SYMBOLIC LOGIC (PHL1 C03)
I SEMESTER
Core Course
MA PHILOSOPHY (2019 Admission Onwards)
UNIVERSITY OF CALICUT School of Distance Education Calicut University P.O., Malappuram, Kerala, India - 673 635
190403 School of Distance Education
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
Study Material
I SEMESTER 2019 Admission Onwards
MA PHILOSOPHY
Core course (PHL1 C03)
SYMBOLIC LOGIC
Prepared by:
Sri. Manoj K. R, Assistant Professor on contract, School of Distance Education, University of Calicut.
Scrutinized by:
Dr. Sheeja O.K, Assistant Professor of Philosophy, Sree Kerala Varma College, Thrissur
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UNIT – I
INTRODUCTION
1.1 What is Logic
Etymologically logic can be defined as the science of thought expressed in language. We get knowledge through our thought process or through our act of thinking. If we substitute reasoning with the word thought, it may be stated that the concern of logic is reasoning as expressed in certain language with a subsidiary process. The study of logic is a normative discipline. The reasoning process which go e s on in the mind is not visible, instead it is the verbal expression of such thoughts that are analysed.
Copy and Cohen define logic as the study of methods and principles of distinguishing correct reasoning from incorrect reasoning. Logic is actually the method and principles in evaluating good reasoning from bad reasoning or correct reasoning from incorrect reasoning. Logic is considered the basic science of sciences, since correct reasoning is the basis of scientific study in any field.
Logic is concerned with norms to distinguish between correct and incorrect reasoning; hence it is called normative study of reasoning. In our day-to-day life, we know the importance of ability to reason s i n c e it is through this ability that we draw appropriate conclusion from given evidence. It is source of most of our knowledge. Since most of our knowledge is not from direct observation instead it is through inference. We must know how to infer correctly. Primarily, Logic is about inferring, about reasoning in particular. It is the study of what constitutes correct reasoning.
By studying logic one can recognize and use certain common forms of correct logical inference hence avoid making common logical errors. According to Creighton, logic is the science which treats the operations of the human mind in its search for truth. From this definition there are three facts that are stated here, firstly it states that logic is a science, then operations of human mind, then a concern for truth.
Logic can be called science because, Logic shares the following characteristics of science a) Science deals with a particular department of study, like botany is the study of plants. b) Science is systematic and organised body of knowledge. c) Science renders true and exact knowledge through special means.
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Similarly, logic gives us systematic knowledge regarding correct thinking and knowledge given is correct and precise. Secondly the operations of the human mind with which logic is concerned is three processes of thinking known as conception,judgement and reasoning. Conception is the function of the human mind by which an idea or a concept is formed in the mind. Another function of the mind is judgement by which relation between things is established. It is the process of comparing concepts or ideas. Judgement is the process of affirmation or denial. When the idea of man and mortality is related, we have the idea of man is mortal. Reasoning is the process of passing from certain known judgement to a new judgement. The process by which one proposition is arrived on the basis of other proposition is called inference. Through the act of reasoning mind draws a new truth from given truth.
From the given judgements a conclusion is reached. All men are mortal
Socrates is a man
------Socrates is mortal
The above three processes conception, judgement, inference expressed in language is called term, proposition and argument. Logic in primarily concerned with reasoning, reasoning presupposes concept and judgement, thought means processes and product of thinking. It is the is well established truth expounded by great minds which is the subject matter of logic. Creighton states that truth is the goal of logic, it can be either formal or material. In formal truth there is agreement of thought among themselves. In material truth there should be agreement to corresponding objects to the world outside.
The positive science studies the nature of things or otherwise called natural science, it deals with things actual and real. While for normative science tells us about how a thing ought to be in order to agree with deal before us. Normative science is one which sets a norm to which the facts under study must conform. While positive science deals with things as they are. Examples of normative sciences are Ethics, Aesthetics etc. Regarding how our conduct ought to be to reach the ideal of goodness is the concern of ethics. Normative science is concerned with an ideal, there is a standard to which its material is directed and its value is estimated with reference to that standard then it is called regulative science. Logic teaches us how our thought ought to be to reach the ideal of truth. Here truth is made the ideal.
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1.2 The Nature of Argument
Logic is directly concerned with thought, but language is an indispensable medium to express what we think, arguments are expressed in language, and we should use the correct language to express the arguments in correct form. Verbal expression of judgement is termed proposition. Proposition is the basic unit of logical analysis; Proposition should be either true or false. A declarative sentence is part of the language to which it is spoken or written, while a proposition is not peculiar to the language in which it is expressed. Eg :- ‘The book is on the table’ in French and German language has the same propositional content, irrespective of the language in which it is spoken. While the same sentence can be uttered in different contexts to have different propositions. A statement can be expressed using different words. Utterance of a sentence in different contexts yields different statement. Statement and Propositions are not exact synonyms. An argument is comprised of a set of sentences consisting of one or more premises, which contain the evidence and a conclusion which is supposed to follow the premises.
An argument may be defined as any group of propositions or statements, of which one is claimed to follow from the others, which are alleged to provide grounds for the truth of that one.
Assertion do not cite reason for a particular event. There are s o m e differences between assertions and an argument. Assertion is only an opinion or a belief. In an argument there will be certain reason provided from which we have to infer the conclusion. The terms premise and conclusion are usually employed in analyzing the structure of the argument. The proposition which is affirmed on the basis of other propositions of the argument is the conclusion of that argument.
Premises of an argument are sentences or claims containing the evidence, while conclusion is the claim that is supposed to follow the premises.
An argument is a set of propositions arranged in terms of their relationship as premises and conclusion. The propositions of an argument are either the premises or the conclusion.
Hence, an argument is defined as ‘a relational arrangement of premises and conclusion’. The conclusion of an argument is always the one proposition that is derived from one or more supporting propositions called premises. In other words, the propositions, which substantiate the conclusion, are the premises. The proposition which is drawn on the basis of the premises is a conclusion.
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Premise 1 - All men are mortal. Premise 2 - Socrates is a man.
Conclusion - Therefore, Socrates is mortal.
Raghu will win the game if he worked hard for it. Raghu worked hard for the game.
Therefore, Raghu won the game
1.3 Truth and Validity
Truth and Falsity apply only to propositions but never to arguments, while validity and invalidity is the logical property of deductive arguments, but not of propositions. There is relation between validity or invalidity of argument and truth or falsity of its premises and conclusion which are propositions. An argument can be valid even if one or more of its premises are false. Some valid arguments can contain only true premises
In the following argument the premises and the conclusion are true
All mammals have lungs. All whales are mammals.
Therefore, all whales have lungs.
For a valid deductive argument, it is impossible to have the premises true and the conclusion false. Validity of an argument depends upon the form of the argument. The validity does not depend upon the content of the argument, validity is function of the form of the argument. An argument is sound when it is factually correct and is valid.
Even with false premises and false conclusion the argument can remain valid. All mammals have wings.
All reptiles are mammals.
------Therefore, all reptiles have wings.
In the above argument all of the premises and conclusion is false still the argument is valid.It makes us clear that truth or falsity of the conclusion of the argument does not by itself determine its validity or invalidity of that argument. The argument being valid does not guarantee the truth of the conclusion. But in a valid argument if the premises are true then necessarily the conclusion must be also be true.
An argument with false premises and true conclusion may be invalid as in the following example:
All mammals have wings. All whales have wings.
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Therefore, all whales are mammals.
There are invalid arguments whose premises and conclusions are all false, as in the
following example:
All mammals have wings. All whales have wings.
Therefore, all mammals are whales.
From the above examples it is clear that the truth or falsity of an argument’s conclusion does not by itself determine the validity or invalidity of that argument. And the fact that an argument is valid does not guarantee the truth of it conclusion. But the falsehood of its conclusion does guarantee that either the argument is invalid or one of its premises is false. Thus, a valid deductive argument is one 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 all true and its conclusion to be false, that argument is invalid.
1.3.1 Induction and deduction
Based on the method and type of inference, logic is divided into two - deductive logic or deduction and inductive logic or induction. In deduction, the form determines the validity of inference. Being the basis of formal logic, deductive reasoning concern is with the form of the argument rather than its content. deductive argument, the conclusion cannot be wider than the premises, but in induction In, the conclusion is equal to or wider than the premises.
In deduction, the conclusion necessarily follows from the given premises. Hence, their relationship is of implication or entailment. See for example,
All humans are mortal. Aristotle is a human being.
Therefore, Aristotle is mortal.
Induction refers to the process of drawing conclusion from observed instances that give specific evidence to support the inference. Inductive inferences are evaluated as sound or unsound by considering not only form but also the content or matter.
Aristotle is human and mortal. Bacon is human and mortal. Descartes is human and mortal.
------.
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------. Therefore, all humans are mortal.
In deduction, the premises form the necessary ground for the conclusion. In induction, the conclusion is always probable. Hence, an inductive argument is neither true nor false, but only sound or unsound. Deduction and induction are not considered as opposites by logicians instead it is considered as complementary process of reasoning. Both being the the basis for hypotheticodeductive method. As stated earlier deductive reasoning is concerned with the form of the argument rather than its content. In a deductive argument, the conclusion is already contained in the premises.
Hence, in formal logic the material truth or falsity of the premises is not important. The following deductive argument is invalid because although the premises are true the conclusion is false:
All lions are four-legged animals.
All tigers are four-legged animals.
Therefore, all tigers are lions.
The evaluation of inductive inferences as sound or unsound is based upon not only form but also the content or matter. Hence, material logic is concerned with the content of the argument and hence it is based more on inductive reasoning. For example, from all the reported instances - ‘Crows are black’,
we infer the conclusion - All crows are black. Yet, this inductive inference is only probable because as soon as we come across the material evidence for a non-black crow it becomes invalid. In a deductive argument the premises provide absolutely conclusive grounds for the conclusion. While in an inductive argument the premises provide only probable grounds for the conclusion.
1.4 CATEGORICAL PROPOSITIONS
In the deductive argument, premises provide conclusive grounds for the truth of the conclusion.
In the following argument the premises and the conclusion are categorical propositions
No Athletes are unhealthy
All football players are athletes
No football players are unhealthy
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It can be said that such propositions is about classes, such that whether class S is either affirmed or denied by P, completely or partly. A class being collection of objects that have some characteristics in common. If we consider the example of a class of ellipses and another class of square both have no members in common. One class is said to be included in another class when all the members of one class is member of the second class. Also, if only some members of one class belong to another class, then it is partially contained.
In the above argument there are three classes, class of Athletes, football players and unhealthy.
There are four different standard forms of categorical propositions
All athletes are tall No athletes are tall Some athletes are tall
Some athletes are not tall
All S is P
In the first proposition class of all athletes in contained in the class of tall, every member of first class belong to the second class, hence schematically this can be written as
It is called universal affirmative proposition
The second proposition
No S is P
Is a universal negative proposition, which states that no member of the first belong to the second, it means first class is completely excluded from the other class, this can be schematically written as
No athletes are tall
Some S is P
The next proposition is particular affirmative proposition, which states that some members of the first class are also members of the second class, it means they have some members in common. This proposition neither affirms nor denies that all athletes are tall, literally it does not deny that some athletes are not tall, but in some context, it does suggest it. The word some is indefinite, in the customary usage it means at least one member of the class ‘athletes’ is a member of the class ‘tall’. Here the relation of inclusion holds but it is not universal, instead it is only partial.
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Some Athletes are tall
Some S is not P
It is called the particular negative proposition. It means at least one member designated by the subject term (class S) is excluded from the whole class designated by predicate term(P).
Some Athletes are not tall
Categorical Propositions either affirms or denies something about a class of things unconditionally
MAIN CHARACTERISTICS ARE –
• It is always in indicative mood
• It is always in present tense
• Copula must be separate from the subject and the object
The thing about which affirmation for negation is made is the subject S The attribute that is affirmed or denied of the subject is the predicate P There are four kinds of copula , is , is not, are , are not
S + Copula + P
• Words are part of grammatical sentences, terms are constituents of logical propositions
• A term is word or group of words which serves as either subject or predicate of a proposition
• Propositions have only two terms namely the subject and the predicate
• Propositions have only two terms subject and predicate, while sentences can have any number of words Quantity and Quality of Proposition • Quality of a proposition indicates whether the proposition is affirmed or denied of the subject. • Quantity of a proposition is determined by the extent of generalization of the subject. • Proposition is universal if it refers to the whole class of objects • It is particular, if the subject refers to only part of the class • Quantity indicators – All, No and Some
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Based upon quantity and quality we can classify proposition in to four
A,E,I,O
A - All S is P E - No S is P
I - Some is P
O - Some is not P
CONNOTATION AND DENOTATION OF A TERM
• Every term has either connotation and denotation
• Every term connotes a quality an denotes a quantity
• Connotation refers to the quality of propositions
• It is the set of qualities possessed by objects referred to that term
• Denotation refers to the objects that possess the given qualities, denotation of the term human being is any particular individual like Socrates
Distribution of Terms
• When a certain attribute is predicated to the whole of the subject term, the subject is said to be distributed.
• If only a part of it is referred to, then the term is characterized as undistributed
• In Universal Propositions the subject term is distributed and in particular propositions the subject term is undistributed
• In affirmative propositions predicate term in undistributed, while in negative proposition the predicate term is distributed.
PROPOSITON SUBJECT PREDICATE
A Distributed Undistributed
E Distributed Distributed
I Undistributed Undistributed
O Undistributed Distributed
Symbolic Logic 11 School of Distance Education We can also use Mnemonic formula to represent the distribution of terms,
AsEbInOp
A proposition distributed the subject term
E proposition distributes both subject and predicate term
I proposition distributed neither subject nor predicate
O proposition distributes predicate term
FOUR TYPES OF CATEGORICAL PROPOSITION
A proposition - Universal Affirmative, A l l S is P E proposition - Universal Negative, No S is P
O proposition - Particular Affirmative, Some S is P
I proposition - Particular Negative, Some S is not P
The term about which affirmation or denial is made is the subject, the term which is affirmed or denied is the predicate. The connecting link is the copula, they are is, is not, are, are not
CATEGORICAL PROPOSITION AND CONDITIONAL PROPOSITION
In a Categorical proposition the relation between the subject and predicate terms is unconditional.
The conditional proposition asserts the relation on the basis of some conditions, there are two types of conditions namely hypothetical and Disjunctive
example for hypothetical -If I win the match, I will celebrate
example for disjunctive - either I shall telephone him or write to him
Symbolic Logic 12 School of Distance Education Distribution of Terms in a Proposition with Eulers Circles
All S is P
S
P
No S is P
P S
Some S is P
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1.5 THE SQUARE OF OPPOSITION • The truth relations between the four categorical proposition is termed ‘Opposition’ • Traditional logicians have explained it by means of diagram called ‘Square of Opposition’ • In the Square of Opposition, when we infer the Opposite proposition from the given proposition, the subject and predicate of the implied proposition is the same as the implying one • Two propositions are said to be opposite if they differ either in quality or quantity or both quantity and quality
A Contrary E
Sub-altern
Sub-altern
I Sub-Contrary O
Symbolic Logic 14 School of Distance Education In the Square of Opposition, we infer the opposite proposition from the given proposition, subject and predicate of the implied proposition is same as the implying one.
Two categorical propositions are said to be opposite if they differ in quality or quantity or both quality and quantity
A and E are contrary propositions
I and O are sub - contrary
A is super altern to I and E is super altern to O I is sub altern to A and O is sub altern to E
A and O are contradictories, similarly E and I are contradictories
If the truth and falsehood of any one of the four-standard form of proposition is known, truth of some or all other can be inferred immediately.
A given true E is false I is true O is false E given true A is false I is false O is true I given true A is undetermined E is false O is undetermined O given true A is false E is undetermined I is undetermined A given false I is undetermined E is undetermined O is true E given false A is undetermined I is true O is undetermined I given false A is false E is true O is true O given false A is true E is false I is true Immediate Inferences:
There are three kinds of immediate inferences: - i) Conversion
ii) Obversion
iii) Contraposition
In Conversion, there is interchange of the position of the subject and predicate terms of the proposition.
If we take the proposition, It is an A proposition
All dogs are mammals
The converse of it is, All mammals are dogs
First proposition is true while second, which is not true, instead here there is conversion through limitation, here there is a combination of sub alternation and conversion, Subject and Predicate positions are interchanged and changing the quantity of the proposition from universal to particular.
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If we convert the A proposition
All S is P Some P is S( by limitation) For E and I proposition conversion is perfectly valid
Some S is P Some P is S No S is P No P is S Some S is non P ( not valid)
The next immediate inference is Obversion
If we take the example
All residents are club members
No residents are non- club members
Obversion
Obvertend Obverse All S is P All non S is non P Some S is P Some S is not non P No S is P All S is non P
Some S is not P Some S is non P
Contraposition
To form the contrapositive ,the subject term of the given proposition is replaced by the compliment of the predicate term and the predicate term is replaced by the compliment of its subject term.
The contrapositive of
All S is P
All non-P is non-S
All Staff are members
All non-members are non-staff
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Here we can see that the contrapositive is valid for A proposition, it introduces nothing new. It is valid for both A and O propositions
Premise Contraposition
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 I: Some S is P Not Valid
O: Some S is not P O: Some non P is not non S
Symbolism and Diagram for Categorical Propositions
To deny that S is empty is by saying that S has members, by denying S = 0 we say that S is not empty, then it is symbolised as S ≠ 0.
In Standard form Categorical Syllogisms there are two classes
Suppose S is the class of Sportsperson and P the class of Intellectuals, the class of thing which belong to both of them can be represented by SP ≠ 0. The product of the classes is the class which belongs to both of them.
If we want to state the product of two is empty, we can state it as an E proposition. No S is P
Some S is P states that atleast one member of S belongs to P, this can be symbolised as
SP ≠ 0
[Ṕ -Here it is read as P complement, Ś– Here it is read as S complement]
In order to represent A and O propositions we need to introduce complement of a class. T
A proposition, All S is P can be symbolised as SṔ = 0
While the O proposition can be symbolised as SṔ ≠ 0
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Some S is not P
SṔ = 0 SP = 0
SP ≠ 0 SṔ ≠ 0
A and O propositions are contradictories, similarly E and I proposition are contradictories
If we represent the propositions through diagrams, the class S can be represented as a circle,
x
S = 0 S ≠ 0
The above figure represents the class S, the first figure represents an empty class, in the second figure there is an x inside the figure which shows there is at least one x which
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belongs to the class S.
Usually, to diagram standard categorical proposition two figures are required, if their subject term is S and Predicate term P, two intersecting circles are drawn
The part of S that is not overlapping P can be symbolised asSṔ = 0 s i m i l a r l y the part of P that does not overlap with S can be symbolised as ŚP = 0. The intersecting part can be symbolised as SP, it is the product of the classes S and P. ŚṔ is the part of the diagram that is external to both S and P.
Diagrams of the four standard form categorical propositions are given below
All S is P --- SṔ = 0
No S is P --- SP = 0
Some S is P --- SP ≠ 0
Some S is not P --- SṔ ≠ 0
1.6 Categorical Syllogism
Any deductive argument in which the conclusion is drawn from two premises is a
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categorical Syllogism. It is a mediate inference in which two premises jointly imply the conclusion. A categorical syllogism is a deductive argument consisting of three terms each of which occurs in the two constituent propositions. The premises and conclusion being standard form categorical propositions are arranged in standard order.
There are three terms which occur in the two constituent propositions
In categorical syllogism all propositions are categorical.
All men are mortal
All kings are men
All kings are mortal
The validity and invalidity of a syllogism(whose constituent propositions are contingent)depends upon it’s form and is completely independent of its specific content or subject matter.
If a syllogism is invalid, any other syllogism of the same form is invalid. To understand the fallacious character, we can check it by constructing another argument of the same form whose invalidity was apparent.
All M is P All S is M
All S is P
Eg:-
All mammals are animals
All cats are mammals
All cats are animals
The above argument is valid
The following argument is invalid
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All Cheetahs are fast runners
Some Rats are fast runners
Some Rats are Cheetahs
The form of the above argument is given below
All P is M Some S is M
Some S is P
In Categorical Propositions there are three terms
Minor term - S, Middle term - M Major term - P
The premise in which minor term occurs is called minor premise, the premise in which
major term occurs is called major premise. Middle term is the term which occurs in both premises, but not in the conclusion. Middle term is common to both major and minor premise.
Forms of the standard form categorical propositions determine the mood of the standard form -syllogism. The representation of it is through 3 letters, the first one names the major premise, the second one minor premise and the third one conclusion.
A standard form syllogism of a mood AII can have different forms.
All P is M
Some S is M
Some S is P
All M is P Some M is S
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Some S is P
Here though both the syllogism are of the same mood, there is difference in the position of the middle term, In the first syllogism the middle term is predicate term of major and minor premise. While in the second syllogism the middle term is the subject term of major and minor premise. Therefore, syllogisms having same mood may differ in their form depending upon the relativeposition of their middle terms. Is Complete description of form of syllogism is by stating itsmood and figure. Figure indicates the position of the middle term in the premises.
FIGURE OF A SYLLOGISM
The figure of the syllogism means the form of a syllogism which is determined by position of the middle term in its two premises. Accordingly, there are four possible arrangements of middle term(M) in two premises, and the figures of the syllogism are as follows –
1st figure 2nd figure 3rd figure 4th figure MP SM PM SM MP MS PM MS
SP SP SP SP
FIGURE OF A SYLLOGISM Special Canons -
In the first figure the middle term is subject of the major premise and predicate of minor premise
In the second figure the middle term is the predicate of both minor and major premise
In the third figure the middle term is the subject of both minor and major premise
In the fourth figure middle term is predicate of the major premise subject of minor premise
MOODS OF A SYLLOGISM
Mood of a proposition is determined by the quantity and quality of the constituent propositions. If all the three propositions of a syllogism are A propositions, the mood of the
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syllogism is AAA.
Valid mood of the first figure
MP SM
------
SP
Rule : Major premise should be affirmative and universal There are four valid mood for the first figure, they are AAA BARBARA
AII DARII
EAE CELARANT EIO FERIO
Valid mood of the 2nd figure
PM SM
------SP
Rule : 1. One premise must be negative
2. Major premise must be universal
The four valid moods of 2nd figure are –
AEE - CAMESTRES AOO - BAROCO EAE - CESARE
EIO - FESTINO
Valid mood of the 3rd figure
MP MS
------
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SP
Rule: 1. Minor premise must be affirmative
2. Conclusion must be particular The 6 valid moods of 3rd figure are – AAI - DARAPII
AII - DATISI
EAO - FELAPTON EIO - FERISON IAI - DISAMISS OAO – BOCARDO
Valid mood of the 4th figure
PM MS
------SP
Rule : 1. One premise is negative,major premise must be universal
2. If the major premise is affirmative, minor premise must be universal
The 5 valid moods of 4rth figure are –
AAI - BRAMANTIP AEE - CAMENES EAO - FESAPO
EIO - FRESISON
IAI - DIMARIS
Venn Diagrams:
For representing the syllogism in the Venn Diagram, three overlapping circles are drawn, there are three terms in the two premises, minor term, major term and middle term, we abbreviate it as S, P and M
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The two premises are represented using the Venn diagram, the circles are named in the order of S, P, M. With the circle S, the diagram of S and S complement can diagrammed. Similarly, with the overlapping circles S and P, four classes SP, ŚP, PŚ, ŚṔ can be diagrammed.
So, with the three overlapping circles we can diagram 8 different classes as shown above in the figure
By Using this diagram, we can check the validity of Categorical Syllogisms
All football players are athletes
All Rugby players are athletes
All Rugby players are football players
To test the validity of the syllogism, we have to make a diagram of three overlapping circles, with names in the usual order S, P and M. Where S denotes the class of all Rugby Players, P the class of all football players and M the class of all athletes.
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In the above figure the shaded region shows the representation of the first and second premises. The shaded areas are SṔḾ, SPḾ, ŚPḾ as per the first two premises. But the area SṔM is unshaded, which means the diagram does not satisfy the conclusion hence the argument is invalid. Which means conclusion says something more than said by the premises, which means it does not imply the premises.
Rules and Fallacies
To establish the conclusion is the purpose of the syllogism there are certain reasons for the syllogism failing to establish the conclusion. It is by setting certain rules that cogency of the argument is made possible, it enables them to avoid fallacies. Manifestation of rules is easily explicit, as a syllogism can be evaluated as to whether it conform to those rules.
1. A valid standard form Categorical syllogism must contain three terms and these three terms should be used in the same sense throughout the argument.
The assertion of the relationship of terms of conclusion to the same third term by the premises makes the justification for conclusion. When there are more than three terms in a categorical syllogism. It is termed invalid and is said to commit the fallacy of four terms (latin:quaternio terminorum).When the same term is used in different senses in the argument it is said to commit the fallacy of equivocation.
2. In valid standard form syllogism, the middle term must be distributed in atleast one of the premises.
All Apples are sweet
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Therefore, All Mangos are Apples
In the above argument the middle term ‘sweet’ is not distributed in any of the premises,
therefore, it commits the fallacy of undistributed middle, hence the argument is invalid.
In the conclusion there is a connection between it terms. The terms in the conclusion is related to each other through a third term called middle term. The premises justify in asserting the connection only if there is a connection through the middle term. In the above syllogism the middle term does not connect to minor and major term.
3. In valid standard form categorical syllogism, if either term is distributed in the conclusion, then it must be distributed in the premises.
The argument is invalid if the conclusion illegitimately goes beyond what is stated by the premises. When the term in the conclusion is distributed, which was not distributed in the premises then conclusions states more than what was intended in the premises. This makes the argument invalid.
All scooters are vehicles
No cars are scooters
------No cars are vehicles
Here major term ‘vehicles’ in undistributed in the major premise, while distributed in conclusion, which makes the argument invalid. Here it is the fallacy of illicit process of the major term.
4. No standard form categorical syllogisms have two negative premises are valid.
A negative proposition denies class inclusion, All or some of the class is excluded from the other. When the premises are negative, whether complete or partial, no relation between the terms (say S and P) of the conclusion can be inferred.
Symbolic Logic 27 School of Distance Education 5. If either of premises of the valid categorical syllogism is negative, then conclusion must be negative.
We get affirmative conclusion when the premises are also affirmative, but if the one of the premises is negative then the conclusion should also be negative. If this rule is broken then it commits the fallacy of drawing affirmative conclusion from negative premises.
6. No valid form standard categorical syllogism can have particular conclusion from two universal premises
Since here the premises are without existential import. So, to assert objects of specific kinds from two universal premises is to illegimately go beyond what is warranted by the premises.
Any syllogism that violates this rule is said to commit existential fallacy. These 6 rules apply only to standard form categorical syllogism, any syllogism that breaks these rules is invalid.
1.7 Disjunctive and Hypothetical Syllogisms
The kinds of propositions determine what name the Syllogism takes. A categorical syllogism contains categorical propositions exclusively. Categorical proposition is termed simple in contrast to propositions that contain other propositions as its constituents. Compound propositions contain other propositions as its components.
A Disjunctive proposition usually contain two components which are called disjuncts eg:- Either Raju crossed the road or Raju got hit by car
Raju did not cross the road
Raju got hit by a car
The above syllogism is valid
Here there are two components which are called disjuncts, either of these disjuncts is not categorically affirmed, it says at least one of it is true.
Symbolic Logic 28 School of Distance Education The disjunctive proposition does not categorical affirm the truth of either one of its judgements, it consists of two disjuncts, it carries the possibility that both may be true. It says at least one of it may be true.
Either Raju crossed the road or Raju got hit by car
Raju crossed the road
Raju did not get hit by car
The above a syllogism is not valid, since truth of one disjunct does not imply the falsity of the other, it might be Raju crossed the road and got hit by the car. Hence, we have valid disjunctive syllogism when the categorical premise contradicts one of the disjuncts of disjunctive premise and the conclusion affirms the other disjunct of the disjunctive premise.
Either Raghu is in Delhi or Raghu is in Madras
Raghu is in Delhi
Raghu is in not in Madras
In the above syllogism conclusion follows validly, here the one of the disjuncts is affirmed by the categorical premise and conclusion contradicts the other disjunct.
Hypothetical Syllogism
A statement of the form
If he wins the race then he will be promoted
If he is promoted, he will celebrate
If he wins the race, he will celebrate
In the conditional proposition containing two components, the one following if is called the antecedent and the component following then is called consequent. It can be seen that the antecedent of the first premise same as the antecedent of the conclusion and the consequent of the conclusion is the same as the consequent of the second premise. The consequent of the first
Symbolic Logic 29 School of Distance Education premise is same as the antecedent of the second premise.
Hypothetical syllogism whose component parts of the premises and conclusion are so related is a valid argument.
The rules of pure hypothetical syllogism are as follows:
1) Both of the premises should have one common categorical proposition.
2) This common proposition is the antecedent in one premise and consequent in other premise.
3) The conclusion should not have this common term, but instead it should contain the antecedent of one premise as antecedent (other than the common term) and consequent other premise as consequent (other than the common term)
Symbolic form
p q
⊃ q r
⊃ p r
∴ ⊃
1.8 SYMBOLIC LOGIC
Systems of logic had existed from the times of Aristotle (384-322 B.C). In symbolic logic just like mathematics advantages of using symbols is made to use, since it is easy for one to manipulate symbols, it provides overall structure of the sentence. Aristotle had used certain abbreviations in order to facilitate his investigations. Using symbols, we can make new developments in logic especially in dealing with complex arguments. It is economical to use symbols for scientific purposes since lot of time and space is needed to report in long sequences of familiar words. And when long sentences are involved it is more difficult to grasp the meaning.
Symbolic Logic 30 School of Distance Education K x K x K x K x K x K x K x K = L x L x L
Can be symbolised as K8 = L3
The modern symbolic logic used some special symbols, the difference between old and modern logic is of degree rather than of kind. The modern logic has devised its own technical language hence it has become an efficient tool for analysis and deduction. The logical structures of arguments can be displayed with more clarity with the special symbols used while in ordinary language it may be more obscure. The special symbolic language makes us easily identify valid and invalid arguments. Also, the ambiguity and vagueness of expressions are checked. Symbolic language enables better drawing of inferences over ordinary language. With the advent of symbolic logic there has been explosion of knowledge in this area.
h The advent of Symbolic logic was during 20t century though there were certain systems of logic in the times of Aristotle. From 1840’s we could see the development of symbolic logic from two branches of history, firstly George Boole (1815-1864)applying algebraic notations to non- mathematical kind of arguments. The algebraic notation and methods to first symbolise and then to validate arguments.
The other branch to which the development may be pointed to was the work of Augustus De Morgan (1806-1871) and Charles Sanders Pierce (1839-1914) who were involved in developing precise notation for relational arguments.There were many stalwarts like Gottlob Frege (1848-1925), Guiseppo Peano (1858-1932), Alfred North Whitehead ((1861-1947) those of whom had made magnificent contribution and finally the work of Bertrand Russell(1872- 1970)‘Principia Mathematica’ which was eventually a landmark in the history of Symbolic logic.
Symbolic Logic 31 School of Distance Education
UNIT - II
ARGUMENTS CONTAINING COMPOUND STATEMENTS
2.1 SIMPLE AND COMPOUND STATEMENTS
Compound statements contain other statements as its component parts. Compound proposition is that contain two or more propositions as its components. A simple statement do not contain other statements as its component.
Raju is brave and Hari is intelligent
The statement Raju is brave cannot be further split into component propositions
For a part of a statement to be component of a larger statement two conditions must be satisfied –
I) Firstly, it must be a statement in its own right.
II) If the part is replaced in the larger statement by any other statement, the replaced statement must be meaningful.
Conjunction:
Conjunction is a compound statement in which the word ‘and’ is inserted between the two statements. The symbol for the conjunction operator is dot(.)If suppose first component of the compound statement is p and second component is q then their conjunction is written as p.q. Truth value of the compound statement is determined by truth value of its parts. For the truth value of the compound statement to be true both of its conjuncts need to be true. If any of the component becomes false then the truth value of A.B becomes false.
Hence conjunction is termed truth functional compound statement.
The truth table for conjunction can be given as –
Symbolic Logic 32 School of Distance Education
p q p.q T T T T F F F T F F F F Disjunction:
Disjunction is a compound statement in which ‘or’ is inserted between the component statements, the two statements are called disjuncts. The word or can be used in both exclusive and inclusive sense.
Logicians recognize two kinds of disjunctions, inclusive disjunction and exclusive disjunction. A disjunction containing non-exclusive alternatives is called inclusive disjunction. Example,
‘Ramesh is either sick or lazy’. The sense of ‘or’ in inclusive disjunction is ‘at least one, both may be’.
A disjunction containing exclusive alternatives is called exclusive disjunction. For example,
‘Today is either Saturday or Sunday’. Another example, Either tea or Coffee. The sense of ‘or’
in exclusive disjunction is ‘at least one, but not both’.
The symbol ‘v’ is called wedge, in Latin it means the inclusive sense of ‘or’. It is the first letter of the word ‘vel’
The truth table for ‘inclusive disjunction’ is as follows:
P q pvq T T T T F T F T T F F F
Symbolic Logic 33 School of Distance Education Negation :
The symbol ‘~’ called “curl” or “tilde” is used to form the negation of a statement. The truth table for negation is as follows
If A symbolizes the statement ‘Cat is on the mat’ then ~ A states that it is not the case that
cat is on the mat. It is false that Cat is on the mat.
p ~ p
T F
F T
Just like we put brackets and braces in mathematics, in symbolic logic the importance of punctuation is the same, when there are many compound statements which itself gets combined to form complicated compounds, in order to resolve the ambiguity of such statement’s punctuation becomes essential.
A statement of the form A.Bv C becomes ambiguous regarding whether it is A conjunction of
B v C or disjunction of A.B with C
If we put brackets to it there are two senses associated with it. (A.B) v C
A. (BvC)
If A and B are false and C is true
Then the first statement is true while the second statement is false.
So, the above statement signifies the importance of punctuation, just like in mathematics symbolic logic we use parentheses, brackets and braces for punctuation.
If we symbolise statements, Either Ramu or Sanju will win the game with its negation then it will be symbolised as
~ (R v S)
Symbolic Logic 34 School of Distance Education The negation of the statement Neither Ramu and Sanju will win the game can be symbolised as
~(R.S)
Ramu and Sanju will both not win the game ~(R). ~(S)
Ramu and Sanju will not both win the game ~ (R.S)
If the simple statements with the repeated use of truth functional connectives form the compound statement, then it is the truth value of the simple statements that determine the truth value of the compound statement.
If we take an example –given M and N are true statements and P and Q false
P v [M.(QvN)]
Then the truth value of the above statement will be
P v M . Q v N F T F T
T T
T
From the above diagram it is clear that the truth value of the statement is true.
Now to symbolise a statement like, Either Team E or Team F will win the tournament but will both not win the tournament.
(E v F). ~(E.F)
Symbolic Logic 35 School of Distance Education 2.3 Conditional Statements
In the conditional statement of the form e.g.: - If the bus is late then I may not reach office on time, here the component between if and then is called the antecedent and the component after then is called the consequent.
There can be different kinds of implications that conditional statements can express like, if all Rabbits likes Carrots, Chotu is a Rabbit, then Chotu likes carrots, here the consequent logically follows from the antecedent. If the figure is square then it has four sides, here the consequent is implied by the antecedent.
So various examples can be cited for conditional statements, there can be different senses of if then phrase, we try to identify some common partial meaning of these conditionals.
For the conditional to be true, the negation of conjunction of antecedent and negation of consequent should be true.
~ (P. ~Q) should be true
When we open the brackets
~P v Q
Any conditional with true antecedent and false consequent must necessarily be false. For the conditional to be true the indicated conjunction (of the antecedent and negation of the consequent) must be false. Which means the negation of the conjunction must be true.
This relation is shown by a new symbol ‘ↄ’, which is called horseshoe. It represents the partial meaning of all conditional statements. It is a truth functional connective,
It is therefore an abbreviation for ~(p. ~q)
P ↄ q
If p then q
Symbolic Logic 36 School of Distance Education P q ~q p. ~q ~( p. ~q) p ↄ q
T T F F T T
T F T T F F
F T F F T T
F F T F T T
So in the above table there are Six columns , if we look in to the first two columns we can find the possible truth values applicable to P and Q(component statements). Then the other columns contain the stages in determining the truth value of the conditional statement.
The horseshoe symbolise a weak kind of implication called material implication.It is a special concept and not to be equated with usual implication.
The different ways of expressing conditional statement P ↄ Q are If P then Q, P implies that Q,
P entails that Q, P is a sufficient condition that Q,Q is a necessary condition that P.
‘ↄ’ is the symbolization for the weak implication and is called material implication, it means it should not be confused with other implications. There are implications which are neither causal, definitional nor logical, if we take a condition in which there is no real connection in which antecedent a n d which consequent states. Such a statement contains ridiculously a false statement as the consequent such that it denies truth of the antecedent.
E.g.: - If team A wins the game then colour of rose flower will change
Most conditionals express more than material implications between antecedent and the consequent but the symbolisation ’ↄ’ abstracts from the meaning of conditional statements. Regarding the expression of conditionals as material implication in valid arguments this symbolisation is justified.
2.4 Argument forms and truth tables Validity and invalidity are based upon the formal characteristics of the argument. Two argument of the same form is either both valid or both invalid whatever its subject matter may be.
Symbolic Logic 37 School of Distance Education The condition for validity of argument being t h e r e should be no instance in w h i c h the conclusion is false with premises being true. An argument is said to be invalid if exactly another argument of the same form can be constructed with true premises and false conclusion. Remember that an argument is valid if and only if its form is valid, and an argument form is valid if and only if it has no counterexample-no instance in which all the premises are true but the conclusion is false. The terms valid and invalid is applicable to both argument and argument forms.
Argument form may be defined as an array of symbols which contains statement variables, such that when statement is substituted for statement variables, same statement being substituted for every occurrence of the statement variable throughout the result is an argument.
Suppose if we have to check the validity of the argument,
Electrical wirings should be insulated or there is chance of electrocution
Electrical wirings not insulated
There is chance of electrocution
Which can be symbolised as
E v C
~E
C
The following has the same form
P v q
~ p
Therefore q
Symbolic Logic 38 School of Distance Education We can check the Validity of the disjunctive syllogism by constructing truth table
P q pvq ~ p
T T T F T F T F F T T T
F F F T
The whole class of substitution instances is being given in each rows, the T’s and F’s in the initial columns is the truth value of statements that can be substituted to statement variables P and Q of the argument form. These truth values determine the truth value of other columns.
Here first premise is the 3rd column and second premise 4th column
And conclusion the second column
We check all rows of premises and conclusion for true premises and false conclusion. If there is no substitution instance with true premises and false conclusion then the argument is valid.
So, in order check the validity of the syllogism, we look for the substitution instance having both premises true, so in the 3rd row we have substitution instance with both premises true for which the conclusion is also true. It shows that the argument is valid. In order to determine the validity or invalidity of an argument we have look in to all substitution instances and check whether there are true premises and false conclusions.
Hence truth table provides an effective method to determine the validity and invalidity of
Symbolic Logic 39 School of Distance Education argument
Modus Tollens p Ɔ q
~ q
~ p
p q pƆ q ~ p ~ q
T T T F F
T F F F T
F T T T F
F F T T T
If we check the rows of the premises, column 3 and column 5, there is a substitution instance of premises being true in the 4th row, when we check the conclusion of it in the 4th row of the 4th column which also true. This proves that the above argument form is valid. There is no substitution instance with true premises and false conclusion hence the argument is valid.
Now to check the validity of following argument form –
(p Ɔ q).(p Ɔ r)
p
Therefore, q v r
Here we have three statement variable, the number of rows is given by the equation 2n
, where n is the number of variables, hence for 3 variables it will be 8
Symbolic Logic 40 School of Distance Education p q R p Ɔ q pƆ r q v r (p Ɔ q).(p Ɔ r) T T T T T T T T T F T F T F
T F F F F F F T F T F T T F F T T T T T T F F T T T T T F T F T T T T F F F T T F T The premises are 1st and the 7th column and the conclusion the 6th column, so here we check whether true substitution instance gives false conclusion.
In the above truth table, there is no substitution instances with true premises and false conclusion, hence the argument is valid.
Statement forms:
We define a statement form to be any sequence of symbols containing statement variables, such that when statements are substituted for the statement variables—the same statement being substituted for every occurrence of the same statement variable throughout—the result is a statement. (page 27, im copi , symbolic logic)
A statement form is any sequence of symbols containing statement variables but no
statements, such that when statements are substituted for the statement variables-the same statement being substituted for the same statement variable throughout- the result is a statement. Thus, ~p is called a negation form or denial form, p v q is a statement form called disjunctive statement form, p . q is called conjunctive statement form and p Ͻ q is conditional statement form. Any statement of a certain form is said to be a substitution instance of that statement form.
The specific form of a given statement is defined as that statement form from which the statement results by substituting a different simple statement for each different statement variable. For example, p Ͻ q is the specific form of the statement A Ͻ B.
Symbolic Logic 41 School of Distance Education Tautologous, Contradictory, and Contingent statement forms:
We determine whether a given proposition is tautology, contradictory or contingent by looking at the truth tables.
Tautology:
A statement is a tautology if the column under its main connective is ‘True’ on every row of a complete truth table. Now consider the statement - ‘it is raining or it is not raining’, which is symbolized as
‘p v ~ p
The truth table for p v ~ p is represented as follows:
p ~ p p v ~p T F T F T T
Contradictory:
A statement is a contradictory if the column under its main connective is ‘False’ on every row of a complete truth table. Now consider: ‘it is raining and it is not raining’ which is symbolized as ‘p . ~ p
The truth table for p . ~ p is represented as follows:
P ~ p p. ~p T F F F T F
Here we get false as conclusion for all substitution instances, hence the statement being contradictory.
Contingent:
A statement is contingent if it is neither a tautology nor a contradiction; i.e. if it is T on at
least one row and F on at least one row. Now consider the statement: ‘if it is raining then
Symbolic Logic 42 School of Distance Education the roads are wet’ which is symbolized as P Ͻ Q. The truth table for implication is as follows:
p q p Ͻ q
T T T
T F F
F T T
F F T
So, there is at least one F and one T, it makes the statement contingent.
BICONDITIONAL:
The symbol tribar’≡’ stands for material equivalence or biconditional. Biconditional is a compound proposition in which simple statements are connected with phrase if and only if. Eg:-
The vehicle will operate if and only if there is enough fuel.
Material invalid implication is a special technical concept that logician introduces such that it aids him and simplifies the task of discriminating between valid and arguments.
When truth value of two statement is the same then the statements is said to have material equivalence. The symbol for material equivalence is ‘≡’, this symbol should be read if and only if. Statement forms of the pattern P ≡ Q is called biconditional.
Two statements are said to be logically equivalent when the biconditional that expresses their material equivalence is a tautology. (page 29, im copi)
Eg:-
p ≡ ~~p
p q P ≡ q
Symbolic Logic 43 School of Distance Education T T T
F T F
T F F
F T
Now to check using truth table that whether the following statements are equivalent
(p Ͻ q) ≡ (~q Ͻ ~p)
The above statements are logically equivalent.
Truth table for (p q) ∙ ~ q
p q p q ~ q (p q). ~q
T T T F F
T F F T F
Symbolic Logic 44 School of Distance Education F T T F F
F F T T T
The above statement is contingent
UNIT - III
METHODS OF DEDUCTION
Symbolic Logic 45 School of Distance Education 3.1 Formal Proofs of Validity
It is not always possible to test the validity of arguments by using truth tables when the arguments contain more that three different simple statements. Another method is to deduce the conclusion from premises by sequence of shorter elementary statements that are known to be valid. The premises and statements are listed as proofs of validity in one column and the justification for the latter is written in the adjacent to it. The rule of inference and previous statement by which the statement in question is deduced is mentioned as justification
Formal proof of validity for a given argument is defined to be a sequence of statements, each of which is either a premiss of that argument or follows from preceding statements by an elementary valid argument, and such that the last statement in the sequence is the conclusion of the argument whose validity is being proved.
This definition must be completed and made definite by specifying what is to count as an ‘elementary valid argument’. We first define an elementary valid argument as any argument that is a substitution instance of an elementary valid argument form. Then, we present a list of just nine argument forms that are sufficiently obvious to be regarded as elementary valid argument forms and accepted as Rules of Inference (page 33, Symbolic logic, im copi)
3.2 Nine rules of inference
Modus Ponens
P Ɔ q
P
q
P Ɔ q
q
Symbolic Logic 46 School of Distance Education ~ p
∴
Hypothetical syllogism
P Ɔ q
Q Ɔ r
p Ɔ r
∴ Disjunctive Syllogism
P v q
~ p
q
(p Ɔ q). (r Ɔ s)
P v r
q v s
∴
Destructive Dilemma
(p Ɔ q). (r Ɔ s)
Symbolic Logic 47 School of Distance Education ~q v ~s
~p v ~r
∴
Simplification
p.q
P
∴
Conjunction
P
q
p.q
∴
Addition
P
P v q
∴
State the rules of inference from which the following conclusion is reached
1. (M Ɔ ~N).( ~O Ɔ P)
(M Ɔ ~N)
∴
Symbolic Logic 48 School of Distance Education Answer : Simplification
2.( ~ (A.B) Ɔ ~C).(D Ɔ ~E)
~(A.B) v D
~ C v ~ E
∴ Answer : Constructive Dilemma
3. A Ɔ (B≡ ~ C) ------1
(B≡ ~ C) Ɔ D ------2
A Ɔ D ------3
∴ Answer : 1,2 Hypothetical Syllogism
Construct a formal proof of validity of each of the following arguments
1. P Ɔ Q R Ɔ S
(~Q v ~ S).( ~P v ~Q)
~P v ~R
∴ Answer:
P Ɔ Q ------1
R Ɔ S ------2 (~Q v ~ S).( ~P v ~Q)------3
Symbolic Logic 49 School of Distance Education ~P v ~R
∴
(P Ɔ Q).(R Ɔ S) 1,2 conjunction------4
(~Q v ~ S) 3 simplification ------5
~P v ~R 4,5 Destructive Dilemma---6
2. (A Ɔ ~B).(C Ɔ ~D) ( E Ɔ ~ F).(G Ɔ ~H) (C Ɔ F).(D Ɔ B)
E v A
~ C v ~D
∴ Answer
(A Ɔ ~B).(C Ɔ ~D) ------1 ( E Ɔ ~ F).(G Ɔ ~H) ------2 (C Ɔ F).(D Ɔ B) ------3
E v A ------4
E Ɔ ~ F 2 simplification-----5
A Ɔ ~B 1 simplification-----6
( E Ɔ ~ F). (A Ɔ ~B) 5,6 conjunction -----7
~ F v ~B 7,4 constructive dilemma----8
~ C v ~D 3,8 destructive dilemma
Construct a formal proof of validity of the following argument, using the abbreviations suggested:
Symbolic Logic 50 School of Distance Education 1. If the king does not castle and the pawn advances, then either the bishop is blocked or
the rook is pinned. If the king does not castle, then if the bishop is blocked, then the game is
a draw. Either the king castles or if the rook is pinned, then the exchange is lost. The king does not castle and the pawn advances. Therefore, either the game is a draw or the exchange is lost.
(C: The king castles. W: The pawn advances. B: The bishop is blocked. P: The rook is pinned. D: The game is a draw. E: The exchange is lost.)
So we have to first symbolize the above argument
~C.W Ɔ B v P
~C Ɔ (B Ɔ D) C v (P Ɔ E)
~C.W
D v E So, in order to solve the argument, first we have to number it ∴
~C.W Ɔ B v P ------1
~C Ɔ (B Ɔ D) ------2
C v P Ɔ E ------3
~C.W ------4
D v E
∴
Symbolic Logic 51 School of Distance Education
B v P 1,4 modus ponens -----5
~C 4 simplification ------6
B Ɔ D 2,6 Modus ponens------7
P Ɔ E 3,6 Disjunctive Syllogism -----8 (B Ɔ D).(P Ɔ E) 7,8 Conjunction ------9
D v E 9,5 Constructive Dilemma
The nine rules of inference may not be able to prove the validity of certain truth functional arguments. The rule of replacement or the principle of extensionality states when we replace a part in the compound truth functional statement with an expression which is its logical equivalent, then the resultant statement has the same truth value as the original statement.
The result of the replacement of parts of the compound statements or as a whole by logically equivalent statements can be inferred using the replacement rules.It is considered the additional rule of inference, so the following are the additional rules of inference continuing after the nine rules of inference.
10. De Morgan’s theorem
~(p.q) ≡ ~p v ~q
~(p v q) ≡ ~ p . ~q
11. Commutation p v q ≡ q v p p.q ≡ q.p
12. Association
P v (qvr) ≡ (p v q) v r p.(q.r) ≡ (p.q).r
Symbolic Logic Page 52 School of Distance Education 13. Distribution
p.(qvr) ≡ (p.q) v (p.r)
p v(q.r)≡ (pvq).(pvr)
14. Double Negation
P ≡ ~~P
15. Transposition
p Ɔ q ≡ (~q Ɔ ~p)
16. Material Implication p Ɔ q ≡ (~p v q)
17. Material Equivalence
( p≡q )≡[ (p Ɔ q).(q Ɔ p)]
( p≡q )≡ (p.q) v (~p. ~q)
18. Exportation
(p.q) Ɔ r ≡ p Ɔ (q Ɔ r)
19. Tautology
p ≡ p.p
p ≡ p v p
So, in order to prove the validity of arguments using truth table method is cumbersome, when the number of rows listed may get large. Truth table method is mechanical, while to prove the validity of arguments using the 19 rules of inference, questions like where to begin and the way to proceed needs to get solved. Here though it is not a mechanical method like truth table, formal proof of validity is much easier than to write truth tables with many columns and rows.
The difference between the first 9 rules of inference and rules of replacement is that rules
Symbolic Logic Page 53 School of Distance Education of replacement can be applied to whole of lines or part of lines.
Not only p Ɔ q can be replaced as ~p v q (whole of a line)
(p Ɔ q). r can be written as (part of a line) (~p v q).r
Wherever there are logically equivalent statements it can be replaced as whole or part, but first nine rules of inference can be applied only to line as a whole.
According to the rules of replacement, the logical equivalence of statements to be replaced are given as per the rules 10 through 19. In the case of substitution of statements in the argument form all the statements substituted in the statement variable needs to be the same, while in the case of replacement one occurrence of the statement variable can change without other not being replaced.
One start first by deducing the conclusions from the given premises according to the given rules of inference, after these further deductions are made from the sub-conclusions which act as premises then one understands how the conclusion of the argument is proved. According to the rules of inference we can apply it for elimination of some statements in the premises not in the conclusion. Like if we apply the rule of hypothetical syllogism then the middle term q is eliminated, like
P Ɔ q q Ɔ r p Ɔ r
also, by applying simplification p.q
p
∴ Here q is eliminated
By commutation left side conjunct can be shifted to ride side. Another rule is addition, addition of a statement that occur in the conclusion but not in the premises. We can also infer by looking backwards from the conclusion regarding which premises or pair of premises from which we can deduce conclusion based upon the rules. For each of the following arguments, state the rules by which conclusion follows from the premises
Symbolic Logic Page 54 School of Distance Education
1. (~E Ɔ F ).(G v ~H)
(~E Ɔ F).(~H v G)
∴ Answer : Commutation
2.(~A v B).(C v ~D)
( A Ɔ B).(C v ~D)
∴ Answer : Material implication
3. A Ɔ ~(B v ~ C)
M Ɔ ~B.~~C
∴
Answer : De Morgan’s Theorem
4. (G.H) Ɔ {I. [(J. K). L]}
(G.H) Ɔ {I. [(K.J). L]}
∴ Answer: Commutation
5. [A Ɔ (B v C)] v [ A Ɔ (B v C)]
[A Ɔ (B v C)]
∴ Answer: Tautology
a) Construct formal proof of validity for the following argument
1. A Ɔ (B Ɔ C) C Ɔ ~ C
(D Ɔ A).(E Ɔ B)
Symbolic Logic Page 55 School of Distance Education
D Ɔ ~E
∴
A Ɔ (B Ɔ C) ------1
C Ɔ ~ C ------2
(D Ɔ A).(E Ɔ B) ------3
D Ɔ ~E
∴
(A.B) Ɔ C 1 exportation------4
~ C v ~ C 2 Material implication------5
~C 5 Tautology ------6
~(A.B) 4,6 Modus Tollens------7
~A v ~B 7 De Morgan ------8
~D v ~ E 3,8 Destructive Dilemma ------9
D Ɔ ~ E 9 Material implication
2. (A v B) v (C.D)
(~ A .D).~(~A.B)
~ A . C
∴
(A v B) v (C.D) ------1 (~ A .D).~(~A.B)------2
~ A . C
∴
Symbolic Logic Page 56 School of Distance Education (~A.D) .( A v ~ B) 2, Demorgan---3
~ A .(D.A v ~ B) 3 Association ------4
~A------4 simplification –5
A v (Bv (C.D)) ------1 Association------6
[B v (C.D)] ------6, 5 Disjunctive Syllogism----7
(B v C).(B v D )------7 Distribution ------8
B v C ------8 Simplification ------9
~ A . (B v C) ------5,9 Conjunction ------10
~A.B v ~A.C ------10 Demorgan ------11
~(~A.B). (~ A .D) ------2 Commutation------12
~(~A.B) ------12 Simplification------13
~ A.C ------11,13 Disjunctive Syllogism.
b) Construct formal proofs of validity of the following arguments with the suggested notation
Either the Mayor and the councillor will both run for re-election or the primary race will be wide open and the party will be torn by dissension. The Mayor will not run for re-election. Therefore the party will be torn by dissension. (M,C,W,D)
(M . C) v W.D ------1
~ M ------2
D
∴
Symbolic Logic Page 57 School of Distance Education ~M v ~C 2 addition------3
~(M.C) 3 DeMorgan----4
W.D 1,4 Disjunctive Syllogism----5
D.W 5 commutation------6
D
3.4 The Rule of Conditional Proof
There is another rule that can be applied to the method of deduction called rule of conditional proof. It can be applied to arguments whose conclusion is conditionals. If an argument has conditional statements as its conclusion say if it is M Ɔ N then the conjunction of premises is H, for the argument is valid only if the conditional is a tautology
H Ɔ (M Ɔ N)
Arguments usually contain conditional statements, the antecedents of it being the conjunction of the premises and conclusion being the consequent. So we need to deduce the conclusion (M Ɔ N) from the premises conjoined in H through the sequence of elementary valid arguments proving the argument to be valid and conditional H Ɔ (M Ɔ N) being a tautology.
By using the exportation (Rules of replacement)
(H.M) Ɔ N
The premise of this argument has all the premises of the first argument and antecedent of the conclusion of the first. We have to prove this to be a tautology. We deduce N using the premises conjoined in H.M, by sequence of elementary valid arguments. Through it we prove it to be a tautology.
(H v I) Ɔ ( J.K) ------1
(K v L) Ɔ M ------2
Symbolic Logic Page 58 School of Distance Education
H Ɔ M
∴ H / M (C.P)------3
∴
H v I ------3 addition---4
J.K ------1,3 Modus Ponens----5
K.L ------5 commutation ------6
K ------6 simplification------7
K v L ------7 addition ------8
M ------2,8 Modus Ponens-----9
Hence it proves the validity of the above argument.
3.5 The Strengthened Rule of Conditional Proof
In order to strengthen the rule of conditional truth for situations in which the conclusion is not an explicit conditional thus providing wider applicability. A new method of writing proofs is used using conditional methods.
Earlier we used conditional proof as the method for proving the validity of arguments in which the conclusion which is a conditional statement, the antecedent of the conditional is made the premise as an assumption, then deducing the conditional’s consequent.
When an assumption is made in the conditional proof of validity its scope is limited not extending to the end of the argument. Inorder to mark the scope of the assumption, in the new method a bent arrow is used. Head of the arrow pointing to the assumption from left, and the shaft run down along all line till the scope of the assumption benting inwards marking end of the scope.
Symbolic Logic Page 59 School of Distance Education
(L v M) Ɔ [(N vO) Ɔ P] / L Ɔ [(N.O) Ɔ P] ------1
∴ L ------2
L v M 2 addition------3 (NvO) Ɔ P 1,3 Modus Ponens------4
N.O ------5
N 5 Simplification ------6
N v O 6 addition ------7
P 4,7 Modus Ponens------8
N.O Ɔ P 5,8 Conditional Proposition------9
L Ɔ (N.O) Ɔ P 2, 9 conditional Proposition------10
Where the scope of an assumption ends then assumption is said to be discharged. Justification of reference to the assumption belongs only to the lines lying between the limited scope and the line that discharges it.
After discharge of an assumption based on limited scope is made another assumption can bemade and then discharged or a second assumption of a limited scope may be written within the scope of the first. It can be that one scope may be contained in another or scopes of different assumptions may follow each other.
Not only any assumptions of limited scope can be made. Even negation of arguments conclusion can be assumed in strengthened rule of conditional proof.
3.6 The Rule of Indirect Proof:
In the indirect proof of validity of an argument, the argument is constructed where the negation of the conclusion is assumed as an additional premise and, and deriving explicit contradiction from the augmented set of premises. This method is also called method of proof by Reductio ad absurdum .In elementary geometry where the opposite of what one wants
Symbolic Logic Page 60 School of Distance Education to prove is assumed. If that assumption leads to contradiction then the theorem that we wanted to prove is true.
Arguments having tautology as conclusion can be verified using truth table method. If the tautologous conclusion of an argument is not a conditional statement and the premises are consistent with each other and quite irrelevant to the conclusion then argument cannot be proved valid using the usual methods, for which indirect proofs need to be used.
For the following arguments construct indirect proof
1. (A Ɔ B).(C Ɔ D) ------1
(B v D) Ɔ E ------2
~ E ------3
~(A v C)
∴
A v C Indirect Proof------4
B v D 1,4 Constructive Dilemma ----5
E 2, 5 Modus Ponens ------6
~E.E 3,6 conjunction ------7
So step 7 is a contradiction, which means our negation of the conclusion is false. Hence the argument is valid.
2. (A Ɔ ~B).(C Ɔ D) ------1 (~B Ɔ E).(D Ɔ ~M)------2 (E Ɔ ~N).(~M Ɔ O)------3
(A.C)------4
Symbolic Logic Page 61 School of Distance Education ~ N.O
∴ ~(~ N.O) ------Indirect Proof --5
~~N v ~O De Morgan ------6
~E v ~~M 3,6 Destructive Dilemma---7
~~ B v ~ D 2,7 Destructive Dilemma ---8
~ A v ~ C 1,8 Destructive Dilemma---9
~(A.C) 8 De morgan ------10 (A.C).~(A.C) 4, 9 Conjunction ------11
So here the step 12 is a contradiction which proves that the negation of the conclusion is false, hence the argument valid.
If this was proved using formal proof of validity without using indirect proof the
There would have been longer steps and proof would have been more tedious.
Symbolic Logic Page 62 School of Distance Education UNIT - IV
Quantification Theory
4.1 Singular Propositions and General Propositions
All Humans are mortal
Aristotle is human
Aristotle is mortal
The inner logical structure of non-compound statements have to be analysed since validity of such arguments depend upon it. There should be methods for it. Any subject term has its attribute designated by the predicate term. Individuals described here are not only persons it can be things, planets, nations etc. Attributes can be not only adjectives it can be nouns or verbs. Singular propositions are symbolised using
‘a through w’ using the first letter of individuals name to denote that individual. It is
also called individual constants.
In the above example, mortal may be symbolised as M, and Humans as H. The predicate term is written left to the subject term. Aristotle is human can be symbolised as Ha and Aristotle is mortal Ma
So if there are symbolisation for singular propositions having the same predicate term like
Socrates is human - Hs Bacon is human - Hb Spinoza is human - Hs
All these propositions have a attribute symbol H followed by an individual constant. Since all these propositions have same attribute symbol and different constants
We can write pattern common to all singular propositions as Hx. Hx is called propositional function. Here x is called the individual variable. Hs, Hb, Hs are either true or false while Hx is neither true or false. When the individual variable in Hx is replaced by an individual constant like Hs they become individual propositions.
Symbolic Logic Page 63 School of Distance Education When the substitution of the individual constant in the individual variable in the propositional function results in singular proposition, this singular proposition can regard as the substitution instance of the propositional function. The process of obtaining a proposition from a propositional function by substituting a constant for a variable is called ‘instantiation’.
In the case of general propositions, it is quantification, they can result from propositional functions but here it is not instantiation.
Everything is mortal can be written as
We can use x in the place of pronoun it
Given any x , if x is mortal
x written as universal quantifier is (x) The above proposition can be symbolised as
(x)Mx
We can paraphrase
‘Something is mortal’ as
There is at least one x such that it is Mx
There is one x such that is symbolised using (Ǝx) is called the existential quantifier. So the above proposition can be symbolised as
(Ǝx)Mx
It can be seen that in front of the propositional function Mx , the symbolisation (Ǝx) is placed to form a general proposition. So, a general proposition is formed when either a universal or existential quantifier is placed before the propositional function. Here the existential quantification of the propositional function is true if at least one substitution instance of the propositional function is true. For Universal quantification to be true all of its substitution instances need to be true.
Now if we consider the negations of the above two propositions, ie., The symbolisation of it is
Symbolic Logic 72 Page 64 School of Distance Education Nothing is mortal - ( x) ~Mx
Something is not mortal – (Ǝx)~Mx
If we use the attribute symbol φ ‘phi’, then we can represent the relation between existential and Universal generalisation using the square array
The top two propositions are contraries, which means both can be false but both cannot be true, while for the two propositions in the bottom both can be true but both cannot be false. For the propositions on the sides, the lower proposition is implied by the proposition above it. Regarding the opposite corners they are called contradictories, ie., when one is true other must be false.
Symbolic Logic Page 65 School of Distance Education The four types of subject predicate propositions in traditional logic may be listed as- All humans are mortal
No humans are mortal
Some humans are mortal
Some humans are not mortal
If we symbolise it using propositional functions and quantifiers Given any individual thing whatever, if it is human then it is mortal Given any x if x is human then x is mortal
(x)(Hx Ɔ Mx)
Now for the next E proposition
No humans are mortal, can be paraphrased as
Given any x if x is human then x is not mortal, can be symbolised as
(x) (Hx Ɔ ~Mx)
For the I proposition
There is at least one x if x is human, then x is mortal, can be symbolised as
Ǝx(Hx.Mx)
And finally, for the O proposition
There is at least one x if x is human, then x is not mortal, can be symbolised as
Ǝx(Hx. ~Mx)
If φ(phi) and ψ(psi) are used to represent the attributes, then the four general subject predicate propositions of traditional logic can be represented in the following square array
φ and ψ are Greek letters
Symbolic Logic Page 66 School of Distance Education
(x)(φx Ɔ ψx) (x)(φx Ɔ ~ψx)
A E
I O
(Ǝx)(φx. ψx) (Ǝx)(φx. ~ψx)
(x)(φx Ɔ ψx) and (x)(φx Ɔ ~ψx) has only true substitution instances,where φx has no true substitutional instances, regardless of what attribute is symbolised by ψ.This is so because all their substitution instance is a conditional statement with false antecedents.
Here A and E are not contraries instead, here A and E are true.
Now φx having no true substitution instances, whatever the value taken by ψx
φx. Ψx and φx. ~ψx have only false substitution instances. Since it is conjunction and their first conjunct is false.
Thus, I and O propositions are false. Also, they are not subcontraries. A and E are true and I and O are false, it means the universal do not imply the particular.
If we assume there is one individual then (x) (φx Ɔ ψx) implies Ǝx (φx Ɔ ψx), we should remember that here Ǝx (φx Ɔ ψx) is not the I proposition. The symbolisation Ǝx (φx Ɔ ψx) means there is atleast one object either having attribute ψ and with not having the attribute φ. The I proposition symbolised as Ǝx(φx.ψx), it means there is atleast one thing with the attributes φ and ψ.
Symbolic Logic Page 67 School of Distance Education All the members are either Artists or Engineers
Can be symbolised as
(x)[ Mx Ɔ (Ax v Ex)]
Some Students are either intelligent or hardworking
Can be symbolised as
Ǝx [Sx .( Ix v Hx)]
Iron and Copper are Metals
(x)[Ix Ɔ Mx]. (x)[Cx Ɔ Mx] Or
(x)[(Ix v Cx) Ɔ Mx]
It does not mean that anything that is metal are both iron and copper so the following symbolisation is wrong
(x)[(Ix.Cx) Ɔ Mx]
4.2 Proving Validity: Preliminary Quantification Rules
When there are arguments comprised of propositional functions and quantifiers there are certain rules of inference that are listed for its formal proofs of validity.
There are four rules concerning quantification.
If all the substitution instance of a propositional function is true then the universal quantification of the propositional function is true. The principle that Inference from universal quantification can result in valid substitution instance of the propositional function can be listed as a rule of inference. It is called the principle of universal instantiation, it is abbreviated as UI
(x)(φx)
Symbolic Logic Page 68 School of Distance Education φn
All humans are mortal
Aristotle is human
Aristotle is mortal
(x)[Hx Ɔ Mx] ------1
Hv / Mv ------2
Hv Ɔ Mv ------1 U.I------3
Mv ------3,2 M.P
2. Universal Generalisation:
If k is an arbitrarily selected individual, then a special attribute of k that can be generalised, what is it that makes true of the arbitrary selected individual true of other individuals. In Universal instantiation φk follows validly from (x)φx. It means what is true of all individuals is true of an arbitrary individual k. It can also be true in the reverse order, what is true of arbitrary individual can be true of all individuals.
Here the principle that universal quantification of the propositional function can be validly inferred from a substitution instance k, can be added to the rules of inference.It is called the principle of universal generalisation written as ‘UG’.
The symbolic expression can be written as
φk
------
؞(x)φx
(where k is an arbitrary selected individual, and φk is not within the scope of any assumption
containing the special symbol ‘k')
Symbolic Logic Page 69 School of Distance Education 3. Existential Generalisation:
If and only if there exists one true substitution instance in t he propositional function then only existential quantification of the propositional function is true. Then it can be added to the list of rules of inference, the principle that existential quantification of the propositional function can validly be inferred from any substitution instance of the propositional function. This rule allows inference of general propositions that are existentially quantified. It is called the principle of existential generalisation.
The symbolic formulation is given below –
φk
------
؞Ǝx φx
4. Existential Instantiation:
The existential quantification of a propositional function asserts that there exists at least one individual the substitution of whose name for the variable x in that propositional function will yield a true substitution instance of it. Here we use any individual constant which has no prior occurrence. Knowing that there is an individual, which has no prior occurrence and having agreed to denote it by a symbol ‘w’ instead of y. From the substitution instance of the symbol
’w’ in the propositional function x’, we can infer the existential quantification of the propositional function, the substitution instance of that propositional function with respect to the individual symbol w.
Ǝx φx
------
؞φv
(where v is an individual constant which has no prior occurrence in the context)
Symbolic Logic Page 70 School of Distance Education a) The translation of sentences to quantified propositional function is given below
1. Only Dancers have flexibility
(Dx: x is a dancer, Fx: x is flexibility)
(x)( Dx Ɔ Fx)
2. No furniture is protected, unless it has been polished
( Fx : x is furniture, Ox: x is polished) (x)[Fx Ɔ (~Px v Ox)]
3. None but the winner deserves the credit
( Wx: x is winner, Dx: x is deserves credit)
(x) [ Wx Ɔ Dx]
B) Constructing formal proof of validity for the following argument -
i) All cars have four wheels
Some vehicles are cars
Some vehicles have four wheels
Which can be symbolised as
(x)( Cx Ɔ Fx) ------1
(Ǝx)(Vx.Cx) ------2
(؞ (Ǝx)( Vx.Fx /
Vw.Cw ------2,E.I------3
Cw Ɔ Fw ------1,U.I------4
Cw.Vw ------3 Commutation—5
Symbolic Logic Page 71 School of Distance Education Cw ------5 Simplification—6
Fw ------4,6 Modus Ponens-----7
Vw ------3, Simplification------8
Vw.Fw ------8,7 Conjunction ------9
Ǝx( Vx.Fx)------9, Existential generalisation----10
ii) Mangoes are edible . Only Items of food are edible.All items of food are good, therefore All
Mangoes are good.(Mx,Ex,Gx,Fx)
MxƆ Ex ------1
Ex Ɔ Fx ------2
Fx Ɔ Gx ------3
؞ Mx Ɔ Gx /
My Ɔ Ey 1 UI------4
Ey Ɔ Fy 2 UI------5
Fy Ɔ Gy 3 UI------6
My Ɔ Fy 4,5 H.S------7
My Ɔ Gy 7,6 H.S------8
Mx Ɔ Gx 8 U.G ------9
C) Formal proof of validity for the following arguments using the rule of conditional proof
(x)(Ax Ɔ Bx) ------1
Symbolic Logic Page 72 School of Distance Education (x)[(Ax.Bx) Ɔ Cx] ------2
[؞ (x)[Ax Ɔ Cx /
Ay ------3
Ay Ɔ By 1 U.I ------4
By 4,3 M.P ------5
(Ay.By) Ɔ Cy 2, U.I ------6
Ay.By 3,5 conjunction------7
Cy 6,7 M.P ------8
Ay Ɔ Cy 3,8 Conditional Proof-----9
(x)[Ax Ɔ Cx] 9 U.G ------10
Symbolic Logic Page 73 School of Distance Education
References:
Introduction to Logic, 8th edition, Irving M Copi and Carl Cohen, Macmillan Publishing Company, NewYork,1990
Irving M Copi, Symbolic Logic,Macmillan Publishing Company, New York,1979
Virginia Klenk, Understanding Symbolic Logic, 5th edition,Prentice Hall, 2007
Logic and Scientific method, Self-Learning Material,School of Distance Education, University of Calicut
Symbolic logic and Informatics, Self-Learning Material, School of Distance Education, University ofCalicut
Essentials of Symbolic Logic, Self-Learning Material, School of Distance Education, University of Calicut
Essentials of formal logic, Self-learning material, School of Distance Education, University of Calicut.
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