Chapter 1. Introduction: Logic and Language

Irving M. Copi, Symbolic Logic, New York: Macmillan Publishing, 1979, pp. 1-7.

 

(1) What Is Logic?

① Charles Peirce: ‘It will, however, generally be conceded that its central problem is the classification of arguments, so that all those that are bad are thrown into one division, and those which are good into another....'

The study of logic, then, is the study of the methods and principles used in distinguishing correct (good) from incorrect (bad) arguments.

 

② Logic as the science of reasoning: 

(a) Reasoning is that special kind of thinking called inferring, in which conclusions are drawn from premisses.

(b) As thinking, however, reasoning is not the special province of logic, but part of the psychologist’s subject matter as well. Psychologists who examine the reasoning process find it to be extremely complex and highly emotional, consisting of awkward trial and error procedures that are illuminated by sudden—and sometimes apparently irrelevant—flashes of insight. These are all of importance to psychology.

(c) Logicians, however, are not interested in the actual process of reasoning, but rather with the correctness of the completed reasoning process.

(d) Their question is always: Does the conclusion that is reached follow from the premisses used or assumed? If the premisses provide adequate grounds for accepting the conclusion, if asserting the premisses to be true warrants asserting the conclusion to be true also, then the reasoning is correct. Otherwise, the reasoning is incorrect. Logicians methods and techniques have been developed primarily for the purpose of making this distinction clear. Logicians are interested in all reasoning, regardless of its subject matter, but only from this special point of view.

 

(2) The Nature of Argument 

① Inferring is an activity in which one proposition is affirmed on the basis of one or more other propositions that are accepted as the starting point of the process. The logician is not concerned with the process of inference, but with the propositions that are the initial and end points of that process, and the relationships between them.

추론이란 그 과정의 시작점에서 받아들여진 하나 또는 여러 명제들을 기반으로 하여 한 명제를 확언하는 활동이다. 논리학자들은 추론의 과정에 관심을 갖지 않으며, 그 과정의 시작과 마지막에 해당하는 명제들과 그 명제들 사이의 관계에 관심을 갖는다.

② Propositions are either true or false, and in this they differ from questions, commands, and exclamations. Grammarians classify the linguistic formulations of propositions, questions, commands, and exclamations as declarative, interrogative, imperative, and exclamatory sentences, respectively. These are familiar notions.

(a) Declarative sentences / Propositions 

Declarative sentences	The propositions they may be uttered to assert
A declarative sentence is always part of a language, the language in which it is spoken or written.	propositions are not peculiar to any of the languages in which they may be expressed.
the same sentence may be uttered in different contexts to assert different propositions. (For example, the sentence 'I am hungry’ may be uttered by different persons to make different assertions.)

(b) Sentences / Statement

Sentences	Statements
The same statement can be made using different words, and the same sentence can be uttered in different contexts to make different statements.

(c) Proposition/ Statement

Proposition	Statements
The terms proposition and ‘statement are not exact synonyms, but in the writings of logicians they are used in much the same sense.

 

③ An argument

(a) Corresponding to every possible inference is an argument, and it is with these arguments that logic is chiefly concerned.

(b) An argument may be defined as any group of propositions or statements, of which one is claimed to follow from the others, which are alleged to provide grounds for the truth of that one.

(c) Every argument has a structure, in the analysis of which the terms ‘premiss’ and ‘conclusion’ are usually employed.

(d) The conclusion of an argument is that proposition which is affirmed on the basis of the other propositions of the argument.

(e) These other propositions, which are affirmed as providing grounds or reasons for accepting the conclusion, are the premisses of that argument.

(f) We note that ‘premiss’ and ‘conclusion’ are relative terms, in the sense that the same proposition can be a premiss in one argument and a conclusion in another. Any proposition can be either a premiss or a conclusion, depending upon its context. It is a premiss when it occurs in an argument in which it is assumed for the sake of proving some other proposition. And it is a conclusion when it occurs in an argument that is claimed to prove it on the basis of other propositions that are assumed.

 

 

④ Deductive and Inductive arguments

(a) It is customary to distinguish between deductive and inductive arguments. All arguments involve the claim that their premisses provide some grounds for the truth of their conclusions, but only a deductive argument involves the claim that its premisses provide absolutely conclusive grounds.

(b) The technical terms ‘valid’ and ‘invalid’ are used in place of ‘correct' and 'incorrect' in characterizing deductive arguments. A deductive argument is valid when its premisses and conclusion are so related that it is absolutely impossible for the premisses to be true unless the conclusion is true also.

(c) The task of deductive logic is to clarify the nature of the relationship that holds between premisses and conclusion in a valid argument, and to provide techniques for discriminating valid from invalid arguments.

(d) Inductive arguments involve the claim only that their premisses provide some grounds for their conclusions. Neither the term 'valid' nor its opposite ‘invalid’ is properly applied to inductive arguments. Inductive arguments difler among themselves in the degree of likelihood or probability that their premisses confer upon their conclusions. Inductive arguments are studied in inductive logic. In this book, however, we shall be concerned only with deductive arguments, and shall use the word ‘argument’ to refer to deductive arguments exclusively.

 

(3) Truth and Validity

① Truth and falsehood characterize propositions or statements and may also be said to characterize the declarative sentences in which they are formulated.

② Arguments, however, are not properly characterized as being either true or false but as valid or invalid.

③ There is a connection between the validity or invalidity of an argument and the truth or falsehood of its premisses and conclusion, but the connection is by no means a simple one.

④ Validity and Soundness

Some valid arguments contain true propositions only, as, for example,

All bats are mammals.
All mammals have lungs.
Therefore, all bats have lungs.

An argument may contain false propositions exclusively and still be valid, as, for example,

All trout are mammals.
All mammals have wings.
Therefore, all trout have wings.

This argument is valid because if its premisses were true, its conclusion would have to be true also, even though, in fact, they are all false. These two examples show that although some valid arguments have true conclusions, not all of them do. The validity of an argument does not, therefore, guarantee the truth of its conclusion.

When we consider the argument

If I am President, then I am famous.
I am not President.
Therefore, I am not famous.

we can see that although both premisses and conclusion are true, the argument is invalid. Its invalidity becomes obvious when it is compared with another argument of the same form:

If Rockefeller is President, then Rockefeller is famous.
Rockefeller is not President.
Therefore, Rockefeller is not famous.

This argument is clearly invalid because its premisses are true but its conclusion is false. The two latter examples show that although some invalid arguments have false conclusions, not all of them do. The falsehood of its conclusion does not guarantee the invalidity of an argument. But the falsehood of its conclusion does guarantee that either the argument is invalid or at least one of its premisses is false.

∴ 논증의 타당함과 타당하지 않음은 결론의 참/거짓을 보증하지 않는다 (그 역도 마찬가지로 성립한다).  

 An argument must satisfy two conditions to establish the truth of its conclusion. It must be valid, and all of its premisses must be true. Such an argument is termed ‘sound’.

∴  건전한 논증은 타당한 논증이면서 그 전제가 모두 참인  (따라서 결론도 참인) 논증이다. 

To determine the truth or falsehood of premisses is the task of scientific inquiry in general, since premisses may deal with any subject matter at all. But determining the validity or invalidity of arguments is the special province of deductive logic.

The logician is interested in the question of validity even for arguments that might be unsound because their premisses might happen to be false. A question might be raised about the legitimacy of that interest. It might be suggested that logicians should confine their attention to arguments that have true premisses only. It is often necessary, however, to depend upon the validity of arguments whose premisses are not known to be true. Scientists test their theories by deducing from them conclusions that predict the behavior of observable phenomena in the laboratory or observatory. The conclusion is then tested directly by observation of experimental data and if it is true, the results confirm the theory from which the conclusion was deduced. If the conclusion is false, the results disconfirm or refute the theory. In either case, the scientist is vitally interested in the validity of the argument by which the testable conclusion is deduced from the theory being investigated, for if that argument is invalid, his whole procedure is without point. Although an oversimplification of scientific method, our example shows that questions of validity ai e impoi taut even for arguments whose premisses are not true.

 

(4) Symbolic Logic

① It has been explained that logic is concerned with arguments and that arguments contain propositions or statements as their premisses and conclusions.

② These premisses and conclusions are not linguistic entities, such as declarative sentences, but are, rather, what declarative sentences are typically uttered to assert.

③ The communication of propositions and arguments, however, requires the use of language, and this complicates our problem. Arguments formulated in English or any other natural language are often difficult to appraise because of the vague and equivocal nature of the words in which they are expressed, the ambiguity of their construction, the misleading idioms they may contain, and their pleasing but deceptive metaphorical style. The resolution of these difficulties is not the central problem for the logician, however, for even when they are resolved, the problem of deciding the validity or invalidity of the argument still remains.

To avoid the peripheral difficulties connected with ordinary language, workers in the various sciences have developed specialized technical vocabularies. The scientist economizes the space and time required for writing his reports and theories by adopting special symbols to express ideas that would otherwise require a long sequence of familiar words to formulate. This has the further advantage of reducing the amount of attention needed, for when a sentence or equation grows too long, its meaning is more difficult to grasp. The introduction of the exponent symbol in mathematics permits the expression of the equation

A × A × A × A × A × A × A × A × A × A × A × A = B × B × B × B × B × B × B

more briefly and intelligibly as

A¹² = B⁷

A like advantage has been obtained by the use of graphic formulas in organic chemistry. The language of every advanced science has been enriched by similar symbolic innovations.

④ Notation

A special technical notation has been developed for logic as well. Aristotle made use of certain abbreviations to facilitate his own investigations. Modern symbolic logic augmented this base by the introduction of many more special symbols. The difference between the old and the new logic is one of degree rather than of kind, but the difference in degree is tremendous. Modern symbolic logic has become immeasurably more powerful a tool for analysis and deduction through the development of its own technical language. The special symbols of modern logic permit us to exhibit with greater clarity the logical structures of arguments that may be obscured by formulation in ordinary language. It is easier to divide arguments into the valid and the invalid when they are expressed in a special symbolic language, for with symbols the peripheral problems of vagueness, ambiguity, idiom, metaphor, and amphiboly do not arise. The introduction and use of special symbols serve not only to facilitate the appraisal of arguments, but also to clarify the nature of deductive inference.

The logician's special symbols are much better adapted to the actual drawing of inferences than is ordinary language. Their superiority in this respect is comparable to that of Arabic over the older Roman numerals for purposes of computation. It is easy to multiply 148 by 47, but very difficult to compute the product of CXLVIII and XLVII. Similarly, the drawing of  inferences and the evaluation of arguments is greatly facilitated by the adoption of a special logical notation. To quote Alfred North Whitehead, an important contributor to the advance of symbolic logic:

. . . by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.

 

⑤ History of Logic

(a) Although this book treats symbolic logic systematically rather than historically, a few historical remarks may be appropriate at this point. Since the 1840s, symbolic logic has developed along two different historical paths. One of them began with the English mathematician George Boole (1815-1864). Boole applied algebraic notations and methods first to symbolize and then to validate arguments of the kind studied by Aristotle in the fourth century b.c. This route may be characterized as an effort to apply mathematical notations and methods to traditional, nonmathematical kinds of arguments.

(b) The other path began with the independent efforts of the English mathematician Augustus De Morgan (1806-1871) and the American scientist and philosopher Charles Peirce (1839-1914) to devise a very precise notation for relational arguments. The earlier logic had largely ignored this type of argument, which, nevertheless, plays a central role in mathematics. This historical path may be characterized as an effort to create a new quasi-mathematical kind of logical notation and analytical technique for use in mathematical derivations and demonstrations.

(c) These two historical paths coalesced in the brilliant works of the German mathematician and philosopher Gottlob Frege (1848-1925), the Italian mathematician Guiseppe Peano (1858-1932), and the English philosophers Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), whose Principle Mathematica was an important landmark in the history of symbolic logic. Some of Boole’s contributions are reported in the first two sections of Chapter 7 and in Appendix B. The contributions of the others have become so thoroughly incorporated into modern symbolic logic that only occasional references to their more distinctive ideas are appropriate.