A Course in Symbolic Logic

Haim Gaifman

Philosophy Department Columbia University

Copyright c

_{°1992 by Haim Gaifman}



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1.1.2 Types and Tokens . . . 4

1.1.3 Vagueness and Open Texture . . . 9

1.1.4 Other Causes of Truth-Value Gaps . . . 13

1.2 Some Other Uses of Declarative Sentences . . . 14

2 Sentential Logic 17 2.0 . . . 17

2.1 Sentences, Connectives, Truth-Tables . . . 19

2.1.0 . . . 19

2.1.1 Negation . . . 20

2.1.2 Conjunction . . . 21

2.1.3 Truth-Tables . . . 23

2.1.4 Atomic Sentences in Sentential Logic . . . 27

2.2.1 Some Basic Laws Concerning Equivalence . . . 32

2.2.2 Disjunction . . . 36

2.2.3 Logical Truth and Falsity, Tautologies and Contradictions . . . 40

2.3 Syntactic Structure . . . 42

2.3.0 . . . 42

2.3.1 Sentences as Trees . . . 50

2.3.2 Polish Notation . . . 53

2.4 Syntax and Semantics . . . 53

2.5 Sentential Logic as an Algebra . . . 54

2.5.0 . . . 54

2.5.1 Using the Equivalence Laws . . . 58

2.5.2 Additional Equivalence Laws . . . 64

2.5.3 Duality . . . 69

2.6 Conditional and Biconditional . . . 72

3 Sentential Logic in Natural Language 80 3.0 . . . 81

3.1 Classical Sentential Connectives in English . . . 85

3.1.1 Negation . . . 85

3.1.2 Conjunction . . . 86

3.1.3 Disjunction . . . 92

4.0 . . . 105

4.1 Logical Implication . . . 105

4.2 Implications with Many Premises . . . 110

4.2.0 . . . 110

4.2.1 Some Basic Implication Laws and Top-Down Derivations . . . 112

4.2.2 Additional Implication Laws and Derivations as Trees . . . 118

4.2.3 Logically Inconsistent Premises . . . 125

4.3 Fool-Proof Method . . . 127

4.3.1 Validity and Counterexamples . . . 127

4.3.2 The Basic Laws . . . 129

4.3.3 The Fool-Proof Method . . . 134

4.4 Proofs by Contradiction . . . 137

4.4.0 . . . 137

4.4.1 The Fool-Proof Method for Proofs by Contradiction . . . 139

4.5 Implications of Sentential Logic in Natural Language . . . 143

4.5.0 . . . 143

5 Mathematical Interlude 153

5.0 . . . 153

5.1 Basic Concepts of Set Theory . . . 154

5.1.1 Sets, Membership and Extensionality . . . 154

5.1.2 Subsets, Intersections, and Unions . . . 159

5.1.3 Sequences and Ordered Pairs . . . 165

5.1.4 Relations and Cartesian Products . . . 166

5.2 Inductive Definitions and Proofs, Formal Languages . . . 173

5.2.1 Inductive definitions . . . 173

5.2.2 Proofs by Induction . . . 184

5.2.3 Formal Languages as Sets of Strings . . . 188

5.2.4 Simultaneous Induction . . . 194

6 The Sentential Calculus 197 6.0 . . . 197

6.1 The Language and Its Semantics . . . 197

6.1.0 . . . 197

6.1.1 Sentences as Strings . . . 198

6.1.2 Semantics of the Sentential Calculus . . . 202

6.1.3 Normal Forms, Truth-Functions and Complete Sets of Connectives . . . 206

6.2 Deductive Systems of Sentential Calculi . . . 217

6.2.1 On Formal Deductive Systems . . . 217

6.2.2 Hilbert-Type Deductive Systems . . . 219

6.2.5 Gentzen-Type Deductive Systems . . . 235

7 Predicate Logic Without Quantifiers 241 7.0 . . . 241

7.1 PC0, The Formal Language and Its Semantics . . . 244

7.1.0 . . . 244

7.1.1 The Semantics of PC0 . . . 246

7.2 PC0 with Equality . . . 250

7.2.0 . . . 250

7.2.1 Top-Down Fool-Proof Methods For PC0 with Equality . . . 253

7.3 Structures of Predicate Logic in Natural Language . . . 261

7.3.1 Variables and Predicates . . . 261

7.3.2 Predicates and Grammatical Categories of Natural Language . . . 264

7.3.3 Meaning Postulates and Logical Truth Revisited . . . 267

7.4 PC∗0 , Predicate Logic with Individual Variables . . . 269

7.4.0 . . . 269

7.4.1 Substitutions . . . 271

7.4.2 Variables and Structural Representation . . . 273

8 First-Order Logic 277 8.1 First View . . . 277

8.2 Wffs and Sentences of FOL . . . 279

8.2.0 . . . 279

8.2.1 Bound and Free Variables . . . 283

8.2.3 Substitutions of Free and Bound Variables . . . 288

8.2.4 First-Order Languages with Function Symbols . . . 291

8.3 First-Order Quantification in Natural Language . . . 295

8.3.1 Natural Language and the Use of Variables . . . 295

8.3.2 Some Basic Forms of Quantification . . . 297

8.3.3 Universal Quantification . . . 302

8.3.4 Existential Quantification . . . 307

8.3.5 More on First Order Quantification in English . . . 309

8.3.6 Formalization Techniques . . . 314

9 FOL: Models, Truth and Logical Implication 323 9.1 Models, Satisfaction and Truth . . . 323

9.1.0 . . . 323

9.1.1 The Truth Definition . . . 325

9.1.2 Defining Sets and Relations by Wffs . . . 331

9.2 Logical Implications in FOL . . . 334

9.2.0 . . . 334

9.2.1 Proving Non-Implications by Counterexamples . . . 336

9.2.2 Proving Implications by Direct Semantic Arguments . . . 338

9.2.3 Equivalence Laws and Simplifications in FOL . . . 341

9.3 The Top-Down Derivation Method for FOL Implications . . . 345

9.3.0 . . . 345

9.3.1 The Implication Laws for FOL . . . 345

9.3.2 Examples of Top-Down Derivations . . . 350

Logic is concerned with the fundamental patterns of conceptual thought. It uncovers struc-tures that underlie our thinking in everyday life and in domains that have very little in common, as diverse, for example, as mathematics, history, or jurisprudence. A rather rough idea of the scope of logic can be obtained by noting certain keywords: object, property, con-cept, relation, true, false, negation, names, common names, deduction, implication, necessity, possibility, and others.

Symbolic logic aims at setting up formal systems that bring to the fore basic aspects of reasoning. These systems can be regarded as artificial languages into which we try to translate statements of natural language (e.g., English). While many aspects of the original statement are lost in such a translation, others are made explicit. It is these latter aspects that are the focus of the logical investigation.

Historically, logic was conceived as the science of valid reasoning, one that derives solely from the meaning of words such as ‘not’, ‘and’, ‘or’, ‘all’, ‘every’, ‘there is’, and others, or syntactical constructs like ‘if ... then ’. These words and constructs are sometimes called logical particles. A logical particle plays the same role in domains that have nothing else in common.

Here is a traditional very simple example. From the two premises:

All animals are mortal. All humans are animals. we infer, by logic alone:

All humans are mortal.

The inference is purely logical; it does not depend on the meanings of ‘animal’, and ‘human’, but solely on the meaning of the construct

all .... are i

The same pattern underlies the following inference, in which all non-logical concepts are different. From:

All uncharged elementary particles are unaffected by electromagnetic fields. All photons are uncharged elementary particles.

we can infer:

All photons are unaffected by electromagnetic fields.

The two cases are regarded in Aristotelian logic as instances of the same syllogism–a certain elementary type of inference. The particular syllogism under which the two examples fall is the following scheme, where the premises are written above the line and the conclusion under it:

(1)

All Ps are Qs All Rs are Ps All Rs are Qs Our first example is obtained if we substitute:

‘animal’ for ‘P’, ‘mortal being’ for ‘Q’, ‘human’ for ‘R’.

The second is obtained if we substitute ‘uncharged elementary particle’ for ‘P’, ‘thing unaf-fected by electromagnetic fields’ for ‘Q’, ‘photon’ for ‘R’.

Had we substituted in the first case ‘immortal being’ for ‘Q’, (instead of ‘mortal being’) we would have gotten the inference:

All animals are immortal. All humans are animals. ---All humans are immortal.

Here the first premise is false, and so is the conclusion. But the inference is correct. Its correctness does not require that the premises be true, but that they stand in a certain logical relation to the conclusion: they should logically imply it.

The use of schematic symbols is a first step in setting up a formalism. Yet, there is a long way from here to a fully fledged formal language. As will become clear during the course, there is much more to a formalism than the employment of formal symbols.

Either Jill went to the movie, or she went to the party. If Jill went to the party, she met there Jack.

Jill did not go to the movie.

we can infer:

Jill met Jack at the party.

The logical particles on which this last inference is based are:

either...or , if... then , not

The same scheme is exemplified by the following inference. (Again, its validity does not mean that the premises are true, but only that they imply the conclusion.) From:

Either Ms. Hill invented her story, or Mr. Thomas harassed her. If Mr. Thomas harassed Ms. Hill, then he lied to the Senate. Ms. Hill did not invent her story.

we can infer:

Mr. Thomas lied to the Senate.

The scheme that covers both of these inferences can be written in the following self-explanatory notation, where the premises (here there are three) are above the line and the conclusion is below it: (2) A or B If B then C not A C

Note that the schematic symbols of (2) are of a different kind than those of (1). Whereas in (1) they stand for general names: ‘human’, ‘mortal being’, ‘photon’, etc., they stand in (2) for complete sentences, such as ‘Jill went to the party’ or ‘Mr. Thomas harassed Ms. Hill’. The part of logic that takes complete sentences as basic units, and investigates the combining of sentences into other sentences, by means of expressions such as ‘not’, ‘and’, ‘or’, ‘if ... then ’, is called sentential logic. The sentence-combining operations are called sentential operations, and the resulting sentences (e.g., those in (2)) are sentential combinations, or

sentential compounds. Sentential logic is a basic part that is usually included in other, richer systems of logic.

The logic that treats, in addition to sentential operations, the attribution of properties to objects (‘Jill is not tall, but pretty’), or relations between objects (‘Jack likes Jill’) is predicate logic. If we add to predicate logic means for expressing general statements, like those formed in English by using ‘all’, ‘every’, ‘some’ (‘All humans are mortal’, ‘Jill likes some tall man’, ‘Everyone likes somebody’), then we get first-order quantification logic, known also as first-order predicate logic, or, for short, first-order logic.

The examples schematized by (1) and by (2), are rather simple. In general, the recasting of a sentence as an instance of some scheme is far from obvious. It amounts to an analysis: a way of getting the sentence from components, which displays a certain logical structure. The choice of components and how, in general, we combine them is crucial for the development of logic, just as the choice of primitive concepts and basic assumptions is crucial for any science. Logic was bogged down for more than twenty centuries, because it was tied to a particular way of analyzing sentences, one that is based on syllogisms. It could not accommodate the wealth of logical reasoning that can be expressed in natural language and is apparent in the deductive sciences. To be sure, valuable insights and sophisticated analyses were achieved during that period. But only at the turn of the century, when the basic Aristotelian setup has been abandoned in favour of an essentially different approach, did symbolic logic come into its own.

Examples such as (1) and (2) can serve as entry points, but they do not show what symbolic logic is about. We are not concerned merely with schematizing this or that way of reasoning, but with the construction and the study of formal languages. The study is a tool in the investigation of conceptual thought. It aims, and has rich implications, beyond the linguistic dimension.

Logic is also not restricted to the first-order case. Other logical formalisms have been designed in order to handle a variety of notions, including those expressed by the terms ‘possible’, ‘necessary’, ‘can’, ‘must’, and many others. There are also numerous systems that treat a great variety of reasoning methods, which are quite different from the type exemplified in first-order logic.

A Bit of History

The beginning of logic goes back to Aristotle (384 – 322 B.C.). Aristotle’s works contain, besides non-formal or semi-formal investigations of logical topics, a schematization and a systematic theory of syllogisms. This logic was developed and enlarged in the middle ages but remained very limited, owing to its underlying approach, which bases the logic on the grammatical subject-predicate relation. Other parts of logic, namely fragments of sentential logic, have been researched by the Stoic philosophers (330 – 100 B.C.). They made use of

schematic symbols, which did not however amount to a formal system. That stage had to come much later.

The decisive steps in logic have taken place in the second half of the nineteen and the beginning of the twentieth century, in the works of George Boole (1815 – 1864), Charles Peirce (1839 – 1914), Giuseppe Peano (1858 – 1932), and–fore and foremost–Gottlob Frege (1848 – 1925), whose work Begriffschrift (1879) presents the first logical system rich enough for expressing mathematical reasoning. The other most significant event was the publication, in three volumes, of the Principia Mathematicae by Russell and Whitehead (1910 — 1913), an extremely ambitious work from which the current formalisms derive directly. (Frege’s work was little noticed at the time, though Russell and other logicians knew it.) The big bang in logic was directly related to new developments in mathematics, mostly in the works of Dedekind and Cantor. It owed much to the creation–by the latter– of set theory (1874). After the work of Russell and Whitehead, it has been taken much further by mathematicians and philosophers, among whom we find Hilbert, Ackerman, Ramsey, L¨owenheim, Skolem, and–somewhat later–Gentzen, Herbrand, Church, Tarski and many others. It owes its most important results to Kurt G¨odel–landmark results of deep philosophical significance.

“Critical Thinking”, what Symbolic Logic Is Not

In the middle ages logic was described as the “the art of reasoning”. It has been, and often still is, viewed as the discipline concerned with correct ways of deriving conclusions from given premises. Much of medieval logic was concerned with the analysis and the classification of arguments to be found in various kinds of discourse, from everyday life to politics, philos-ophy, and theology. In this way logic was related to rhetoric: the art of convincing people through talk. The medieval logicians devoted considerable effort to the uncovering of invalid arguments, known as fallacies, of which they classified a few dozens.

Perhaps a vestige of this tradition, or a renewal of it, is a logic course given in some curricula under “Critical Thinking”. In certain cases, I am told, the name is a euphemism for teaching text comprehension, filling thereby a high school lacuna. But, to judge by the textbooks, the course commonly comprises an assortment of topics that have to do with inference-drawing: deductive, inductive, statistics and probability, some elements of symbolic logic, and a discus-sion of various fallacies. There is no doubt that there is value to such an overview, or to the analysis of common fallacies. pretense of the title should be discounted. Critical thinking, without the scare quotes, is not something that can be taught directly as an academic subject. (Imagine a course called “Thinking”.)

Thinking, like walking, is learned by practice. And good thinking, clear, rigorous, critical, is what one acquires in life, or through work in particular disciplines. Observations and corrective tips are useful, but they will not get you far, unless incorporated in long continuous experience. Clear thinking is a lifelong project.

A course in symbolic logic is not a course in “critical thinking”. What you will learn may, hopefully will, affect your thinking. You will study a certain formalism, using which you will learn, among the rest, to analyze certain types of English sentences, to reconstruct them and to trace their logical relationships. These should make you more sensitive to some aspects of meaning and of reasoning. But any improvement of your thinking ability will be a consequence of the mental effort and the practice that the course requires. Thinking does not arrive by learning a set of rules, or by following this or that prescription.

The Research Program of Symbolic Logic

Symbolic logic is an enterprise that takes its “raw material” from the existing activity of reasoning, displayed by human beings. It focuses on certain fundamental patterns and tries to represent them in an artificial language (also called calculus). The formal language is supposed to capture basic aspects of conceptual thought.

The enterprise comprises discovery as well as construction. It would not be accurate to say that the investigator merely uncovers existing patterns. The structures are revealed and constructed at the same time. Having constructed a formal system, we can go back and see how much of our reasoning is actually captured by it. The formalism is philosophically interesting, or fruitful, in as much as it gives us a handle on essential aspects of thought. It can be also of technical interest. For it can provide tools for language processing and computer simulation of reasoning processes. In either respect, there is no a priori guarantee of success. We should keep in mind that, even when the formal system represents something basic or important, its significance may be tied to some particular segment of our cognitive activity. There should be no pretense of “reducing” the enormous wealth of our thinking to an artificial symbolic system. At least there should be no a priori conviction that it can be done. To what extent can human reasoning be captured by a formal system is an intriguing and difficult question. It has been much discussed in the context of artificial intelligence (not always with the best results).

Let us compare investigations in logic to investigations of human body-postures. A medical researcher can use x rays and other scans, or slow motion pictures, in order to find out how human bodies function in a range of activities. He will accumulate a vast amount of data. In order to organize his data into a meaningful account, he may classify it according to “human types”, establish certain regularities and formulate some general rules. Here he is already deviating from the “raw material”; he is introducing an abstract system in as much as his types are idealized constructs, which actual humans only approximate. (Not to mention the fact that in the very acquiring of data he is already making use of some theoretical system.) Our investigator may also arrive at conclusions concerning the correct ways in which humans should walk, or sit in order to preserve healthy tissue, minimize unnecessary tension, etc. His research will establish certain norms; not only will it reveal how humans use their bodies,

but also how they ought to. He may even conclude that most people do not maintain their bodies as they should. In this way the descriptive merges into the normative. And there is a feedback, for the normative may provide further concepts and further guidelines for the descriptive. Our investigator’s conclusions may be subject to debates, objections, or revisions. Here, as well, the descriptive and the normative are interlaced. Finally, it is possible that certain recommendations become highly influential, to the extent of being incorporated in the educational system. They would thus become part of the culture, determining, to an extent, the actual behaviour of humans, say, the way they hammer, or the kind of chairs they prefer. All of these aspects exist when we are concerned with human thinking. Here, as well, the descriptive merges into the normative. Having started by investigating actual thinking, we might end by concluding how thinking ought to be done. Furthermore, the enterprise might influence actual thinking habits, projecting back on the very culture within which it was carried out.

First-Order Logic

The development of symbolic logic has had by now far reaching consequences, which have affected deeply our philosophical outlook. Coupled with certain technological developments, it has also affected our culture in general. The basic system that the project has yielded is first-order logic. The name refers to a type of language, characterized–as stated in the first section–by a certain logical vocabulary. First-order logic serves also as the core of many modifications, restrictions and–most important–enlargements.

Although first-order logic is rather simple, all mathematical reasoning (derivations used in proving mathematical results) can be reproduced within it. Since it is completely defined by precise formal rules, first-order logic can itself be treated and investigated as a mathematical system . Mathematical logicians have done this, and they have been able to prove highly interesting theorems about it; for example, theorems that assert that certain statements are unprovable from such and such axioms. These and other theorems about the system are known as metatheorems; for they are not about numbers, equations, geometrical spaces, or algebras, but about the language in which theorems about numbers, equations, geometrical spaces, or algebras are proven. They enlighten us about its possibilities and limitations. In philosophy, the development of symbolic logic has had far reaching effects. The role of logic within the general philosophical inquiry has been a subject of debate. There is a wide spectrum of opinions, from those who accord it a central place, to those who restrict it to a specialized area. The subject’s importance varies with one’s interests and the flavour of one’s philosophy. In any case, logic is considered a basic subject, knowledge of which is required in most graduate and undergraduate philosophy programs.

The Wider Scope of Artificial Languages

Historically, the idea of a comprehensive formal language, defined by precise rules of math-ematical nature, goes back to Leibniz (1646 — 1716). Leibniz thought of an arithmetical representation of concepts, and dreamed of a universal formal language within which all truth could be expressed. A similar vision was also before the eyes of Frege and Russell. The actual languages produced by logicians fall short of any kind of Leibnizian dream. This is no accident, for by now we have highly convincing reasons for discounting the possibility of such a universal language. The reasons have been provided by logic itself, in the form of certain metatheorems (G¨odel’s incompleteness results).

As noted, there is by now a wide variety of logical systems, which express many aspects of reasoning and of conceptual organization. In the last forty years the enterprise of artificial lan-guages has undergone a radical change due to the emergence of computer science. Computer scientists have developed scores of languages of types different from the types constructed by logicians. Their goal has not been the investigation of thought, but the description and the manipulation of computational activity. Computer languages serve to define the functioning of computers and to “communicate” with them, to “tell” a computer what to do. A major consideration that enters into the setting up of programming languages is that of efficiency: programs should be implementable in reasonable run time, on practical hardware. Usually, there is a trade-off between a program’s simplicity and conceptual clarity, on one hand, and its efficiency on the other.

At the same time we have witnessed a marked convergence of some of the projects of pro-gramming languages and those of logic. For example, the propro-gramming language LISP (and its many variants) is closely connected with the logical system known as the λ calculus, devel-oped in the thirties by Church. The calculus and its variants have been the focus of a great amount of research by logicians and computer scientists. There has been also an important direct effect of symbolic logic on computer science. The clarity and simplicity of first-order logic suggested its use as a basis for a programming language. Ways were found to imple-ment portions of first-order logic in an efficient way, which led to the developimple-ment of what is known as logic programming. This, by now, is a vast area with hundreds, if not thousands, of researchers. Logic enters also, in an essential way, into other areas of computer science, in particular, artificial intelligence and automated theorem proving.

The Goals and the Structure of the Course

The main purpose of the course is to teach FOL (first-order logic), to relate it to natural language (English) and to point out various philosophical problems that arise thereby. The level is elementary, in as much as the course does not include proofs of the deeper results, such as G¨odel’s completeness and incompleteness theorems. Nonetheless, the course aims at

providing a good grasp of FOL. This includes an understanding of formal syntactic structures an understanding of the semantics (that is, of the notion of an interpretation of the language and how it determines truth and falsity), the mastering of certain deductive and related techniques, and an ability to use the formal system in the analysis of English sentences and informal reasoning.

The first chapter, which is more of a general nature, is intended to clarify the concepts presuppositions that underlie the project of classical logic: the category of declarative sentence and the classification of all declarative sentences into true and false. Various problems that arise when trying to apply this framework to natural language are discussed, among which are indexicality, ambiguity, vagueness and open texture. This introduction is also intended to put symbolic logic into a wider and more concrete perspective, removing from it any false aura of a given truth.

We get down to the actual course material in chapter 2, which provides a first view of sentential logic, based on a semantic-oriented approach. Here are defined the connectives, a variety of syntactic concepts (components, main connective, unique readability and others), truth-tables and the concept of logical equivalence. The chapter contains also various simplification techniques and an algebraic perspective on the system.

Having gotten a sufficient grasp of the formalism, we proceed in chapter 3 to match the formal setup with English. The chapter discusses, with the aid of many examples, ways of expressing in English the classical connectives, the extent to which English sentences can be rendered in sentential logic and what such an analysis reveals.

Chapter 4 treats logical implications and proofs. After defining (semantically) the concept of a logical implication, the chapter presents a very convenient method of deciding whether a purported implication, from a set of premises to a conclusion, is valid. The method combines the ideas of Gentzen’s calculus with a top-down derivation technique. If the implication is valid it yields a formal proof (which can be rewritten bottom-up), if not–it produces a counterexample, thereby establishing the non-validity. In the last section we return to natural language and consider possible recastings of various inferences carried in English into a formal mode. Here we discuss also some concepts from the philosophy of language, such as implicature.

Chapter 5 provides some basic mathematical tools that are needed for the rigorous treatment of logical calculi, in particular, for defining interpretations (models), giving a truth-definition, and for setting up deductive systems. These tools consist of elementary notions of set theory, and the basic techniques of inductive definitions and proofs.

In chapter 6 the formal language of the sentential calculus is defined with full mathematical rigor, together with the concept of a deductive system. Here the crucial distinction between syntax and semantics is clarified and the relation between the two is established in terms of soundness and completeness.

Chapter 7 presents predicate logic (without quantifiers), based on a vocabulary of predicates and individual constants. The equality predicate is introduced and the top-down method for deciding logical implications is extended so as to include atomic sentences that are equalities. This chapter treats also predication in English. In the second half of that chapter the system is extended by the introduction of variables and steps are taken towards the introduction of quantifiers.

In chapter 8 the fully fledged language of first-order logic is defined, as well as the basic syntactic concepts that go with it: quantifier scope, free and bound variables, and legitimate substitutions of terms. Emphasis is placed on an intuitive understanding of what first-order formulas express and on translations from and into English.

The general concept of a model for a first-order language is presented in chapter 9, as well as the definitions of satisfaction and truth. Based on these we get the concepts of logical equivalence and logical implication. The top-down derivation technique of sentential logic is extended to first-order logic. As before, the method is guaranteed to yield a proof of any valid implication. A proof of this claim–which is not included in this chapter–yields immediately the completeness theorem.

Declarative Sentences

1.0

Symbolic logic is concerned first and foremost with declarative sentences. These are sentences that purport to make factual statements. They are true if what they state is the case, and they are false–if it is not.

‘Grass is green,’ ‘Every prime number is odd,’ ‘Not every prime number is odd,’ ‘The moon is larger than the earth,’ ‘John Kennedy was not the first president of the USA to be assassinated,’ ‘Jack loves Jill, but wouldn’t admit it’,

declarative sentences. The first the third and the fifth are true. The second and the fourth are false. The last is true just in this case: (i) Jack loves Jill and (ii) Jack does not admit that he loves Jill.

You can see what distinguishes declarative sentences by comparing them with other types. Interrogative sentences, for example, are used to express questions:

‘Who deduced existence from thinking?’ ‘Did Homer write the Odyssey?’

Such sentences call for answers, which–depending on the kind of question–come in several forms; e.g., the first of the above questions calls for a name of a person, the second–for a ‘yes’ or a ‘no’.

Commands are expressed by means of imperative sentences, such as:

‘Love thy neighbour as thou lovest thyself,’ ‘Do not walk on the grass’. 1

Given in the appropriate circumstance, by someone with authority, they call for compliance. None of these, or of the other kinds of sentence, is true or false in the same sense that a declarative sentence is. We can say of a question that it is to the point, important, interesting, and so on, or that it is irrelevant, misleading or ill-posed. A command can be justified, appropriate, or illegitimate or out of place. But truth and falsity–in the basic, elementary sense of these terms–pertain to declarative sentences only. Sentences are used in many ways to achieve diverse purposes in human interaction. To question and to command are only two of a great variety of linguistic acts. We have requests, greetings, condolences, promises, oaths, and many others. What is then, within this picture of human interaction, the principal role of declarative sentences? It is–first and foremost–to convey information, to tell someone that such and such is the case, that a certain state of affairs obtains.

But over and above their use in human communication, declarative sentences constitute de-scriptions (or purported dede-scriptions) of some reality: a reality perceived by humans, but perceived as existing in itself, independently of its being described. A logical investigation of declarative sentences can serve as tool that clarifies the nature of that reality. By uncovering certain basic features of our thinking it may also uncover basic features of the world that the thinking organizes. One can appreciate already, at this stage, the potential that the logic of declarative sentences has for epistemology–the inquiry into the nature of knowledge, and for ontology–the inquiry into the nature of reality.

For this reason, when sentences are the target of a philosophical inquiry, the declarative ones play the most important role. Formal methods are not restricted to declarative sentences; formal systems have been designed for handling other types, such as questions and commands. But symbolic logic is mostly about declarative sentences, and it is with these that we shall be concerned here.

Henceforth, I shall use ‘sentence’ to refer to declarative sentences, unless indicated otherwise.

1.1

Truth-Values

1.1.0

A declarative sentence is true or false, according as to whether what it states is, or is not the case. It is very convenient to introduce two abstract objects, TRUE and FALSE, and to mark the sentence’s being true by assigning to it the value TRUE, and its being false–by assigning to it the value FALSE. We refer to these objects as truth-values.

Truth-values are merely a technical device. They make it possible to use concise and clear formulations. One should not be mystified by these objects and one should not look for hidden meanings. To say that a sentence has the value TRUE is just another way of saying that it is

true, and to say that it has the value FALSE is no more than saying that it is false. Any two objects can be chosen as TRUE and FALSE. For the only thing that matters about truth-values is their use as markers of truth and falsity.

Notation: We use ‘T’ and ‘F’ as abbreviations for ‘TRUE’ and ‘FALSE’.

While the introduction of truth-values is a technical convenience, the very possibility of classi-fying sentences into true and false is a substantial philosophical issue. Does every sentence fall under one of these categories? Little reflection will show that in our everyday discourse such a classification is, to a large extent, problematic. The problem is not of knowing a sentence’s truth-value; we may not know whether Oswald was Kennedy’s only assassin, or whether 232₊₁ is a prime number, but we find no difficulty in appreciating the fact that, independently of our knowledge, ‘Oswald was Kennedy’s only assassin’ is either true or false, and so is ‘232+ 1 is prime’. The problem is that in many cases it is not clear what the conditions for truth and falsity are and whether the classification applies at all. Perhaps certain sentences should on various occasions be considered as neither true nor false; which means, in our terminology, that neither T nor F is their value.

The logic we are going to study, which is classical two-valued logic, assumes bivalence: the principle that every sentence has one of the two values T or F. This principle makes for systems that are relatively simple and highly fruitful at the same time. Logicians have, of course, been aware of the problems surrounding the assignment of truth-values. But in order to get off ground, an inquiry must start by focusing on some aspects, while others are ignored. Later it may broadened so as to handle additional features and other situations. The art is to know what to focus on and what, initially, to ignore. Classical two-valued logic has been extremely successful in contexts where bivalence prevails. And it serves also as a point of reference for further investigations, where problems of missing truth-values can be addressed. In short, we are doing what every scientist does, when he starts with a deliberately idealized picture.

In the coming sections of this chapter I shall highlight the main situations where the as-signment of definite truth-values is called into question. This will also be an occasion for discussing briefly some major topics regarding language: context-dependency, tokens and types, indexicals, ambiguity and vagueness.

1.1.1

Context Dependency

The same sentence may have different truth-values on different occasions of its use. Consider, for example:

Jack: I am tall, Jill: I am tall.

If Jack is not tall, but Jill is, then–in Jack’s mouth–the sentence is false, but in Jill’s mouth it is true. This shows that we are dealing here with two kinds of things: the entity referred to as sentence, which is the same in the mouth of Jack and the mouth of Jill, and its different utterances. The distinction is fundamental; it, and some of its hinging phenomena, will be now discussed.

1.1.2

Types and Tokens

Linguistic intercourse is based on the production of certain physical items: stretches of sounds, marks on paper, and their like, which are interpreted as words and sentences. Such items are called tokens. When you started to read this section you encountered a token of ‘linguistic’, which was part of a token of the opening sentence. And what you have just encountered is another token of ‘linguistic’, this time enclosed in inverted commas.

Of course, “token” is meaningful only in as much as it is a token of something, a word, a letter, a sentence, or–in general–some other, more abstract entity. This other entity is called type. By a sentence-token we mean a token of a sentence, that is, a token of a sentence-type. Note that our terms ‘letter’, ‘word’, or ‘sentence’, are ambiguous. Sometimes they refer to types and sometimes to tokens. This is shown clearly in situations that involve counting. How many words are there on this page? The answer depends on whether you count repetitions of the same word. If you do, then you interpret “word” as word-token, if you don’t–you interpret it as type. Usually the number of tokens exceeds the number of word-types; for we do, as a rule, repeat.

Our ability to use language is preconditioned by our ability to recognize different tokens as being tokens of the same type. This “sameness” relation is often indicated by the physical similarity of tokens. Thus, the two tokens of ‘ability’ in the first sentence of this paragraph have exactly the same shape. But on the whole, what counts as being tokens of the same type is a matter of convention; similarity is not necessary. Think of the different fonts one can use for the same letters, and of the enormous variety of handwritings. (Reading someone’s written words is often impossible without knowing the language, even when the alphabet is known.) And to clinch the point, note that the same words are represented by tokens in different physical media: the acoustic and the visual.

Things would have been considerably simpler if we could disregard the difference between tokens of the same type. But this is not so; for, as the last example shows, different tokens of the same type may have different truth-values.

Indexicals and Demostratives

An indexical is a word whose reference depends–in a systematic way–on certain surroundings of its token, e.g., the token’s origin, its time, or its place. Such is the pronoun ‘I’, which refers to its utterer, and such are the words ‘now’ and ‘here’, which refer to the utterance’s time and place. The shift of reference may results in a truth-value change. Indexicals are, indeed, the most common cause for assigning different truth-values to different tokens of the same sentence. In the last example the difference in truth-value is caused by the indexical ‘I’, which denotes Jack, in the mouth of Jack, Jill–in the mouth of Jill. Quite often the indexicals are implicit. In

(1) It is raining,

the present tense indicates that the time is the time of the utterance. And, in the absence of an explicit place indication, the place is the place of the utterance. When (1) is uttered in New York, on May 17 1992 at 9:00 AM, it is equivalent to:

(10_{) It is raining in New York, on May 17 1992 at 9:00 AM.}

It is not difficult to spot indexicals, once you are aware of their possible existence. Besides ‘now’ and ‘here’, we have also the indexicals ‘yesterday’, ‘tomorrow’, ‘last week’, ‘next room’ and many others.

Demonstratives, like indexicals, have systematically determined token-dependent references. They usually require an accompanying demonstration–some non-linguistic act of pointing. Such are the words ‘that’ and ‘this’. The use of ‘you’ involves a demonstrative element (the act of addressing somebody), as do sometimes ‘he’ and ‘she’. (It is not always easy to describe what exactly the demonstration is, but this is another matter.) Sometimes a distinction is made between pure indexicals–which, like ‘I’, require no demonstration–and non-pure ones. And sometimes ‘indexical’ is used for both indexicals and demonstratives.

Some Kinds of Ambiguity

Many, perhaps most, proper names denote different objects on different occasions. ‘Roosevelt’ can mean either the first or the second USA president of this name, ‘Dewey’ can refer either to the philosopher or to the Republican politician, ‘Tolstoy’ can refer to any of several Russian writers. First and second names, or initials, can help in avoiding confusion (thus, we distin-guish between Teddy Roosevelt–the man who was fond of speaking softly while carrying a big stick, and Franklin Roosevelt–the second world war leader in the wheel chair). Additional names reduce the ambiguity, but need not eliminate it. A glance in the telephone directory under ‘Smith’, or–in New York–under ‘Cohen’, will show this. Other distinguishing marks

can be used: ‘Dewey the philosopher’ versus ‘Dewey the politician’, ‘Johan Strauss the father’ versus ‘Johan Strauss the son’.

Above all, a name’s denotation is determined by the context in which the name is used. (If I ask my daughter: has Bill telephoned? it is unlikely that she will take me to have referred to Bill Clinton.) But there are no clear-cut linguistic rules that regulate this. Various factors enter: what has been stated before, the topic of the discussion, and what is known of the interlocutor’s knowledge and intentions. Proper names behave quite differently from indexicals; the latter are subject to systematic rules (‘you’ refers to the person addressed, ‘now’ refers to the time of the utterance, etc.), the former are not.

Besides indexicals and proper names, linguistic expressions in general may have different denotations, or meanings, on different occasions. The “same word” might mean different things, e.g., tank–a large container for storage, and tank–an armored vehicle on caterpillar treads. But here we should be careful, for the very difference of meaning is often taken to constitute a difference of words (i.e., of types). Homonyms are different words written and pronounced in the same way; their difference rests solely on difference in meaning. When ‘tank’ is split into homonyms, it is no longer a single ambiguous word. Accordingly,

(2) John jumped into the tank,

is, strictly speaking, not an ambiguous sentence (which has different truth-values on different occasions) but an ambiguous expression that can be read as more than one sentence: a sentence containing the as-container’-homonym, and a sentence containing the ‘tank-as-armored-vehicle’-homonym. The context in which (2) occurs (e.g., sentences that come before and after it) may help us to decide the intended reading.

By contrast, different tokens of ‘It is now raining here’ are tokens of the same sentence. For ‘now’ and ‘here’ do not constitute different words when used at different times, or at different places. A child learning to speak does not coin a new English word, when he uses ‘I’ for the first time. We can however say that the English language gained a new word when ‘tank’ (already in use as a name of certain containers) was introduced as a name of certain armored cars1_{. Many cases of ambiguity–where the meanings are linked–do not deserve} to be treated as homonyms. ‘Word’ can mean word-type or word-token, but this does not constitute sufficient ground for distinguishing two homonyms. We would do better, one feels, to regard it as a single ambiguous word.

Ambiguous terms are not the only source of sentential ambiguity; often the sentential structure itself can be construed in more than one way.

(3) Taking the money out of his wallet, he put it on the table.

1_{By the same reasoning, no new word is coined when a new baby is given a current name, like ‘Henry’.}

Was it the money or the wallet he put on the table? That depends on the syntactic structure of (3); it is the first, if ‘it’ goes proxy for ‘the money’, the second–if it goes proxy for ‘his wallet’. Syntactic ambiguity takes place when the same sequence of words lends itself to different structural interpretations. The truth-value can depend on the way we structure the sentence, or–in more technical terminology–on the way we parse it. Here, again, the context can decide the intended parsing.

We can have a concept of “sentence” according to which different parsings determine different sentences; if so, (3) is to be regarded in the same light as (2): an expression representing more than one sentence. But on the usual, everyday concept of sentence, (3) is a single syntactically ambiguous sentence.

In symbolic logic the artificial language is set up in a way that bars any ambiguity. Ev-ery sentence has a unique syntactic structure and all referring terms have unique, context-independent references. Therefore a translation from natural language into symbolic logic involves an interpretation whereby, in cases of ambiguity, a particular reading is chosen. As a preparatory step, we can try to paraphrase the sentences of natural language, so as to eliminate various context dependencies. This is the subject of the next subsection.

Eliminating Simple Context Dependencies

Dependencies on context, which are caused by indexicals or by ambiguity, can be eliminated by replacing indexicals and ambiguous terms by terms that have unique and fixed denotations throughout the discussion.

For example, each occurrence of ‘word’ can be replaced by ‘word-type’ or by ‘word-token’, depending on whether the first or the second is meant; and when either will do, we can make this explicit by writing ‘word-type or word-token’. Sometimes we resort to new nicknames ‘The first John’ for our old school mate, ‘The second John’ for the new department chief. And, to be clear and succinct, we can introduce ‘John1’ and ‘John2’. The same policy can be used to eliminate homonyms. To be sure, ‘John1’ is not an English name, but a newly coined word. Our aim, however, is not to preserve the original phrasings, but to recast them into forms more suitable for logical analysis.

Indexicals can be eliminated by using names or descriptive phrases with fixed denotations. Thus (1)–when uttered in New York at 9:00 AM, May 17 1992–is rephrased as (10_{). And ‘I} am tall’–when uttered by Jill on May 10 1992–is recast as:

(4) Jill is tall on May 10 1992.

Here ‘is’ is to be interpreted in a timeless mode, something like is/was/will-be. Note the different degrees of precision in the specifications of time. The weather may change from hour to hour (hence we have ‘9:00 AM’ in (10_{)), but presumably Jill’s being tall is not subject to}

hourly changes.

In this way sentence-tokens that involve context-dependency are translated into what Quine named eternal sentences, that is: sentences whose truth-values do not depend on time, loca-tion, or other contextual elements.

Note: We are not concerned here with a conceptual elimination of indexicals. The time scale used in (10_{) and (4) is defined by referring to the planet earth, and ‘earth’ is defined by} a demonstrative: this planet, or the planet we are now on. We aim only to eliminate context dependency that can cause trouble in logical analysis. And this is achieved by paraphrases of the kind just given.

Note also that, for local purposes: if we are concerned only with a particular discourse, we have only to replace the terms whose denotations vary within that discourse. If ‘today’ refers to the same day in all sentence-tokens that are relevant to our purpose, we need not replace it.

The situation is altogether different when it comes to ambiguities in general. If my daughter tells me ‘Bill telephoned an hour ago’ , I shall probably guess correctly who of the various Bill’s it was. But all I can appeal to is an assortment of considerations: the Bill I was expecting a call from, the Bill likely to call at that time, the Bill that has recently figured in our social life, etc. Considerations of this kind are classified in the philosophy of language under pragmatics. The resort to pragmatics, rather than to clear-cut rules, is of great interest for linguistic theory and the philosophy of language; but is of no concern for logic, at least not the logic that is our present subject. For our purposes, it is enough that there is a paraphrase that eliminates context-dependency. Logic takes it up from there. How we get there is another concern.

The cases considered thus far are the tip of the iceberg. The real game of ambiguity and context-dependency starts when adjectives, descriptive phrases, and verbs are brought into the picture. This subject–a wide area of linguistic and philosophical investigations–is not part of this course. A few observations may however give us some idea of the extent of the problems. Consider attributes such as

small, big, heavy, dark, high, fast, slow, rich,

and their like. You don’t need much reflection to realize that they are relative and highly context-dependent.

(5) Lionel is big.

You may deduce from (5) and (6) that Lionel is bigger than Kitty. Not so if it is known that Lionel is a cat and Kitty is a lioness. In that case the ‘big’ in (5) should read: ‘a big cat’, or ‘big as cats go’, and the ‘small’ in (6)– as ‘a small lioness’. If we apply the strategy suggested above for ambiguous names, we shall split ‘big’ and ‘small’ into many adjectives, say ‘bigx’ and ‘smallx’ where ‘x’ indicates some kind of objects; the ‘big’ in (5) is thus read as ‘bigc’: big on the scale of cats, and the ‘small’ in (6)–as ‘smalll’: small on the scale of lions. Another, better strategy is to provide for a systematic treatment of compounds such as ‘big as a ...’, ‘rich as a ...’, where ‘...’ describes some (natural) class.

Systematic treatments do not apply however when the adjective must be interpreted by refer-ring to a particular occasion. ‘The trunk is heavy’ can mean that the trunk is heavy, when I do the lifting, or when you do the lifting, or when both of us do the lifting. And occasionally there is nothing precise or explicit that we can fall back on.

(7) Jack Havenhearst lives in a high building on the outskirts of Toronto.

How high is “high”? A high building in Jerusalem is not so high in Manhattan. The context may decide it, or it may not. Perhaps the speaker has derived his statement from some vague recollection. In cases like this, when ambiguity is tied up with vagueness, the very possession of a definite truth-value is put into question.

Before proceeding, note that the problems just mentioned concern attributes of the “neutral” kind. We have not touched on evaluative terms such as

‘beautiful’, ‘ugly’, ‘tasteful’, ‘repulsive’, ‘nice’, ‘sexy’, ‘attractive’, and their like, which involve additional subjective dimensions, nor on:

‘important’, ‘significant’, ‘marginal’, ‘central’, not to mention the ubiquitous ‘good’ and ‘bad’.

1.1.3

Vagueness and Open Texture

Some people are definitely bald, some are definitely not. But some are borderline cases, for whom the question: Is he bald? does not seem to have a yes-or-no answer. The same applies to every type of statement you might think of in the context of everyday discourse. For example, is it raining now? Sometimes the answer is yes, sometimes no, and sometimes neither appears satisfactory (does the present very light drizzle qualify as “rain”?)

Are we now in the USA? That type of question has almost always a well-defined answer, even when we don’t know it; for international borders–things of extreme significance–are very

carefully drawn. But what if somebody happens to straddle the border-line? There is first the problem of pinpointing one’s location, and second the problem of pinpointing the border; and in both the “pinpointing” has limited precision. Even the question: Is the date now May 17 1992? may, on some occasion, lack a yes-or-no answer; for the time-point defined by the utterance of ‘now’ is determined with no more than a certain precision, surely not up to a millisecond, say.

In everyday discourse we often handle borderline cases by employing a more refined classifi-cation. For example, we can use ‘quite bald’ and ‘hairy’ for the clear cases, and ‘baldish’ for those in between. This provides for more accurate descriptions. But it leaves us in the same situation when it comes to drawing the line between bald (in the old sense) and non-bald. And if we were to ban our original adjective, allowing only the refined ones, there would be still borderline cases for each of the new attributes.

Cases in which neither T nor F is to be assigned are characterized as truth-value gaps, or for short, as gaps. The cases considered before–those of indexicals and ambiguous terms–are not genuine gaps, in as much as they can be resolved by removing the ambiguity. In cases of vagueness the gaps are for real (or so many philosophers think).

Vagueness inheres in our very conceptual fabric. It does not arise because we are missing some facts. Knowing all there is to know about the hairs on Mr. Hairfew’s head: their number, distribution and length, may not determine whether he is bald or not. There is no point to insisting on a yes-or-no answer. The concept is simply not intended for cases like his. If you think of it you will see that the phenomenon is all around. Only mathematics is exempt, and some theoretical parts of exact science. It appears whenever an empirical element is present.2 Vagueness has been often regarded as a flaw, something to get rid of–if possible. But it has a vital role in reducing the amount of processed information. In principle, we could–instead of using ‘young’, ‘rich’, ‘bald’, and their like–use descriptions that tell us one’s exact age, one’s financial assets to the last penny, or one’s precise amount of cranial hair. All of which would involve colossal waste of valuable resources. For in most cases a two-fold classification into young and not-young, rich and not-rich, bald and not-bald, will do. And additional information can be obtained if and when needed. The efficiency thereby achieved is worth the price of borderline cases with truth-value gaps.

A deeper reason for vagueness is that every conceptual framework gives us only a limited purchase on “reality” or “the facts”. There is always a place for surprise, for something turning up that resists classification, something that defies our neatly arranged scheme. The examples considered so far are relatively simple borderline cases. In these situations a certain classification does not apply, yet an alternative exhaustive description is available.

2_{It is not a priori impossible that some experiment will turn up a particle for which the question: is it an}

electron? has no clear-cut answer. The theory rules this out; but the theory may change, for its authority is established by empirical criteria. Here however we are confronted with open texture rather than with simple vagueness.

There is no mystery about the financial status of Ms. Richfield. In principle a full list of her assets–calculated to the last cent–can be drawn. The difficulty in deciding whether she is rich stems solely from the vagueness of ‘rich’. But there are situations where no alternative description is available, situations that involve more than occasional borderline cases.

(8) Jeremy, the chimpanzee, knows that Jill will feed him soon.

Can we say that a monkey “knows” that something is going to happen in the near future? Granting the way we apply ‘know’ to people (itself a knotty issue and a subject of a vast philosophical literature) can we apply it, in some instances, to animals? All we can do is speculate on the monkey’s mode of consciousness, dispositions, state of mind or state of brain. And it is not even clear what are the factors that are relevant for deciding the status of (8). Surely there will be conflicting opinions. Cases of this kind display the undecidedness of our conceptual apparatus, the fact that it is open-ended and may evolve in more than one way. They are known as open texture. As the example above shows, open texture involves quite common concepts. Think of generosity, freedom, or sanity.

Vagueness of Generality

General statements convey quantitative information regarding some class (or multitude) of objects. They are usually expressed by words such as ‘all’, ‘every’, ‘some’, ‘most’, and their kin. For example:

All human beings have kidneys and lungs.

In classical logic generality is expressed by quantifiers, which have precise unambiguous in-terpretations. But in natural language the intended extent of generality is often ambiguous, as well as vague. ‘Everyone’ can cover many ranges, from one’s set of acquaintances to every human on earth. Consider, for example:

(9) Everyone knows that Reagan used to consult astrologers. (10) Everyone wants to be rich and famous.

(11) Everyone will sometime die.

Only in (11) can we interpret ‘everyone’ as meaning every human being– the way it is con-strued in symbolic logic. In (9) and in (10) the intended interpretation is obviously different. In (9) ‘everyone’ refers to a very small minority: people who are knowledgeable about Reagan. (9) is just another way of saying that the item in question had some publicity. (10) covers a wider range than (9), but falls short of the generality of (11). Even when the range covered

by ‘everyone’ or ‘everything’ is explicit, the strength of the assertion can vary. For example, when the teacher asserts

(12) Everyone in class passed the test,

she will be taken literally; her assertion would be misleading even if a single student had failed. But a casual remark:

(13) Everyone in college is looking forward to the holidays season,

means only that a large majority does; it would not be considered false on the ground of a few exceptions. How large is the required majority? This is vague.

Such phenomena are even more pronounced when the general statement–usually expressed by means of the indefinite plural–is intended to express a law, or a rule. For rules may have exceptions. (And the exceptions to this last rule are in mathematics, or some of the exact sciences, or in statements like (11).) The amount of tolerated exceptions is vague. Consider, for example:

(14) Women live longer than men,

(15) When squirrels grow heavy furs in the autumn, the winters are colder, (16) Birds fly.

Statistical data (e.g., average life span) can be cited in support of (14); but under what conditions the is sentence true? This is vague. (15) sums up a general impression of past experience; presumably, statistics can be invoked here as well. (16), on the other hand, is better viewed as a rule that determines the “normal” case: If something is known to be a bird, then–in the absence of other relevant information–presume that it flies.

The general principles concerning ambiguity and vagueness apply also here. We may have to give up precise systematic prescriptions and settle for pragmatic guidelines. And we should accept the possibility of borderline cases, where the assignment of any truth-value is rather arbitrary.

Cases of the types just given can, of course, be handled by using mathematical-like systems. (14) and (15) call for statistical analysis, with all the criteria that go with it. (16), on the other hand, indicates reasoning based on normalcy assumptions, where one’s conclusions are retracted, if additional information shows the case to be atypical (in the relevant way). Little reflection is needed to see that almost all our decision making involves reasoning of that kind. With no information to the contrary, the usual order of things is presupposed. To do otherwise would freeze all deliberate action. In recent years a great deal of research, by computer scientists, logicians and philosophers, has been devoted to systems within which

reasoning that involves retractions can be expressed. They come under the general term of non-monotone logic.

1.1.4

Other Causes of Truth-Value Gaps

Non-Denoting Terms

Declarative sentences may contain descriptive expressions that function as names but lack denotations. The standard, by now worn-out example is from Russell:

(17) The present king of France is bald.

(It is assumed that (17) is uttered at a time when there is no king of France. If needed the time-indexical can be eliminated by introducing a suitable date.) Proper names, as well, may lack denotation: ‘Pegasus’, or ‘Vulcan’–either the name of the Roman god, or of the non-existent planet.3

Frege held that declarative sentences containing non-denoting terms have no truth-value. This view was later adopted, for different reasons, by Strawson. Russell, on the other hand, proposed a rephrasing by which these sentences get truth-values; (17), for example, is recon-structed as:

(170_{) There is a unique person who is a King of France, and whoever is a King of} France is bald.

Therefore the sentence is false. Also false, by Russell’s reconstruction, is: (18) The present king of France is not bald.

But

(19) It is not the case that the king of France is bald.

is true. (The difference between (18) and (19) is accounted for by giving the negation different scopes with respect to the descriptive term ‘the king of France’–a point that we shall not discuss here.)

3_{‘Neptune’, ‘Pluto’, and ‘Vulcan’ were introduced as names of planets whose existence was deduced on}

theoretical grounds from the observed movements of other planets. Neptune and Pluto were later observed directly. ‘Vulcan’ did not make it. The effects attributed to Vulcan were later explained by relativity theory.

As far as logic is concerned the question is more or less settled–not by a verdict in favour of one of the views, but by having the issue sufficiently clarified, so as to reduce it to a choice between well understood alternatives. It boils down to what one considers as fitting better our linguistic usage. Intuitions may vary. Nonetheless, the different resulting systems are variants within the general framework of classical logic.

Category Mistakes

In the usual order of things, almost every attribute and every relation is associated with a certain type of objects. When the objects do not fit the attribute, we get strange, though grammatically correct, sentences; for example,

(20) The number 3 is thirsty.

This is a category mistake; numbers are not the kind of things that are thirsty, or non-thirsty. Some may want to treat (20) as neither true nor false. Alternatively, (20) and its kin can be regarded as false. This policy can be extended so as to handle negations and other compounds. As in the case of non-denoting terms, the ways of dealing with such examples are well-understood and can be accommodated, as variants, within the general framework of classical logic. Non-denoting terms and category mistakes, interesting as they are when it comes to working out the details, do not pose a foundational challenge to the framework of classical logic. But vagueness and open texture do.

1.2

Some Other Uses of Declarative Sentences

Declarative sentences have other uses, besides that of conveying information, or describing the world. I do not mean their misuse, through lying, or by misleading. Such misuses are direct derivatives of their ordinary use. I mean uses that are altogether different. They have been extensively studied by philosophers and linguists, and are worth noting for the sake of completeness and in order to give us a wider perspective.

Fictional Contexts

In a play, or in a movie, the players utter declarative sentences, much as people in “real life” do; but what goes on is obviously different. Compare, for example, an exclamation of ‘Fire!’ that is part of the play, with a similar exclamation, by the same actor in the same episode, when he observes a real fire breaking out in the hall. We say that the utterances in the play are not true assertions, or are not performed in an assertive mode. They are pretended assertions within a make-belief game.

Yet, within the game, they are subject to the same logic that applies to ordinary statements. Furthermore, truth-values can be meaningfully assigned to certain statements about fictional characters. ‘Hamlet killed Polonius, and was not sorry about it’ will be regarded as true, while ‘Hamlet intended to kill Polonius’ will be regarded as false. This merely reflects what is found in the play. The pretense reaches its limits easily: ‘Hamlet had blood type A’ is neither true nor false, or–if we adopt Russell’s method–false by some legislation in logic. Consider, for contrast, ‘Shakespeare had blood type A’, which has a determinate truth-value; even though we do not, and probably never will, know what this value is. No logical legislation can settle this.

The declarative sentences that appear in novels, poetry, or jokes, achieve a variety of effects: they can amuse, entertain, evoke an aesthetic experience, a feel or a vision. Some can enlighten us, but not in the way that ‘The earth turns around the sun’ does.

Metaphors, Similes, and Aphorisms

Consider the following.

Skepticism is the chastity of the intellect. Santayana To deny to believe and to doubt well, are to a man

as a race is to a horse. Pascal

Those who can–do, those who cannot–teach. Shaw

Taken literally, the first is trivially false, or a category mistake (skepticism is not the chastity of something and ‘chastity’ does not apply to intellects). The second is trivially true or trivially false–depending on whether the claimed likeness is indefinitely wide (any two things are alike in some respect) or narrow and precise. The third–as a plain general statement–is false on any of the usual criteria. Evidently, the points of the sayings have little to do with their literally determined truth-values.

Many have found in metaphors (of which the first is an example) hidden meanings, which can be approximated–though not captured–by non-metaphorical rephrasings. Others have argued that the value of metaphor–what is transmitted, or evoked–is outside the scope of linguistic meaning. And yet a metaphor can be misleading in a way that a joke, or a poem cannot. The same can be said of similes, which achieve their effect through a somewhat different mechanism. Finally there are sayings like the third, which are not to be evaluated literally, but are neither metaphors nor similes. Their point is to underline some noteworthy feature, to focus our attention on a certain pattern.

To sum up: in this chapter a brief overview was given of declarative sentences, their basic role in conveying factual information, the ways they function in natural language and the problems of assigning them truth-values. We have also noted some other uses of declarative sentences. To most of this we shall not return. But you should be aware of the wider picture and of the perspective within which logic has been, and still is being developed.

We shall often emphasize the relations between symbolic logic and natural language. One should remember, however, that symbolic logic is not concerned with language per se. Its aim is not the discovery of linguistic structures, or laws; that is the job of the linguist. Logic and language are closely related because in symbolic logic we try, following linguistic guidelines, to express in a precise, structured way some of the things expressed in natural language. Many aspects of linguistic usage are not representable in a system of logic. Even with respect to conveying factual information, a statement will often resist fruitful formalization, either because it is too vague or confused, or because it is too complex or subtle.

Sentential Logic: Some Basic

Concepts and Techniques

2.0

English sentences are usually made from nouns, verbs, adjectives, etc. But sometimes the smaller components are themselves sentences. For example:

(1) Jack went to the movie and Jill went home

is a compound made of two sentences: ‘Jack went to the movies’, and ‘Jill went home’, joined by the word ‘and’. We can also combine sentences into bigger sentences by ‘or’:

(2) Jack will get a job or Jill will get a job.

In principle, any two sentences can be combined, using ‘and’ or ‘or’, (Recall that, from now on, ‘sentence’ means a declarative sentence.)

We can also make a sentence from a single sentence by negating it: (3) Jack did not graduate last year

can be seen as the negation of: (4) Jack graduated last year.

Strictly speaking, (4) is not a part of (3); not in the same way that ‘Jack went to the movie’ is a part of (1). The forming of negation involves insertions and possibly additional changes, and it varies from language to language. In English and French we usually use an auxiliary verb (‘do’ or ‘is’–in English, ‘avoir’ or ‘etre’–in French), in Hebrew we do not. In English the auxiliary is succeeded by ‘not’, in French it is placed between ‘ne’ and ‘pas’. Grammatical details like these are abstracted away when we set up the system of symbolic logic. What is essential is that every sentence can be negated.

Natural languages provide more than one way of forming negations. Instead of (3), we can use the following as a negation of (4):

(30_{) It is not the case that Jack graduated last year.}

(Here, indeed, (4) appear as a part of the negated sentence.)

Sentential logic is concerned exclusively with the making of sentences from sentences–in ways analogous to the ones just illustrated. Predicate logic, to which we shall come later, provides, in addition to the apparatus of sentential logic, a finer analysis–whereby sentences are made from parts analogous to nouns and verbs.

The sentences of the system we are going to study do not belong to any natural language. But the system is meant to bring forth patterns that underlay language in general, in as much as language expresses logical thinking. For this purpose we posit abstract entities, in the role of sentences, and we postulate certain properties; just as in geometry we presuppose that the points, lines and planes satisfy certain axioms.

Sentential logic includes certain operations, by which sentences can be combined into sen-tences. We shall refer to these operations as sentential connectives, or for short connectives. One connective, called conjunction, corresponds to the operation effected in English by using ‘and’ (as in (1)). Another connective corresponds to the operation of combining two sen-tences by using ‘or’ (as in (2)). For the moment we leave unspecified the exact nature of the sentences. We only assume that they are given together with the connective operations, and that they satisfy certain properties, which we shall state as we go along.

Note: ‘Connective’ suggests the joining of more than one sentence, but it covers also operations on single sentences, such as negation. A connective is called binary if it operates on pairs of sentences. It is called monadic if it operates on single sentences.

The system to be studied here is based on one monadic connective: negation, and on several binary ones. In principle one can consider connectives that operate on more than two sen-tences. But we shall not require them, because we shall be able to express whatever is needed by repeated applications of the connectives we have.1

1_{For each connective, the number of sentences it combines is fixed. One can, however, generalize the notion}