CHAPTER 1: NATURE OF LOGIC

1. NATURE OF LOGIC

A) Traditional and Modern definitions of Logic
B) Basic features of Inductive and Deductive reasoning. Their uses in Law Courts
C) Some basic logical concepts –Form, Content, Truth , Validity, Inference, Implication.

Logic is a science of valid reasoning.

Logic is a word that comes from the Greek word λογική pronounced as logikē that means, the study of reasoning.

All the places where reasoning is needed, logic is needed. The more accurately we use our reasoning, the more effective is our work in that area. So all those who think, use logic knowingly or unknowingly.

Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic falls in the area of epistemology, which asks: "How do we know what we know?" In mathematics, it is the study of valid inferences within some formal language.

Logic has origins in several ancient civilizations, including ancient India, China and Greece. In west, Logic was established as a discipline by Aristotle, who established its fundamental place in philosophy.

The study of logic was part of the classical trivium. Averroes defined logic as "the tool for distinguishing between the true and the false." Richard Whately, defined logic as "the Science, as well as the Art, of reasoning." Frege, defined logic as "the science of the most general laws of truth."

The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.

Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are good examples of informal logic.

Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two branches: propositional logic and predicate logic.

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

A) Traditional and Modern definitions of Logic

Traditional Logic is the type of logic propagated by Aristotle. This is popularly known as traditional formal logic. This is because here, the form of statements used in arguments is given total importance. The traditional formal logic is general designation for the systems of deductive logic that do not involve the use of formal languages, or the apparatus of mathematical logic. The basis of traditional logic is syllogistic reasoning.

Traditional logic is defined as “a system of formal logic mainly concerned with the syllogistic forms of deduction that is based on Aristotle and includes some of the changes by contemporary logicians.”

Modern Logic on the other hand contains more form based relationships in the logical thinking. So, the modern logic is not limited to syllogism based arguments, but it goes much beyond them. It becomes mathematical and symbolic. This is the reason why after the modern logic was developed, it started being used practically in every science where thinking in the right way is needed.

Modern logic is defined as “logic where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics.” The development of the modern "symbolic" or "mathematical" logic is the most significant in the history of logic, and in human intellectual history.

B) Basic features of Deductive and Inductive reasoning.

Logic is divided into two types, these are inductive and deductive reasoning.

The Deductive Reasoning is a reasoning where the conclusion stays within the scope of its supporting statements. It is said that deductive reasoning involves in drawing logical conclusions from definitions and axioms. We can also say that deductive Reasoning involves in deriving known conclusions from known facts. As a result, the conclusions of deductive reasoning are certain.

The Inductive reasoning is a reasoning where conclusion goes beyond the scope of its supporting statements. Many say that inductive reasoning involves in drawing general conclusions from specific examples. We also say that Inductive Reasoning involves in deriving unknown conclusion from known facts. This is the reason why conclusions of Inductive reasoning are probable and not certain.

Though induction and deduction are the two types of logical reasoning, they are not watertight compartments. The general statements that we use as supporting statements in deductive arguments, actually are result of inductive inferences.

So, we find that Induction is of two types, perfect and imperfect.

In perfect induction, we get conclusion from supporting statements, i.e. premises, this conclusion goes beyond the scope of premises, and then we verify and test this conclusion by using some methods that are similar to deduction.

In imperfect induction, we get conclusion from the supporting statements, i.e. premises, this conclusion says something beyond the scope of premises, but then we do not verify and test this conclusion. We just leave it as it is. So, the conclusion of imperfect induction is always probable.

Deduction versus Induction

Aristotle used to classify the type of arguments using syllogism. But all arguments cannot be fitted in the type of syllogism. We can use a simple test for inductive and deductive arguments. The premises in a deductive argument guarantee the truth of the conclusion, so, if the premises are true, the conclusion must be true. The premises in an inductive argument provide some degree of support for the conclusion, but it is possible to have true premises false conclusion.

Aristotle views inductive syllogism as scientific induction and therefore as a more rigorous form of inductive argument. The logical form of the inductive syllogism, after the convertibility maneuver, is the same as the deductive syllogism.

In this sense, induction and deduction possess the same (final) logical form. But, of course, in order to successfully perform an induction, one has to know that convertibility is possible, and this requires an act of intelligence which is able to discern the metaphysical realities between things out in the world. We discuss this issue under non-discursive reasoning below.

Uses of logical reasoning in Law Courts:

In the field of law, we need reasoning in order to present the matter of any litigant effectively, so that we can help him get justice in the existing frames of law. But even the opposite side litigant who has a contrary view also wants justice, so the aim of a good lawyer is always to disclose the truth and just situation. Logical reasoning is absolutely necessary for this as without logical reasoning we cannot find out the truth behind the stories told by the litigants.

C) Some basic logical concepts – Form, Content, Truth , Validity, Inference, Implication.

Logic as we saw, is a science of valid reasoning. In order to know what is valid reasoning, we must first have some concepts clarified. Also, Logic can be best best understood if we understand the basic concepts of logic. So, let us see the definitions of some basic concepts in logic:

Word, is a meaningful group of alphabets used in any language.

Syncatagormatic word, are the words that are used to enhance the meaning of words that can stand on their own. So, the Syncatagormatic words do not make any meaning on their own. They depend on other words for their meaningfulness.

Catagormatic word, is a word that has its own meaning, so it can stand on its own in the process of expressing a meaningful concept.

Sentence, is a meaningful group of words used to convey any meaning.
Statement, is a sentence that asserts some affirmative or negative fact.
Proposition, is a statement used in logical arguments. This means, in logic, a statement is called a proposition.
Form, stands for the relationship of various parts of a statement within itself and in a set of statements called argument.
Content, is the matter of facts mentioned in the statement or argument.
Truth, is the agreement of facts mentioned in an argument with reality.
Validity, is the appropriateness of relationship between various parts of argument.
Inference, is a set of propositions or statements where, on the basis of one or more statements one statement is obtained as a conclusion.

Implication, is a type of statement where the truth of one component is indicated or suggested by the truth of another. Here, the component on which the truth of another component depends is the first component called antecedent and the component that follows from the first component is the consequent.

converting a statement into logical from

There are four standard forms of categorical propositions such as A, E, I and O-propositions having the structure of the form, 'All S is P' 'No S is P’, 'Some S is P' and 'Some S is not P' respectively. Thus, we know that the logical structure of any categorical proposition exhibits the following four items in the order as given below.
Quantifier (Subject term) copula (predicate term)
Here the first item is the 'quantifier' (or more precisely the words expressing the quantity of the proposition). It is attached to the subject term only. The second item in any logical proposition is the subject term. The predicate term, that expresses something about the subject, comes after the copula. The copula is placed in between the subject and predicate term.
Further, the quality of the proposition is expressed in and through the copula. The copula and the predicate term are respectively the third and fourth logical elements of a categorical proposition. Thus, a categorical proposition which is in standard form must exhibit explicitly the subject, the predicate, the copula, its quality and quantity. Let us call a categorical proposition regular if it is in its standard form, otherwise it is called irregular.
In our ordinary language most of the categorical propositions are irregular in nature. Even though there are irregular categorical propositions they can be put in their regular form. It should be noted that while reducing an irregular categorical proposition into its standard form, we should pay enough attention to the meaning of the proposition so that the reduced proposition is equivalent in meaning to its irregular counterpart.
Before describing the method of reduction of irregular propositions into their regular forms, it is profitable to understand the reasons for irregularity of a categorical proposition: The irregularity of any categorical proposition may be due to one or more of these following factors.
(i) The copula is not explicitly stated; rather it is mixed with the main verb which forms the part of the predicate
(ii) Though the logical ingredients of a categorical proposition are present in the sentence yet are not arranged in their proper logical order.
(iii) The quantity of a categorical proposition is not expressed by a proper word like 'all', 'no' (or none), 'some' or it does not contain any word to indicate the quantity of the proposition.
(iv) All exclusive, exceptive and interrogative propositions are clearly irregular.
(v) The quality of the proposition is not specified by attaching the sign of negation to the copula.
Keeping these factors in mind, let us describe systematically the method of reduction of an irregular categorical proposition into its standard form (or into a regular proposition). Below we describe the method of reduction.
I. Reduction of categorical propositions whose copula is not stated explicitly
In our ordinary use of language, very often the copula is not explicitly or separately expressed but is mixed with the main verb. The main verb in such a case forms the part of the predicate. The moment copula is identified; the other items of a logical proposition are brought out in a usual manner. We know that the copula of any logical proposition must be in present tense of the verb "to be" with or without the sign of negation.
Now let us consider an example of an irregular proposition, where the copula is not explicitly stated. "All sincere students deserve success". This is an irregular proposition, as the copula is clearly mixed with the main verb of the proposition. The method of reducing such irregular sentences into regular ones is as follows. The subject and the quantifier of the irregular proposition should remain as they are, while the rest of the proposition may be converted to a class forming property (i.e. term) which would be our logical predicate.
In our above example 'All' is the quantifier attached to the subject 'sincere students'. We should not touch the quantifier nor the subject term of the proposition, they should remain where they are. On the other hand, the rest of the proposition 'deserve success' should be converted into a class forming property 'success deserving'. This should be our logical predicate. Then we link the subject term with the predicate term with a standard copula. Thus,
"All sincere students deserve success." Irregular proposition.
"All sincere students are success deserving." A - Proposition.
"All people seek power." Irregular proposition.
"All people are power seekers." A - Proposition.
"Some people drink Coca Cola." Irregular proposition.
"Some people are Coca Cola drinkers." I - proposition
II. Irregular propositions where the usual logical ingredients are all present but are not arranged in their logical order.
Consider the following examples of irregular propositions. "All is well that ends well" and "Ladies are all affectionate." In these cases, first we have to locate the subject term and then rearrange the words occurring in the proposition to obtain the regular categorical proposition. Such reductions are usually quite straight forward. Thus we reduce the above two examples as given below.
"All is well that ends well." Irregular proposition
"All things that end well are things that are well." A - Proposition
"Ladies are all affectionate." Irregular proposition
"All ladies are affectionate." A - Proposition
III. Statements in which the quantity is not expressed by proper quantity words. Some propositions do not contain word like 'All', 'No', 'some' or contain no words to indicate the quantity. We reduce such a type of irregular proposition into its logical form as explained below.
Here we have to consider two sub-cases : sub-case (i) where there is indication of quantity but no proper quantity words like 'All', 'No', on 'Some' are used and Sub­ case (ii) where the irregular proposition contains no word to indicate its quantity.
Sub-case (i): Affirmative sentences that begin with words like 'every', 'any', 'each' are to be treated as A-propositions, where such words are to be replaced by the word "all" and rest of the proposition remains as it is or may be modified as necessary. The followings are some of the examples of this type.
"Every man is liable to commit mistakes." Irregular proposition.
"All men are persons who liable to commit mistakes." A - Proposition.
"Each student took part in the competition." Irregular proposition.
"All students are persons who took part in the competition." A - Proposition.
"Any one of my students is laborious." Irregular proposition.
"All my students are laborious." A - Proposition.
A negative sentence that begins with a word like 'every', 'any', 'each', or 'all' is to be treated as an O-proposition. Any such proposition may be reduced to its logical form as shown below.
"Every man is not honest". Irregular proposition
"Some men are not honest." O - Proposition
"Any student cannot get first class." Irregular proposition.
"Some students are not persons who can get first class." O - Proposition.
"All is not gold that glitters." Irregular proposition.
"Some things that glitter are not gold." O - Proposition.
Sub- Case (ii):
"Sentences with singular term or definite singular term without the sign of negation are to be treated as A-proposition. For example, "Ram is mortal.", "The oldest university of Orissa is in Bhubaneswar." are A-propositions.
Here the predicate is affirmed of the whole of the subject term. On the other hand, sentences with singular term or definite singular term with the sign of negation are to be treated as E-propositions. For example, "Ram is not a student" and "The tallest student of the class is not a singer" are to be treated as E-propositions. These are cases where the predicate is denied of the whole of the subject term.
IV. “Sentences beginning with the words like 'no', 'never', 'none' are to be treated as E-propositions. The following sentence is an example of this type.
"Never men are perfect." Irregular proposition
"No man is perfect." E - Proposition
V. Affirmative sentences with words, like 'a few', 'certain', 'most', 'many' are to be treated as I-propositions, while negative sentences with these words are to be treated as
O-propositions. Since the word 'few' has a negative sense, an affirmative sentence beginning with the word 'few' is negative in quality. A negative sentence beginning with the word 'few' is affirmative in quality because it involves a double negation that amount to affirmation. The following are examples of above type.
"A few men are present." Irregular proposition.
"Some men are present." I - proposition.
"Certain books are good." Irregular proposition.
"Some books are good." I - proposition.
"Most of the students are laborious." Irregular proposition.
"Some students are laborious." I - proposition.
Here we may note that 'most' means less then 'all' and hence it is equivalent to 'some'.
"Many Indians are religious." Irregular proposition.
"Some Indians are religious." I - proposition.
"Certain books are not readable." Irregular proposition
"Some books are not readable." O - Proposition
"Most of the students are not rich." Irregular proposition.
"Some students are not rich." O - Proposition
"Few men are above temptation." Irregular proposition
"Some men are not above temptation." O - Proposition
"Few men are not selfish." Irregular proposition
"Some men are selfish.' I
VI. Any statement whose subject is qualified with words like 'only', 'alone', 'none but', or 'no one else but' is called an exclusive proposition. This is so called because the term qualified by any such word applies exclusively to the other term. In such cases the quantity of the proposition is not explicitly stated.
The propositions beginning with words like 'only', 'alone', 'none but' etc are to be reduced to their logical form by the following procedure. First interchange the subject and the predicate, and then replace the words like 'only', 'alone' etc with 'all'. For example,
"Only Oriyas are students of this college." Irregular proposition.
"All students of this college are oriyas." A - Proposition.
"The honest alone wins the confidence of people." Irregular Proposition.
"All persons who win the confidence of people are honest." A-proposition.
VII. Propositions in which the predicate is affirmed or denied of the whole subject with some exception is called an exceptive proposition. An exceptive proposition may be definite or indefinite. If the exception is definitely specified as in case of "All metals except mercury are solid" then the proposition is to be treated as universal and if the exception is indefinite, as in case of "All metals except one is solid", the proposition is to be treated as particular.
"All metals except mercury are solid." is a universal proposition which means
"All non-mercury metals are solid."
Now let us consider an example where the exception is indefinite. For example, "All students of my class except a few are well prepared", it is to be reduced to an I-proposition as given below.
"All students of my class except a few are well prepared." Irregular proposition.
"Some students of my class are well prepared." I - proposition.
VIII. There are impersonal propositions where the quantity is not specified. Consider for example, "It is cold", "It is ten O'clock". In such cases propositions in question are to be reduced to A-proposition because the subject in each of these cases is a definite description.
"It is cold". Irregular proposition
"The whether is cold." A - Proposition.
"It is ten O'clock." Irregular proposition.
"The time is ten O'clock." A - Proposition.
There are some propositions where the quantity is not specified. In such cases we have to examine the context of its use to decide the quantity. For example, consider following sentences (1) "Dogs are carnivorous", (2) "Men are mortal", (3) "Students are present." In first two examples, the quantity has to be universal but in the third case, it is particular. Thus, their reductions into logical form are as follows.
"Dogs are carnivorous." Irregular proposition.
"All dogs are carnivorous." A - Proposition.
This is so because we know that "being carnivorous' is true of all dogs.
"Men are mortal." Irregular proposition.
"All men are mortal." A - Proposition
Here 'being mortal' is generally true of men. But in the proposition "Students are present", we mean to assert that some students are present". So the proposition "Men are mortal" is reduced to "All men are mortal" But in the example "Students are present", 'being present' is not generally true of all students.
So the proposition "Students are present" is reduced to "Some dents are present" which is an I-proposition. Thus the context of use of a proposition determines the nature of the proposition.
IX. Problematic propositions are particular in meaning. For example "The poor may be happy" should be treated as a particular proposition, because what such a proposition asserts is that it is sometimes true and sometimes false.
Thus, "The poor may be happy" is reduced to "Some poor people are happy", which is an I-proposition
X. Similarly, there are propositions where the quantity is not specified but their predicates are qualified by the words like 'hardly', 'scarcely', 'seldom'. Such propositions should be treated as particular negative. For example, "Businessmen are seldom honest", is an irregular proposition. It is reduced to "Some businessmen are not honest". If such a proposition contains the sign of negation that these proposition is to be treated as an I-proposition.
For example, "Businessmen are not seldom honest." is to be reduced to "Some businessmen are honest", which is an I - proposition. This is so because it involves a double negation which is equivalent to affirmation.

Proposition Relationships in EDUCTION @ glance

Proposition Relationships in EDUCTION @ glance

Relationship	Changed	Type	Original
Original = S-P			All S is P	No S is P	Some S is P	Some S is not P
			A	E	I	O
Obverse =	All S is non P	A		All S is non P
S-P	No S is non P	E	No S is non P
	Some S is non P	I				Some S is non P
	Some S is not non P	O			Some S is not non P
Converse =	All P is S	A				X
P-S	No P is S	E		No P is S		X
	Some P is S	I	Some P is S		Some P is S	X
	Some P is not S	O				X
Obv Conv=	All S is non P	A		All S is non P		X
P-S	No S is non P	E				X
	Some S is non P	I				X
	Some S is not non P	O	Some S is not non P		Some S is not non P	X
Part Inv =	All non S is P	A			X	X
S-P	No non S is P	E			X	X
	Some non S is P	I		Some non S is P	X	X
	Some non S is not P	O	Some non S is not P		X	X
Full Inverse =	All non S is non P	A			X	X
S-P	No non S is non P	E			X	X
	Some non S is non P	I	Some non S is non P		X	X
	Some non S is not non P	O		Some non S is not non P	X	X
Part Con +ve	All non-P is S	A			X
P-S	No non P is S	E	No non P is S		X
	Some non P is S	I		Some non P is S	X	Some non P is S
	Some non P is not S	O			X
Full Con +ve	All non P is non S	A	All non P is non S		X
P-S	No non P is non S	E			X
	Some non P is non S	I			X
	Some non P is not non S	O		Some non P is not non S	X	Some non P is not non S

Aristotelian Syllogistic Division
 

Kinds of Division

• Logical division divides a class into its subclasses
– E.g., mammals into monotremes, marsupials & placentals
– Division is useful for
• determination of exact relationships among related things
• formulation of definitions

• Other kinds of division

– Physical division divides a whole into its parts
• E.g., a complex machine into its simple mechanical parts

– Metaphysical division divides an entity into its qualities, 

• e.g.,a species into its genus & difference
– man into animality & rationality
• a substance into its attributes
– sugar into color, texture, solubility, taste, &c.
• a quality into its dimensions
– sound into pitch, timbre, volume

How to Divide

• Logical Division

– begins with a summum genus
– proceeds through intermediate genera
– ends at the infimae species
– NB: It does not continue to individuals

• The results of division should meet these criteria:

1. The subclasses of each class should be coextensive with the
original class.
2. The subclasses of each class should be mutually exclusive.
3. The subclasses of each class should be jointly exhaustive.
4. Each stage of a division should be based on a single principle.

Kinds of Classification

• Classification is the technique of inquiry in which similar individuals and classes are grouped into larger classes.
– E.g., how are steam, diesel, & gasoline engines related to one another?

Natural Classification

• Natural classification is a scheme that provides theoretical understanding of its subject matter
– E.g., classification of living things into monerans, protistans, plants, fungi and animals
• The concept “monerans” is now obsolescent because it does not provide sufficient theoretical clarity.

Artificial Classification

• Artificial classification is a scheme established merely to serve some particular human purpose
– E.g., classification of plants as crops, ornamentals, and weed

Classification and Division Compared

• The result of a classification will look like the result of a division.
• Classification begins with a individuals or small classes and works
towards a summum genus.
– i.e., it works in the direction opposite to that of division
• Classification begins with a set of apparently related things found in
the world (i.e., it is based on experience) and builds from there.
– Hence, it is well-suited to natural objects.
– But it will work with any kind of object.

Two Overly Ambitious Ideals

• Pure division
– begins with the summum genus and
– divides on the basis of a priori considerations
• i.e., it is based on logical possibility, not experience

• Dichotomous division
– divides on the basis of the presence or absence of a particular feature
• (NB: Classification can also be dichotomous.)
• Striving for these ideals
– works well with mathematical objects, &c.
– does not work well with natural objects (e.g., kinds of animals)
– guarantees a division that meets criteria (2) – (3)
– sometimes provides more insight than alternative divisions.
• But “ dichotomous division is often difficult and often impracticable”—Aristotle, Parts of Animals I.2-3
• Sometimes, class Rules notification (a bottom-up approach) is more practical.

 RULES OF DIVISION:

When we are using logical division, we need to follow certain rules. thesde are as follows:

1. One division must follow only one criteria. It must be either physical or metaphysical.
2. The division criteria must be mutually exclusive and collectively exhaustive.
3. All the parts of an entity being explained must be covered by the division.
4. No extra members must be suggested as parts of the entity explained during the process of division.

FALLACIES OF DIVISION:

 

When we fail to follow the above rules, we end up in committing the following fallacies:

1. Division by cross criteria: When we divide something by using two or more criteria at the same time, we commit this fallacy. e.g. when we divide Indians into "Hindus, Muslims, Christians, Sikh, Rich, poor, Tall, short, Fair, Dark, introverts and extroverts"; we are committing this fallacy as we are using many criteria, both of physical as well as metaphysical divisions at the same time. at the same time. 
2. Too narrow division: when we exclude some of the members from the group or some qualities of the entity being explained, we commit this fallacy. e.g. Quadrilateral into, square and rectangle. Here we exclude many other types of quadrilaterals and so the division becomes too narrow as it leaves out many other members that actually belong to this group.
3. Too wide division: when we include some members that actually do not belong to the group as we are dividing, our division becomes too wide. e.g. birds into single coloured & multi-coloured. Here, many other single coloured and multi-coloured things and beings get indicated as part of the group of bired, so it is a too wide division.

Deduction versus Induction

We cannot fully understand the nature or role of inductive syllogism in Aristotle without situating it with respect to ordinary, “deductive” syllogism.  

Aristotle’s distinction between deductive and inductive argument is not precisely equivalent to the modern distinction.  

Contemporary authors differentiate between deduction and induction in terms of validity.  (A small group of informal logicians called “Deductivists” dispute this account.)  

According to a well-worn formula, deductive arguments are valid; inductive arguments are invalid. 

The premises in a deductive argument guarantee the truth of the conclusion: if the premises are true, the conclusion must be true.  The premises in an inductive argument provide some degree of support for the conclusion, but it is possible to have true premises and a false conclusion.  

Although some commentators attribute such views to Aristotle, this distinction between strict logical necessity and merely probable or plausible reasoning more easily maps onto the distinction Aristotle makes between scientific and rhetorical reasoning (both of which we discuss below).  

Aristotle views inductive syllogism as scientific (as opposed to rhetorical) induction and therefore as a more rigorous form of inductive argument.

We can best understand what this amounts to by a careful comparison of a deductive and an inductive syllogism on the same topic.  

If we reconstruct, along Aristotelian lines, a deduction on the longevity of bileless animals, the argument would presumably run: All bileless animals are long-lived; all men, horses, mules, and so forth, are bileless animals; therefore, all men, horses, mules, and so forth, are long-lived.  

Defining the terms in this syllogism as: Subject Term, S=men, horses, mules, and so forth; Predicate Term, P=long-lived animals; Middle Term, M=bileless animals, we can represent this metaphysically correct inference as:  Major Premise: All M are P.  Minor Premise: All S are M.  Conclusion: Therefore all S are P.  (Barbara.)  

As we already have seen, the corresponding induction runs: All men, horses, mules, and so forth, are long-lived; all men, horses, mules, and so forth, are bileless animals; therefore, all bileless animals are long-lived.  Using the same definition of terms, we are left with:  Major Premise: All S are P.  Minor Premise: All S are M (convertible to All M are S).  Conclusion: Therefore, all M are P.  (Converted to Barbara.)  

The difference between these two inferences is the difference between deductive and inductive argument in Aristotle.

Clearly, Aristotelian and modern treatments of these issues diverge.  As we have already indicated, in the modern formalism, one automatically defines subject, predicate, and middle terms of a syllogism according to their placement in the argument.  

For Aristotle, the terms in a rigorous syllogism have a metaphysical significance as well.  In our correctly formulated deductive-inductive pair, S represents individual species and/or the individuals that make up those species (men, horses, mules, and so forth); M represents the deep nature of these things (bilelessness), and P represents the property that necessarily attaches to that nature (longevity).  Here then is the fundamental difference between Aristotelian deduction and induction in a nutshell.  

In deduction, we prove that a property (P) belongs to individual species (S) because it possesses a certain nature (M); in induction, we prove that a property (P) belongs to a nature (M) because it belongs to individual species (S).  Expressed formally, deduction proves that the subject term (S) is associated with a predicate term (P) by means of the middle term (M); induction proves that the middle term (M) is associated with the predicate term (P) by means of the subject term (S).  

Aristotle does not claim that inductive syllogism is invalid but that the terms in an induction have been rearranged.  In deduction, the middle term joins the two extremes (the subject and predicate terms); in induction, one extreme, the subject term, acts as the middle term, joining the true middle term with the other extreme.  This is what Aristotle means when he maintains that in induction one uses a subject term to argue to a middle term.  

Formally, with respect to the arrangement of terms, the subject term becomes the “middle term” in the argument.

Aristotle distinguishes then between induction and deduction in three different ways.  First, induction moves from particulars to a universal, whereas deduction moves from a universal to particulars.  The bileless induction moves from particular species to a universal nature; the bileless deduction moves from a universal nature to particular species.  Second, induction moves from observation to language (that is, from sense perception to propositions), whereas deduction moves from language to language (from propositions to a new proposition).  

The bileless induction is really a way of demonstrating how observations of bileless animals lead to (propositional) knowledge about longevity; the bileless deduction demonstrates how (propositional) knowledge of a universal nature leads (propositional) knowledge about particular species. 

Third, induction identifies or explains a nature, whereas deduction applies or demonstrates a nature.  The bileless induction provides an explanation of the nature of particular species: it is of the nature of bileless organisms to possess a long life.  The bileless deduction applies that finding to particular species; once we know that it is of the nature of bileless organisms to possess a long life, we can demonstrate or put on display the property of longevity as it pertains to particular species.

One final point needs clarification.  The logical form of the inductive syllogism, after the convertibility maneuver, is the same as the deductive syllogism.  In this sense, induction and deduction possess the same (final) logical form.  

But, of course, in order to successfully perform an induction, one has to know that convertibility is possible, and this requires an act of intelligence which is able to discern the metaphysical realities between things out in the world.  We discuss this issue under non-discursive reasoning below.

Laws of Thought

During the 18th, 19th, and early 20th Century, scholars who saw themselves as carrying on the Aristotelian and Medieval tradition in logic, often pointed to the “laws of thought” as the basis of all logic.  One still encounters this approach in textbook accounts of informal logic.  The usual list of logical laws (or logical first principles) includes three axioms: the law of identity, the law of non-contradiction, and the law of excluded middle.  (Some authors include a law of sufficient reason, that every event or claim must have a sufficient reason or explanation, and so forth.)  It would be a gross simplification to argue that these ideas derive exclusively from Aristotle or to suggest (as some authors seem to imply) that he self-consciously presented a theory uniquely derived from these three laws.  The idea is rather that Aristotle’s theory presupposes these principles and/or that he discusses or alludes to them somewhere in his work.  Traditional logicians did not regard them as abstruse or esoteric doctrines but as manifestly obvious principles that require assent for logical discourse to be possible.

The law of identity could be summarized as the patently unremarkable but seemingly inescapable notion that things must be, of course, identical with themselves.  Expressed symbolically: “A is A,” where A is an individual, a species, or a genus.  Although Aristotle never explicitly enunciates this law, he does observe, in the Metaphysics, that “the fact that a thing is itself is [the only] answer to all such questions as why the man is man, or the musician musical.” This suggests that he does accept, unsurprisingly, the perfectly obvious idea that things are themselves.  If, however, identical things must possess identical attributes, this opens the door to various logical maneuvers.  One can, for example, substitute equivalent terms for one another and, even more portentously, one can arrive at some conception of analogy and induction.  Aristotle writes, “all water is said to be . . .  the same as all water  . . .  because of a certain likeness.” If water is water, then by the law of identity, anything we discover to be water must possess the same water-properties.

Aristotle provides several formulations of the law of non-contradiction, the idea that logically correct propositions cannot affirm and deny the same thing:

“It is impossible for anyone to believe the same thing to be and not be.” 

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect.”

“The most indisputable of all beliefs is that contradictory statements are not at the same time true.”

Symbolically, the law of non-contradiction is sometimes represented as “not (A and not A).”

The law of excluded middle can be summarized as the idea that every proposition must be either true or false, not both and not neither.  In Aristotle’s words, “It is necessary for the affirmation or the negation to be true or false.”  Symbolically, we can represent the law of excluded middle as an exclusive disjunction: “A is true or A is false,” where only one alternative holds.  Because every proposition must be true or false, it does not follow, of course, that we can know if a particular proposition is true or false.

Despite perennial challenges to these so-called laws (by intuitionists, dialetheists, and others), Aristotelians inevitably claim that such counterarguments hinge on some unresolved ambiguity (equivocation), on a conflation of what we know with what is actually the case, on a false or static account of identity, or on some other failure to fully grasp the implications of what one is saying.

INDUCTION:

Induction is a type of inference where we go from known to unknown or from less general to more general. Here, from the things that are known, we say something about things that are not known. This is the reason why in induction we always say something more than what we already know of. So, Induction, a form of argument in which the premises give grounds for the conclusion but do not make it certain. Induction is contrasted with deduction, in which true premises imply a definite conclusion, the conclusion of Induction is always probable. The probability rate changes as per strength of evidence. Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true. 

Induction is of two types, perfect and imperfect. Perfect induction takes support of deduction in later stages to establish a certain conclusion, while imperfect induction does not do this.

There are two main types of imperfect induction. they are, Simple enumeration and Analogy. 

Simple enumeration is a method of arriving at a generalization on the basis of uniform uncontradicted observation of something. This conclusion can be disproved by observing just one single contrary instance. Yet, the conclusion by simple enumeration is highly probable when the number of observed instances is really high. But if one is arriving at a conclusion on the basis of very limited observation, the conclusion is less probable and hence, it is termed as hasty generalization or illicit generalization.

Analogy is a type of imperfect induction where we are comparing two things, persons, groups or classes. while doing so, we observe some similarities and on the basis of these, we infer some further similarity, as we find an additional quality in one of the two compared things, persons, groups or classes. Here, if the observed similarities are relevant to the additional quality, then our conclusion is likely to be true and we may say that Analogy is good Analogy. But if the observed qualities are not relevant to the additional quality, then our conclusion about predicting the additional similarity is not likely to be true, so, we say that such an analogy is Bad Analogy.

In law, we need to use simple enumeration and Analogy to infer things from circumstantial evidence. Of them analogy is more useful in legal matters. Also, while using precedent law, we use analogy to indicate the support of past decided cases  in our matter.

When we see a person following some pattern of behavior or thinking or actions, while talking of the Modus Operandi of that person, we are using simple enumeration as we talk of the generalized pattern of behavior of that person. This is the method followed by criminal investigators quite often. They determine the Modus Operandi of a criminal to find out the criminal and / or to track the criminals. This is a very common practice used by the police in registering the crime record of certain criminals while maintaining their files.

While contesting any matter, the lawyers use analogy in arguing about similar matters, or actions done by an individual in similar situations, to infer about the truth of the statement given by any witness. For example, if it is shown that the witness had reacted in a particular way in the past in similar situations, or has reacted in a particular way in similar situation created in court, then, one can infer that he must have reacted exactly in same way when the actual event had happened that the witness was witnessing. This type of inference adds to the weight-age in argument in court.

Similarly, when we are arguing any matter, we may come across previously decided matters of same type in the same court, or higher court or another court. We use the citation of these matters as case law or precedent law to lead the judge to the conclusion we want, and the procedure of inductive argument that we use in this type of matter is of analogy. This is why is is said that Analogy is of great use in legal arguments.