Logical Reasoning Part2: Arguments

Logical Reasoning

3.2 Arguments

Premises are a few statements that can justify a conclusion.

An argument is a process to make a conclusion based on a few given statements (premises).

Premise 1: ...

Premise 2: ...

Conclusion: ...

Deductive argument.

Deductive argument and Inductive argument

• Deduction : process of making a specific conclusion based on general premises. The conclusion is certainly TRUE if all premises are true.

• Induction : process of making a general conclusion based on specific premises. The conclusion might be false even though all premises are true.

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Deductive argument and Inductive argument

• Example: Premise 1: All pupils of class 4A are fluent in English. Premise 2: Aims is a pupil of class 4A. Conclusion: Aims is fluent in English. (Deductive argument)

• Example: 1² + 3² = 4, 2² + 3 = 7, 3² + 3 = 12, ... Thus, the number sequence 4, 7, 12, ...can be written as n² + 3 where n = 1, 2, 3, ... (Inductive argument)

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Multiple Choice

Determine whether the following arguments is a deductive argument or an inductive argument.

All integers with unit digit 5 are divisible by 5.

Unit digit of 1245 is 5.

Hence, 1245 is divisible by 5.

• deductive argument

• inductive argument

Multiple Choice

Determine whether the following is a deductive argument or inductive argument.

(a¹)² = a², (a²)³ = a^6, (a³)⁴ = a^{12. Thus,} (a^m)ⁿ = a^mxn

• deductive argument

• inductive argument

Determining whether a deductive argument is sound

• A deductive argument is valid if its conclusion is certain to follow from the premises.

• Example: Premise 1: All graduate teachers have a degree. Premise 2: Kevin is a graduate teacher. Conclusion: Kevin has a degree.

• From the premises, it is certain that Kevin has a degree. Thus, deductive argument is valid.

• Premise 1: All graduate teachers have a degree. Premise 2: Kevin has a degree. Conclusion: Kevin is a graduate teacher.

• From the premises, it is not certain that Kevin is a graduate teacher. Kevin might be a doctor, an accountant and so on.

Multiple Choice

Determine whether the following argument is valid , sound.

Premise 1: if x is an odd number, then 2x is an even number.

Premise 2: x is not an odd number.

Conclusion: 2x is not an even number.

• valid and sound

• valid and not sound

• not valid and not sound

Multiple Choice

Determine whether the following argument is valid, sound.

Premise 1: If 4 is a factor of N, then 2 is a factor of N

Premise 2: 4 is a factor of 32

Conclusion: 2 is a factor of 32

• valid and sound

• valid and not sound

• not valid and not sound

Forming a valid deductive argument for a situation.

• A true conclusion can be made for a valid deductive argument , based on the true two given premises.

• Premise 1: All factors of 9 are odd numbers. Premise 2: x is a factor of 9. Conclusion: x is an odd number.

Strength of an Inductive Argument

• An inductive argument is strong when there is a high probability its conclusion is true.

• A strong inductive argument has firm premises and high reliability level such that a logic and convincing conclusion can be made and the argument is cogent.

• A weak inductive argument has doubtful premises causing the conclusion made to be not logical and not convincing. The argument is not cogent.

• In general, an inductive argument (a) can be strong but not cogent (b) not cogent if there is a false premise or conclusion.

Multiple Select

Determine whether the following argument is strong or weak. Hence determine and justify whether the argument is cogent or not cogent.

Premise 1: 6 is divisible by 3.

Premise 2: 12 is divisible by 3.

Premise 3: 18 is divisible by 3.

Conclusion: All multiples of 6 are divisible by 3.

• The argument is strong

• The argument is weak

• The argument is cogent because all the premises are true and the conclusion is true.

• The argument is not cogent because the conclusion is false.

Multiple Select

Determine whether the following argument is strong or weak. Hence, determine if the argument is cogent or not cogent.

Premise 1: 24 is a multiple of 12.

Premise 2: 48 is a multiple of 12.

Premise 3: 64 is a multiple of 12.

Conclusion: All multiples of 12 are even numbers.

• The argument is strong.

• The arguement is weak.

• The argument is cogent.

• The argument is not cogent.

Multiple Select

Determine whether the following argument is strong or weak. Hence determine whether the argument is cogent or not cogent.

Premise 1: 2 is a prime number.

Premise 2: 3 is a prime number.

Premise 3: 5 is a prime number.

Conclusion: All prime numbers are odd numbers.

• The argument is strong.

• The argument is weak.

• The argument is cogent.

• The argument is not cogent.

Forming a Strong Inductive Argument for a situation.

• A strong and cogent inductive argument depends on its premises and conclusion which are true.

• The given premises provide the proof and support for the conclusion made.

Multiple Choice

The number pattern 4, 7, 10, 13, ... can be concluded generally by

• 4 + 3n, n = 0, 1, 2, 3, 4, ...

• 4(1 + 3n), n = 0, 1, 2, 3, 4, ...

• 4n + 3, n = 0, 1, 2, 3, 4, ...

• 4n - 3, n = 0, 1, 2, 3, 4, ...