1st Year Logic (English) - Logic PDF

 CHAPTER - 1 NATURE AND SCOPE OF LOGIC 1.1 The Discipline of Logic : We, human beings, are rational animals. As rational beings we seek reasons for our beliefs. It is of course true that we do not always provide reasons...

 CHAPTER - 1 NATURE AND SCOPE OF LOGIC 1.1 The Discipline of Logic : We, human beings, are rational animals. As rational beings we seek reasons for our beliefs. It is of course true that we do not always provide reasons for our convictions. But we do recognise the importance of providing reasons or justifications in support of what we claim to be true. The activity of providing reasons for a claim is called reasoning. This reasoning, when expressed in langauge, is called argument. In other words, an argument is a piece of reasoning which consists of a claim together with the reasons in support of that claim. An argument is a sequence of sentences, where one of the sentences is the conclusion (or the claim), and the sentences which provide the reasons for the claim are the premises. Note that the sentences expressing reasons are called premises. And the sentence expressing the claim is called the conclusion. Thus, a sequence of premises together with a conclusion is called an argument. Consider the following example: (1) If logic were difficult then many students would not like logic. (2) In fact, many students like logic. (3) Therefore, logic is not difficult. This is an argument consisting of three sentences of which (1) and (2) are premises and (3) is the conclusion. In this argument (3) expresses the claim “Logic is not difficult” and in support of this claim the two other sentences (1) and (2) are advanced as evidences. The sentences (1) and (2) together provide the justification for accepting the truth expressed by the sentence (3). 2 + 2 LOGIC, Part-I Our intellectual or cognitive activity is argumentative in nature. Scientific activities, as paradigm cases of intellectual activities, are essentially reason-governed. Arguments are the heart and soul of every science or discourse. By the method of rational enquiry science discovers truths, establishes hypothesis and formulates laws. No genuine scientific activity can afford to neglect arguments in the pursuits of knowledge. Like that of scientific activities, in our everyday life also, we normally seek reasons for our beliefs and actions. When our beliefs and actions are not grounded on proper reasons our life often becomes miserable. Suppose someone argues that Ashok is honest because he is well-dressed. This is not a good argument, as the evidence cited in favour of the premise does not provide justification of the claim. Being well-dressed is not at all a reason for believing in the honesty of Ashok, because, there are many well-dressed persons who are not honest. Thus, one who trusts Ashok to be honest on such a wrong ground might land himself in trouble. We have already noticed that an argument is a sequence of sentences of which one sentence is claimed to be true on the basis of the rest of the sentences. We also noted that the sentence whose truth is being claimed is the conclusion and the sentences that provide the justification for the claim are the premises of the argument. Here we are using the word argument in a board sense to include both good and bad arguments, In other words, whenever some claim is made on the basis of some evidence, we call it an argument. If the evidences justify the claim, we call it a good argument, otherwise it is not. Logic is a study of the methods and principles of correct or good arguments. It teaches us how to construct good arguments and detect mistakes in our arguments. The knowledge of logic enables us to increase clarity, consistency and cogency of reasoning in our intellectual as well as everyday life. It also helps us to recognise fallacies or errors in our speech and writings. The knowledge of the principles of good reasoning helps us to avoid logical errors that otherwise would creep into our thoughts causing confusions and puzzles. From this we cannot however claim that a logician or a reasonably good logician cannot commit any logical error. A logician, as a human being, may commit logical mistake or even argue wrongly. This is quite possible. What  NATURE AND SCOPE OF LOGIC 3 we are claiming is that a person with the knowledge of logic is better equipped to avoid errors in arguments and argue more efficiently than what he or she would have done without knowledge of logic. Logical thinking is a skill that can be acquired by the study of logic. Logic promotes rational thinking, critical attitude and thereby help us to form a scientific world view. Therefore, it is desirable that one should know the basic principles of good argument. The knowledge of this would place one in a comparatively better position to understand the situation, evaluate beliefs and take correct decisions. Logic also teaches us to appreciate the good arguments and criticise the bad ones advanced by others. Since logic, in general, deals with arguments, let us continue our analysis of the notion of an argument. 1.2 Structure of Argument : It has been already said that logic is concerned with the analysis and evaluation of arguments. For this we should be able to recognise and identify arguments. It is a fact that our arguments are not always available in a neatly stated form. Thus, to recognise a passage as expressing an argument, we should look at the context of its occurrence. We come across arguments in debates, in a law court, in legislative chambers, in mathematical proofs, etc.. One is also confronted with arguments in ordinary day-to-day life. As a student of logic, one should be able to determine whether a passage does or does not express any argument. So, an enquiry into the distinction between argumentative and non- argumentative passages will be instructive. Firstly, if in a given passage no statement is connected with the other then surely the passage is not argumentative. Hence the minimum requirement of any passage to express an argument or a series of arguments is that the statements should be connected in such way that they collectively justify or support the truth of a claim. Consider the following passage. “The Moon goes round the Earth. All happy men are virtuous. All great scholars are eccentric.” Here no statement is connected with the other. Hence it is not an argumentative passage. Arguments should be distinguished from narrative passages which may consist of loosely connected set of 4 + 2 LOGIC, Part-I statements. Consider the following passage. Dasarath was the king of Ayodhya. He had three queens and four sons. Ram was his eldest son. Ram was very kind to everybody. Sita, the princess of Mithila, was his wife. Here we have narrated several statements but no claim has been made either explicitly or implicitly about any one of them on the basis of the rest. Hence, no argument is involved in the above passage. On the other hand consider the following passage. All teachers deserve our respect because they are our seniors and our seniors deserve our respect. This passage is clearly argumentative. All the sentences are well connected. It is claimed that all teachers deserve our respect. This is the conclusion of the argument. In support of this conclusion reasons have been given. The reasons are stated in the two statments: (1) Our seniors deserve our respect and (2) Teachers are our seniors. These two statements are the premises which together provide reasons for the conclusion. Secondly, to identify an argument, we have to identify its premises as well as the conclusion. Usually in an argument the conclusion is preceded by an expression such as “so”, “hence”, “thus”, “therefore”, “as a result”, “for this reason”, “It is proved that” etc. We call such expressions conclusion indicators. A conclusion is a sentence which begins with any of the conclusion indicator words or phrases. An argument may have also premise indicators. The expressions such as “since”, “because”, “for”, “as”, “follows from”, “as shown by” etc. are called premise-indicators. Usually premises of an argument begin with or are preceded by the premise indicator expressions. For example, if we assert “P because Q” then Q being preceded by a premise indicator signals that Q is the premise of the argument. The same is the case with conclusion-indicators. Indicator words do not always signal the presence of an argument. For instance, in the sentence “Sita is living in Bhubaneswar since her marriage to Ashok” the word “since” indicates a temporal connection rather than a premise in any argument. In the sentence “Ram resigned from his job because of his illness” the word ‘because’ indicates a causal connection, not an argument. It  NATURE AND SCOPE OF LOGIC 5 should be further noted that the non-occurance of premise or conclusion indicator ia a passage does not indicate that the passage is not argumentative. In other words, a passage might be argumentative even when the indicator words or phrases are absent. To decide the nature of the passage we have to look at the context of stating the passage. Thirdly, if a passage consists of just one statement then it does not express an argument. Because, an argument consists of at least one premise and a conclusion. Usually an argument consists of a set of statements which are the premises and another statement which is the conclu- sion. An argument has the following general from: P ( a set of premises) Therefore, Q (the conclusion) Fourthly, there can be passages that are explanatory in nature without being argumentative. If our interest is to establish the truth of a statement say ‘Q’ on the basis of another statement ‘P’ then “ Q, because P” states an argument. On the contrary, if the truth of “Q” is unproblematic and we have no intention to justify “Q” on the basis of “P”, then the formulation “Q because P” is an explanation of why Q occured. Therefore, the difference between an argumentative and explanatory passage is really dependent on our interest or purpose of stating or using the passage in question. 1.3 Definition of Logic : Logic can be defined as the systematic study of the methods and principles of correct reasoning or arguments. Logic teaches us the techniques and methods for testing the correctness of different kinds of reasoning. It helps us to detect errors in reasoning by examining and analysing the various common fallacies in reasoning. Let us the examine some proposed definitions of logic. Some logicians define logic as an art of reasoning. According to this view since logic develops the skill or ability to reason correctly, it is an art. As an art, logic provides the method and technique for testing the correctness or incorrectness of arguments. Music, dance, cooking are instances of art. They aim to develop our skills. In these disciplines practice makes a person more skillful. A student of logic is required to 6 + 2 LOGIC, Part-I work out the exercises as a part of his or her learning the subject. So logic is an art. Some define logic as both science and art of reasoning. A science is a systematic study of phenomena which are within the area of its investigation. It undertakes to formulate the laws or principles which holds good without exception. Logic is a science as it is a systematic study of the methods and principles of correct reasoning. Logic also studies and clarifies the different types of fallacies which are committed in correct reasoning. A distinction can be drawn between positive and normative sciences. A positive science describes how the facts in its area of investigation actually behave. It arrives at general laws by the methods of observation and experiment. A normative science, on the other hand, investigates the norms as standard that should be applied. Logic is not a positive science since it does not report how people actually reason or argue. Since it deals with the standardas principles of correct thinking, it is a normative science. The use of the word ‘reasoning’ in the above definitions may be misleading. The term ‘reasoning’, may signify a mental process or a mental product. In logic, we are not concerned with the actual process of reasoning rather with arguments which is a product. A thought when expressed in language becomes an argument. Thus, the statement that logic is an art and at the same time science of reasoning gives important insights into the nature of logic but as a definition, it is not very accurate. Some logicians claim that logic is the science of laws of thought. But such a view is not correct because all reasoning involves thinking but all thinking cannot be called reasoning. Logic deals with correct reasoning and not with all types of thinking. There are many mental proceses such as remembering, imagining, day-dreaming etc. which can be instances of thinking without involving any reasoning. Psychology studies all these phenomena, but logic deals only with reasoning. Further logic does not discover any descriptive laws but it formulates the principles of correct reasoning. We can sum up by stating that logic helps one to improve upon the quality of reasoning. It provides technique to strengthen and polish the skill of reasoning. It aims at providing a solid foundation by which one can distinguish between correct and incorrect reasoning. 1.4 Sentence and Proposition : In the above discussion, we have used both words “sentence” and “proposition” indiscriminately while characterizing arguments. We first said that argument consists of sentences. Subsequently in the previous section we remarked that arguments consist of propositions. This  NATURE AND SCOPE OF LOGIC 7 might create confusion in your mind. You may wonder, does an argument consist of sentences or of propositions? There should not be any confusion on this point. The sentences, which figure in arguments, express propositions. Strictly speaking propositions are the constituents of arguments. One should, however, be clear about the distincton between sentence and proposition. Firstly, all sentences do not express propositions. Only declarative or indicative sentences express propositions. Questions, (viz., How old are you?, “What is your father?”, ‘Are you a student?)’, commands (‘Go there’, ‘Get out’, ‘Take whatever available’) and exclamation (viz, Oh! What a book!’) are sentences but they do not express any proposition. Such sentences do not have any truth value as they do not assert or deny anything. Secondly, a sentence is a linguistic entity belonging to a specific language, whereas propositions are logical entities having no specific allegiance to any particular language. Of course, to express a proposition we always need a sentence but a proposition is different from a sentence. Two or more sentences belonging to the same or to different languages may express the same proposition. For example, “Rama killed Ravana” and “Ravana was killed by Rama” are two different sentences in English but both express the same proposition. Because, the state of affair described by the first sentence is the same as that of the second senence. So far as the proposition is concerned these two sentences express the same proposition. Similarly, the sentence “Ram killed Ravan” can be translated into any other language like Odia, Hindi or Sanskrit, and the corresponding sentences in these languages would express the same proposition. Thirdly, the same sentence may express different proposition uttered at different times and in different places. For example, the sentence “The present Prime Minister of India is a bachelor” uttered in the year 1994 would express a false propositon whereas the same sentence uttered in the year 2002 would express a true proposition. In other words, the state of affair expressed by the two utterances of the same sentence at different times are different. Even if the sentence is the same, the propositions expressed by the sentence at difference times would be different. Thus propositions are distinct from sentences. 1.5 Kinds of Argument: Deductive and inductive We know that the conclusion of an argument asserts a claim on the basis of the premises. In general, an argument exhibits a relational tie or a relation between premises and the conclusion. 8 + 2 LOGIC, Part-I Logic as a system of reasoning aims at characterising this relation. On the basis of the nature of this relational tie we can broadly distinguish between two kinds of arguments viz., deductive and inductive. In deductive arguments, the premises conclusively justify or support the conclusion. The truth claim expressed by the conclusion is fully supported by the truth claim expressed by the premises. In other words, in case of deductive arguments the truth of the premises absolutely ensures the truth of conclusion. In this sense we call a correct deductive argument demonstrative. If the premises are true, the conclusion must necessarily be true. This means that in a correct deductive argument the premises and conclusion are so related that it is impossible for the premises to be true and the conclusion to be false. Consider the following example, (1) If logic is interesting then many students like it. (2) In fact, logic is interesting. (3) Therefore, many students like it. Here it is impossible that premises are true and conclusion is false. This is a valid deductive arguement in which the premises provide conclusive grounds for the truth of the conclusion. In an inductive argument the premises do not absolutely or conclusively ensure the truth of the conclusion. If the premises of an inductive argument are all true and the reasoning is good then it is reasonable to believe in the truth of the conclusion. But here we cannot be absolutely sure of the truth of the conclusion. For example, consider the following argument. (1) Ram is mortal (2) Hari is mortal (3) Sita is mortal (4)..................... (5).................... Therefore, all men are mortal. This is an inductive argument. Here even if all the premises are true and the reasoning is good yet the truth of the conclusion cannot be asserted conclusively or with certainty. Because,  NATURE AND SCOPE OF LOGIC 9 even if all the premises are observed to be true and nothing contrary has been observed so far, yet the conclusion being a general proposition cannot be observed to be true as it includes future and unobserved cases. Therefore, the conclusion of an inductive argument is always prone to revision. Hence an inductive argument may be evaluated as better or worse according to the degree of support or backing given to the conclusion by the premises. Inductive arguments are of great importance for establishing scientific laws and propositions expressing empirical conjectures about the world. Most of our beliefs are based on induction. They cannot be justified by deductive arguments as such cases are empirical generalisations based on uncontradicted experience. For example, we believe that eating rice nourishes us whereas taking arsenic will be poisonous. These beliefs are established by inductive method. Let us examine some of the misunderstandings usually associated with the distinction between inductive and deduction. It is claimed that induction is a process from particular to general, whereas deduction is a process from general to particular. This is illustrated in the following example. All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Here the conclusion is a particular proposition and one of the the premises, precisely the major premise, is a general proposition. Of course the reasoning involved in the above argument is correct. This is an instance of valid deductive argument. Therefore, it has been said that deduction is a process from general to particular. This is not always true. Because there are valid deductive arguments whose premises are all general propositions and the conclusion is also a general proposition. Consider the following example : All men are mortal. All kings are men. Therefore, all kings are mortal. 10 + 2 LOGIC, Part-I Similarly, there are valid deductive arguments whose premises as well as the conclusion are all particular propositions. Consider the following arguments: (1) If Ram is honest then Ram is virturous, In fact, Ram is honest, Therefore, Ram is virtuous. (2) Some Odias are scientists. Therefore, some scientists are Odias. So, it is not correct to characterise deduction as a process from general to particular. Similarly, we cannot in general claim that in an inductive argument the premises are particular but the conclusion is general. Because, there are inductive arguments whose premises as well as the conclusion are general propositions. Consider the following example, All cows are mammals and have hearts. All whales are mammals and have hearts. All horses are mammals and have hearts. All humans are mammals and have hearts. Therefore, all mammals have hearts. Likewise, we may have a good inductive arguments that may have particular propositions for its premises as well as for its conclusion. This is illustrated in the following inductive argument. During the last ten years maximum temperature in summer in Rourkela has exceeded 400 C., so this year also it will exceed 400 C. The above examples make it clear that it is not correct to characterise deduction as a process from general to particular and induction as a process from particular to general. The fundamental difference between induction and deduction lies on the nature of the relation between premises and the conclusion. In case of deduction, the premises conclusively support the conclusion in the sense that no additional information (or premise) is relevant (i.e. it cannot increase or decrease the validity of a deductive argument). Validity never admits of degree. On the other  NATURE AND SCOPE OF LOGIC 11 hand, the relation between premises and the conclusion in an inductive argument admits of degrees. Even in the best inductive argument premises render the conclusion highly probable. The premises of a good inductive argument never conclusively support the conclusion in the sense that it is possible to discover some additional facts concerning the world that may upset the truth claim made by the conclusion of a well-established inductive argument. Thus, only deductive arguments can be characterised as valid or invalid. Inductive arguments are either strong or weak depending upon the amount of support the premises provide to the conclusion. We know that probability is the essence of any inductive argument i.e. the conclusion of an inductive argument is probable. Note that mere presence of the word “probability”, “probable” etc. in the conclusion never ensures that the argument in question is inductive. Because, there are deductive arguments about the probabilities themselves. Hence we may conclude that an argument is deductive if and only if the conclusion conclusively follows from or is completely determined by its premises, whereas in case of induction, the conclusion is claimed to follow from its premises only with probability. 1.6 The Form and Content of Argument : We notice from our day-to-day experience that all objects are made of some matter and at the same time also have some form. Consider the example of a table. It is made of wood. Wood is its matter or the content. It has a particular form that ordinarily refers to the shape and size of the table. The shape and size distinguish two or more objects made of wood. For example, though a chair and a table are both made of wood yet they are different in form. The form of chair is different from the form of the table. In other words, both chairs and tables are made of wood i.e. their content or matter is the same but their respective forms distinguish one from the other. An analogous distinction can be made with regard to arguments. For any given argument, it is possible to make a distinction between its from and content. (i) Two arguments may share the same form or structure yet differ in content. Consider the following examples: 12 + 2 LOGIC, Part-I Example 1 : All men are mortal. All kings are men. Therefore, all kings are mortal. Example 2 : All mammals have lungs. All rabbits are mammals. Therefore, all rabbits have lungs. These two arguments have the same form namely All M is P. All S is M. Therefore, all S is P. Here M, P and S respectively denote the terms, ‘man’, ‘mortal’ and ‘kings’ in the example 1 and “mammals”, “lungs” and “rabbits” in example 2. The form of the above mentioned arguments may be exhibited diagrammatically as follows : All M P. All S M.  All S P. Clearly the content or the subject matter of these two arguments are different. Example 1 is about men, mortality and kings and example 2 is about mammals, lungs and rabbits, Thus there are arguments with the same structure having different subject matters. On the other hand, (ii) The content may be same but the form or structure of the argument may be different. To show this, let us consider these examples. Example 3 : No heroes are cowards. All soldiers are heroes. Therefore, no soldiers are cowards.  NATURE AND SCOPE OF LOGIC 13 Example 4 : No cowards are heroes. All soldiers are heroes. Therefore, no soldiers are cowards. The content of these two arguments is the same since both are about heroes, cowards and soldiers. But these two arguments have different structures that may be exhibited as follows: The Structure of example 3 is : No M is P, All S is M, Therefore, no S is P, where M, P and S respectively denote heroes, cowards and soldiers. Diagrammatically, it has the following form : No M P. All S M.  No S P. On the other hand, example- 4 is of the form : No P is M. All S is M. No S is P, where M, P and S denote ‘hero’, ‘coward’ and ‘soldier’ respectively. Diagrammatically it is of the following form : No P M All S M  No S P Clearly the structure of the arguments stated in examples 3 and 4 are different though their content remains the same. Thus, we can notice that content and form are really different and 14 + 2 LOGIC, Part-I independent of each other. Now we may end this section by taking note of the fact that logic deals with the form or the structure, not with the content of an argument. 1.7 Truth and Validity : Truth and validity are two different notions. Truth is predicated of propositions whereas validity is predicated of arguments. Propositions are either true or false. Deductive arguments are either valid or invalid. We have noted earlier that a deductive argument claims to provide conclusive proof for its conclusion. A deductive argument is valid if and only if the premises provide conclusive proof for its conclusion. This notion of validity of deductive argument can also be expressed in either of the following two ways. (i) If the premises of a valid argument are all true, then its conclusion must also be true. (ii) It is impossible for the conclusion of a valid argument to be false while its premises are true. Any deductive argument that is not valid, is called invalid. So, a deductive argument is invalid if its premises are all true but the conclusion is false. Note that in some cases, even if the premises and the conclusion are all true yet the argument may be invalid. In all cases of invalid arguments some of our rules of inference are violated. The above remarks on deductive validity shows the connection between validity of an argument and the truth or falsity of its premises and conclusion. But the connection is not a simple one. Of the eight possible combinations of truth or falsity of premises and the conclusion and validity or invalidity of arguments, only one is completely ruled out. The only thing that cannot happen is that the premises are all true, the conclusion is false and the argument is deductively valid. Given below are the other seven combinations of true and false premises and conclusion with example. (i) There are valid arguments whose premises as well as the conclusion are all true.  NATURE AND SCOPE OF LOGIC 15 Example: All men an mortal. All kings are men. Therefore, all kings are mortal. (ii) There are valid arguments whose premises as well as the conclusion are all false. Example: All cats are six-legged. All dogs are cats. Therefore, all dogs are six-legged. (iii) There are valid arguments where the premises are all false but the conclusion is true. Example: All fishes are mammals. All whales are fishes. Therefore, all whales are mammals. (iv) An argument may have true premises and a true conclusion and nevertheless the argument may be invalid. Example: All men an mortal. All kings are mortal. Therefore, all kings are men. (v) There are invalid arguments whose premises are false but the conclusion is true. Example: All mammals have wings. All rabbits have wings. Therefore, all rabbits are mammals. (vi) There are invalid arguments in which premises and conclusion are all false. Example: All cats are biped. All dogs are biped. Therefore, all dogs are cats (vii) Lastly, an argument in which the premises are true and the conclusion is false will be invalid. 16 + 2 LOGIC, Part-I Example: All Oriyas are Indians. Nehru is not an Oriya. Therefore, Nehru is not an Indian. We can summarize our findings in the following tabular way. Premise Conclusion Validity of the argument T T Valid Invalid T F xxx Invalid F T Valid Invalid F F Valid Invalid The above examples show that invalid arguments allow for all possible combinations of true or false premises and true or false conclusion. We cited examples of valid arguments with false conclusion as well as invalid arguments with true conclusions. Thus, it can be noticed that the truth or falsity of the conclusion does not by itself determine the validity or invalidity of the argument. So also the validity of an argument does not by itself guarantee the truth of its conclusion. We also noticed that valid arguments may have only three out of the four possible truth contributions. A valid argument cannot have true premises and a false conclusion. In other words, if an argument is valid and its premises are true, then we can be sure that the conclusion is true. 1.8 Sound and Unsound Argument : At this stage we can draw a distinction between sound and unsound arguments. An argument is called sound if and only if it is valid and all its premises are true. Otherwise, the argument is called unsound. The following is an example of a sound argument.  NATURE AND SCOPE OF LOGIC 17 All mammals have lungs. All rabbits are mammals. Therefore, all rabbits have lungs. Here all the premises are true and the argument is valid. Hence, it is a sound argument. On the other hand, an argument is unsound if it is either invalid or some of its premises are false. For example, No mammals have lungs. No whales are mammals. Therefore, no whales have lungs. Here the argument is invalid and the premises are also false. Hence it is unsound. Further, even if an argument is valid but some or all of its premises are false then also the argument is unsound. Consider the following example: No insects have six legs. All spiders are insects. Therefore, no spiders have six legs. Here both the premises are false but the argument is valid. Hence, it is also an unsound argument. Thus mere validity of an argument does not make the argument sound, because there are valid arguments that are not sound. To say that an argument is unsound amounts to the claim that the argument is either invalid or some of its premises are false. Thus the soundness of an argument implies validity as well as the truth of all its premises. But the unsoundness of an argument does not imply invalidity, because there are unsound arguments that are valid. At this stage the following question may be asked. Why logicians should not confine their attention only to sound arguments? The answer is, we cannot study only sound arguments though it is interesting. Because, to know an argument to be sound we must know that all its premises are true. But knowing the truth of the premises is not always possible. Further, we are often involved in arguments whose premises are not known to be true. For example, when a scientist verifies a scientific hypothesis or even a theory, he or she very often deduces consequences from the hypothesis or the theory in question and compares these consequences with the data and if 18 + 2 LOGIC, Part-I the result tallies then the hypothesis or the theory is verified to be true. Here the investigator does not know the truth of the hypothesis or the theory prior to the process of testing. If the truth of the theory or the hypothesis was known to the scientist prior to the verification, the verification would be pointless. So, to confine our attention to sound arguments only would be self-defeating. But this does not make sound arguments logically uninteresting. Because, if by some means, we know that an argument is sound then we may infer the truth of its conclusion. 1.9 The Fundamental Principles of Logic : We have noted that the task of logic is to study the principles underlying the validity of deductive arguments and the strength of inductive arguments. Since not all deductive arguments are valid, we need to know the principles that ensure a valid argument to be valid and invalid argument to be invalid. It has been suggested that the arguments that satisfy or conform to the laws or principles of logic are valid and arguments that do not do so are invalid. In other words, validity amounts to not violating any law of logic. Logic deals with these principles and also with their interrelation. Out of the various laws of logic there are three fundamental principles, namely, (I) the law of identity, (ii) the law of contradiction (or the law of non-contradiction) and the law of excluded middle. These are known as the laws of thought or fundamental principles of logic. In calling these as laws of thought, there is a danger of interpreting them as psychological laws concerning mental processes of thinking. This would be a misunderstanding of their true nature. These are not descriptive laws. They do not tell us how people think. Rather these are prescriptive in nature. They tell us how one should think or, more precisely, how one should reason. So instead of calling them laws of thought, it is better to call them principles of logic. These three laws are considered as fundamental or basic in the sense that any correct or good argument must conform to these laws. This means that these laws are presuppositions of any good argument. Let us state and explain these laws. The Law of Identity : The laws of identity states that everything is identical with itself, i.e. a thing is what it is. In other words, a thing is not other than itself. Symbolically we may say that for anything x, x is x is always true, For example, accepting this law, we assert “table is a table”, “Chair is a chair”,  NATURE AND SCOPE OF LOGIC 19 “Man is a man” and so on. It does not assert anything about the nature of x, it does not tell us whether x in question is white or heavy or soft. It does not tell us about any particular character of the world. It gives us a very useful instruction concerning the use of concepts occurring in an argument. It tells us that in any good argument or in any process of good reasoning, every concept occurring in it must be used in the same sense thoughout the argument. In other words, the meaning of concepts occurring in an argument should remain constant throughout the argument. This law can also be taken as an instruction for assigning truth-values to propositional variables in classical logic. In this sense, it states that distinct occurrences of the same propositional variable always receive the same truth value throughout the argument. In propositional sense it states that every proposition implies itself. Symbolically the law states that for any proposition P, (P P) holds. (Read ‘P P’ as ‘if P then P’.) This means that if a proposition is used in an argument to state something, then whenever this proposition occurs in the argument it is used to state that thing. It states that the same proposition should not state different things in the same argument. One cannot accept and reject a given proposition at the same breath in any given argument. If we accept and reject a proposition P at the same time in an argument then the very use of P becomes pointless. Hence, for any discourse or argument to be possible, we have to accept the law of identity. The fundamental nature of the law of identity can be understood in the following way. For example, if we deny this law i.e. if we deny x is x then it implies that there is a denial. Hence by use of the law of identity we have denial is a denial. It is of the form d is d, where d stands for ‘denial’. Thus the law of identity is back. The very denial of this law implies its presence. Therefore, without presupposing this law we can not even state or assert anything. For any discourse or argument to be possible we have to accept the law of identity. In this sense, it is a fundamental or basic principle of logic. The Law of Contradiction (or the Law of Non-contradiction) : Like that of the law of identity, the law of contradiction, (otherwise called the law of non- contradiction) admits various formulations in different contexts of its use. It states that two contradictory qualities or predicates cannot be asserted with regard to anything at one and the same time. In other words, for anything, it is not the case that it possesses a property and does 20 + 2 LOGIC, Part-I not possess that property at the same time. Aristole says, “the same cannot belong and not belong together to the same under same respect”. For example let A stand for “to be honest” and B stand for “not to be honest”. Then A and B will never belong to the same thing. In other words, it is not possible to assert and deny the same. In its propositional formulation, it asserts that two contradictory statements are not true together. It also suggests that no proposition is both true and false. Symbolically, the law of contradication is represented by the formula (p · p) where p stands for any proposition (Read (p · p) as ‘not both p and not p’). This law, like the law of identity, also suggests the method of assigning truth values to propositions. It says that while assigning truth values to the propositions, we should not assign both truth and falsehood to the same proposition. This law is one of the minimum conditions of any good argument. It says that if something is a table, then it is not the case that it is not a table. A thing cannot both be a table and not a table at the same time. Any arguments violating this law would be inconsistent and thereby becomes pointless as it would claim nothing. When we say that a good argument must conform to this law, it implies that it should be consistent. The law of Excluded Middle : The law of excluded middle, like the other two above laws, is also a fundamental law in the sense that every good argument must conform to this law. It asserts that everything is either A or not A, where A stands for any quality. A and not-A exhaust the entire discourse. In other words, a thing can be either A or not-A but it cannot be neither. Hence there cannot be an intermediary between contradictory properties. Let A stand for “to be good” and B stand for “not to be good”. Then ‘either A or B’ will belong to everything. In the propositional interpreta- tion the law of excluded middle asserts that every propositon is either true or false. Symbolically, it can be formalised as (p   p) is always true for any proposition P. (Read ‘p  p’ as ‘p or not p’). For any proposition p, p admits either the truth value T ( for truth) or F (for falsehood). P cannot be neither i.e. there is nothing in between T and F. The third or intermediary value between T and F is excluded. These three laws in their propositional interpretation are all tautologies and hence are  NATURE AND SCOPE OF LOGIC 21 logically true. Since they are all tautologies, they are equivalent to each other. For that matter they are all equivalent to any of the tautologies. But logicians consider these laws as having a very special status. They are basic or fundamental principles to which any good or correct argument must conform. Further, these laws cannot be proved. Because, to prove them amounts to constructing valid arguments in which each such law must occur as conclusion. Since any valid argument, in general, must conform to these laws, the proofs of such law (if any) as a form of valid argument must also conform to these three laws. This means the proof of these laws would involve the fallacy of pititio principii (i.e. the fallacy of assuming what we wish to prove). Thus we may say that these laws are presuppositions of any good argument. Furhter, the fundamental nature of these laws can be seen in relation to the construction of truth tables. These laws provide us the necessary instructions for assigning truth values to the propositions in classical logic. Each proposition is assigned the value T or F ( in accord with the law of excluded middle) but not both (in accord with the law of contradiction) and distinct occurrences of the same variable always receive the same truth value through out the expression (in accord with the law of identify). So these laws are fundamental, self evident and unavoidable for providing consistent arguments in any field of human knowledge. ******** 22 + 2 LOGIC, Part-I SUMMARY Logic is the study of the methods of evaluating arguments. An argument is a set of propositions consisting of one or more premises and a conclusion. The premises claim to provide reasons in support of the conclusion. Arguments are different from mere narration of facts and explanations. Propositions are typically expressed by declarative sentences. There are two types of arguments — deductive and inductive. A deductive argument claims to provide conclusive support for its conclusion. An inductive argument claims to provide partial support for its conclusion; it cites evidence which makes the conclusion somewhat reasonable to believe. Deductive arguments are either valid or invalid. A valid argument has the essential feature that it is impossible for its conclusion to be false while its premises are true. An argument is invalid if and only if it is not valid. An argument is deductively sound if and only if it is valid and has all true premises. An unsound argument is one which is either invalid or has at least one false premise. Since logic is a systematic study of methods and principles of correct reasoning and it teaches us the technique of testing the correctness of arguments, it can be viewed both as a science and an art. There are three fundamental laws of logic. These are (1) the law of identity, (2) the law of contradiction, and (3) the law of excluded middle. These are basic principles of correct thinking which are presupposed in any logical thinking. **********  NATURE AND SCOPE OF LOGIC 23 MODEL QUESTIONS Objective -type I. Point out in each case whether it is true or false. 1) Logic is the study of methods for evaluating arguments. 2) If a valid argument has only false premises, then it must have a false conclusion. 3) Every valid argument with a false conclusion has at least one false premise. 4) Some arguments are true. 5) Every sound argument is valid. 6) Every invalid argument is unsound. 7) A sound argument can have a false conclusion 8) If an argument is valid and has only true premises, then its conclusion must be true. 9) If all the premises of an argument are true, then it is sound. 10) Every valid argument with a true conclusion is sound. 11) Every valid argument has a true conclusion. 12) If the premises of a valid argument are false, then its conclusion is also false. 13) Every unsound argument is invalid. 14) If an argument has one false premise, then it is unsound. 15) If all the premises of a valid argument are true, then its conclusion is also true. 16) Some arguments are false. 17) Deductive logic is concerned with tests for validity and invalidity of arguments. 18) If a deductive argument has all true premises and a false conclusion, then it is invalid. 19) Every valid argument has true premises and only true premises. 20) If an argument is invalid, then it must have true premises and a false conclusion. 24 + 2 LOGIC, Part-I II. Answer the following questions by selecting the correct option: 1. Which one of the following sentences expresses a proposition? a) What is your name? b) The sky is blue. c) Please close the door. d) May God bless you! 2. Logic deals with a) arguments b) law c) morality d) explanations 3. Which of the following is a conclusion indicator? a) for the reason that b) which implies that c) in view of the fact that d) inasmuch as 4. Which of the following is not a premise-indicator? a) thus b) since c) because d) as 5. Which of the following is a premise indicator? a) Therefore b) Hence c) Because d) So  NATURE AND SCOPE OF LOGIC 25 6. Which one of the following correctly expresses the difference between deduc- tive and inductive arguments? a) In an inductive argument, the premises provide some support for the conclusion, but a deductive argument, if it is valid, provides conclusive support for the conclusion. b) Inductive arguments reason from the general to the particular, while deductive arguments reason from the particular to the general. c) Deductive arguments reason from the general to the particular, while inductive arguments reason from the particular to the general. d) Deductive arguments consist of true propositions, while inductive arguments consist of only false proposition. 7. If the purpose of a passage is to account for some proposition, then the passage is probably: a) attempting to define key terms. b) an argument. c) going to fail, since there is no way to account for most propositions. d) an explanation. 8. In correct reasoning: a) all the propositions are true. b) the conclusion supports the premises. c) the conclusion is never false. d) the premises support the conclusion. 9. A valid argument must have a. only true premises. b. a true conclusion. c. both true premises and true conclusion. d. none of the above. 26 + 2 LOGIC, Part-I 10. If the premises and the conclusion of an argument are all true, then a. the argument must be valid. b. the argument must be sound. c. the argument must be valid and sound. d. none of the above 11. An invalid argument may have a. false premises and false conclusion. b. false premises and true conclusion. c. true premises and true conclusion. d. any of the above. 12. If an argument is valid then a. it must be sound. b. it cannot be sound. c. it is sound only if all its premises are true. d. it is sound but has false premises. 13. If an argument is valid then a. it is impossible for its premises to be true when the conclusion is false. b. it cannot be sound. c. it is possible for its premises to be true when its conclusion is false. d. none of the above is true. 14. A sound argument a. may have a false premise. b. may fail to be valid. c. may have a false conclusion. d. none of the above.  NATURE AND SCOPE OF LOGIC 27 III. Match the descriptions given in the right with the words given in the left. 1. Valid A.It is either true or false. 2. Deductively sound B. An argument that is either invalid or has a false premise 3. Proposition C. The part of logic concerned with tests for validity and invalidity of arguments. 4. Deductive logic D. A valid argument with (all) true premises. 5. Inductive logic E. The part of logic concerned with tests for strength and weakness of arguments. 6. Logic F. The study of methods for testing whether the premises of an argument adequately support its conclusion. 7. Argument G. A set of propositions consisting of a conclusion and one or more premises in support of the conclusion. 8. Deductively unsound H. It is impossible for the conclusion to be false while the premises are true. Essay-type questions: 1. What is an argument? Point out the distinction between deductive and inductive arguments. 2. Explain the nature of Logic. 3. Distinguish between: (a) truth and validity (b) sentence and proposition (c) valid and invalid arguments (d) arguments and explanations 4. State and explain the fundamental principles of logic. 5. What are the principles of logic? Explain. *** 28 +2 LOGIC, Part-I CHAPTER - 2 LOGIC AND LANGUAGE 2.1 Uses of Language : Logic, as we have said earlier, deals with the analysis and evaluation of arguments. Since arguments are expressed in language, the study of arguments requires that we should pay careful attention to language in which arguments are expressed. If you reflect on how language is used in everyday life, you can notice that our ordinary language has different uses. Language has a variety of functions. By using language we do various things like stating facts, reporting events, giving orders, singing songs, praying God, making requests, cutting jokes, asking questions, making promises, greeting friends and so on. These are wide varieties of language uses. We will not make any attempt to provide an exhaustive list of language uses. Rather we shall discuss here a broad classification of some of the important uses of language. There are three important uses of language that we shall discuss here. These are: (a) Descriptive, (b) Emotive, and (c) Direc- tive uses of language. (a) Descriptive Use of Language : Language is often used to describe something or to give information about something. So the descriptive use of language is also called informative use of language. When a sentence is used descriptively it reports that something has some feature or that something lacks some feature. Consider the following two sentences: 1. Birds have feather. 2. Birds are not mammals.  LOGIC AND LANGUAGE 29 The first sentence reports that having feather is a feature of birds. The second sentence reports that birds do not have some essential qualities found in mammals. In either case it pro- vides information about the world. Both affirmation and denial about things in the world are examples of descriptive use of language. The following are some more examples of language functioning descriptively. 1. Crows are black. 2. Puri is not the capital of Orissa 3. A spider has eight legs. 4. Logic is the study of correct reasoning. 5. The 15th of August is Indian Independence Day. All these above statements happen to be true statements. However, it should be noted that not only true sentences are instances of informative use of language, but also false sentences are instances of informative use of language. “A spider has six legs” is a false statement since spiders in fact have eight legs. Yet the statement “A spider has six legs”, even though false, is nonetheless an example of descriptive use of language. When language functions informatively we can sensibly ask whether what is asserted is true or false. In other words, the question “Is it true?” can be meaningfully asked of all such instances. When language is used to affirm or deny any proposition, its function is informative. Language used to present arguments serves informative function. All descriptions of things, events, and their properties and relations consist of informative discourse. The language of science is a clear instance of descriptive use of language. (b) Emotive Use of Language : Language is often used to express our feelings, emotions or attitudes. It is used either to express one’s own feelings, emotions or attitudes, or evoke certain feelings, emotions or attitudes in someone else, or both. When one expresses feelings while alone, one is not expressing it to evoke feelings in others. But very often we attempt to move others by our expressions of emotions. In all such cases language is used emotively. Consider the following utterances: 30 +2 LOGIC, Part-I 1. Jai Hind! 2. Cheers! 3. It is disgusting! 4. It is too bad! 5. It is wonderful! 6. Let us win this game! In appropriate contexts all these can count as instances of language functioning emotively. If a sentence is followed by an exclamation mark, then very likely it is used emotively. The language of poetry also provides examples of language serving the expressive function. Emotive use is different from descriptive use of language. Emotive or expressive discourse is neither true nor false. When language is used emotively, it cannot be characterized as true or false. We can, however, respond to it by asking questions such as “Is the person sincere?” and “How should I feel?” Expressive use of language is also different from directive use of language. (c) Directive Use of Language : Language is often used to give direction to do or not to do something. Commands, requests, instructions, questions are instances of directive use of language. Consider the following examples: 1. Finish your homework. 2. Wash your clothes. 3. You should wear helmet when riding a scooter. 4. Don’t smoke. 5. Are you feeling well? 6. Will you please help me? In all these above examples language is functioning directively. Anyone who utters any of these sentences, in a typical situation, is directing someone to do something or to respond in an appropriate manner. In all instances of language functioning directively, we can meaningfully ask the question “Should I respond?” You will notice that directive discourse, like emotive discourse, is neither true nor false. But directive discourse, specially the imperative statements, can figure in  LOGIC AND LANGUAGE 31 some arguments. A command such as “Close the window”, or an advice such as “You should wear helmet while riding scooter” is either obeyed or disobeyed, but it is neither true nor false. Though commands, advices, and requests are neither true nor false, these can be reasonable or unreasonable, proper or improper. These characterisations of imperative statements are somewhat analogous to characterisation of informative statements as true or false. Moreover, imperative statements often imply or presuppose the truth of some propositions. If I request you to close the window, my request presupposes the truth of the proposition that the window is open. Since reasons can be cited for or against imperative statements, such statements do occur in imperative arguments. We are not going to discuss the logic of imperatives in this book. In our study of logic we shall restrict our discussion to arguments that are stated in the language that functions informatively. The study of logic is concerned with language that functions informatively. So it is important to distinguish language that is informative from language that serves other functions. There is, however, no mechanical method for distinguishing informative use of language from language that serves other functions. Grammatical structure of a sentence often provides a clue to its function, but there is no necessary connection between function and grammatical form. We can determine whether the language in a particular context is functioning informatively or not by asking “Is this instance of language being used to make an assertion that is either true or false?” If the answer is “yes” then it is an instance of informative use of language. It should be noted that language, in particular contexts, very often functions in more than one way. One and the same sentence might have more than one function. For effective communication language is often used deliberately to serve multiple functions. Language used to serve expressive function might contain some relevant information. So also language that is primarily informative may make use of other functions as well. Most discourses in our ordinary communication contain elements from all the three uses of language enumerated above. In logic we restrict our attention to those cases where our discourse is at least partly informative or descriptive. 32 +2 LOGIC, Part-I 2.2 Words and Terms : An argument consists of declarative sentences. We have also noted that the declarative sentences express propositions. As a sentence consists of words, so also a proposition consists of terms. As words are the constituent of sentences, so also terms are the constituent of propositions. Consider, for example, the proposition: “Crows are black”. It consists of a subject term “crows” and a predicate term “black” joined together by a copula “are”. Similarly, the proposition “All men are mortal” has a subject term “man” and predicate term “mortal”. A categorical proposition expresses a relation between a subject term and a predicate term. A term consists of one or many words. For example, in the proposition “All scientists who are famous are sincere and hardworking persons”, the subject term is “scientists who are famous” and the predicate term is “sincere and hardworking persons”. Here a group of words together constitute the subject term. The predicate term too consists of several words in this example. The terms in this proposition are many-worded terms. A term constitutes a unit of meaning in the proposition. It expresses a single idea. The terms in a categorical proposition can be best understood as being about classes and individuals. A term signifies either a class of individuals or an individual. For example, in the proposition “Man is mortal” it is asserted that the class “man” is included in the class of “mortals”. Here both the terms are class terms and they can be said to be expressing general ideas. The proposition “Asok is mortal” states that an individual (Asok) belongs to the class of mortals. Here “Asok” expresses a singular idea. Some words can be independently used as a subject or a predicate term in a proposition. A word, which by itself can designate a term, is called acategorematic word. Names of individuals and things, and words designating classes and properties are categorematic. For example, ‘Socrates’, ‘man’, ‘mortal’ are categorematic words, since these words can be independently used as subject or predicate term. Some words cannot be independently used as a term, but can occur as a part of a many-worded term. Words like ‘the’, ‘a’, ‘of’ and ‘on’ belong to this  LOGIC AND LANGUAGE 33 category of words. These are called syncategorematic words. Words like ‘hurray!’ and ‘oh!’ cannot be used to signify a term either independently or in conjunction with other words. These are acategorematic words. It should be noted that unless a word is used in a proposition, its nature cannot be determined. Looking to its use in a proposition, it can be determined as categorematic, syncategorematic or acetogorematic. For the same word can be used differently. i) He is intelligent (here intelligent is used categorematically) ii) Intelligient boys are rewarded. (here intelligent is used syncategorematically) iii) How intelligent ! he showed his talent in the competition. (here intelligent is used acetogorematically) Thus looking to the use of a term in a proposition, the nature of the term can be determined. Usually words used as noun, pronoun and adjective are categorematic, and words used as verb, adverb, adjective, preposition and conjuction are syncategorematic and word used as interjection are acategorematic. Generally categorematic words are treated as terms, but syncategorematic and acetogorematic words are not treated as terms. We can now sum up our discussion. A term is always a component in a proposition. In a categorical proposition a term is either a subject or a predicate of the proposition. It can be single-worded or many-worded. In a categorical proposition, a term signifies a thing or individual, or a class of things or individuals or a property. We shall have further occasion to discuss the nature of subject and predicate terms in greater detail in Chapter 4. 2.3 Denotation and Connotation of Terms : Class terms or general terms, such as ‘man’, ‘dog’, ‘river’ etc., have two different kinds of meaning – they apply to things and signify some properties. The things to which a term applies in the same sense constitute its denotation and the common and essential properties signified by the term constitute its connotation. 34 +2 LOGIC, Part-I (a) Denotation : A general term applies to several things. In one sense of ‘meaning’, the things to which a general term applies constitute its meaning. This is meaning in the sense of reference. The things to which a general term applies are called the denotation of the term. This sense of ‘meaning’ is called the denotative, or referential meaning of the term. Let us elucidate the idea of denotational meaning with an example. Consider the term ‘month’. The term ‘month’ applies to January, February, March, April, etc.  to all the twelve months. The term ‘month’ applies equally in the same sense to all these months. When we say that a month has less than 32 days, part of what we mean is that January has less than 32 days, February has less than 32 days and so on. Thus the months January, February, March, etc. constitute the denotation of the term ‘month’. The denotation of ‘boat’ is the set of all boats, the denotation of ‘dog’ is the set of all dogs, and the denatotion of ‘chair’ is the set of all chairs that are (or ever have been or ever will be) in the world. The collection of all the objects, individuals or events to which a term applies constitutes the denotation of the term. The denotation of a term is the collection of individual things to which the term correctly applies. One might think that since in course of time old things are destroyed and new things are created, denotation of a term does not always remain constant. For example, one might reason that since people die and babies are born, denotation of ‘human beings’ would decrease with every death and increase with every birth. But this view is based on a mistake. The denotation of the term ‘human’ consists of all human beings – living, dead, and the unborn. So, particular deaths and births do not change the denotation of ‘human beings’. (b) Denotation and Extension : A distinction is sometimes drawn between ‘denotation’ and ‘extension’ of terms. It has been suggested that these two should not be used as synonymous expressions. It is pointed out that while denotation consists of all the members of the class-term, its extension consist of all the subclasses included within it. Thus, for example, the denotation of the term ‘book’ will comist of all the individual books to which the term correctly applies; extension of the term ‘book’ will  LOGIC AND LANGUAGE 35 consist of its sub-classes like ‘logic books’, ‘physics books’, ‘story books’ etc. Hence extension of a term consist of classes and denotation of a term consists of individual members. It can be noticed that a term lacking denotation may still have extension. For example, the term ‘unicorn’ lacks denotation since there are no such animals in the world, but it still can have extension in the form of having sub-classes such as ‘white unicorns’, ‘black unicorns’, ‘brown unicorns’ and so on. Similarly, terms like proper names lack extension but have denotation. (c) Connotation : A general term signifies some properties or qualities on the basis of which we know how to apply the term correctly. All the objects denoted by a given term share some common and essential attributes. Connotation of a term consists of these common and essential qualities. By ‘common quality’ we mean the quality shared in common by all the members. By ‘essential quality’ we mean those qualities without which the term will not apply to something. On the basis of these common and essential attributes we are able to decide whether the term applies to a given object or not. In other words, the set of features shared by all and only those things to which a term applies is called the connotation of that term. The connotation of the term ‘triangle’ consists of those attributes common to all triangles and found only in triangles. ‘Triangle’ means a plane figure bound by three straight lines. ‘A plane figure bound by three straight lines’ constitutes the connotation of ‘triangle’. The connotation of a term is the set of all and only those properties that a thing must possess for that term to apply to it. The word intension is also used as synonym of connotation. The word ‘connotation’ is ambiguous. It has multiple senses. There are at least three different senses in which words intension and connotation have been used. These are called subjective, objective, and conventional connotation of a term. ‘Subjective connotation’ of a term is the set of attributes the user associates with that term. It is the set of all the attributes the user believes to be true of the objects denoted by the term. Subjective connotation is psychological in character. It varies from individual to individual and from context to context. It may also vary from time to time for the same person. So, subjective interpretations cannot provide any reliable 36 +2 LOGIC, Part-I guidance to the meaning of a term. By ‘objective connotation’ we mean the list of all the properties found in the individuals belonging to the concept. But many of these properties may not be essential for the application of the concept. Let us, for the sake of argument, suppose that all human beings have two legs. Then ‘being biped’ will be a part of the objective connotation of ‘human beings’. But this does not make it an essential property of human beings. If someone loses a leg in an accident, he or she does not cease to be human. Moreover, there are also other biped creatures, e.g. birds, apart from human beings. There is another problem with the idea of objective connotation. Since objective connotation will include all the features common to the objects denoted by the term, it would not be humanly possible to know all of these. Human beings are not omniscient. If meaning is identified with objective connotation, then we shall have to concede that one never knows the complete meaning of the terms one is using. So, objective connotation cannot be the public meaning of a term. By conventional connotation we mean only those properties conceived to be the necessary and sufficient for ascribing objects to the term. Conventional connotation makes it easy for us to pick out the objects falling under the class term. In logic when we talk of connotation we mean to use it in the sense of conventional connotation. Conventional connotation is the same as logical connotation. (d) Relation between Denotation and Connotation : Connotation and denotation are closely interrelated. The following points should be noted with regard to their relation. (1) Connotation of a term determines its denotation, but denotation of a term does not uniquely determine its connotation. One may know the connotation of a term without knowing its denotation. For instance, even though there are no unicorns in the world, people know the meaning of the word “unicorn” in the sense of knowing its connotation. ‘Unicorn’ means ‘a horse like animal with one long horn on its forehead’. So, one who knows this, knows the connotation of the term, although no one has ever come across  LOGIC AND LANGUAGE 37 any unicorn in the world. Further, two terms having different connotations may have the same denotation. For example, consider the two terms ‘equilateral triangle’ and ‘equiangular triangle’. The term ‘equilateral triangle’ means a plane figure enclosed by three straight lines of equal length. The term ‘equiangular triangle’ means a plane figure enclosed by three straight lines that intersect each other to form equal angles. Notice that these two terms have different connotative meaning but both denote exactly the same set of figures. So, connotation of a term determines its denotation. In other words, the connotation of a term provides us a set of criteria for deciding whether an object falls within the extension of that term. For example, when we come across an animal in a zoo, we decide whether or not it belongs to the class of leopards by seeing whether or not it has the relevant features of a leopard. (2) Connotation and denotation vary inversely. When connotation increases denotation decreases, and when denotation increases connotation decreases. Some logicians call it the law of inverse variation. Consider the following sequence of terms: 1. ‘animal’, 2. ‘aquatic animal’, and 3. ‘aquatic animal with fins’. These terms have been arranged in the order of increasing connotation. The connotation of ‘aquatic animal’ is greater than that of ‘animal’. Aquatic animals have all the qualities common and essential for something to be an animal and in addition to those properties they have the property of living in water. Similarly, the connotation of ‘aquatic animal with fins’ is greater than that of ‘aquatic animal’, since aquatic animals with fins have all the properties of aquatic animals plus the property of having fins. So you can see that the three terms are in the order of increasing connotation. You will also notice that these terms are in the order of decreasing denotation. The total number of aquatic animals is less than that of animals, and the total number of aquatic animals with fins is less than that of aquatic animals. Thus when we arranged the terms in an order of increasing connotation, the terms also automatically got arranged in an order of decreasing denotation. 38 +2 LOGIC, Part-I Similar relation of inverse variation can also be noticed if we arrange terms in an order of decreasing connotation. 1. ‘aquatic animal with fins’ 2. ‘aquatic animal’ 3. ‘animal’ In the above sequence of terms connotation progressively decreases while there is a progressive increase in the denotation of the terms. Thus we have noticed that if we increase the connotation in a series of general terms by including more features, the denotation of the corresponding terms in the series tends to decrease, and if we decrease the connotation in any series of terms, the denotation os the corresponding terms tends to increase. There are, however, some exceptions to this relation of inverse variation. In some cases an increase in connotation does not result in the decrease in denotation. We observed earlier that the two terms having different connotation might have the same denotation. We cited the case of the two terms ‘equilateral triangle’ and ‘equiangular triangle’ to illustrate that point. The same example can be cited to illustrate an exception to the ‘law of inverse variation’. Although the term ‘equilateral and equiangular triangle’ has greater connotation than that of the term ‘equilateral triangle’, there is no variation in the denotation of these terms. This illustrates the point that increase in connotation is not always accompanied by decrease in denotation. (e) Relation between Extension and Connotation : We can notice that there is also a relation of inverse variation between extension and connotation of terms. From the point of extension a higher class (genus) includes a lower class (species). But from the point of view of connotation the lower class (species) includes the connotation of the higher class (genus). Thus as we pass from a species to its genus the extension increases but the connotation decreases. On the other hand, when we pass from a genus to its species there is increase in connotation but decrease in extension. For example, if we increase the connotation of ‘humans beings’ adding a new proprerty ‘educated’ then the new term ‘educated human being’ will have greater connotation but its extension will be less than that of the extension  LOGIC AND LANGUAGE 39 of ‘human being’. Similarly, if we decrease the connotation of ‘human beings’ by taking out the property ‘rationality’ from it we get a new term ‘animal’ with greater extension, since the term ‘animal’ contains more subclasses than that of ‘human beings’. In the above examples we notice that increase in connotation leads to decrease in extension and decrease in connotation leads to increase in extension. Analogously we can notice that increase or decrease in extension will also lead to corresponding decrease or increase in connotation. (f) Genus and Species : As already pointed out earlier, genus is a higher class in relation to a sub-class called species. A species is a subordinate class of a higher class called its genus. Genus and species, thus are relative terms. For example, animal is a subordinate class in relation to all living beings, so animal is the species of the genus living beings. But animal is the genus in relation to its subordinate class human beings. Human being is the genus in relation to its spices intelligent human beings. A term having the widest extension which is never a subordinate class to anything, is called Summum genus. Similarly a term having the narrowest extension which is never a genus is called infima species. *** 40 +2 LOGIC, Part-I SUMMARY Ordinary language has a variety of functions which can be broadly classified into descriptive, emotive and directive uses of language. Logic is concerned with language that functions descriptively or informatively. Grammatical form of a sentence is not a sure guide to its function. Language is very often used to serve multiple functions. As words are the constituents of sentences, terms are the constituents of propositions. Terms are expressed by one word or many words. A term signifies an individual a property or a class of individuals. General terms have both denotation and connotation. Denotation of a term consists of the things to which the term applies. Connotation of a term consists of the set of common and essential attributes shared by the objects denoted by the term. The denotation of a term is determined by its connotation, but connotation is not determined by its denotation. Generally, denotation and connotation of terms vary inversely when the terms are arranged in accordance with their subordination. Increase or decrease in connotation of a term results in another term with a corresponding decrease or increase in denotation. So also, increase or decrease in denotation results in corresponding decrease or increase in connotation. A similar relation of inverse variation can also be noticed between extension and connotation of terms. ******  LOGIC AND LANGUAGE 41 MODEL QUESTIONS Objective type I. Fill in the blanks by selecting the appropriate word from the options given in the bracket. 1. When someone asserts or denies a fact, it is an instance of ____________ use of language. (informative, expressive, directive) 2. “Please close the door.” This sentence illustrates the ___________ use of language. (informative, expressive, directive) 3. The set of attributes shared by all and only those objects to which the term refers to is called the ____________ of that term. (extension, connotation, denotation) 4. Terms having different connotation may have the same _________. (denotation, meaning, intension) 5. When a question is posed in order to request an answer, it is a case of _______ discourse. (descriptive, expressive, directive) 6. The _________ use of language includes both correct and incorrect information. (descriptive, expressive, directive) 7. The __________ connotation of a word for a speaker is the set of all attributes the speaker believes to be possessed by objects denoted by that word. (objective, subjective, conventional) 8. The __________ connotation of a word is the total set of characteristics shared by all objects denoted by that word. (objective, subjective, conventional) 9. The __________ of a term increases when attributes are added to it. (extension, connotation, denotation) 10. The class term “unicorn” has no _____________. (extension, connotation, denotation) II. Point out in each case whether the statement is true or false. 1. The extension of a term refers to its subclasses. 2. The connotation of “town” includes Aska, Burla, Chowdwar, and Dhenkanal. 3. The term “sky flower” has connotation but no denotation. 42 +2 LOGIC, Part-I 4. The only function of language is the communication of ideas. 5. Expressive discourse is used to manifest the feelings of the speaker or to evoke them in others. 6. Directive discourse consists of true propositions. 7. “That’s too bad” is a directive use of language. 8. The grammatical division of sentences into declarative, interrogative, imperative, and exclamatory corresponds precisely with the functions of sentences. 9. Two words can have the same literal meaning but different emotive meanings. 10. The collection of objects to which a term may be applied constitutes its denotation. 11. The denotation of a term determines its connotation. 12. To talk about the intension of a term is the same as referring to its connotation. 13. Terms with different denotations cannot have the same connotation. 14. Adding more attributes to the connotation of a term may greatly narrow the range of its denotation. 15. Adding additional attributes to the connotation of a term will certainly not increase its denotation. Essay type 1. Distinguish between different functions of language. Explain the function of language that is important from the point of view of logic. 2. What is meant by denotation of a term? Distinguish between denotation and extension. 3. Distinguish between denotation and connotation of terms. Is there any term which has connotation but no denotation? Discuss. 4. Explain with suitable example the relation of inverse variation between connotation and extention of terms. Can you think of any exception to this relation? Discuss. 5. Distinguish between subjective, objective and logical connotation of terms. ***  CHAPTER - 3 DEFINITION AND MEANING 3.1 Verbal and Factual Disputes : Language is our principal tool for communication. We engage in arguments by using language. Language consists of sentences and a sentence consists of words. If words are not used carefully, it will affect our communication and our ability to create and communicate correct arguments. Sometimes the key words used in our communication are ambiguous or excessively vague. A word or term is ambiguous when it has two or more distinct meanings. For example, the words like ‘pen’, ‘mad’, ‘mouth’ are ambiguous. The word ‘pen’ stands for a writing instrument; but it also refers to an animal enclosure. The word ‘mad’ means insanity; but it is also used to mean anger. ‘Mouth’ can refer to a person’s mouth or the mouth of a river. If the speaker uses a word in one sense but the hearer understands it in another sense, controversy is likely to arise. Consider, for example, the following question: ‘If a tree falls in the forest and nobody is there to hear it, is there a sound?’ Conflicting answers to this question might be due to the ambiguity of the crucial word ‘sound’ in the question. The word ‘sound’ can mean sound waves or the sound sensation. If the word ‘sound’ is used to mean sound waves then there are sounds in the forest when the tree falls, whether or not someone is there to hear it. Alternatively, if the word is used to mean the sensation of sound, then clearly there is no sound when no one is there to experience it. Words in ordinary language are not only ambiguous, they are also often excessively vague. An expression is vague, if borderline cases for its application occur. In other words, an 44 + 2 LOGIC, Part-I expression is vague if it is unclear whether or not the expression is applicable in a particular context. Examples of vague terms include ‘kid’, ‘bald’, ‘old’, ‘happy’, ‘rich’, and ‘thin’. Another way that vagueness can occur is when an expression has several criteria that must be met for its correct application, and there is no specification of how many of the criteria must be satisfied, or to what degree. Such controversies, which are due to either vague or ambiguous use of words, are needless controversies and should be avoided as far as possible. Disputes arising out of either ambiguity or vagueness of words are verbal disputes. A verbal dispute can be avoided by stating the sense in which the key words are used. Once the parties to the dispute make clear the sense in which they are using the terms they will realise that their disagreement is on the meaning of their terms and the parties in the dispute might not be opposed to one another. A verbal dispute would disappear once the parties involved in the dispute distinguish between different meanings of the important terms and agree on the meaning of these terms. We can save a lot of time, sharpen our reasoning ability, and communicate with each other more effectively if we watch for disagreements about the meaning of words and try to resolve them whenever we can. But all disputes are not verbal disputes. Some disputes are genuine disputes in which there is some genuine disagreement between parties. Genuine or non-verbal disputes can be of two kinds. Such disputes involve either disagreement in attitude or disagreement in belief. Disagreement in attitude arises when two persons or parties express different feelings or attitudes towards the same situation. For example, some people like eggs and some others do not like eggs. Those who like eggs would say that eggs are delicious, but those who do not like would disagree and say that eggs do not have good taste. Here they are expressing their disagreement in their attitude towards eggs. If two persons give different answer to the question whether eggs are good tasting or not, then the fact is that one of them likes its taste and the other does not like the taste. No verbal dispute is involved here, since both the persons agree on the meaning of the words by means of which they convey their respective feelings. This is a case a genuine disagreement in the sense that they have different feelings towards the same situation.  DEFINITION AND MEANING 45 There is an important type of dispute in which parties to the disputes disagree on what they believe to be true. These are factual disputes. People engaged in a factual dispute often agree on the meaning of the words by means of which they convey their respective positions, but they disagree over the truth of some specific proposition. Suppose one person asserts that a spider has eight legs and another person disagrees and claims that spiders have six legs. Further investigation of the matter will settle their dispute in support of the claim that spiders have eight legs, because spiders in fact have eight legs. There are, however, some factual disputes which are difficult to settle because we are not in a position to verify the facts, but in such cases we can say what would settle the issue. 3.2 Definitions : A definition states the meaning of a term. Since meaning of an expression is explained by its definition, definitions provide a useful method of preventing or eliminating differences in the use of languages. In other words, definitions are essential for preventing ambiguity. We observed earlier that verbal disputes often arise due to use of ambiguous and excessively vague words and phrases. Such disputes can be avoided if we agree on the definition of the key terms in our arguments. Definitions thus help us in correcting mistakes in our reasoning by eliminating ambiguity. The term to be defined is called the definiendum. The word or words used to state the meaning of the term defined is called the definiens. In other words, the term sought to be defined is the definiendum and the expression supplying the definition is the definiens. Consider the definition of ‘triangle’. A triangle is a plane figure enclosed by three straight lines. In this definition the term ‘triangle’ is the definiedum and ‘a plane figure enclosed by three straight lines’ is the definiens. The definiedum and the definiens are said to have the same meaning. Both are claimed to be synonymous expressions and therefore one can be substituted for the other without any loss of meaning. A definition is either reportive or stipulative. When a definition states the meaning of a term as it is used it is called reportive definition. When it states the meaning of a term as it is to 46 + 2 LOGIC, Part-I be used it is called stipulative definition. A reportive definition of a term reports the way in which the term is already used. It tells us how a term is actually used by the native speakers of the language. Such a definition reports the conventional meaning of a term. It reports the long- established usage of a term. Reportive definitions can be true or false, depending upon the accuracy with which it captures the actual usage of the term. Reportive definitions help to eliminate ambiguity. Reportive definitions also help us to increase our vocabulary by learning the meaning of new symbols by the help of symbols we already understand. Most dictionary definitions are examples of reportive definitions. ‘A triangle is a plane figure enclosed by three straight lines’ is a reportive definition. So also the definition of ‘unicorn’ as ‘a horse-like animal having a single straight horn projecting from its forehead’ is a reportive definition. A stipulative definition assigns a new meaning to a term. By stipulative definitions new terms are introduced into a language or old terms are made precise by deliberately assigning new meaning for the terms. A stipulative definition creates a new usage that did not exist prior to the stipulation. Stipulative definitions have the form “Let us use the word ‘…’ to mean ‘…’. Stipulative definitions are neither true nor false. One can however judge a stipulative definition to be useful or cumbersome, clear or obscure, but not as true or false. Since the purpose is to introduce a term by assigning a new meaning, no existing convention or usage can be cited for its correctness or incorrectness. A stipulative definition is a proposal or a decision to use a term in a certain sense. When a term is first introduced into a language it gets its meaning by stipulation. For example, when spacecrafts were made to fly to outer space and persons were specially trained to fly these spacecrafts, there was a need to introduce a technical term to refer to such persons. So the term ‘astronaut’ was coined to mean ‘a person who has been trained to travel in a spacecraft’. When the word was first introduced to mean this it was an instance of stipulative definition. Now, however, the term has already gained wide currency and this definition can be cited as an instance of reportive definition. 3.3 Denotative and Connotative Definitions : We noted earlier that a general term has two kinds of meaning. It applies to certain things and signifies some properties. These two features of a general term are called its denotation and  DEFINITION AND MEANING 47 connotation respectively. Keeping the distinction between denotation and connotation in mind, we can now proceed to discuss the two ways a general term can be defined. (a) Denotative Definition : A denotative definition explains the meaning of a term by identifying the things to which the term applies. A denotative definition of the phrase ‘the past prime ministers of India’ will consist of a complete list of persons who have been prime ministers of India in the past. We thus give a denotative definition simply by listing all the things or individuals who are members of the class. A denotative definition can be either verbal or non-verbal (ostensive). A verbal definition is one that linguistically identifies all objects to which the term can be correctly applied by naming them. For example, to give a verbal definition of the term ‘planets in the solar system’ one can give the names of all the planets that go around the Sun. A verbal denotational definition of ‘past presidents of India’ will consist of the names of all the past presidents of India. There are, however, several limitations to verbal denotational definitions: a. It is not always possible to name all the objects denoted by a word or expression. General terms refer to infinite number of things. Terms like ‘numbers’ and ‘instants of time’ refer to infinite series. Even terms such as ‘chairs’ and tables’ denote not only chairs and tables now existing all over the world but also those which existed in the past and those which would exist in the future. Even when an expression refers to a finite set of things (such as ‘the colleges of Odisha in the year 2003’) it is either cumbersome or inconvenient to name all of the things to which the expression refers in order to elucidate the meaning of the expression. b. Further, most terms do not have names for each and every individual they include within their denotation. For example, all the individual tables that constitute the denotation of the term ‘table’ do not have names. To provide a verbal denotative definition of ‘table’ would require providing a list containing the names of all the individual tables. This is an 48 + 2 LOGIC, Part-I impracticable and impossible task. c. In view of these difficulties, one might suggest that although denotative definition by complete enumeration is not possible, one may try instead to provide a denotative definition by offering a few examples. In practice, we generally follow this procedure to elucidate the meaning a term. This is a very useful method in practice. Citing of examples often assist us to understand the meaning of a term. But the important question is: Can we provide the definition of a term by this method? d. There are some terms for which denotative definition is entirely impossible. The term ‘unicorn’ has an accepted meaning in our language. ‘Unicorns are fictitious’ and ‘There are no unicorns in the world’ are meani