Sanjay Ranade, HoD, DCJ, UoM1 A CONCISE LOGIC William H Halverson.

Sanjay Ranade, HoD, DCJ, UoM 1

A CONCISE LOGICWilliam H Halverson


The subject matter of logic is arguments, that is, bits of reasoning concerning which it is claimed that some statement or statements (the premise or premises) provide evidence for the truth of some other statement (the conclusion).


1.1 Reasoning

This is a particular kind of thinking. What kind? – The kind in which one derives a conclusion from certain alleged facts and/or hypothetical assumptions.


If I say, “The road was not wet last evening but it is wet this morning, so obviously it has rained during the night,” I am reasoning – in this case, from certain alleged facts to the conclusion that it rained during the night.

If I say, “We are going to have higher prices on the stock market now because the Congress has won the election with a majority and stocks always do better when Congress is in power,” I am also reasoning – this time from a fact (a Congress winning) and an alleged general truth (stocks do better when the Congress is in power) to the conclusion that there will be higher prices on the stock market.


It is not the process in the mind or brain while reasoning that is important to a logician.

Logic is concerned with the product, not the process, of reasoning.


1.2 Statements

Reasoning always concerns relations between and among statements.

A statement is a verbal formulation affirming or denying that something is the case.

Statements thus, have the important property of being either true or false. To say that a statement is true is to say that what it asserts to be the case is in fact the case. Otherwise it is false.


In the case of a general statement, to say that it is true is to say that what it asserts to be the case is the case without exception.

If all the houses in the world but one were white, the statement “All houses are white” would be false, not true.

If there were but one white house in the world, the statement “No houses are white” would be false, not true.


A statement may be close to the truth, in the sense that a slight modification of the statement would make it true, but a statement cannot be partly true and partly false, nor can it contain some truth and some falsehood.


Although every statement is true or false, it of course does not follow that we or anyone else can in all cases know whether a given statement is true or false.

The statement “There is a least one place in the universe other than the earth where intelligent life exists,” for example, is either true or false, but we do not know which.


Indeed the whole of humanity’s quest for knowledge might be described as an effort to determine which statements are true and which are false.


1.3 Arguments

In logic, arguments are not disagreements (“The argument concerned who was best qualified to be mayor”) or quarrels (“They had their usual end-of-the-month argument over family finances”).

Logic looks at arguments as instances of reasoning.


Thus, arguments are instances of reasoning concerning which it is claimed that some statement or statements (the premise or premises) provide evidence for the truth of some other statement (the conclusion).


An argument then, contains a minimum of two statements, that is, at least one premise and a conclusion.

Logicians generally distinguish two kinds of arguments….


Deductive argument

This is one for which it is claimed that the premises constitute conclusive evidence for the truth of the conclusion. In a correct deductive argument, if the premises are true, the conclusion must be true. E.g.

All college students are literate.Madhav is a college student._________________________Therefore, Madhav is literate.


This argument does not establish absolutely that Madhav is literate. It establishes that Madhav is literate if the premise of the argument are true.

The strength of the correct deductive arguments are that their conclusions follow necessarily from their

premises.

The weakness of such arguments is that their premises are frequently doubtful, if not obviously

false.


Inductive argument

This is one for which it is claimed that the premises merely offer some evidence in support of the conclusion.

It is logically possible for the premises of a correct inductive argument to be true and the conclusion false; the truth of those premises establishes only that there is some probability that the conclusion is true.


College student 1 is literate College student 2 is literate College student 3 is literate ……………….. ……………….. ……………….. College student n is literate

___________________________

Therefore, all college students are literate


The advantage that inductive arguments have over many deductive arguments is that their premises, as in the above example, are typically assertions of observable fact that can be easily verified. One can often be reasonably sure that the premises of such an argument are true.

The weakness of inductive arguments is that their conclusions do not follow necessarily from their premises: the conclusion may be false even if the premises are true.


Inductive and deductive reasoning are often combined in what looks like a single argument

A recent poll of a group of voters believed to be representative of the entire electorate yielded the followings findings:

58 per cent of those polled favoured candidate Mahajan.37 per cent of those polled favoured candidate Mohanty.5 per cent of those polled were undecided.

We can reasonably conclude, then, that Mahajan will receive approximately 58 per cent of the votes cast on election day. Any candidate who receives 58 per cent of the votes cast on election day will be elected. Therefore, Mahajan will be elected.


This is essentially an inductive argument, as is evident from the fact that the conclusion follows from the premises not with certainty but only with some degree of probability.

The premises, if true, make it likely that approximately 58 per cent of those who vote on election day will vote for Mahajan. Since 58 per cent constitutes a majority of the votes cast, and since a majority of the votes cast guarantees election, it is concluded Mahajan will be elected.


Look at the following statements:

A recent poll shows 58 per cent of those polled favouring candidate Mahajan, 37 per cent candidate Mohanty, and 5 per cent undecided.

Approximately 58 per cent of those who vote on election day will vote for candidate Mahajan.

Any candidate who receives approximately 58 per cent of the votes cast on election day will be elected.

Candidate Mahajan will be elected.


Statement 2 is an inducted inference from the evidence summarised in statement 1.

Statement 4 is a deductive inference from statements 2 and 3.

Statement 2 in the example is therefore the conclusion of an inductive argument and the premise of a deductive argument.

What appears initially to be a single argument is in reality two arguments, one inductive and one deductive.


Exercise 1.1-1.3

Each of the following passages contains at least one argument. For each passage a) identify the premise(s) and the conclusion of the argument and b) state whether the argument is deductive or inductive.


Our bank is committed to serving you. We have made a multimillion rupee investment in data processing equipment. We are open more hours each week than any of our competitors. Our automatic teller is open twenty-four hours a day, seven days a week. And our staff consists only of trained professionals.


Inflation is caused by an excess of money in circulation. Deficit spending by government results in too much money being in circulation. Thus federal deficit spending is the root cause of inflation. If the federal government would balance its budget, inflation would cease.


Jayant deserves this award for outstanding public service. He is a leader in this community, a deacon in his church, a member of the school board, and chairman of the United Fund.


I expect my new Pulsar to give me much better mileage than my old motorcycle, because according to recent tests the Pulsar gives the best mileage of any car sold in this country.


It is necessary that the land and the surrounding waters have the figure which is cast by the shadow of the earth, for during an eclipse it projects on the moon the circumference of a perfect circle. Therefore the earth is not a plane…or a cylinder…but is it is perfectly round - Copernicus


The student body always elects as homecoming queen the most beautiful girl on campus. Some years ago the students at a certain university elected a young cow as homecoming queen. It is clear, therefore, that the young cow was the most beautiful girl on campus that year.


No month can possibly have more than 23 weekdays, since the longest months contain only 31 days, and in any 31-day period there must be at least four Saturdays and four Sundays.


Evidently working with computers is good for one’s health, for according to a recent study, nearly all of the people who have ever worked with computers are still living.


Vasant Rajaram Bhosale became president in 1932. During his campaign for the presidency, he wore leg braces a result of his bout with polio, so it is clear that he had polio at some time before 1932.


As more and more nuclear power plants have started up, more and more nuclear accidents have occurred. Ten years from now, according to present plans, there will be twice as many nuclear plants as there are today. Thus it is inevitable that the frequency of such accidents will increase.


1.4 Logical assumptions

If we assume a) all crows are black, b) there are some birds in Agartala and c) at least 50 per cent of the birds in Agartala are crows, what are we entitled to conclude about the truth or falsity of the following statements?


1. Some birds in Agartala are black.

2. Some birds in Agartala are not black.

3. Some crows in Agartala are not black.

4. Some birds in Agartala are not crows.

5. Some noncrows are not black.

6. No crows in Agartala are nonblack.


Statements are sometimes related in such a way, that if one knows or assumes the truth or falsity of some statements, one can infer the truth or falsity of certain other statements. This relationship is called logical implication.

Deductive logic is concerned exclusively with this relationship.

Logical implication is thus a relationship between and among statements such that we can infer the truth or falsity of certain statements on the assumption that a given statement or set of

statements is true or false.


1.5 Probable Inference

Suppose that you are driving down the express highway in your car, and two things happen:

a) The engine sputters and stops and

b) You observe with dismay that the fuel gauge is on EMPTY.

Can you, on the basis of these facts, infer that the fuel tank is empty?


The reasoning involved in any case in which the premises are clearly relevant to the conclusion of the argument but the conclusion cannot be rigorously deduced from those premises is called probable inference. All inductive arguments involve such inferences.


The type of reasoning involving probable inference in the example mentioned earlier may be called an argument leading to a presumption of determinate fact: from certain observed determinate facts we infer an unobserved determinate fact.

For e.g. that the thermometer registers 100 degree Celsius we infer that we have fever.


A second type of reasoning involving probable inference is that in which we infer an empirical generalisation.

This is a general statement about a class of objects made on the basis of observation of some members of the class.

If a biologist says that “all female salmon, when they are about to lay their eggs, return to the fresh-water streams where they were hatched,” the biologist is stating an empirical generalisation.


A third type of argument involving probable inference is one in which the conclusion of the argument is a theoretical hypothesis.

Consider the wave theory of light – this theory is not an empirical generalisation for it is a not a statement about all instances. One cannot observe the wavelike character of some representative rays of light, whereas one can observe the spawning of salmon.


1.6 Logic

Against the background of the foregoing discussion and definitions, we can now define logic asThe science of distinguishing between correct and

incorrect reasoning or arguments.

The subject matter of logic is reasoning, arguments, the drawing of inferences.

The specific task of logic is to provide methods for distinguishing between good and bad reasoning, correct and incorrect arguments, allowable and

unallowable inferences.


In pursuing this task, logicians typically engage in three distinct activities: analysis, classification, and assessment.


Logic begins with analysis because before one can get on with the task of distinguishing between correct and incorrect arguments, one must determine what is and what is not an argument.

As logicians study the enormous variety of arguments that they encounter in both spoken and written discourse, they observe certain similarities and differences between and among the arguments. These similarities and differences form the basis for the classification of arguments into types and subtypes.

Analysis and classification, are however, only preludes to the chief business of logic, which is the assessment of arguments, the determination of their correctness or incorrectness.


1.7 The Uses of Logic

An understanding of the principles of logic is likely to reduce the likelihood of being deceived by the faulty arguments of others.

An understanding of logic should enhance one’s ability to clarify one’s beliefs.

An understanding of logic may be expected to aid us in the quest for knowledge.


Exercises 1.4-1.7 Sandhya is taller than Chitra and Bina is taller than Sandhya.

Therefore, a. Bina is taller than Chitrab. It is not the case that Chitra is taller than Bina.c. It is not the case that Chitra is taller than Binad. It is not the case that Chitra is taller than Sandhyae. Chitra is shorter than Sandhyaf. Chitra is shorter than Binag. It is not the case that Chitra is shorter than Binah. Chitra is shorter than Bina and Sandhyai. Chitra and Sandhya are shorter than Binaj. Either Chitra is shorter than Sandhya or Bina is shorter than

Sandhya.


Indicate what probable inference, if any , you would be inclined to draw from the following premise. State whether the inference is a presumption of a determinate fact, an empirical generalisation, a theoretical hypothesis or a probable inference of some other kind.

In a recent road test, five Marutis achieved mileage readings of 34, 31, 32, 34 and 33 km per litre respectively, whereas five comparably equipped Hondas in the same weight class achieved mileage readings of 37, 35, 39, 36 and 35 km per litre. Therefore…

Returning home from a recent vacation, the Sonawanes discovered that their back door was smashed, clothes and other items were strewn about in several rooms, and a number of valuables – including jewelry, silverware, and photographic equipment – were missing. It seems evident that….

On the way home from work the next day, Mr Sonawane stopped at a pawnshop to find out whether anyone had tried to pawn the items that were missing. As he walked into the pawnshop, a teenaged boy whom he recognised as his son’s best friend was attempting to pawn a camera of the same make as one that was missing from Mr Sonawane’s house. Upon seeing Mr Sonawane, the boy dropped the camera and ran out of the pawnshop. Mr Sonawane would appear to be justified in concluding…

Mr Sonawane inspected the camera that the boy had left behind, and determined that although it was similar to one of the items missing from his house, it was not in fact his camera. He then concluded that …


2. Language


2.1 Functions of Language

Language is used for four basic purposes –

a) To make assertions – assertive function

b) To ask questions – interrogative function

c) To direct behaviour – directive function

d) To express feeling – expressive function


Language is being assertorically when it is used to state that something is or is not the case. Such discourse is also said to express a truth claim, since to assert that something is or is not the case is to claim that what is asserted (the statement) is true.


The asking questions is a second basic function of language. It is not necessary that an interrogative question end with a question mark. Language may serve an interrogative function even in the absence of an interrogative sentence.


The directive function of language may be seen in the issuing of commands, requests, instructions or recommendations. The function is often served by sentence in the imperative mood.


The expressive function of language is exemplified by ejaculations as “Wow”, “great”.


A single sentence can serve more than one function.

One cannot always determine the function of a piece of discourse simply on the basis of the grammatical form. One must pay attention to the apparent meaning of what is being said, and make a reasoned judgement about its function.


Arguments consist of statements, and statements are without exception instances of language used assertorically,it is obvious that it is language in its assertoric function that is of primary interest to logic.


Identify the function or functions served by each of the following examples of discourse. A given piece of discourse may serve more than one function


Exeter lies in a surpassingly beautiful slice of southern New Hampshire, where the Atlantic, tall pine forests, and pre-Revolutionary homes have joined to create a background for a mystery – a mystery as fantastic and sweeping as the legend of Sleepy Hollow, mixed with the most vivid trappings of science fiction.


Future shock is a time phenomenon, a product of the greatly accelerated rate of change in society. It arises from the superimposition of a new culture on an old one. It is culture shock inn one’s own culture.


The only possible way out of chaos is for us to come once more under the control of ideals of true civilisation through the adoption of an attitude toward life that contains those ideals. But what is the nature of the attitude toward life in which the will to general progress and to ethical progress are alike founded and in which they are bound together? It consists in an ethical affirmation of the world and of life.


Johnny’s first explorations of the external world were in the form of pictures – graphic art. Some of his paintings still hang on the walls in the house in Madison, Connecticut. The violence with which a child sees nature! The brilliant savagery of the struggles already precipitated in a infant’s mind!


It’s a little late to be proving that a behavioral technology is well advanced. How can you deny it? Many of its methods and techniques are really as old as the hills. Look at their frightful misuse inn the hands of the Nazis! And what about the technique of the psychological clinic? What about education? Or religion? Or practical politics? Or advertising and salesmanship? Bring them all together and you have a sort of rule-of-thumb technology of vast power.


One need not be a partisan of the provincial governments to argue that Mr Trudeau is making a grave mistake in trying to railroad through London a bill of rights the Canadian provinces cannot agree on … Imposing constitutional provisions from outside… can only aggravate the already bitter conflicts in Canada about language, energy and the balance of federal and provincial powers.


It’s a little late to be proving that behavioural technology is well advanced. How can you deny it? Many of its methods and techniques are really as old as the hills. Look at the frightful misuse in the hands of the Nazis! And what about the technique of the psychological clinic? What about education? Or religion? Or practical politics? Or advertising and salesmanship? Bring them all together and you have a sort of rule-of-thumb technology off vast power.


2.2 Language and Metalanguage

When we talk ‘about’ cars we speak in terms of ‘speed’, ‘mileage’, ‘torque’ etc.

When we discuss thunderstorms we speak in terms of ‘frightening’, ‘awesome’, destructive’ etc.


The world is a physical entity, a complex system of objects and events occurring in space and

time.


Language is a linguistic entity and its most obvious components are not objects and events but words.


Consider these two statements..

Lightning is a discharge of electrical energy in the atmosphere.

Lightning is a noun.

Are both these statements true? If yes, it would follow that that there is something in this world that is both a discharge of electrical energy in the atmosphere and a noun!


Statement 2 is about language. It is about the word lightning, not about a phenomenon or event called lightning that is in the natural world.

Let us call a language being spoken about an object language and a language being used to speak about it a metalanguage.

Metalanguage thus, must have some way of indicating that the objects to which it is referring are words in the object language and not entities in the physical world. The most common way is to enclose words in quotation marks.


Thus, to avoid confusion, the statement 2 should be written as –

‘Lightning’ is a noun.


Exercise 2.2 Identify the statements in the following list that can most reasonably be interpreted as being statements in a metalanguage. Insert quotation marks as necessary to make the statements thus identified correct and meaningful.

Wisdom, courage, temperance, and justice were considered by the Greeks to be the principal virtues.

Justice is sometimes represented pictorially as a blindfolded woman holding a scale.

Justice comes from the Latin justitia. Nouns have cases, verbs have tenses. Go, went, and will go are, respectively, the present, past and

future tenses of the verb to go. I am firm, you are stubborn. Firmness and stubbornness are the same thing; its just a

question of who you’re talking about. Firm means stubborn.


2.3 Extension and Intension of words

The principal components of a language are words. From one point of view, a word is merely a noise made by a human being or a mark on a piece of paper.

What distinguishes words from other noises, such as the babbling of an infant or a random series of letters of the alphabet, is the fact that they have meaning.

Sanjay Ranade, HoD, DCJ, UoM1 A CONCISE LOGIC William H Halverson.

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