Chapter 1: Thinking Critically

Assignment #4 – Unit 1D

Analyzing Arguments

Pages 53 – 70

  1. Summarize the differences between deductive and inductive arguments. Give an example of each type.
    An Inductive argument makes a case for a general conclusion from more specific premises.

Example:

Premise: Birds fly up into the air but eventually come back down.

Premise: People who jump into the air fall back down.

Premise: Rocks thrown into the air come back down.

Premise: Balls thrown into the air come back down.

Conclusion: What goes up must come down. (This is formed by generalizing from more specific premises.)

A deductive argument makes a case for a specific conclusion from more general premises.
Example:

Premise: All politicians are married.

Premise: Senator Harris is a politician.

Conclusion: Senator Harris is married. (We deduce a specific conclusion from more general premises).

  1. Briefly explain the idea of strength and how it applies to inductive arguments. Can an inductive argument prove its conclusion true? Can an inductive argument be valid? Can it be sound?
    We evaluate an inductive argument in terms of its strength.
    A strong argument makes the case for its conclusion seem quite convincing, even though it does not prove the conclusion. A weak argument is one in which the premises do not seem to lend much support to the conclusion. Note that evaluating strength involves personal judgment. An Argument that one person finds strong might appear weak to someone else. Note also that the strength of an inductive argument is not necessarily related to the truth of its conclusion.
    An inductive argument cannot prove its conclusion true, so it can be evaluated only in terms of its strength. An argument is strong if it makes a compelling case for its conclusion. It is weak if its conclusion is not well supported by its premises.
  1. Briefly explain the ideas of validity and soundness and how they apply to deductive arguments. Can a valid deductive argument be unsound? Can a sound deductive argument be invalid? Explain.
    The first questions above deals with the validity of the argument. A deductive argument is valid if its conclusion follows necessarily from its premises. Note that validity is concerned only with the logical structure of the argument. Validity involves no personal judgment and has nothing to do with the truth value of the premises or conclusions.
    If a deductive argument is valid and its premises are true, then we say that the argument is sound. Soundness represents the highest test of reliability of a deductive argument because, at least in principle, a sound argument proves its conclusion true beyond all doubt.
  2. Describe the procedure used to test the validity of a deductive argument with a Venn Diagram.
    To test the validity of a deductive argument with a Venn diagram:
    - Draw a Venn diagram that represents all the information contained in the premises.
    - Check to see whether the Venn diagram also contains the conclusion. If it does, then the argument is valid. Otherwise, the argument is not valid.
  3. Create your own example of each of the four basic conditional arguments. Then explain why your argument is valid or invalid.

Four Basic Conditional Arguments
Affirming the Hypothesis / Affirming the Conclusion / Denying the Hypothesis / Denying the Conclusion
Structure / If p, then q
p is true / If p, then q
q is true / If p, then q.
p is not true / If p, then q
q is not true
Validity / q is true
Valid / p is true
Invalid / q is not true
Invalid / p is not true
Valid
  1. What is a chain of conditionals? Give an example of a valid argument made from such a chain.
    Premise: If p, then q
    Premise: If q, then r
    Conclusion: If p, then r
    If Maria Lopez is elected to the school board, then the school district will raise academic standards.
    If the school board raises academic standards, then my children will benefit.
    If Maria Lopez is elected to the school board, then my children will benefit.
  2. Can inductive logic be used to prove a mathematical theorem? Explain.
    Perhaps more than any other subject, mathematics relies on the idea of proof.
    A mathematical proof is a deductive argument that demonstrates the truth of a certain claim or theorem. A theorem is considered proven if it is supported by a valid and sound proof. Although mathematical proofs use deduction, theorems are often discovered by induction.
  3. How can inductive testing of a mathematical rule be useful? Give an example.

The principle of seeking inductive evidence can be very useful when you are having difficulty remembering whether a particular theorem or mathematical rule applies. It often helps to try a few test cases and see if the rule works. Although test cases can never constitute a proof, they often are enough to satisfy yourself of a rule’s truth. However, the rule cannot be true if even one test case fails.

Exercises in your textbook: pgs. 66 – 69 19 – 24 ; 25 – 32 ; 33 – 38 ;
39 – 46 ; 47 – 54 ; 55 – 58 ; 59 - 62