Deductive Reasoning and Evaluation

Goal of Reasoning

The goal of reasoning is to preserve truth. Ideally, if the premises are true, they offer complete or partial support for the truth of the conclusion.

Types of Arguments

  • Deduction: Necessary reasoning. Conclusion must be true if premises are true. (Non-ampliative conclusion.)
  • Induction: Probabilistic reasoning. Conclusion is likely true based on premises. (Ampliative conclusion.)

Ampliation occurs when the conclusion contains more information than the premises.

Examples

  • Deductive Example:
    P1) No plants are tacos.
    P2) Some green things are plants.
    C) Some green things are not tacos.
  • Inductive Example:
    P1) Fridays are often the busiest day.
    P2) Today is a Friday.
    C) Today will likely be busy.

Evaluating Deductive Arguments

  • Valid: It is not possible for premises to be true while the conclusion is false.
  • Invalid: It is possible for premises to be true while the conclusion is false.
  • Sound: A valid argument with all true premises.
  • Unsound: Either invalid, or has at least one false premise.

Assessing Deductive Arguments

  • Valid deduction with true premises → Sound
  • Valid deduction with false premises → Unsound
  • Invalid deduction → Unsound

Types of Deductive Arguments

1. Argument by Definition

Based on the definitions agreed upon for a term.
Example: Lily is a philologist. Therefore, she studies the development and history of texts.

2. Logical Terms and Reasoning

  • Categorical Reasoning: Based on group inclusion/exclusion. (Indicators: \"all\", \"no\", \"some\", \"some are not\", \"only\", \"except\", \"not any\")
    Example: All humans are mammals. All mammals are vertebrates. Therefore, all humans are vertebrates.
  • Disjunctive Reasoning: Based on \"either-or\" logic.
    Example: Either you eat pie or cake. You don't eat cake. Therefore, you eat pie.
  • Hypothetical Reasoning: Based on \"if-then\" conditional structures.
    Example: If there are no clouds, it won’t rain. There are no clouds. Therefore, it won’t rain.

Types of Conditions

  • Necessary Condition: A requirement that must be satisfied for a statement to be true.
    Example: Passing a driver’s test is necessary to get a driver's license.
  • Sufficient Condition: A guarantee for a statement's truth.
    Example: Being a dog is sufficient for being an animal.

Active Learning Prompt: Identify Necessary vs Sufficient

  • Being 18 years old is a ______ condition for being older than 16 years old.
  • Being at least 35 years old is a ______ condition for being President of the U.S.
  • Being an opossum is a ______ condition for being a marsupial.
  • Being a mythical creature is a ______ condition for being a unicorn.

3. Mathematical Reasoning

Based on relations in math, algebra, arithmetic, or geometry.
Example: Dividing 24 students evenly into 3 classrooms results in 8 students per room.

4. Syllogisms

Two premises and one conclusion structured logically.

Active Learning Prompts: Identify Types of Deductive Reasoning

  • If I wish to withdraw from a class, then I’ll complete a Registration Form. I do not complete the form. Therefore, I do not wish to withdraw.
  • Some cats are not gray. All gray things are colored. Therefore, some cats are not colored.
  • A circle is not the same type of shape as a square. A circle is not a polygon; a square is a polygon.
  • Bernie or Amir cook tonight. Bernie does not cook. Therefore, Amir cooks tonight.
  • All garlic is tasty. Some plants are not tasty. Therefore, not all plants are garlic.
  • If feet had wings, pigs could fly. If pigs could fly, I’d own a farm called “Where Pigs Fly.” Thus, if feet had wings, I’d own a farm called “Where Pigs Fly.”
  • If it snowed, the school is closed. It snowed. Therefore, the school is closed.
  • Triangle A is congruent with Triangle B. Triangle A is isosceles. Therefore, Triangle B is isosceles.
  • Adele trespassed. Therefore, she intentionally entered someone’s land without consent or necessity.

Counterexample Method

Test if it's possible for premises to be true and conclusion false.
Example: P: All cities with fewer than 1/4 million inhabitants are small. C: All cities are small.
Counterexample: New York City.

Active Learning Prompt: Identify the Correct Counterexample

P1) If it snowed, the school is closed.
P2) The school is closed.
C) Therefore, it snowed.

  • a) It did not snow but school is closed for the holidays.
  • b) It did not snow but the school is closed because of flooding.
  • c) It did snow but the school is closed because of flooding.
  • d) A and B
  • e) A, B, and C

Counterargument Method

Substitute new terms while preserving logical structure to expose invalidity.
Example:

  • Original: Some fruit are green. Some fruit are apples. Therefore, some fruit are green and apples.
  • Counter: Some fruit are bananas. Some fruit are pears. Therefore, some fruit are bananas and pears.

Active Reflection:

Create your own counterargument for a similar logical form.