Learn & Review: INTRODUCTION to PROPOSITIONAL LOGIC

Introduction to Propositional Logic and English Sentence Translation

This summary outlines the fundamental concepts of propositional logic, focusing on statements, propositions, and their translation into and from English sentences using logical connectives.

1. Statements and Propositions

• Statement: A declarative sentence that can be definitively classified as either true or false.
  • True is represented as 1.
  • False is represented as 0.
  • This concept is closely related to Boolean logic.
• Examples of Statements:
  • "Milk is white." (Generally true)
  • "The cardinality of the empty set is equal to zero." (True)
  • "Humans are just fish with legs." (False)
• Non-Statements: Sentences that cannot be true or false.
  • Questions: "Will you go to the store for me?"
  • Imperatives (Commands): "Kick me."
• Propositions: Closely related to statements, propositions capture the general idea of a statement. For the purpose of this course, the distinction is not critical.
  • Propositions are typically denoted by capital letters (e.g., P, Q, R) when they have specific English translations.
  • Lowercase letters (e.g., p, q, r) are used for general propositions without specific meanings, often used in proofs.

2. Syntax of Propositional Logic: Well-Formed Formulas (WFFs)

A well-formed formula (WFF), also known as a "woof," represents a syntactically correct expression in propositional logic.

• Basic WFFs: A single statement or proposition on its own is a WFF.
• Forming Complex WFFs using Connectives:
  • Negation (NOT): If 'p' is a WFF, then ¬p (or "not p") is a WFF.
    • Symbol: ¬ (or a half-box shape)
  • Conjunction (AND): If 'p' and 'q' are WFFs, then p ∧ q (or "p and q") is a WFF.
    • Symbol: ∧ (also called a "carrot")
  • Disjunction (OR): If 'p' and 'q' are WFFs, then p ∨ q (or "p or q") is a WFF.
    • Symbol: ∨ (also called a "wedge")
  • Implication (IF...THEN): If 'p' and 'q' are WFFs, then p → q (or "if p then q") is a WFF.
    • Symbol: →

3. Translating Between English and Well-Formed Formulas

3.1. Translating Well-Formed Formulas into English

• Process:

  1. Identify the propositions and their English translations from a given key.
  2. Identify the logical connectives used in the WFF.
  3. Read the WFF aloud, substituting the English phrases for the propositions and using the English equivalents for the connectives.
  4. Pay attention to the structure and grouping implied by the connectives.

• Example:

  • Key:
    • r: "I write an exam"
    • p: "I cheat"
    • q: "I will get caught"
    • s: "I will fail"
  • WFF: (r ∧ p) → (q ∧ s)
  • Translation: "If I write an exam and I cheat, then I will get caught and I will fail."

• Note: The resulting English sentence does not necessarily need to make logical sense; the focus is on accurate translation based on the provided key and WFF structure.

3.2. Translating English Sentences into Well-Formed Formulas

• Process:

  1. Identify the main logical structure of the English sentence (e.g., "if...then," "and," "not").
  2. Identify all the simple declarative statements within the sentence.
  3. Define propositions (using capital letters) for these simple statements. Crucially, define propositions in their affirmative form. Negations should be handled by the ¬ connective.
  4. Substitute the defined propositions and connectives back into the structure identified in step 1.
  5. Verify the translation by converting the resulting WFF back into English.

• Example:

  • English Sentence: "If James does not die, then Mary will not get any money and James' family will be happy."
  • Identification of Connectives: "if...then" (→), "not" (¬), "and" (∧).
  • Defining Propositions (Affirmative Form):
    • p: "James dies"
    • q: "Mary will get money"
    • r: "James' family will be happy"
  • Constructing the WFF:
    • "James does not die" translates to ¬p.
    • "Mary will not get any money" translates to ¬q.
    • "James' family will be happy" translates to r.
    • The sentence structure is: "If (¬p), then (¬q and r)".
    • WFF: ¬p → (¬q ∧ r)

• Key Reminders:

  • Always highlight connectives in the English sentence.
  • Ensure propositions are stated affirmatively before applying the negation connective.
  • Double-check translations by converting back to English.

The next steps involve understanding the truth conditions of these connectives through truth tables.