INTRODUCTION TO LOGIC

Dr. Mangala pirya
Wat suthivararam
sathorn, yannawa.
Bangkok, 10120.
thailand

INTRODUCTION TO LOGIC
………………….
INTRODUCTION
Nature and scope of Logic
1. Definitions of Logic :-
There are definition of Logic as follow :-
(1) Logic is the study of the methods and principles used to distinguish good (correct) from bad (incorrect) reasoning.
This definition must be taken to imply that only the student of Logic can reason well or correctly. The study of Logic will give students techniques and methods for testing the correctness of many different kind of reasoning, including their own, and when errors are easily detected they are loss likely to be allowed to stand.
The aim of the study of Logic is to discover and made available those criteria that can be used to test arguments for correctness.
(2) Logic is the science of the Laws of thought.
This definition although it gives a clue to the nature of Logic is not accurate.
In the first place thinking is studied by psychologists , Logic cannot be the science of the Law
Of thought because psychology is also a science that deals with Laws of thought (among other things) and Logic is rot a branch of psychology it is a separate and distinct field of study.
In the second place, if “thought” refers to any process that occur in people’s rained not all thought is an object of study for the Logician. All reasoning is thinking, but not all thinking is reasoning.
(3) Logic is the science of reasoning.
This definition is much better but it also will not do. Reasoning is a special kind of thinking in which inference takes place, that is in which conclusions are drown from premises.
The distinction between correct and incorrect reasoning is the control problem with which Logic deals. The Logician’s method and techniques have been developed primarily for the purpose of making this distinction clear. All reasoning (regardless of its subject matter) is of interest to the logician. But this special concern for its correctness is the logical focus.
(4) Logic is the study of the general conditions of valid inference.

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2. Characteristics of Logic :-
(1) Propositions or statements or judgments any sentence either true or false Eg:- All Elephants are Animals
(2) The nature of Logic is of theory and practical
(3) The nature of Logic is fallacies or errors Eg:- All pages are made up of paper
Some men are pages
Some men are made up of paper ----------- ๏ ----------
3. Relationship between Logic and other branches of knowledge:-
(1) Logic is related with philosophy
(2) Logic is related with psychology
(3) Logic is related with Linguistics
(4) Logic is related with mathematics
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4. Uses of Logic :-
(1) It is used for distinguishing a correct reasoning from a incorrect reasoning
(2) It is used for Argumentation
(3) One can detect fallacies
(4) It is used for detecting Deduction and Induction ----------- ๏ ----------
Difference between Traditional Logic and Symbolic Logic :-
Traditional Logic
Symbolic Logic
1. It is also know as classical logic Aristotlean-Logic formal logic
Other names are mathematical Logic, modern Logic, Algebra Logic
2. Aristotle is the father of traditional Logic
Leibning, Book are primarily responsible
3. Relatively long history since sristotle
Relatively short history since Leibning
4. Limited in scope and application
Wider in scope and application
5. Variable in Traditional Logic has limited role
Variable in symbolic Logic has greafor role
6. As restricted of Roman nation
Eg:- I. II,III, IV, V
As flexible or Arabic notation
Eg:- 1,2,3,4,5

TRUTH AND VALIDITY
It is possible to establish the truth or falsehood of any given proposition but it is not possible with the arguments we can only have the attributes of validity and invalidity with regards to arguments. There is a connection between validity and invalidity of an argument and the truth or falsehood of its premises and conclusions are given below.
(1) Some valid arguments have true proposition.
Eg:- All mammals have Lungs
All euhales are mammals
All euhales have Lungs
(2) A valid argument can have false proposition
Eg:- All spiders have ten Legs
All Ten legged creatures have wings
All spiders have wings.
(3) An invalid argument can have false premises and true conclusion.
Eg:- All mammals have wings
All whales have wings
All whales are mammals
(4) A valid argument can have false premises and a true conclusion.
Eg:- All fishes are mammals
All whales are fishes
All whales are mammals
(5) An invalid argument can have false premises and false conclusion.
Eg:- All mammals have wings
All whales have wings
All whales are mammals
(6) An invalid argument can have false premises and false conclusion.
Eg:- If I own all the gold in south Africa
I would be wealthy
I do not own all the gold in south Africa
I am not wealthy

CONCLUSION

Truth and falsity characterize proposition, but never arguments. Validity and invalidity characterize deductive arguments, but never propositions.
A valid deductive argument is not in which the conclusion cannot possibly be false if all the premises are true.
If it is possible for the premises of a deductive. Argument to be true and its conclusion to be false, that argument is invalid.
If a deductive argument is invalid, it may be constituted by any combination of true and/or false premises and a true or false conclusion.
If a deductive argument is valid and its conclusion is true it may have any combination of true and/or false premises.
But if the conclusion of a valid deductive argument is false, at least one of its promises must also be false.
If a deductive argument is valid and at least one of its premises is false, the remainder of its premises and its conclusion may be true or false in any combination.
But if all the premises of a valid deductive argument are true, its conclusion must also be true.
A deductive argument is sound if and only if it is valid and all of its premises are true.
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DEDUCTION AND INDUCTION
Arguments are traditionally divided into two different types. Deductive and inductive, Every argument involves the claim that its promises provide some grounds for the truth of its conclusion.
DEDUCTION
Only a deductive argument involves the claim that its premises provide conclusive grounds for its conclusion. When the reasoning in a deductive argument is correct we call that argument valid when the reasoning of a deductive argument is incorrect we call that argument invalid.
We may therefore define validity as follows :-
(1) A deductive argument is valid when its promises, it true, do provide, conclusion grounds for the truth of its conclusion
(2) A deductive argument is one whose conclusion is claimed to follow from its promises with absolute necessity.
In a valid deductive argument (but not in an inductive argument) premises and conclusion are so related that it is absolutely impossible for the premises to do true unless the conclusion is true also.
In every deductive argument either the premises succeed in providing conclusive grounds for the truth of the conclusive or they do not succeed. Therefore every deductive argument is either valid or invalid.
This is a paint of some importance if a deductive argument is not valid. It must be invalid. If is not invalid. It must be valid.
In the realm of deduction Logic. The control task is to clarify the relation between promises and thus to allow us to discriminate valid from invalid arguments.
The theory of deduction including both traditional Logic and symbolic Logic.

INDUCTION
An inductive argument is one whose conclusion is claimed to follow from its premises only with probability.
An inductive argument makes a very different claim not that its premises give conclusive grounds for the truth of its conclusion, but only that its premises provide some support for that conclusion. Inductive arguments, therefore cannot be “valid” or “invalid” is sense in which these terms are applied to deductive argument.
Of course, inductive arguments may be a valuated as better or worse, according to the degree of support given to their conclusions by their premises, thus, the greater the likelihood or probability, that its premises confer on its conclusion the greater the merit of an inductive argument. But that likelihood even when the premises are all true, must fall short of certainty.
A popular way of distinguishing between the deductive and the inductive reasoning is this. The deductive arguments begins with universalization and ends in particularization, on the other hand the inductive reasoning begin with particularization and end in universalization.
This way of distinguishing between deduction and induction does not always work because of the following reasons.
(1) Sometimes a valid deductive argument can have universal premises as well as a universal conclusion.
Eg:- All animals are mortal
All humans are animal
All humans are mortal
(2) Sometimes a valid deductive argument can have particular prepositions for its premises as well as for its conclusion.
Eg:- If Socrates is human
Then Socrates is mortal
Socrates is human
Socrates is mortal
(3) Some similarly on inductive argument need not rely on particular premises but may fare universal prepositions for its premises as well as its conclusion.
Eg:- All crows are mammals and have lungs
All wholes are mammals and have lungs
All humans are mammals and have lungs
Probably all mammals and have lungs
(4) Sometimes an inductive argument can have only particular prepositions.
Eg:- Hitler was a dictator and was ruthless
Stalin was a dictator and was ruthless
Castro is a dictator
Castro is ruthless
CONCLUSIOIN
The care of the difference between deductive and inductive arguments lies in the strength of the claim that is made about the relation between the premises of the argument and its conclusion
In deductive arguments the conclusion is claimed to follow from its premises with absolute necessity, in inductive arguments. The conclusion is claimed to follow from its pelisses only with some degree of probability.
A deductive argument is valid if its premises do provide conclusive proof of its conclusion, otherwise it is invalid. But the terms “validity” and “Invalidity” do not apply to inductive arguments, which are appraised with other terms.
The addition of now premises may alter the strength of an inductive argument, but a deductive argument, if valid cannot be made more valid or invalid by the addition of any premises.
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DATEGOPICAL PREPOSITION
All prepositions are sentences. But not all sentences are prepositions, prepositions are either true or false, either accepted or denied questions, commands, euslaimations are not prepositions, preposition are also know as judgements or statements
The process of inference is most basic in the field of Logic inference is the process in which one preposition is arrived at and affirmed on the basis of one or more prepositions accepted as the starting paint of the process. The process of inference involves an argument where a conclusion follows from premises. Both premise and conclusion are relative terms. The same preposition can be a premises in one argument and a conclusion in the other. No preposition by itself is a premise or a conclusion. They become so in the course of on argument. Sometimes promises lead to a conclusion other conclusion lead to the promises still other times conclusion may occur between the premises.
The usual premises indicators are since, because, for as in as much as etc. They usual conclusion indicator are therefore Nonce, thus, so , consequently etc.
PROPOSITIONS

Categorical (assertive) Conditional
Eg: All what are mammals
Hypothetical Disjunctive
Eg: If it rains Eg: students are either
then it is a holiday lazy or industrious
CHARACTERISTICS : -
All categorical had a subject and predicate.
Eg: - All football players are athletes
All quantifier Copulas or verb Football player-Subject are, are not, is, is not Subject are – copula Quantifiers Athletes – predicate All, no, some
QUANTITY: -
Quantity refers to the number which is either universal or particular. Universal quantifiers are All and No particular quantifiers is some, some means at least one.
QUANTITY: -
Quality refers to affirmation or negation of the given class.
FOUR KIND OF CATEGORICAL PREPOSITIONS :-
(1) Universal affirmative
(2) Universal negative
(3) Particular affirmative
(4) Particular negative
Universal affirmative
A
All s are p
Sap
All politicians
are liars
Universal negative
E
No s are p
Sep
No politicians
are liars
Particular affirmative
I
Some s are p
Sip
Some politicians
are liars
Particular negative
O
Some s are not p
Sop
Some politicians
are not liars

DISTRIBUTION OF TERM
The technical term “distribution” is introduced to characterized the way in which terms occur in categorical preposition
A term is distributed in a preposition. If it refer to all the member of the class.
A term is undistributed in a preposition. If it refer to same or few member of the class.
(A) Statement
Eg:- All roses are flowers
Subject is distributed
Predicate is undistributed
(E) Statement Eg:- No athletes are vegetarians Subject is distributed Predicate is distributed
(I) Statement Eg:- Some soldiers are cowards Subject is undistributed Predicate is undistributed
(O) Statement Eg:- Some politicians are not punctual Subject is distributed Predicate is undistributed
Statement
Subject
Predicate
A
E
I
O
- Distributed
- Distributed
- Undistributed
- Undistributed
- Undistributed
- Distributed
- Undistributed
- Distributed

Subject Distributed

A:- All s are p

I:- Some s are p


E:- No s are p

O:- Some are not p

Subject Undistributed
Distribution of term-Euler circles
The technical term “distribution” is introduced to characterized the ways in which terms occur in categorical preposition. Both subject and predicate of a preposition are called “terms” A term is distributed in a preposition if it refers to all the member of the class. A term is undistributed in a preposition if it refers to some or few member of the class.
Preposition
Subject
Predicate
A
E
I
O
- Distributed
- Distributed
- Undistributed
- Undistributed
- Undistributed
- Distributed
- Undistributed
- Distributed

(A) Eg:- All roses are flowers
Subject is distribute, because it refer to All roses.
Predicate is undistributed, because it does not refer to all flowers.
S P
(roses) (flowers)

S = O
It has not member or it is imply
(E) Eg:- No athletes are vegetarians.
Predicate is distributed, because it is concluded from athletes.
S (athletes) P (regetorions)
SP = O


It has not member or it is imply
(I) Eg:- Some soldiers are towards
Subject is undistributed, because it is does not refer to all soldiers. Predicate is undistributed, because it does not refer to all towards (there are forward who are not soldiers).
S (soldiers) P (forwards)
X SP O

It is not empty, but it has at least one member.
(O)Eg:- Some politicians are not punctual Subject is undistributed, because it does not refer to all politicians Predicate is distributed because it is conclude from some politician.



S (politician) P (punctual)
X S O

It is not empty, but it has at least one member.
THE MOOD AND THE FIGURE
THE MOOD :-
The mood of a Syllogism is represented by there tatters, the first of the major premise and the second of the miner premise and the third of the conclusion.
Eg :- (A) All men are mortal (I) Socrates is a man (I) Socrates is a mortal Mood = A-I-I
THE FIGURE :-
The figure indicates the position of the middle term in the premises there are four possible figures.
I fig
II fig
III fig
IV fig
M – P
P – M
M – P
P – M
S – M
S – M
M – S
M – S
S - P
S - P
S - P
S - P

I fig Eg:-
(A) All men are mortal (I) Socrates is a man (I) Socrates is a mortal
Mood A-I-I - 1
II fig Eg:-
(A) All dogs are mammals (A) All cats are mammals (A) All cats are dogs Mood A-A-A - 2
III fig Eg:-
(A) All artists are egotists (I) Some artists are paupers (I) Some paupers are egotists Mood A-I-I - 3
IV fig Eg:-
(E) No heroes are cowards (I) Some coward are soldiers (O) Some soldiers are not heroes Mood E-I-O - 4
There are 256 possible combinations of various moods some of them are valid some of them are invalid. The valid moods under each figures are mentioned below.
figures

Mood


1
AAA
AII
EAE
EIO

B RB R
D R
C L R NT
F R
2
AEE
AOO
EAE
EIO
3
AII
IAI
EAO
EIO

D T S
D S M S
F S P
FR S S N
4
AEE
IAI
EAO
EIO

FROM AND MATTER
An argument is consisted of both form and matter. The content of an argument is matter. The content is arranged is form. So form is the manner in which all the constituents of the reasoning (premises as well as conclusion) are arranged. Not only an argument, but every object of the earth is made of matter and it has form.
Form means shape or type.
Matter means material of thing.
Eg:- Table
It is made from wood, iron and steel = matter
If may be square and round = form
Therefore there is no matter, without form or no form without matter. In thought, matter and form are correlated. Some ideas may be expressed in different forms. The true view is that Logic is both formal and material.
Eg:- All men are mortal Socrates is a man Socrates is mortal
Argument is syllogism in form while its matter consists the meaning of the three prepositions constituting it.
Distinction between form and matter of material objects lead to the distinction between form and matter in thought also.
The form may change while the matter remains the same.
Eg:- All men are mortals = U-A No man are immortals = U-N
The preposition is different inform, but the matter is same, because they have the same meaning
The matter may change while the form remain the same.
Eg:- All men are mortals = U-A All dogs are animals = U-A
The both have the same form, but different meaning, so their matter is different.
SQUARE OF OPPOSITION OF PREPOSITIONS
(All s are p) A contraries E (No S are P)




(Some s are P) I sub contraries O (Some s are not P)
RELATIIONSHIP BETWEEN CONTRADICTORIES
Two prepositions are called contradictories if one denies the other is they cannot both be true and cannot both be false. In other words, if one is true other must be false and vice versa.
Thus A and O preposition are contradictories.
Eg:- A : All judges are lawyers O : Some judges are lawyers similarly E and I preposition are contradictories.
Eg:- E : No poets are idles I : Some poets are idles
RELATIONSHIP OF CONTRARIES:-
Two preposition are said to be contraries if they cannot both be true although they might both be false.
Thus A and E preposition are contraries
Eg:- A : All politicians are liars E : No politicians are liars
RELATIONSHIP OF SUBCONTRARIES
Two preposition are sub contraries if they cannot both be false although they might both be true.
Thus I and O preposition are contraries
Eg:- I : Some mangoes are sweet O : Some mangoes are not sweet
RELATIONSHIP OF SUBALTERNATION
In all the earlier relationships there is disagreement sometimes there can be oppositions even when there is no disagreement they agree in quality, but differ in quantity. This relationship is known as sub alternation.
The truth of A preposition necessarily leads to the truth of the I prepositions
Eg:- A : All spiders are eight leggel animal I : Some spiders are eight leggel animal
Likewise the truth of the E prepositions leads to the truth of O prepositions.
Eg:- E : No spiders are insects O : Some spiders are not insects
Thus we learn that the truth lead to the truth of sub alter (I,O), but this cannot be demonstrated in the case of falsehood.
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*Of the subalterns (A,E) necessarily
MEDIATE INFERENCE
CATEGORICAL SY SYLLOGISM :-
A syllogism simply means putting two and two together. All the syllogism are arguments in which a conclusion is derived from two premises a categorical preposition which contain three terms, such occur in twice. All there terms are arranged in a standard form.
In a mediate inference conclusion is derived from more than one premise. Thus mediated inference is different from immediate inference where the conclusion is derived from only one premise.
The three forms in a categorical syllogisms are as follows.
(1) Major term :- The predicate term of the conclusion is called the major term
(2) Miner term :- The subject term of the conclusion is called the miner term.
(3) Middle term :- The term which occurs two times in the premises is called the middle term.
Eg:- All man are moral Socrates is a man Socrates is a mortal
Major term – mortal Miner term – Socrates Middle term – man
The premises containing the major term is called the major premise the premise, containing the minor term is called the miner premise. The major premise occurs first, the miner premise second and the conclusion lost.
Eg:-
All men are mortal Major premise
Socrates is a man Miner premise Socrates is a mortal
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INMEDIATE INFERENCE :-
Given the truth or falsehood of any one preposition, the truth or falsehood or in deter mina nay of the other statements can be interred immediately. The immediate inferred in traditional logic are as follows.
(1) If A is given true, E is false. I is true, C is false.
(2) If E is given true, A is tales, I is false, O is true.
(3) If I is given true, E is false, O is undetermined, A is undetermined.
(4) If o is given true, A is false, E is undetermined, I is undetermined.
(5) If A is given false, O is true, E is undetermined, I is undetermined.
(6) If E is given false, I is true, A is undetermined, O is undetermined.
(7) If I is given false, E is true, A is false, O is true.
(8) If O is given false, A true, E is false, I is true.
1) If the statement. “Some Diamonds are precious stones” is true then.
1. All Diamonds are precious stones is undetermined.
2. No Diamonds are precious stones is false.
3. Some Diamonds are not precious stones is undetermined.
2) If the statement.
No rich men are kind is false, then the statement.
1. All rich men are kind is undetermined.
2. Some rich men are kind is true.
3. Some rich men are not kind is undetermined
THREE KINDS OF IMMEDIATE INFERENCE : EDUCTION
Immediate inferences are divided into three kinds. (1) conversion, (2) obversion (3) contraposition
(1) CONVERSION :-
This preceeds by simply inter-interchanging the subject and the predicate forms of a preposition without altering its meaning.
Examples

prepositions
converse
Examples
All Dogs
are animals
No angels
are man
Some woman are writers
Some animals
Are not dogs
A

E

I

O
All S are P

No S are P

Some S are P

Some S are not P
Some P are S

No P are S

Some P are S

not possible
Some animals are dogs
No man are angels
Some writes are woman
-


(2) OBVERSION :-
In obversion the subject term remains unchanged the quantity is Qpverted and the quality is changed by replacing the predicate by its complement. The complementary class consists of the collection of things which do not belong to the original class.
Examples

prepositions
observe
Examples
All residents
are voters
No umpires
are partisans
Some metals are conductors
Some nations
are not hard working
A

E

I

O


All S are P

No S are P

Some S are P

Some S are not P


No S are non –P

All S are non –P

Some S are not non – P
Some s are non – P

No residents are non – voters
All umpires are non – partisans
Some metals are not non - conducter
Some nations are non-hard working




Examples

prepositions
observe
Examples
All rose
are flowers
No frogs
are mammals
Some students are intelligent
Some politician
are not honest
A

E

I

O

All S are P

No S are P

Some S are P

Some S are not P

No S are non –P

All S are non –P

Some S are not non – P
Some s are non – P
No roses
non flowers
All frogs
non mammals
Some students are not non intelligent
Some politician
are non honest

(3) CONTRAPOSITION :-
In this the subject term is replaced by the compliment of its predicate term and the predicate term is replaced by the compliment of its subject term.
Examples

prepositions
contrapose
Examples
All men
are mortal
No writer
are idles
Some odminis trator are partisans
Some dogs
are not aggressive
A

E

I


O
All S are P

No S are P

Some S are P


Some S are not P

No non –P are non - S
Some –P are non - S
No equatent


Some non P are not non s
All non – mortals are non – men
Some non – idlers are non – writers
-


Some non aggressive animals are not non - dogs

Eg:- (1) All apples are fruits. All non fruits are non apples.
(2) No parrots are reptiles. Some non reptiles are not non parrots
(3) Some pens are expensive _____ (4) Some pens are not expensive Some non – expensive things are not non – pens
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FIAGRAMMATIC REPRESENTATION OF CATEGCRICAL PROPOSITIONS
A: All S are P E: No S are P

S P S P

S = O SP = O
I: some S are P O: Some S are not P
S X P S X P

SP O S O

S S SP P P



S S S P P P

SPM
S M P M

M


M
These are know as Euler circles

STEPS
(1) Find out in miner term (s) the major term (p) and the middle term (m)
(2) Draw three circles one overlapping the other two those three circles represent the miner term the major term and the middle term.
(3) The major premise must be shaded according to Euler representation secondly the minor term the minor premise must be shaded.
(4) We have to find out whether the shading of conclusion is already taken care of in the shading of the premise.
- If yes then the argument must be valid.
- If No then the argument must be invalid.
Validity claimed on the basic that the shading of premises leads to the shading of conclusion.
(5) Sometimes there may a difficulty of shading the particular preposition. If there is a difficulty of placing on X mark because there is an interjecting line in such circumstances X mark must be placed on the interjecting line.
On verification if should be clear whether the X mark is placed completely inside or not
If yes then the argument must be valid.
If No it must be invalid
(6) If there is a universal premise and a particular premise, the universal premise must be shaded first.

VENN DIAGRAMS
Euler circle representation




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รวมคำศัพท์ท้ายเล่ม

definition [N] ; คำนิยาม
easily [ADV] ; อย่างแท้จริง
detect [VT] ; ค้นหา
allow to [PHRV] ; ยินยอมให้
psychologist [N] ; นักจิตวิทยา
deal [VT] ; จัดการ
distinction [N] ; ความแตกต่าง
premiss [N] ; หลักฐาน
true [ADV] ; เป็นจริง
false [ADJ] ; เท็จ
deductive [ADJ] ; ซึ่งหักลบไป
argument [N] ; เหตุผล, การโต้งแย้ง, ข้อพิสูจน์
remainder [N] ; ส่วนที่เหลืออยู่
premiss [N] ; หลักฐาน
validity [N] ; ความมีเหตุผล
control [N] ; การจำกัด
particularize [VT] ; ทำให้เฉพาะเจาะจง [VI] ; แยกรายละเอียด
universality [N] ; ความเป็นสากล