Propositional Logic: An Exhaustive Study Guide

Chapter 6: Propositional Logic

6.1 Symbols and Translation

• Learning propositional logic empowers individuals by using special symbols to simplify the expression of statements and arguments while exposing their underlying 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."

  • This argument, when translated into symbols, can be recognized as a valid form, demonstrating that validity in deductive arguments depends purely on their form.

• Ordinary language can obscure the form of arguments, necessitating the introduction of logical symbols and operators.

  • Earlier chapters introduced letters representing terms in syllogisms while reducing them to standard form.

  • In this chapter, the focus shifts to special symbols called operators (symbols connecting propositions) and connectives (symbols connecting or negating propositions).

• Understanding propositional logic requires distinguishing between simple statements and compound statements:

  • Simple Statement: A statement that 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.

  • Notation: Any uppercase letter can be chosen to represent each simple statement (e.g., F, J, P, B).

Operators and Their Functions

• Operator List and Functions:

  • Tilde (∼): Negation (not, it is not the case that)

  • Dot (·): Conjunction (and, also, moreover)

  • Wedge (∨): Disjunction (or, unless)

  • Horseshoe (⊃): Implication (if…then…, only if)

  • Triple Bar (≡): Equivalence (if and only if)

• Lowercase Letters: Reserved for use as statement variables in more complex expressions.

  • Example Translations:

  • T: It is not the case that T.

  • D and C.

  • Either P or E.

  • If N then F.

  • B if and only if R.

Types of Statements

• Negation (∼): Applies to negated propositions.

  • Examples:

  • "Rolex does not make computers" translates to ∼R. All negated statements must have the tilde (∼) preceding them.

• Conjunction (·): Used to express logical AND. The main operator in conjunctions is a dot.

  • Examples:

  • Tiffany sells jewelry, and Gucci sells cologne translates to T · G.

  • Synonymous expressions of conjunction:

    • "Tiffany sells jewelry, but Gucci sells cologne"

    • "Tiffany sells jewelry; however, Gucci sells cologne."

• Disjunction (∨): Represents logical OR, interpreted as inclusive.

  • Equivalence of Terms:

  • "You won’t graduate unless you pass freshman English" is equivalent to the disjunction "Either you pass freshman English or you won’t graduate."

• Conditional Statements (⊃): Phrases indicating implications such as "if…then…".

  • Example:

  • "If A, then B" translates to A ⊃ B.

  • If a condition specifies "only if", the structure flips:

    • "C only if H" translates to C ⊃ H.

• Biconditional (≡): Represents statements with logical if and only if, indicating both necessary and sufficient conditions.

  • Example:

  • "JFK tightens security if and only if O’Hare does" translates to J ≡ O.

• Necessary and Sufficient Conditions:

  • Definition: A sufficient condition is one where, if the condition occurs, the result follows. A necessary condition must occur for the result to happen.

  • Examples:

  • "Having the flu is a sufficient condition for feeling miserable".

  • "Having air to breathe is a necessary condition for survival".

Parentheses and Logical Structure

• To avoid ambiguity when translating statements involving more than two letters, use parentheses, brackets, or braces.

  • Examples:

  • The statement “Prozac relieves depression, and Allegra combats allergies, or Zocor lowers cholesterol” needs proper parentheses to clarify meaning in symbolic form.

  • Incorrect Without Parentheses:

  • The statement can be misinterpreted without proper structuring.

• Example for Clarity:

  • Original: "If Sanofi and Pfizer lower prices or Novartis downsizes, then Roche will expand production."

  • Use of parentheses helps indicate logical structure clearly.

Well-Formed Formulas (WFFs)

• Definition: A WFF is a syntactically correct arrangement of symbols in logic.

• Importance: Ensures expressions follow logical syntax rules, similar to grammatical structure in English.

• Examples of WFFs:

  • ∼A, A · B, A ∨ B

• Examples of Non-WFFs:

  • ∼, A · B ·

• Rules for Constructing WFFs:

  • Statements must be combined with logical operators.

  • A tilde cannot follow or precede another operator without a statement.

  • Parentheses, brackets, and braces must be used to indicate the operator’s scope clearly.