Chapter 6: Propositional Logic

Chapter 6: Propositional Logic

6.1 Symbols and Translation

Overview

• Learning propositional logic is empowering.

• Introduces special symbols to simplify statements and arguments and expose their structure.

• Example Argument:

  • “If the Greenland ice sheets are melting, then climate change is a reality. Therefore, climate change is a reality since the Greenland ice sheets are melting.”

  • Symbolic translation recognizes it as having a valid form, indicating validity.

Importance of Form

• Validity of a deductive argument is purely a function of its form.

  • Knowing the form allows for immediate assessment of validity.

• Ordinary linguistic usage can obscure this form, necessitating the use of logical symbols.

• Logical procedures introduced in Chapter 5 simplified syllogisms to standard form.

Symbolic Representation

• Propositional Logic defines fundamental elements as whole statements (propositions); noted as different from terms in previous chapters.

• Statements represented by letters; letters combined using operators to form complex symbolic representations.

Distinction Between Statement Types

Simple Statements

• Definition: A simple statement does not contain any other statement as a component.

• Examples:

  • Fast foods tend to be unhealthy.

  • James Joyce wrote Ulysses.

  • Parakeets are colorful birds.

  • The bluefin tuna is threatened with extinction.

• Representation:

  • Any convenient uppercase letter can represent a simple statement.

  • Example Assignments:

    • F: Fast foods unhealthy

    • J: Joyce wrote Ulysses

    • P: Parakeets colorful

    • B: Bluefin tuna threatened

Compound Statements

• Definition: A compound statement contains at least one simple statement as a component.

• Operators/Connectives:

  • Used to translate and connect statements:

    • Tilde (∼): Negation

    • Dot (·): Conjunction

    • Wedge (∨): Disjunction

    • Horseshoe (⊃): Implication

    • Triple Bar (≡): Equivalence

Logical Operators and Their Translations

Negation (∼)

• Translation Examples:

  • It is not the case that the Taliban supports educating women: ∼T

  • Equivalent Expressions: All statements negated are written with the tilde immediately preceding the proposition.

Conjunction (·)

• Translations:

  • “Tiffany sells jewelry, and Gucci sells cologne.”

    • Symbolic representation: T·G

  • Other words like: also, moreover, but may also translate to conjunction.

Disjunction (∨)

• Translations:

  • “Either Alta allows snowboards or Telluride does.”

    • Symbolic representation: A ∨ T

  • “You won’t graduate unless you pass freshman English.”

    • Symbolically: E ∨ ¬G

Implication (⊃)

• Translations:

  • “If Purdue raises tuition, then so does Notre Dame.”

    • Symbolic: P ⊃ N

  • Note: Confusion between sufficient and necessary conditions should be avoided.

    • Sufficient condition A implies B: A ⊃ B

    • Necessary condition B cannot happen without A: B ⊃ A

Equivalence (≡)

• Expression:

  • “A is a sufficient and necessary condition for B.”

    • Symbol: A ≡ B

  • Example Analysis: Statements can be treated as logically equivalent while defining order in biconditional representation.

Proper Expression of Logic

Well-formed Formulas (WFFs)

• Definition: A syntactically correct arrangement of symbols.

  • Examples include correct structures like ∼P, P·Q, and incorrect structures like P·QP.

Key Rules for WFFs

• When combining statements, operators must occur between them.

• Tilde cannot immediately follow a statement but may precede non-operator statements.

• Parentheses, brackets, or braces must be used for clarity and to prevent ambiguity.

Conclusion

• Understanding this structure allows for clearer communication and analysis of logical arguments using propositional logic principles.

• Proper representation and translation of statements is crucial for logical validity assessment.

• The next sections will delve deeper into specific translations and practical applications of these logical forms.