Introduction

“In a Republican nation, whose citizens are to be led by reason and persuasion and not by force, the art of reasoning becomes of the first importance.”-Thomas Jefforson

Higher education is meant to enrich the intellectual prowess of the learners and to promote a respect for reason. It is not enough to know the answers to the questions asked: it is equally important to learn to question the answers. A critical, systematically inquiring mind is the end result of a proper intellectual training. Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning. Study of logic is an effective tool for achieving these goals. Furthermore, mere delivery of information to the students does not teach them how to organize or systematize their thoughts, or how to assess what they have received. While teaching some effective time –tested methods for processing, analyzing and evaluating information, study of logic also teaches how to enrich one’s intellectual life with a logical approach. 

These class notes were prepared for first paper entitled Principles of Logic in Diploma-in-Reasoning of Department of Philosophy, Kurukshetra University, Kurukshetra, to help the students, when I taught in this course last three years. Basically I used collection of materials from different sources viz. printed books as well as internet sources to prepare it. You can find our course syllabus in my post “Philosophy is Nothing without Logic” at The Positive Philosophy page and also in the side links of this page. This is only a first draft, kindly send your suggestions and ideas to dr.sirswal@gmail.com or niyamak.drs@gmail.com, I shall be highly thankful to you and a short list of books are mentioned below of the Table of Contents and reference sites are linked with this  page. This page introduces the basic conceptions of formal logic, informal logic and also Symbolic logic.

 

                                                                                                DESH RAJ SIRSWAL

                                                                                                http://drsirswal.webs.com

Link:

http://niyamaklogic.wordpress.com/2009/05/05/introduction/

Dated: May 5, 2009.

Last Modified: October 08,2009.

Table of Contents

Section-1

Unit-I: Nature of Logic

Unit-II: Nature of Implication

Unit-III: Truth and Validity

Unit-IV: Laws of Thought

Section-2

Unit-V: Nature of Proposition

Unit-VI: Translating sentences into standard form propositions

Section-3

Unit-VII: Immediate Inference-Square of Opposition

Unit-VIII: Immediate Inference-Eduction

Section-4

Unit-IX: Mediate Inference –Categorical Syllogism

Unit-X: Rules and Fallacies for Categorical Syllogisms

Unit-XI: Check the Validity of Syllogism through Venn-Diagrams

Section-5

Unit-XII: Arguments of Ordinary language

Unit-XIII: Types of Arguments

Unit-XIV: Fallacies (Formal & Informal)

Section-6 : Symbolic Logic

Unit-XV: Truth-functional Logic

Unit-XVI: Truth-functional Compound Statements

Unit-XVII: Validity and Invalidity by Truth-table Method

Unit-XVIII: Statement Forms

Suggested Readings

Unit-I: Nature of Logic

Logic: Logic is a branch of Philosophy, dedicated to the study of reasoning. Logic is the study of the methods and principles used to distinguish correct reasoning from incorrect reasoning. The word derives from Greek logike, means “possessed of reason, intellectual, dialectical, argumentative”, from logos equivalent to “word, thought, idea, argument, account, reason, or principle”. The aim of the logic is to provide methods, techniques and devices which help in differentiating right reasoning from wrong reasoning and good reasoning from bad. But it does not mean that only those who study logic can reason correctly. However it is true that those who study logic certainly make less errors while arguing.

Knowledge of logic helps one to face a problem in a more orderly and systematic way and in many cases makes the solution less difficult and more certain. Like any other active field of study, it too has grown in many directions.

Kinds of Logic

Today, logic is both a branch of philosophy and a branch of mathematics. It is applications as well known in the area of artificial intelligence. This page aims primarily to acquaint the readers with the basics of what is known as classical logic or classical first order logic and sometime also called as formal logic, because proponents of this logic mostly believe that statements in natural language have underlying logical forms. In their view, the expression in logic exhibit these latent deep structures or the logical forms. If the deep structures of the form is correct, only then a piece of reasoning in natural language is valid. Here we will also study Informal Logic (Fallacies). Except this there are many other kinds of logic: Many-Valued Logic, Fuzzy Logic, Non-Monotonic Logic, Modal Logic and Paraconsistent Logic. You can find a short note and suggested books about kinds of logic in Chhanda Chakraborti : Logic: Informal, Symbolic & Inductive.

History of Logic

The earliest sustained work on the subject of logic is that of Aristotle, In contrast with other traditions, Aristotelian logic became widely accepted in science and mathematics, ultimately giving rise to the formally sophisticated systems of modern logic.

Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. In India, the Nasadiya Sukta of the Rigveda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti: “A”, “not A”, “A and not A”, and “not A and not not A”. The Chinese philosopher Gongsun Long(ca. 325–250 BC) proposed the paradox “One and one cannot become two, since neither becomes two.” Also, the Chinese ‘School of Names’ is recorded as having examined logical puzzles such as “A White Horse is not a Horse” as early as the fifth century BCE. In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.

Logic in Islamic philosophy also contributed to the development of modern logic, which included the development of “Avicennian logic” as an alternative to Aristotelian logic. Avicenna’s system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic, and inductive logic. The rise of the Asharite school, however, limited original work on logic in Islamic philosophy, though it did continue into the 15th century and had a significant influence on European logic during the Renaissance.

In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century, though it did not survive long into the colonial period. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the Indian tradition of logic.

During the later medieval period, major efforts were made to show that Aristotle’s ideas were compatible with Christian faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.

The syllogistic logic developed by Aristotle predominated until the mid-nineteenth century when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879 Frege published Begriffsschrift which inaugurated modern logic with the invention of quantifier notation. In 1903 Alfred North Whitehead and Bertrand Russell  published Principia Mathematica  on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931 Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues.

The development of logic since Frege, Russell and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments often as a compulsory discipline. (From Wikipedia, the free encyclopedia)

Logical Reasoning:

Reasoning is an art as well as a science; it is something we do as well as understand. The mental recognition of cause –and –effect relationship is called ‘reasoning’. It may be prediction of an event from an observed cause or  the inference of a cause from an observed event. Logical Reasoning is a process of passing from the known to the unknown. It is the process of deriving a logical inference from a hypothesis through reasoning.

Logical Deduction: Another important factor in logical reasoning is logical deduction. Deriving an inference from units of arguments which are called proposition in logic or deducing an inference from statements is called logical deduction. For example:

                        All men are mortal.

                        Raveesh is a man.

                    Therefore, Raveesh is mortal.

From statement (a) and (b) we derive a logical conclusion that Raveesh is mortal.

Deduction & Induction:

In logic, we often refer to the two broad methods of reasoning as the deductive and inductive approaches.

Deductive reasoning works from the more general to the more specific. Sometimes this is informally called a “top-down” approach. We might begin with thinking up a theory about our topic of interest. We then narrow that down into more specific hypotheses that we can test. We narrow down even further when we collect observations to address the hypotheses. This ultimately leads us to be able to test the hypotheses with specific data — a confirmation (or not) of our original theories.

Inductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, we sometimes call this a “bottom up” approach In inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories.

The Idea of Form

The form or logical form of an argument is the representation of its sentences using the formal grammar and symbolism of a logical system to display its similarity with all other arguments of the same type. It consists of stripping out all spurious grammatical features from the sentence (such as gender, and passive forms), and replacing all the expressions specific to the subject matter of the argument by schematic variables. Thus, for example, the expression ‘all A’s are B’s’ shows the logical form which is common to the sentences ‘all men are mortals’, ‘all cats are carnivores’, ‘all Greeks are philosophers’ and so on.

Last updated: 20-08-2009.

Unit-II: Nature of Implication

In any discipline one seeks to establish facts and to draw conclusions based on observations and theories. One can do so deductively or inductively. In inductive reasoning, one starts with many observations and formulates an explanation that seems to fit. In deductive reasoning, one starts with premises and, using the rules of logical inference, draws conclusions from them. In disciplines such as mathematics, deductive reasoning is the predominant means of drawing conclusions. In fields such as psychology, inductive reasoning predominates, but once a theory has been formulated, it is both tested and applied through the processes of deductive thinking. It is in this that logic plays a role.

The other method of reasoning, the deductive method, begins with an accepted generalization–an already formulated or established general truth and applies it to discover a new logical relationship. That is, through deduction we can come to understand or establish the nature of something strange or uncertain by associating or grouping it with something known or understood.

Deductive arguments are formed in two ways:

1. General to Particular. This is the kind most people think of when they think of deduction. For example, the classic syllogism:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

2. General to General. Another kind of deduction arrives at new generalizations through the syllogism. For example:

All trees have root systems.
All root systems need nitrogen.
Therefore, All trees need nitrogen.

But before we get into syllogistic analysis, a little more needs to be said about deduction as a whole. We said earlier that deduction begins with an accepted generalization. Such a statement raises two questions: (1) Where do these generalizations come from and (2) Why are they accepted or assumed to be true?

The generalizations used in deductive thinking come from several sources:

• Inductive thinking
• Other deductive arguments (of the general to general type)
• Revelation
• Assumption (a priori givens that cannot be proved but that are assumed. All knowledge must begin with belief.)

In all four of these cases, the immediate source may be authority rather than personal experience. That is, the inductive conclusion, the deductive argument, the revelation, or the assumption may have been achieved by a third party who presents the generalization to us for acceptance on the basis of authority, in which case we take it on faith. You may not be able to do a large scale inductive experiment to find out whether a certain generalization is true, so you look in a book and accept the generalization of the authority.

 Last Updated:20-08-2009

Unit-III: Truth and Validity

The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in logic refers generally to a property of particular statements and deductive arguments. Although validity and logical truth are synonymous concepts, the terms are used variously in different contexts. Whether or not logical truth is analytic truth is a matter of clarification.

Validity of Arguments

When an argument is set forth to prove that its conclusion is true (as opposed to probably true), then the argument is intended to be deductive. An argument set forth to show that its conclusion is probably true may be regarded as inductive. To say that an argument is valid is to say that the conclusion really does follow from the premises. That is, an argument is valid precisely when it cannot possibly lead from true premises to a false conclusion. The following definition is fairly typical:

An argument is deductively valid if, whenever all premises are true, the conclusion is also necessarily true.

An argument that is not valid is said to be “invalid’’.

An example of a valid argument is given by the following well-known syllogism:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

What makes this a valid argument is not the mere fact that it has true premises and a true conclusion, but the fact of the logical necessity of the conclusion, given the two premises. No matter how the universe might be constructed, it could never be the case that this argument should turn out to have simultaneously true premises but a false conclusion. The above argument may be contrasted with the following invalid one:

All men are mortal.

Socrates is mortal.

Therefore, Socrates is a man.

In this case, the conclusion does not follow inescapably from the premise: a universe is easily imagined in which ‘Socrates’ is not a man but a woman, so that in fact the above premises would be true but the conclusion false. This possibility makes the argument invalid. (Although whether or not an argument is valid does not depend on what anyone could actually imagine to be the case, this approach helps us evaluate some arguments.)

A standard view is that whether an argument is valid is a matter of the argument’s logical form. Many techniques are employed by logicians to represent an argument’s logical form. A simple example, applied to the above two illustrations, is the following: Let the letters ‘P’, ‘Q’, and ‘s’ stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:

All P are Q.

s is a P.

Therefore, s is a Q.

Similarly, the second argument becomes:

All P are Q.

s is a Q.

Therefore, s is a P.

These abbreviations make plain the logical form of each respective argument. At this level, notice that we can talk about any arguments that may take on one or the other of the above two configurations, by replacing the letters P, Q and s by appropriate expressions. Of particular interest is the fact that we may exploit an argument’s form to help discover whether or not the argument from which it has been obtained is or is not valid. To do this, we define an “interpretation” of the argument as an assignment of sets of objects to the upper-case letters in the argument form, and the assignment of a single individual member of a set to the lower-case letters of the argument form. Thus, letting P stand for the set of men, Q stand for the set of mortals, and s stand for Socrates is an interpretation of each of the above arguments. Using this terminology, we may give a formal analogue of the definition of deductive validity:

An argument is formally valid if its form is one such that for each interpretation under which the premises are all true also the conclusion is true.

As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.

Last Updated: 20-08-2009.

Unit-IV: Laws of Thought

Some early thinkers, after having defined logic as “the science of the laws of thought”, went on to assert that there are exactly three basic laws of thought, laws so fundamental that obedience to them is both the necessary and the sufficient condition of correct thinking. In the western tradition, the concept of laws of thought can be traced back to Aristotle (384-322BC) the eminent Greek thinker, who is considered to be pioneer of western logic. As part of his project, Aristotle was trying to describe the basic laws by which human thought (and reasoning) can occur. As examples of foundational laws, he identified the following three laws:

Principle of Identity: This principle asserts that if any statement is true, then it is true. Using our notation we may rephrase it by saying that the principle of identity asserts that every statement of the from pÉp must be true, that every such statement is tautology.

Principle of Noncontradiction: This principle asserts that no statement can be both true and false. Using our notation we may rephrase it by saying that the principle of noncontradiction asserts that every statement of the form p ·~p must be false, that every such statement is self-contradictory.

Principle of Excluded middle: This principle asserts that every statement is either true or false. Using our notation we may rephrase it by saying that the principle pÚ ~p must be true, that every such statement is a tautology.

These laws are sometimes misunderstood. The first does not imply nothing ever changes. The second does not imply that a thing can have only one property. The third does not imply that everything is black or white: it implies only that either everything is black or something is not black. Aristotle identified these laws as the necessary condition of human thought: without them, thought cannot occur. He also held them as laws of thought, i.e. as fundamental principles for human rational thinking. George Boole, one of the greatest mathematicians of 19^th century and one of the founders of mathematical logic, fully supported these Aristotelian notions.

 Last Updated:20-08-2009.

Unit-V: Nature of Proposition

Proposition:

Propositions are the material of our reasoning. A proposition links nouns, pronouns and phrases to other words in a sentence. The word or phrase that the proposition introduces is called the object of the proposition.A proposition is a judgment expressed in a language and a judgment is a mental act in which two or more than two ideas are combined together.

Judgments have two types:

1. Affirmative- Indians are laborious.

2. Negative- Indians are not dull.

A proposition usually indicates the temporal, spatial or logical relationship of its object to the rest of the sentence as in the following examples:

The book is on the table.

The book is beneath the table.

The book is leaning against the table.

The book is beside the table.

She held the book over the table.

She read the book during class.

Components of Proposition:

Every proposition has three parts:

1. Subject- about whom something assert or deny.
2. Predicate-what assert or deny.
3. Copula- Conjunct both terms. Copula will be negative or affirmative.

For example:

Sonia is a good orator.

   S      C                  P 

Difference with Sentences:

1. Propositions are different from sentences. Sentences have many kinds like questions, exclamations. But non of these can be asserted and denied. Truth and Falsity apply always to proposition, but not apply to questions or commands or exclamations.
2. Propositions must be distributed from sentences by means of what they are asserted. Two different sentences consisting the same proposition.
3. Sentences are parts of some language, but propositions are not tied to any given language i.e. “It is raining”, “Barsat ho rahi hai”, both consist the same content.
4. A sentence called as proposition when its both term (subject and predicate) are nouns i.e. Ram is a man. “Flower is beautiful” is not a proposition because its predicate is adjective.
5. Proposition is always in present tense. But sentences are expressed in all terms.
6. Proposition explains quantity and quality but sentence does not  explain it.
7. All propositions are sentences but not all sentences are propsotions.

Types of Proposition

According to the relation of terms proposition has three types:

Categorical Proposition: It is a type of proposition which has no condition for their assertion. – Roshan is a student.

Conditional or Hypothetical Proposition: A type of compound proposition, it is false only when the antecedent is true and the consequent is false.- If Ram will pass, then he will get a cycle.

Disjunctive Proposition: A type of compound proposition; if true, at least one of the component of propositions must be true.-Ram is honest or clever.

Categorical Proposition:

A categorical proposition is simply a statement about the relationship between categories. It states whether one category or categorical term is fully contained with another, is partially contained within another or is completely separate.

A dog is an animal.

Some dogs are friendly.

No dog is a cat.

Propositions may have quality: either affirmative or negative.

They may also have quantity: such as ‘a’, ’some’, ‘most’ or ‘all’. The ‘all’ quantity is also described as being universal and other quantities particular.

Aristotelian Four-fold Classification of Categorical Propositions:

Aristotle classified categorical proposition in four, based on Quality and Quantity distribution:

Universal Affirmative – All S is P. – A Type Proposition– All men are mortal.

Universal Negative     – No S is P. – E Type Proposition–  No men are immortal.

Particular Affirmative – Some S is P. –  I Type Proposition– Some men are wise.

Particular Negative    – Some S in not P. –O Type Proposition– Some men are not wise.

Distribution of Terms

Both subject and predicate of a proposition are called term. A term is a word or group of words which is either a subject or a predicate of a proposition.

A term is said to be distributed if it refers to all the members of a class. In the other words, a term is distributed when it includes or excludes all the members of a class and if a term includes or excludes only some members of a class, then it is undistributed.

In a categorical syllogism the distribution of terms depends on the quantifier:

A Type:      In “All A are B”-propositions the subject (A) is distributed.

E Type:      In “No A are B”-propositions both the subject (A) and the predicate (B) are distributed.

I Type :      In “Some A are B”-propositions neither the subject nor the predicate are distributed.

O Type :     In “Some A are not B”-propositions the predicate is distributed.

Form	Type	Quality	Quantity	Distribution of X	Distribution of Y
All S is P	A	Affirmative	Universal	Distributed	Undistributed
No S is P	E	Negative	Universal	Distributed	Distributed
Some S is P	I	Affirmative	Particular	Undistributed	Undistributed
Some S is not P	O	Negative	Particular	Undistributed	Distributed

Copi and Cohen state two rules about distribution of terms in valid syllogisms:

• 1. The middle term must be distributed in at least one premise.
• 2. If the major term or the minor term is distributed in the conclusion, then it must be distributed    in the premises.

Venn and Boolean Expression of Categorical Proposition:

The modern interpretation of categorical logic also permits a more convenient way of assessing the truth-conditions of categorical propositions, by drawing Venn diagrams , topological representations of the logical relationships among the classes designated by categorical terms. The basic idea is fairly straightforward:

 

   

     

                                                            S                         P

 

 

All S is P. – Universal Affirmative –A Type Proposition

Or S non-P = 0

 

 

S                         P

 

No S is P. – Universal Negative- E Type Proposition

Or S P = 0

 

 

 

Some S is P.-Particular Affirmative-I Type Proposition

Or S P ≠ 0

 

 

Some S is not P.-Particular Negative-O Type Proposition

Or S non-P ≠0

Denotation and Connotation of Terms

Denotation denotes the objects, connotation connotes the characteristics. Denotation of a term refers to  the objects or things which possess the quality. Connotation refers to the set of characteristics essentially possessed by every object denoted by the term. For example, man, Gita, Mohan, Kamal etc. Man means that possess morality and rationality.

Man= Denotation      Morality and rationality = Connotation

Last Updated:20-08-2009.

Unit-VI: Translating sentences into standard form propositions

For reasoning in everyday life, as you know, people do not talk in standard categorical form. Categorical form is much too stilted for writing effective discourse.
There is a need to develop skills of logical translation to standard form categorical propositions in order to minimize errors in evaluating syllogistic arguments. Very often translation into standard form reveals fallacies of equivocation and fallacies of amphiboly in the original text.
1.Translation Rules of Thumb:

The subject and predicate terms must be the names of classes. If the predicate term is a descriptive phrase, make it a substantive (i.e., noun phrase).

Translation must not (significantly) alter the original meaning of the sentence. Categorical propositions must have a form of the verb “to be” as the copula in the present tense.

The quality and quantity indicators are set up from the meaning of the sentences.

Quantity indicators: “All,” “No,” “Some.”

Quality indicators: “No,” “are,” “are not.”

The word order is rearranged according to the sense of the sentence.

This rule requires special care—in some instances, it may well be the most difficult rule to follow. On occasion, we need to divide one sentence into two or more propositions

Before we take up some special cases, let’s look at some typical examples:
The following translations are relatively straightforward.

“Ships are beautiful” translates to
“All ships are beautiful things.”
“The whale is a mammal” translates to
“All whales are mammals.”

“Whoever is a child is silly” translates to
“All children are silly creatures.”
“Snakes coil” translates to
“All snakes are coiling things.”

“All swans are not white” translates to
“Some swans are not white.”

“Nothing ventured, nothing gained” translates to
“No non-ventured things are gained things.”
Or the obverse…
“All non-ventured things are non-gained things.”

2. Singular propositions are to be treated as (but not usually translated into) a universal proposition (i.e., an A or an E).
E.g., “Socrates is a man” is an A proposition, but
“Socrates is not a god” is an E proposition.
3. Exclusive propositions have the cue words “only” or “none but.” The order of the subject and predicate terms must be reversed.
E.g., “None but A is B” translates to “All B is A.”
“Only A is B” translates to “All B is A.”
“None but red trucks are fire engines” translates to
“All fire engines are red things.”

4. Exceptive propositions are compound propositions.
E.g., “All except A is B” translates to “All non-A is B and “No A is B.
E.g., “All except human beings are nonsymbolic animals” translates to …”
“All nonhuman beings are nonsymbolic animals” and
“No human beings are nonsymbolic animals”
(or, of course the obverse, “All human beings are symbolic animals.”)

5. A Compound statement asserts two propositions.
E.g., “There is a time to sow and a time to reap” translates to
“Some occasions are times to sow” and
“Some occasions are times to reap.”

6. Abstract: An inductive strategy for mechanizing translation is illustrated.

We have at this time a kind of “took kit” to work on syllogisms. Our tools include:
obversion, conversion, and contraposition
Venn diagrams
logical analogies
rules and fallacies
various techniques for reducing the number of terms
translation strategies

The following inductive technique can be used for mechanizing translation by isolating the steps for testing the validity of a syllogism. The steps can be itemized as follows:

Identify the conclusion and premises.

Put the syllogism into standard order as best you can.

Supply the suppressed statements, if any.

Reduce the number of terms to three per syllogism.

Translate the statements to standard form.

Test for validity

Let us conclude by evaluating a complex example from A.S. Neill’s Summerhill (New York: Hart, 1960) 94.

“If Summerhill teachers had urged, “Come on lads, get on the field!” sports in Summerhill would have become a perverted thing because only under freedom to play or not to play can one develop true sportsmanship.”

First, let’s do steps 1 and 2

Major Premise: All instances where true sportsmanship develops are times where one has the freedom to play or not play. (Note the reversal of terms due to the nature of the original exclusive proposition.)

Minor Premise: . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion: No occasions at Summerhill are instances where sports becomes a perverted thing.

Once we notice that “instances where true sportsmanship develops” in context is the complementary class of “instances where sports becomes a perverted thing,” then we can match our classes by obverting the conclusion to obtain “All playing times at Summerhill are instances where true sportsmanship develops.”

Next, we can supply the missing minor premise with the classes “playing times at Summmerhill” and “times where one has the freedom to play or not play.”

The resultant argument now becomes…

Major Premise: All instances where true sportsmanship is developed are times where one has the freedom to play or not play.

Minor Premise: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion All playing times at Summerhill are instances where true sportsmanship is developed.

The minor premise contains the minor term “playing times at Summerhill’ and the middle term “times where one has the freedom to play or not play.” For validity, the middle term must be distributed (otherwise the fallacy of the undistributed middle term would occur), and the statement must be affirmative (otherwise the fallacy of drawing an affirmative conclusion from a negative premise would occur).

Hence the minor premise would have to be…”All times where one has the freedom to play or not play are playing times at Summerhill,” but this statement is false.

The true statement “All playing times at Summerhill are times where one has the freedom to play or not play” makes the syllogism invalid by the fallacy of the undistributed middle term. Hence, Neill’s argument is not sound.

Had Neill omitted the word “only” from his argument, the argument could have been logically acceptable.

Unit-VII: Immediate Inference-Square of Opposition

Inference: Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. In intelligence testing, mostly deductive inference ability is judged. Inference is studied within several different fields.

• Human inference (i.e. how humans draw conclusions)is traditionally studied within the field of cognitive psychology.
• Logic studies the laws of valid inference.
• Statisticians have developed formal rules for inference (statistical inference) from quantitative data.
• Artificial intelligence researchers develop automated inference systems.

IMMEDIATE INFERENCE:

Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. If we expand the scope of our investigation to include shared terms and their complements, we can identify logical relationships of three additional varieties. Since each of these new cases involves a pair of categorical propositions that are logically equivalent to each other—that is, either both of them are true or both are false—they enable us to draw an immediate inference from the truth (or falsity) of either member of the pair to the truth (or falsity) the other. Immediate inference has two types mentioned below:

Square of Opposition:

Any logical relation among the kinds of categorical propositions (A, E,I and O) exhibited on the Square of Opposition. There are four ways in which propositions may be “observed” –as Contradictories, Contraries, Sub-contraries and sub-alternation. These are represent with an important and widely used diagram called the Square of Opposition. This is given below:

 

 The four corners of this chart represent the four basic forms of propositions recognized in classical logic:

A propositions, or universal affirmatives take the form: All S are P.
E propositions, or universal negations take the form: No S are P.
I  propositions, or particular affirmatives take the form: Some S are P.
O propositions, or particular negations take the form: Some S are not P.

Given the assumption made within classical (Aristotelian) categorical logic, that every category contains at least one member, the following relationships, depicted on the square, hold:

Contradictory:

A and O propositions are contradictory, as are E and I propositions. Propositions are contradictory when the truth of one implies the falsity of the other, and conversely. Here we see that the truth of a proposition of the form All S are P implies the falsity of the corresponding proposition of the form Some S are not P. For example, if the proposition “all industrialists are capitalists” (A) is true, then the proposition “some industrialists are not capitalists” (O) must be false. Similarly, if “no mammals are aquatic” (E) is false, then the proposition “some mammals are aquatic” must be true.

Contrary:

A and E propositions are contrary. Propositions are contrary when they cannot both be true; if one is true, then other must be false. They can both be false. An A proposition, e.g., “all giraffes have long necks” cannot be true at the same time as the corresponding E proposition: “no giraffes have long necks.” Note, however, that corresponding A and E propositions, while contrary, are not contradictory. While they cannot both be true, they can both be false, as with the examples of “all planets are gas giants” and “no planets are gas giants.”

Subcontrary:

I and O propositions are subcontrary. Propositions are subcontrary when it is impossible for both to be false; it one is false the other must be true. They can both be true. Because “some lunches are free” is false, “some lunches are not free” must be true. Note, however, that it is possible for corresponding I and O propositions both to be true, as with “some nations are democracies,” and “some nations are not democracies.” Again, I and O propositions are subcontrary, but not contrary or contradictory.

Subalternation:

Two propositions are said to stand in the relation of Subalternation when the truth of the first (“the superaltern”) implies the truth of the second (“the subaltern”), but not conversely. A propositions stand in the Subalternation relation with the corresponding I propositions. The truth of the A proposition “all plastics are synthetic,” implies the truth of the proposition “some plastics are synthetic.” However, the truth of the O proposition “some cars are not American-made products” does not imply the truth of the E proposition “no cars are American-made products.” In traditional logic, the truth of an A or E proposition implies the truth of the corresponding I or O proposition, respectively. Consequently, the falsity of an I or O proposition implies the falsity of the corresponding A or E proposition, respectively. However, the truth of a particular proposition does not imply the truth of the corresponding universal proposition, nor does the falsity of an universal proposition carry downwards to the respective particular propositions.

 

Inferences from square of opposition:

A number of very useful immediate inferences may be readily drawn from the information embedded in the traditional square of opposition. Given in the truth, or the falsehood, of any one of anyone of the four standard from categorical proposition, it will be seen that the truth or falsehood of some or all of the others can be inferred immediately.

A being given as True: E is false; I is true; O is false.

E being given as True: A is false; I is false; O is true.

I being given as True: E is false; A and O are undetermined.

O being given as True: A is false; E and I are undetermined.

A being given as  False : O is true , E and I are undetermined.

E being given as False: I is true; A and O are undetermined.

I  being given as False: A is false; E is true; O is true.

O being given as False: A is true; E is false; I is true.

For example:

What can you infer about the truth or falsity of the following if you assume “ Some reptiles are not poisonous” is false?

(1)    All reptiles are poisonous. – True

(2)   No reptiles are poisonous. -  False

(3)   Some reptiles are poisonous.- True

What is the name of the opposition relation in which the categorical statements differ:

1. In quantity only?
2. In both quality and quantity
3. Between A and I.
4. Between I and O.
5. Between E and I.

Unit-VIII: Immediate Inference-Eduction

Eduction:

Eduction is of three types –Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence.

Details are given below:

Conversion

An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid.

Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false.

Steps for Conversion: Reversing the subject and the predicate terms in the premise.

Valid Conversions

Convertend                                       Converse

                                 A: All S is P.                                       I: Some P is S (by limitation)

                                  E: No S is P                                        E: No P is S

                                  I : Some S is P                                    I : Some P is S

                                  O: Some S is not P                              (conversion not valid)

Example:

                        All bags are mangoes.-A

                        Some mangoes are bags.-I

 

                        No men are intelligent.-E

                        No intelligent are men.-E

 

                        Some cows are tables.-I

                        Some tables are cows.-I

 

                      Some students are not cats.

                      (not valid)

                       

Obversion

An inference formed by changing the quality of a proposition and replacing the predicate term by its complement. Obversion is valid for any standard form Categorical proposition.

Obversion is the only immediate inference that is valid for categorical propositions of every form. In each of the instances, the original proposition and its obverse must have exactly the same truth-value, whether it turns out to be true or false.

Steps for Obversion:

1. Replace the quality of the given statements. That is, if affirmative, change it into negative, and if negative, change it into affirmative.
2. Replace the predicate term by its complementary term.

                                                               Valid Obversions

                                          Obverted                                             Obverse

                                            A: All S is P.                                       E: No S is non-P.

                                             E: No S is P                                        A: All S is non-P.

                                            I : Some S is P                                    O : Some S is not non-P.

                                            O: Some S is not P                              I: Some S is non-P.

Example:

                                    All females are perfect beings.-A

                                    No females are non-perfect beings.-E

                                   

                                    No female are perfect beings.-E

                                   All female are non-perfect beings.-A

 

                                  Some female are perfect beings.-I

                                  Some females are not non-perfect beings.-O

 

                                 Some female are not perfect beings.-O

                                 Some female are non-perfect beings.-I

Contraposition

An inference formed by replacing the subject term of a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term. Not all contrapositions are valid.

Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true. Thus, in each of the pairs, both propositions have exactly the same truth-value.

Note: In contraposition the subject of the conclusion is contradictory of the predicate of the premise and predicate of the conclusion is contradictory of the subject of the premise.

Steps for Contraposition:

a. Convert the statement: reverse the subject and the predicate terms.

b. Replace both terms by their complementary terms.

                                                    Valid Contrapositions

                               Premises                                               Contrapositive

                                A: All S is P.                                      A: All non-P is non-S.

                               E: No S is P                                       O: Some non-P is not non-S.

                                                                                             (By limitation)

                              I : Some S is P                                    (Contraposition not valid)

                              O: Some S is not P                            O: Some non-P is not non-S.

Example:

                                    All citizens are voter.-A

                                    All non-voters are non-citizens.-A

 

                                    No politicians are honest.-E

                                    Some-non-honest are not non-politicians.-O

                                    (by limitation)

 

                                    Some applicants are graduate. -I

                                    (cannot be contraposited)

 

                                    Some students are not scholarship holders.-O

                                    Some non-scholarship holders are not non-students.-O

Existential Import

It is time to express more explicitly an important qualification regarding the logical relationships among categorical propositions. You may have noticed that at several points in these two lessons we declared that there must be some things a certain kind. This special assumption, that the class designated by the subject term of a universal proposition has at least one member, is called existential import . Classical logicians typically presupposed that universal propositions do have existential import.

Exercises:

Make the conversion, obversion and contraposition of the following:

1. Some monks are not vegetarians.
2. All journalists are pessimists.
3. No reptiles are warm blooded animals.
4. Some men are happy.
5. No businessmen are philosophers.
6. Artists are professionals.

Link:

http://niyamaklogic.wordpress.com/2009/05/04/unit-viii-immediate-inference-eduction/

Dated: May 4, 2009.

Last Modified: October 08, 2009.