3.2 Arguments
An argument is a process of making a conclusion based on a set of statements known as premises. In mathematical and logical reasoning, arguments are classified into deductive arguments and inductive arguments. Understanding the structure of arguments allows students to determine whether a deductive argument is valid and sound, or whether an inductive argument is strong and cogent, and to formulate valid arguments or general rules to solve problems.
Learning Objectives
- Explain the meaning of an argument and differentiate between deductive and inductive arguments.
- Determine and justify the validity of a deductive argument and hence determine whether the valid argument is sound.
- Form a valid deductive argument for a situation.
- Determine and justify the strength of an inductive argument and hence determine whether the strong argument is cogent.
- Form a strong inductive argument for a certain situation.
- Solve problems involving logical reasoning.
Classification and Evaluation of Arguments
Compare deductive and inductive arguments and how each is evaluated.
Deductive Argument
Inductive Argument
Brief Explanation
What is an Argument?
Argument is a process of making a conclusion based on a set of statements known as premises. A simple argument usually contains at least two premises and one conclusion.
Types of Statements
A specific statement refers to a particular case, while a general statement describes a concept or applies universally.
| Valid Form | Premise 1 | Premise 2 | Conclusion |
|---|---|---|---|
| Form I | All \(A\) are \(B\). | \(C\) is \(A\). | \(C\) is \(B\). |
| Form II | If \(p\), then \(q\). | \(p\) is true. | \(q\) is true. |
| Form III | If \(p\), then \(q\). | Not \(q\) is true. | Not \(p\) is true. |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Determine whether the following argument is deductive or inductive:
Premise 1: The sum of exterior angles of a triangle is \(360^\circ\).
Premise 2: The sum of exterior angles of a quadrilateral is \(360^\circ\).
Conclusion: The sum of exterior angles of each polygon is \(360^\circ\).
State the validity and soundness of the following argument:
Premise 1: All basketballs are spherical in shape.
Premise 2: The Earth is spherical in shape.
Conclusion: The Earth is a basketball.
Determine whether the following inductive argument is strong or weak, and whether it is cogent:
Premise 1: \(11 \times 5 = 55\)
Premise 2: \(12 \times 5 = 60\)
Conclusion: All multiples of \(5\) end with digit \(0\) or \(5\).
Common Mistakes
- Confusing validity with soundness: assuming an argument is invalid simply because one of its premises is false in real life. Validity depends solely on structure/form; soundness requires both validity and true premises.
- Confusing Form II and affirming the consequent: thinking that “If \(p\), then \(q\); \(q\) is true; therefore \(p\) is true” is valid. This is invalid.
- Confusing Form III and denying the antecedent: thinking that “If \(p\), then \(q\); not \(p\) is true; therefore not \(q\) is true” is valid. This is invalid.
- Thinking that adding more premises automatically makes an inductive argument strong. The strength depends on the truth value and probability of the conclusion, not just quantity of premises.
Misconceptions
A valid argument must always have true facts.
An argument can be completely valid in form even if its premises are completely false. For example, “All actresses are good at dancing; Jasmine is an actress; therefore Jasmine is good at dancing” is valid Form I, but unsound because Premise 1 is false.
Deductive arguments and inductive arguments mean the same thing as long as they prove something.
Deductive moves from general premises to a specific conclusion. Inductive moves from specific premises to a general conclusion.
If an inductive argument is strong, it is automatically cogent.
A strong inductive argument is only cogent if all its premises are factually true. If any premise is false, it is strong but not cogent.
Summary
- Argument: A set of statements comprising premises and a conclusion.
- Deductive Argument: General premise(s) \(\rightarrow\) Specific conclusion. Assessed by Validity (conforms to Form I, Form II, or Form III) and Soundness (valid AND all premises are true).
- Inductive Argument: Specific premise(s) \(\rightarrow\) General conclusion. Assessed by Strength (conclusion is true / highly probable) and Cogency (strong AND all premises are true).
- Form I: All \(A\) are \(B\); \(C\) is \(A\); Therefore, \(C\) is \(B\).
- Form II: If \(p\), then \(q\); \(p\) is true; Therefore, \(q\) is true.
- Form III: If \(p\), then \(q\); Not \(q\) is true; Therefore, Not \(p\) is true.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Consider the following argument:
Premise 1: If \(x > 6\), then \(x > 4\).
Premise 2: \(x > 6\).
Conclusion: \(x > 4\).
Which form of valid deductive argument does this represent?
2 An inductive argument has premises that are all true, but its conclusion is false. How is this argument classified?
B. Fill in the Blanks
3 A deductive argument is said to be when it is valid and all its premises and conclusion are true.
4 An argument that makes a general conclusion based on specific premises is called an argument.
C. Drag and Drop: Match the Structure to Its Classification
Drag each description to the correct term. On a phone, tap the answer choice first, then tap the matching box.
