PHIL222 - Formal Logic

PHIL222 - Logic Overview

Week 1

Introduction to Logic

  • Definition of Logic:

    • Logic is the science of truth and reasoning.

  • Purpose of Learning Logic:

    • Enhances reasoning skills related to:

      • Financial matters (bills)

      • Philosophical inquiries (meaning of life, fairness, etc.)

      • Political views and opinions

      • Understanding numerical data

      • Analyzing cause and effect relationships

      • Engaging with fictional scenarios

      • Anticipating future events.

  • Outcome:

    • Mastery in logical reasoning aids in answering questions, settling disagreements, defending views, making discoveries, revising beliefs, and planning for the future.

Propositions

  • Definition:

    • A proposition is a declarative sentence that can be classified as true or false.

    • Propositions describe states of affairs.

  • Characteristics:

    • The hallmark is that they can only be true or false.

    • Not every sentence qualifies; questions, commands, and exclamations are excluded from being propositions.

Examples of Propositions

  • True and False Propositions:

    • "The University of Otago is located on Mars."

    • "Wellington is the capital of New Zealand."

  • Non-Propositions:

    • "What time is it?" (Question)

    • "Shut up." (Command)

    • "Ouch!" (Exclamation)

    • These expressions lack truth values.

Truth and Falsity

  • Basic Principles:

    • Non-Contradiction (Exclusion Principle): A proposition cannot be both true and false.

    • Excluded Middle (Exhaustion Principle): A proposition is either true or false, with no middle ground.

  • Philosophical Debate:

    • Some philosophers argue these principles can be bypassed, suggesting logic can function without them.

Arguments

  • Aristotelian Definition of Reasoning:

    • Reasoning is structured as an argument, leading to conclusions from premises.

  • Components of Arguments:

    • Premises: Propositions reflecting facts or assumptions.

    • Conclusion: The proposition derived from premises.

Examples of Arguments

  • Example 1:

    • Claim: "You’re fired."

    • Premise 1: If you come late again, you’ll be fired.

    • Premise 2: You were late today.

    • Conclusion: Therefore, you’re fired.

  • Importance: Structure clarifies if the conclusion follows logically from the premises.

Logical Consequence

  • Criteria for a Good Argument:

    • An argument is considered good if the conclusion logically follows from the premises as defined by Aristotle.

Validity

  • Definition:

    • An argument is valid if the truth of its premises guarantees the truth of its conclusion.

    • Valid if there is no possible scenario where all premises are true, but the conclusion is false.

    • This property is termed Necessary Truth-Preservation (NTP).

Examples of Valid and Invalid Arguments

  • Valid Argument:

    • If a number is even, it is divisible by 2.

    • Number x is not divisible by 2.

    • Conclusion: Thus, x is not an even number.

  • Invalid Argument:

    • If McDonald’s is open, then I’ll eat a burger.

    • McDonald’s is not open.

    • Conclusion: Therefore, I will not eat a burger.

    • This conclusion could be false if the individual eats at another place.

Validity and Soundness

  • Valid Not Sound Example:

    • If you take PHIL222, you'll learn to bake cakes (which is false).

  • Sound Argument:

    • Dunedin is in New Zealand.

    • New Zealand is in Oceania.

    • Conclusion: Therefore, Dunedin is in Oceania.

    • Since all premises are true, the argument is sound.

Connectives in Propositions

  • Types of Propositions:

    • Basic Propositions: Simple statements with no internal structure.

    • Compound Propositions: Made up from other propositions using connectives.

  • Key Connectives:

    • Negation

    • Conjunction

    • Disjunction

    • Conditional

    • Biconditional

Connective Definitions

  • Negation:

    • The opposite of the original proposition.

    • Example: "There is not an elephant in the room" is the negation of "There is an elephant in the room."

  • Conjunction:

    • States both propositions are true.

    • Example: "We watched a movie and ate popcorn."

  • Disjunction:

    • At least one proposition is true.

    • Example: "Peter is cooking or looking after his kids."

    • Note: Inclusive, allowing both to be true.

  • Conditional:

    • "If... then..." structure indicating a condition.

    • Example: "If it’s raining, then we stay at home."

  • Biconditional:

    • States equivalency between two propositions.

    • Example: "An animal is oviparous if and only if it is born from an egg."

    • True when both propositions share the same truth value.