[This write-up is prepared primarily on the basis of the IGNOU Study material on Logic and certain other materials and is provided here for academic reference for students. No originality, authorship or copyright to the above is claimed.

The post in docx format is available here.]

I. Introduction

Categorical syllogism is the essence of traditional logic. This form of inference is called mediate inference because the conclusion is drawn from two premises.

Further, this is called categorical because all propositions involved are categorical.

Since syllogistic inference is nearly identical with deductive inference, an exhaustive analysis of inference is required as a prelude to a proper understanding of syllogism.

II. Reason & Inference

A. What is Reasoning?

Reasoning consists, essentially, in the employment of intellect, in its ability to ‘see’ beyond, and within as well, what is available to senses.

Reasoning, therefore, is a sort of bridge which connects ‘unknown’ with ‘known’.

B. Inference

Inference is regarded as the process involved in extracting what is unknown from what is known.

Reasoning is essentially a psychological process which is, undoubtedly, not the concern of logic. Therefore, some logicians thought it proper to replace reasoning with inference. However, this replacement did not improve matters much.

The reason is obvious. If all human beings stop thinking, then there will be nothing like inference.

This dependence shows that inference is as much a psychological activity as reasoning is. What is psychological is necessarily subjective. Logic, in virtue of its close association with knowledge, has nothing to do with anything that is subjective.

Implication

Therefore, it was imperative for logicians to discover an escape route. Cohen and Nagel for this particular reason chose to use ‘implication’ instead of ‘inference’. The difference in kind can be understood easily when we look at the usage.

Statements always ‘imply’ but do not ‘infer’. Therefore, implication is in the nature of relation between statements.

On the other hand, I ‘infer’, but I do not ‘imply’. This clearly shows that inference is an activity of mind.

Salmon fell in line with Cohen and Nagel when he said that the very possibility of inference depends upon reasoning. Despite the fact that inference is subjective, logicians like Copi, Carnap, Russell, etc., chose to retain the word inference.

But, all along, they only meant implication. Therefore keeping these restrictions in our mind let us use freely the word ‘inference’.

Reasonableness

Though the use of the word ‘reason’ is not much rewarding, the word ‘reasonableness’ has some weight. We often talk about reasonableness of the conclusion.

In this context reasonableness means ‘grounds of acceptability’. Surely, in this restricted sense, reasonableness is objective just as inference is.

III. Kinds of Inference: Deductive Inference

In a broad sense, there are two kinds of inference; deductive and inductive.

Deductive inference regards the form or structure as primary and therefore it is called formal logic (inference and logic are interchangeable). It remains to be seen what form means. Inductive logic regards matter or content of argument as primary. Some logicians, like Cohen and Nagel, did not regard induction as logic at all. Without considering the merits and demerits of their arguments, let us consider briefly the characteristics of these two kinds.

A. Distinction between Truth & Falsehood Vs. Validity & Invalidity

The study of formal logic begins with the distinction between truth and falsehood on the one

hand, and validity and invalidity on the other. This particular distinction is very prominent.

Only statements are true (or false) whereas only arguments are valid (or invalid).

From the table, we can infer the following:

· A valid argument (1 and 3) may consist of completely true statements or completely false statements.

· An invalid argument (2 and 4), similarly, may consist of statements in exactly the same manner mentioned above.

It shows that truth and validity, on the one hand, and falsity and invalidity, on the other, do not coincide always.

B. Material & Logical Truth

Material truth is what characterizes matter of fact. Logical truth is determined by the structure of argument. We shall consider examples which correspond to four combinations (see table1). Let us call premises p1, p2, etc. and conclusion q.

Arg1:

p1: No foreigners are voters.

p2: All Europeans are foreigners.

q: ∴No Europeans are voters.

Arg2:

p1: Some poets are literary figures.

p2: All play writers are literary figures.

q: ∴Some play writers are poets.

Arg3:

p1: All politicians are ministers.

p2: Medha Patkar is a politician.

q: ∴Medha Patkar is a minister.

Arg4:

p1: 3 is the cube root of – 27.

p2: - 27 is the cube root of 729.

q:∴ 3 is the cube root of 729.

These four arguments apply to arguments 1, 2, 3, and 4 of Table 1 respectively. First and third

arguments have a definite structure in virtue of which they are held to be valid. Second and

fourth arguments have a different structure which makes them invalid.

When an argument is valid, the premise or premises imply the conclusion. If there is no implication, then the argument is invalid.

Validity is governed by a certain rule which is represented in a tabular form. [Let us designate ‘true’ by ‘1’ and ‘false’ by ‘0’ as a matter convention].

We can also say that the premises necessitate the conclusion and when they necessitate the conclusion there is implication. In this case, necessity is of a particular kind, viz., logical necessity. Therefore, when there is implication, conclusion is necessarily true and vice versa.

C. Deductive Logic as Tautology

Very often, deductive logic is identified with mathematical model. It is generally admitted that in both these disciplines information provided by the conclusion is the same as the one provided by the premises. It means that both are characterized by material identity

Deductive logic, therefore, is an example for tautology.

This characterization is highly significant and is in need of some elaboration. If, one can ask, the conclusion does not go beyond premises (it may go below or well within) and no new information is acquired in the process, then why argue and what is the use of arguments?

The answer is very simple. Knowledge is not the same as mere acquisition of information. In other words, novelty is not a measure of knowledge. The legend is that Socrates extracted a geometrical theorem from a slave purported to be totally ignorant of mathematics. The moral is that knowledge is within, not in the sense in which brain or liver is within.

Knowledge is the outcome of critical attitude. It is discovered, not invented and so goes an ancient Indian maxim: eliminate ignorance and become enlightened. If what is said is not clear, then consider this path.

Deductive argument helps us to know what is latent in the premises, i.e., the meaning of the premises.

It is an excursion into the analysis of their meaning or meanings. And the conclusion is an expression of the same. If so, it is easy to see how the denial of the conclusion in such a case amounts to denying the meaning or meanings of the premises which were accepted earlier.

What is called self-contradiction is exactly the same as the combination of the denial of the conclusion and the acceptance of the premises.

Therefore, we say that a valid deductive argument is characterized by logical necessity. If so, a deductive argument is always true. This is the meaning of tautology.

D. Analytic & A Priori

Consider this example: ‘all men with no hair on their heads are bald’. We know that this statement is true in virtue of the meaning of the word ‘bald’; not otherwise. Such a statement is called analytic.

In such statements the predicate term (here ‘bald’) is contained in the subject term (here ‘men with no hair on their heads’).

Knowledge obtained from an analytic statement is necessarily a priori, meaning knowledge prior to sense experience.

In philosophical parlance, all analytic statements are necessarily a priori.

Deductive logic provides knowledge a priori, though the premises and the conclusion considered independently are not analytic. It is the knowledge of the relation between the premises and the conclusion which is a priori. Therefore deductive argument and analytic statement share a common characteristic; in both the cases the denial leads to self-contradiction.

How can we say that deductive logic provides a priori knowledge? Consider an example.

Arg. 5: All saints are pious.

All philosophers are saints.

∴All philosophers are pious.

Evidently, there is no need to examine saints and philosophers to know that the conclusion is true. Indeed, it is not even necessary that there should be saints who are pious as well as philosophers. This being the case, arg. 5 takes the following form without leading to any distortion of meaning.

Arg. 5a: If all saints are pious and all philosophers are saints, then all philosophers are pious.

The argument is transformed into a statement which involves relation. Implication (the present

relation is one such) is such that without the aid of sense experience, but with the laws of formal

logic alone, it enables us to derive the conclusion.

Thus, like an analytic statement, a valid deductive argument provides a priori knowledge and hence it is devoid of novelty. It is this sort of relation that precisely describes the relation between the premises and the conclusion in deductive inference.

This does not mean that deductive argument is absolutely certain. This is because necessity is a logical property whereas certainty is a psychological state. The former is objective and the latter is subjective.

When sense experience takes back seat, reason becomes the prime means of acquiring knowledge. Following the footsteps of Descartes, who is regarded as the father of rationalism,

we can conclude, somewhat loosely, that deductive logic is rational. So we have sketched three

characteristics; logical necessity, a priori and rational. There is a thread which runs through these characteristics. Therefore, one character presupposes another.

E. Valid & Invalid Arguments

Deductive argument is characterized by qualitative difference in opposition to quantitative difference, i.e. the difference between valid and invalid arguments is only in kind but not in degree.

Further, validity is not a matter of degree. A valid argument cannot become more valid in virtue of the addition of premise or premises.

On the other hand, if any one premise is taken out of a valid argument, then the argument does not become ‘less valid’; it simply becomes invalid.

So, an argument is either valid or invalid.

A valid argument is always satiated. In other words, the premises in a valid argument constitute the necessary and sufficient conditions to accept the conclusion. An argument is invalid due to a ‘missing link’ in the class of premises.

Deductive argument, therefore, is regarded as demonstrative argument. Acceptance of premises leaves no room for any reasonable or meaningful doubt.

F. Strawson’s Analysis of Nature of Deductive Logic

PF Strawson’s three aspects of formal logic:

1. Generality:

2. Form; and

3. System

· Generality means that individual is not the subject of formal logic.

· Formal Logic concerns only the relation between systems but not objects.

Futile to embark on study involving objects as such a study has no end. e.g.:

Arg. 6:

P1: The author of Abhijnana Sakuntala was in the court of King Bhoja

P2: Kalidasa is the author of Abhijnana Sakuntala

Q: Therefore, Kalidasa was in the court of King Bhoja

Arg. 7:

P1: The author of Monadology was in the court of the queen of Prussia

P2: Leibniz is the uthor of Monadology

Q: Therefore, Leibniz was in the court of the queen of Prussia.

Comparison of Arg 6 and 7 shows both are identical in form and only different in subject matter. Possible to construct countless such Arg.

Essence of formal logic is in stating p1 & p2 imply/ entail q or q follows from p1 & p2

What is relevant is the implication/ entailment, which is independent of subject-matter. Without even knowing the subject matter of p1 and p2, validity of arguments can be determined.

Arg. 7a:

P1: The author of x was in the court of Z

P2: Y is the author of X

Q: Therefore, y was in the court of Z.

G. Logical Forms

Arg. 7a is a logical form. It has two components:

· Variables: x, y & z

· Consonants: In categorical propositions, words all, some, no and not are constants

Structure of an argument is determined by consonants & not variables. Some points on the dependence of the laws of an argument on consonants:

· Every class of arguments have definite constants

· Structure of one class of arguments is different from another class of such arguments

· When the structure of an argument differs from that of another, the corresponding laws also differ from another.

· See, birds and aquatic creatures example reg. anatomical features

H. Integration of Rules

· Structure of argument and rules are mutually interdependent.

· If it is possible to decide the structure of an argument and the different classes of argument, it is possible to achieve formalization/ systematization.

· Formalisation: Enables make a complete set of rules and classify them so as to correlate with different arguments.

IV. Kinds of Inference: Induction

A. What is Induction?

Induction is a non-demonstrative argument where the premises are not and cannot constitute conclusive evidences of the argument.

Induction: Epogage (Aristotle) or Ampliative (CS Pierce): Conclusion goes beyond the premises.

Experience:

· Provides reasonable grounds for believing and not conclusive evidence. Inductive experience has sense experience.

· Therefore, premises are called as observation-statements.

· Conclusion is not an observation-statement as it overshoots the material provided by observation-statements

· e.g., no matter how many crows have been observed, it cannot be concluded: “All crows are black”.

B. Whether Inductive Arguments Always Produce Universal Statements?

· Misconception that inductive arguments produce universal statements.

· It provides a statement which depends on experience for further verification but in itself is not an experience statement.

· At times, experience vouches for the conclusion

· Inductive leap or generlisation: Leap from observed to unobserved or unobservable.

· All generalisations are not universal statements- possible to construct a universal statement without generalization: e.g., after having looked at all the books in a library, it is possible to conclude that all books in the library are hard bound,

C. Inductive Arguments need not be Future Oriented

· Need not be future oriented. Can also be past-oriented: e.g., history, anthropology and geology.

· Prime characteristics of induction are :

o the conclusion does not necessarily follow the premises, and

o experience precedes inference> induction is a posteriori

· Whatever knowledge is acquired after experience is called a posteriori.

D. Uncertainty, Probability and Inductive Arguments

· Karl Popper questioned how inductive logic can be called empirical.

· Inductive arguments called in philosophy of science as Probability

· Inductive conclusions are only probable: probability is a matter of degree while validity is not.

· Therefore, inductive arguments can be less or highly probable.

V. Deductive Reasoning & Syllogism

When people reason, they use logical pattern as undercurrents. Logicians tend to discover these undercurrents. They have standard form arguments and compares them with arguments.

Logicians generalize argument types> important for deduction, without which it will be merely rhetoric without practical value.

A. Comparison of Lay-Man & Logicians Words

Arg.1

Lay-man’s method: ‘Does God exist? Of course, he does not! No one has ever seen him, heard him, talked to him; has any one?’

Logician’s Method:

All bodies which exist are perceivable. BAP

God is not perceivable. GEP

∴ God is not a body which exists. ∴GEB

Arg.2

Lay-man’s method: ‘Was the Neanderthal a man? Yes he was. In fact we have proof to assert that he made tools, could paint, lived in groups etc.’

Logician’s method:

All beings who make tools, can paint, live in groups, etc. are men. BAM

The Neanderthal was a being who made tools, could paint, lived in groups, etc. NAB

∴The Neanderthal was a man. ∴ NAM

Observations:

· Statements to be proved: Conclusion

· Reasons: Premises

· Order is immaterial but conclusions generally appear in the end and are preceded by therefore, as a result, consequently, etc.

· If conclusion appears at the beginning, these are preceded with because, for, etc.

· At least one of the premises is a universal proposition: if not , the syllogism is not valid.

· Only three terms with each term occurring twice.

· S and P of the conclusion are minor (S) and major terms (P).

· The premise in which the minor occurs is called the minor premise and the premise in which the major occurs is called the major premise.

· One term is common to both the premises. This is called the middle term (M).

· In the first example ‘God’ is minor , ‘bodies which exist’ is major and ‘perceivable’ is middle and in the second example ‘Neanderthal’ is minor, ‘man’ is major and ‘beings who…groups’ is middle.

B. Mediate Inference

Again, order of premises does not matter though, generally, major finds the first place. Aristotle had convincing reason to choose these names. While the major has maximum extension, minor has minimum extension. The middle is so called because its extension varies between the limits set by the minor and the major. Aristotle argued that our inference proceeds from minor to major through middle. This explains the meaning of mediate inference.

VI. Types of Syllogisms

A. Classification & Constants

Classification of syllogisms is based on constants: all ,some, not. They are not dependent on variables. Take the example:

All X are Y

All Y are Z

All X are Z

Even if you replace X, Y & Z by say, P, Q, R or A, B, C, the validity is unchanged.

No X are Y

All Y are Z

All are Z

But if consonants are changed, it might, but will not necessarily, affect logical validity. In this illustration, consonant in the major premise is changed alters the logical validity even though the variables are the same: the syllogism is invalid.

Some X are Y

All Y are Z

Some X are Z

This is an example of a case where change of consonants results in logical validity. In this illustration, consonant in the major premise is changed alters the logical validity even though the variables are the same: the syllogism is invalid.

Therefore, the logical status, that is, validity/ invalidity is not determined by the variables but by the consonants.

B. Simple, General & Compound Propositions in Logic

English sentences are either true or false or neither. Consider the following sentences:

1. Warsaw is the capital of Poland.

2. 2 + 5 = 3.

3. How are you?

The first sentence is true, the second is false, while the last one is neither true nor false.

A statement that is either true or false but not both is called a proposition.

Propositional logic deals with such statements and compound propositions that combine together simple propositions (e.g., combining sentences (1) and (2) above we may say “Warsaw is the capital of Poland and 2+5 = 3”).

Several propositions are compounded using constants. Each constant determines the species which belongs to the sub-class.

Conditional Proposition: A proposition of the form “if p then q” or “p implies q”, represented “p ! q” is called a conditional proposition. For instance: “if John is from Chicago then John is from Illinois”.

VII. Three Kinds of Conditional Syllogisms

A. Pure Hypothetical Syllogism

o All propositions are hypothetical: ‘hypothetical’ because they express a condition

o Words ;if… then’ constitute the condition and also the constant: if no consonant, then it ceases to be hypothetical.

o Statement after ‘if’: antecedent

Statement after ‘then’: consequent

o One statement is common to two premises.

o If quality is constant, then it would appear in one statement as antecedent and in another as constant.

o Both the premises as well as the conclusion are conditionals. For such a conditional to be valid the antecedent of one premise must match the consequent of the other. What one may validly conclude, then, is a conditional containing the remaining antecedent as antecedent and the remaining consequent as consequent..

Affirmative

If A, then B.

If B, then C.

(So) If A, then C

If this party wins, then we shall have a good government:

If we shall have a good government, then we shall prosper:

Therefore, if this party wins, then we shall prosper:

Negative

If A, then not B

If not B, then not C.

(So) If A, then not C

If this party wins, then we shall not have a good government

If we shall not have a good government

B. Mixed Hypothetical Syllogism

What is?

If major premise alone is hypothetical. The minor premise and the conclusion are merely simple or general.

If A then B

Therefore, B

E.g., If I do my duty, then I shall be happy

I do my duty

Therefore, I shall be happy

o Middle Proposition:

§ No middle term

§ But there is middle proposition common to major and minor premise

Modus Ponens: Affirming the Antecedent

§ Modus ponens or modus ponendo ponens (La- mode that by affirming affirms) is not a logical rule but a rule of inference.

§ The mechanism is antique and has been used in Ancient Greece (Theophrastus) and in India

§ Formal notation:

E.g. If today is Monday, Ravi will go to work

Today is Monday

Therefore, Ravi will go to work

Note that the above argument is valid but may not be true in all cases. For the argument to be true, the premises must be true.

Modus Tollens: Denying the Consequent

§ Modus tollendo tollens (La-mode that by denying denies) or modus tollens is an argument form

§ This is an application of the general form that if a statement is true its contrapositive would also be true.

§ Eg.: If the watchdog detects an intruder, it will bark

The watchdog did not bark

Therefore, the watchdog did not detect an intruder

Fallacies

§ Fallacy of Affirming the Consequent (AC)

If p, then q.

E.g., If the watchdog detects an intruder, it will bark

The watchdog barked

Therefore, the watchdog detected an intruder

§ Fallacy of Denying the Antecedent

§ If p, then q.

Not p.

Not q.

§ If the watchdog detects an intruder, it will bark

The watchdog did not detect an intruder

Therefore, the watchdog will not bark

Table:

Valid Forms	AA: Affirming the Antecedent	Modus Pollens	If p, then q. p. q	If today is Monday, Ravi will go to work Today is Monday Therefore, Ravi will go to work
Valid Forms	DC: Denying the Consequent	Modus Tollens	If p, then q. Not q. Not p	If today is Monday, Ravi will go to work Ravi will not go to work Therefore, today is not Monday
Invalid Forms	AC: Affirming the Consequent	Fallacy of Affirming the Consequent	If p, then q. q. p.	If today is Monday, Ravi will go to work Ravi will go to work Therefore, Today is Monday
Invalid Forms	DA: Denying the Antecedent	Fallacy of Denying the Antecedent	If p, then q. Not p. Not q.	If today is Monday, Ravi will go to work Today is not Monday Therefore, Ravi will not go to work

C. Disjunctive Syllogism

The components of a disjunctive proposition--p and q--are called disjuncts. Such a statement does not actually assert that p is true, or that q is, but it does say that one or the other of them is true.

Logical Form:

Either p or q.

Therefore, Not-q.

Either p or q.

Therefore, Not-p.

E.g.,

Either the meeting is in room 302, or it is in room 306.

It is not in room 302.

Therefore, it is in room 306.

So long as we eliminate all the disjuncts but one, that one must be true--assuming, of course, that the disjunctive premise is true to begin with.

Inclusive & Exclusive ‘or’

The disjunctive syllogism proceeds by denying one of the disjuncts.

Is it equally valid to argue by affirming a disjunct? Is the following inference valid? The answer depends on how we are using the conjunction "or."

Exclusive Sense: We sometimes use it in what is called the exclusive sense to mean, "p or q but not both," as in, "Tom is either asleep or reading."

An argument that denies a disjunct is valid in either case, but an argument that affirms a disjunct is valid only if "or" is used in the exclusive sense.

Inclusive Sense: We also use "or" in the inclusive sense to mean, "p or q or both," as in, "If she's tired or busy, she won't call back."

The problem is that nothing in the logical form of the argument tells us which sense is being used. To make it clear that p and q are exclusive alternatives, people sometimes say, "p, or else q."

But, in most cases, we have to decide from the context which sense is intended.

For logical purposes, therefore, we assume that "or" is used inclusively, so that affirming a disjunct is fallacious.

In cases where such an argument seems valid intuitively, it is easy to translate the argument into a different form that makes the validity clear.

Notes: Logic: Categorical Syllogism

Comments

Post a Comment

Popular posts from this blog