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Logic & Critical Thinking

Session 81 March, 2014

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Theory of Deduction• A method of reasoning from the general to the specific. In a

deductive argument, a conclusion follows necessarily from the stated premises.

• A deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion .

• The fundamental property of a deductively valid argument is this: If all of its premises are true, then its conclusion must be true.

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Contd….

• Every deductive argument is either valid or invalid. If it is valid, it is impossible for its premise to be true without its conclusion also being true.

• The theory of deduction aims to explain the relationship of valid premises and conclusion.

• Everything made of copper conducts electricity. (Premise)This wire is made of copper. (Premise)This wire will conduct electricity. (Conclusion)

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Classes and Categorical Propositions• Classical logic deals mainly with arguments based on relations of

classes of objects to one another.

• By a class we mean a collection of all objects that have some specified characteristics in common

• The classes can be related in at least of the three ways:– All of one class may be included in all of another class (All dogs are mammals)– Some but not all, of the members of one class may be included in another

class (Some chess players are females)– Two classes may have no members in common (No triangles are circles)

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Contd….

• These 3 relations may be applied to classes, or categories, of every sort.

• In a deductive argument, we present propositions that state the relations between one category and some other category.

• These propositions are called categorical propositions.

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Four Kinds of Categorical Propositions• The four kinds of categorical propositions include:– All S is P (A)– No S is P (E)– Some S is P (I)– Some S is not P (O)

• Where “S” refers to subject and “P” refers to predicate

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All politicians are liars

Subject Predicate

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Contd….• All politicians are liars (A)

– Universal affirmative proposition

• No politicians are liars (E)– Universal negative proposition

• Some politicians are liars (I)– Particular affirmative proposition

• Some politicians are not liars (O)– Particular negative proposition

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Universal Affirmative PropositionAll politicians are liars

• It is about two classes, the class of all politicians and the class of all liars, saying that the first class is included or contained in the second class.

• A universal affirmative proposition says that every member of the first class is also a member of the second class. In the present example, the subject term “politicians” designates the class of all politicians, and the predicate term “liars” designates the class of all liars.

• Any universal affirmative proposition is written as: All S is P

where the terms S and P represent the subject and predicate terms, respectively. It is denoted by “A”.

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Politicians(S)

Liars(P)

The diagram for the A proposition, which asserts that all S is P, shows that portion of S which is outside of P

shaded out, indicating that no members of S that are not members of P.

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Universal Negative Proposition

No politicians are liars• It denies of politicians universally that they are liars.• Concerned with two classes, a universal negative proposition

says that the first class is wholly excluded from the second, which is to say that there is no member of the first class that is also a member of the second.

• Any universal negative proposition may be written as:No S is P

where the terms S and P represent the subject and predicate terms, respectively. It is denoted by “E”

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Politicians(S)

Liars(P)

The diagram for the E proposition, will exhibit this mutual exclusion by having the overlapping portion of the two circles representing the classes S and P

shaded out.

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Particular Affirmative PropositionSome politicians are liars

• Clearly, what the present example affirms is that some members of the class of all politicians are (also) members of the class of all liars.

• But it does not affirm this of politicians universally: Not all politicians universally, but, rather, some particular politician or politicians, are said to be liars.

• This proposition neither affirms nor denies that all politicians are liars; it makes no pronouncement on the matter.

• Any particular affirmative proposition may be written as:Some S is P

where the terms S and P represent the subject and predicate terms, respectively. It is denoted by “I”

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Contd…..• The word “some” is indefinite. Does it mean “at least one,” or

“at least two,” or “at least one hundred?” In this type of proposition, it is customary to regard the word “some” as meaning “at least one.”

• The name “particular affirmative” is appropriate because the proposition affirms that the relationship of class inclusion holds, but does not affirm it of the first class universally, but only partially, of some particular member or members of the first class.

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Politicians(S)

Liars(P)

The diagram for the I proposition indicates that there is at least one member of S that is also a member of P by placing X in the region which the two circles overlap.

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Particular Negative PropositionSome politicians are not liars

• This example, like the one preceding it, does not refer to politicians universally but only to some member or members of that class; it is particular.

• But unlike the third example, it does not affirm that the particular members of the first class referred to are included in the second class; this is precisely what is denied.

• A particular negative proposition, schematically written as:Some S is not P

• Above statement states that at least one member of the class designated by the subject term S is excluded from the whole of the class designated by the predicate term P.

where the terms S and P represent the subject and predicate terms, respectively. It is denoted by “O”

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Politicians(S)

Liars(P)

The diagram for the O proposition indicates that there is at least one member of S that is not a member of P by

lacing X in the region S that is outside P.

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Standard-Form Categorical Propositions

Proposition Form Name Type ExampleAll S is P A Universal Affirmative All lawyers are

wealthy peopleNo S is P E Universal Negative No criminals are

good citizensSome S is P I Particular Affirmative Some chemicals

are poisonous Some S is not P O Particular Negative Some insects are

not pests

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Quantity, Quality and Distribution

• Quantity refers to the amount of members of the subject class that are used in the proposition.

• If the proposition refers to all members of the subject class, it is universal.

• If the proposition does not employ all members of the subject class, it is particular. For instance, an I-proposition ("Some S are P") is particular since it only refers to some of the members of the subject class.

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Contd….• Quality refers to whether the proposition affirms or denies

the inclusion of a subject within the class of the predicate.

• The two possible qualities are called affirmative and negative.

• For instance, an A-proposition ("All S are P") is affirmative since it states that the subject is contained within the predicate.

• On the other hand, an O-proposition ("Some S are not P") is negative since it excludes the subject from the predicate.

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All Humans Are Mammals

Quantifier Subject Copula Predicate

Distribution• Distribution is an attribute of the terms (subject and predicate)

of propositions. • A term is said to be distributed, if the proposition makes an

assertion about every member of the class denoted by the term; otherwise, it is undistributed.

• In other words, a term is distributed, if and only if the statement assigns (or distributes) an attribute to every member of the class denoted by the term.

• Thus, if a statement asserts something about every member of the S class, then S is distributed; otherwise S and P are undistributed.

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Sentence Standard Form

Attribute Distribution

All apples are delicious A All S is P Universal affirmative S only

No apples are delicious E No S is P Universal negative S and P

Some apples are delicious I Some S is P Particular

affirmative Neither

Some apples are not delicious O Some S is not P Particular

negative P only

The Square of Opposition

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Contradictories• (A and O; E and I): have opposite truth values.• The propositions on the diagonals of the square are

CONTRADICTORIES: they DENY each other TOTALLY.• BOTH cannot be true at the same time• BOTH cannot be false at the same time.• Two propositions are contradictory if one is the denial or negation of

the other: that is, they cannot both be true and cannot both be false.

• A and O propositions are contradictory, as are I and E... one of the pair MUST be true and the other MUST be false. If the statement "All S are P",(A) is true, then the statement "some S are not P", (O) must be false. Example: If "All dogs are animals" is true, then "Some dogs are not animals" must be false.

Contraries• “A” statements and “E” statements.• Both CANNOT be true at the same time. But could both be false.• If one is TRUE, the other is FALSE.• If one is FALSE, the other MAY be true or false,• THEREFORE, if one is FALSE, the other is UNKOWN.• Two propositions are contrary if they cannot both be true but they

might both be false. A and E are contrary. It can't be that "all dogs are animals" and "no dogs are animals" at the same time, but it may be that only some dogs are animals, making both Universal statements false.

Subalternates• “A” and “I”, and “E” and “O” statements.• the superaltern (the universal) implies the subaltern (the

particular)• If the UNIVERSAL is true, the PARTICULAR is true.• If the PARTICULAR is false, the UNIVERSAL is false.• A and I propositions are related by subalteration. Subalterns are a

different sort of 'opposition', because a subalternation does not imply a contradiction at all. The truth of I may be inferred by the truth of A. If "All S are P" is true, then we can be certain that "Some S are P" must be true. The reverse, from I to A, is invalid. The same goes for the negative propositions E and O. One can infer the truth of O from the validity of E, but not vice versa

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Subcontraries• Two propositions are subcontraries if they cannot

both be false, although they both may be true.• I (Some S is P) and O (Some S is not P) propositions-

which are both particular but differ in quality-are subcontraries unless one is necessarily false.

For example: • Some dogs are cocker spaniels.• Some dogs are not cocker spaniels.

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Immediate Inferences • Conversion• Contraposition • Obversion

• These operations give us rules to create logically equivalent claims and determine in some cases if two categorical claims are logically equivalent.

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Conversion

• The converse of a claim is created by switching positions of subject and predicate terms.

• E: No S are P = No P are S

• I: Some S are P = Some P are S

ConvErsIon - Valid for E & I

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Contd….• E: No metal is house = No house is metal

• I: Some country is pop = Some pop is country

• Avoid the common mistake of converting an A-claim!

The fact that all H are W does not imply that all W must be H. For example, it is true that all employees are human, but it is not true that all humans are employees.

• And avoid the similar mistake of converting an O-claim!

If it is true that some managers are not leaders, that does not imply that some leaders are not managers.

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Contraposition• The contrapositive of a claim is created by:

(1) switching positions of subject and predicate terms, and; (2) replacing both terms with their complements

• A: All S are P = All non-P are non-S• O: Some S are not P = Some non-P are not non-S

• ContrApOsition - Valid for A & O

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Obversion• The obverse of a claim is created by:

(1) changing affirmative to negative or vice-versa, and; (2) replacing predicate term with its complement

• A: All S are P = No S are non-P E: No S are P = All S are non-P

I: Some S are P = Some S are not non-P O: Some S are not P = Some S are non-P

• Obversion - Valid for ALL

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Thank you