises thought to be implicit, and when not? If, to save an argument, you would need to put into the mouth of its author beliefs he probably does not or would not hold, then the best course of action is to declare the argument invalid.

Though much of the above explanation may seem abstract, you will find that you already have strong intuitions concerning which premise(s), or even which conclusion, is missing. On its face, for example, argument (25) is deductively invalid.

25. Only children are admitted free. So none of our executives is admitted free.

However, were someone to propose (25), we would probably realize that he was assuming an unstated connection between the premise and the conclusion. No formula exists for finding such connections; but in this example we readily see that some connection must be made between children and executives: namely, ‘Our executives are not children.’ If we add this statement as a missing premise, the argument is deductively valid.

Exercise 1.4 Missing premises and conclusion

2. Either I pay Jerry off, or he publishes those photos; but if I pay him off, I’ll be broke. So either I will be broke or I will lose my job.