Propositional Logic: Definitions, Connectives, and Argument Analysis

Administrative Reminders and Logistics

• Tutorials: Students must sign up for a tutorial immediately if they have not already done so. Tutorials begin next week, not the current week.
• Course Assessments: Despite any potential discrepancies in the paper outline, all tests are scheduled for Wednesdays. These are short tests conducted within the lecture theatre. Specific dates for these tests are available on Moodle.
• Lecture Recordings: Students should verify if the recordings from previous sessions (e.g., yesterday's lecture) are working correctly. Technical issues with microphones in specific lecture theatres may occur, but recordings should generally be functional.

Introduction to Propositions

• Definition of a Proposition: A proposition is a statement that has a definite truth value; it must be either true or false.
• Sentences that are NOT Propositions:
  • Questions: Any sentence ending in a question mark is not a proposition because it does not assert a truth value.
    • Example: "Is it cold today?"
  • Orders/Imperatives: Sentences that use imperative verbs or give commands are not propositions.
    • Example: "Put your hands up."
  • Incomplete Sentences: A phrase lacking a verb or proper structure is not a proposition.
    • Example: "Blank space" is not a proposition. However, modifying it to "The space is blank" creates a proposition.
• Ambiguity in Language: Natural language can often be open to interpretation.
  • Example: "Today was a fairy tale." This is generally considered a proposition as it evaluates to a truth value.
  • Example: "You need to calm down." This could be argued as a proposition (it is either true or false that you need to calm down), but in practice, it often functions as an order ("Calm down"), which is a non-proposition.
  • Example: "It could be over now." This is a question, but changing it to "It is over now" turns it into a proposition.

Logical Connectives: Negation, Conjunction, and Disjunction

Negation ($\neg$)

• Function: Negation takes a statement and reverses its truth value.
• Safest Method for Negation: To negate a statement, explicitly add the word "NOT" into the sentence.
  • Example: If $P$ is "The fridge is empty," the negation $\neg P$ is "The fridge is not empty."
• Negation vs. Opposite: A negation is not always the semantic opposite.
  • Case Study: If the statement is "My car is white," the negation is "My car is not white" (or "My car is a color other than white"). Using an opposite like "My car is black" is incorrect because if the car is blue, both "white" and "black" are false, whereas a true negation must be true whenever the original statement is false.

Conjunction ($\land$)

• The AND Rule: The conjunction of two propositions is true if and only if both propositions are true.
• Truth Table for Conjunction ($P \land Q$):
  • $True \land True = True$
  • $True \land False = False$
  • $False \land True = False$
  • $False \land False = False$
• Correction Note: In a previous lecture, there was an error stating $False \land True = True$. This is incorrect; it always evaluates to $False$.

Disjunction (OR)

In English, "OR" is ambiguous, leading to two types of logic:

• Inclusive OR ($\lor$): At least one statement is true (could be one, or both).
  • Rule: In mathematics and logic, the word "or" is assumed to be inclusive by default unless specified otherwise.
  • Example: "If your mother or father has high blood pressure, you will too." This is inclusive because the outcome still applies if both parents have high blood pressure.
  • Clarifying Phrase: "At least one of $P$ and $Q$ is true."
• Exclusive OR ($\oplus$ or XOR): Exactly one statement is true, but not both.
  • Clarifying Phrase: "Either $P$ or $Q$ is true, but not both of them."
  • Example: "Either you know it or you don't." In practice, you cannot simultaneously know and not know something ($A \land \neg A$), so the distinction doesn't change the truth value here.
  • Example: "Coffee or tea?" or an airplane meal choice "Chicken or beef?" Usually, these imply you cannot have both, making them exclusive.
• Contextual Interpretations:
  • Help from a lecturer or tutor: Viewed as inclusive because getting help from one does not preclude getting help from the other.
  • "And/or": A phrase used specifically to emphasize the inclusive nature of a disjunction.

Logical Implication and Equivalence

Implication ($\rightarrow$ or $\implies$)

• The "If-Then" Structure: Represented as $P \implies Q$, where $P$ is the premise and $Q$ is the conclusion.
• Logic of the Truth Table:
  • If the premise ($P$) is true and the outcome ($Q$) is false, the implication is false ($True \implies False = False$).
  • If the premise ($P$) is false, the whole statement is considered true by default (vacuously true), regardless of the outcome ($Q$).
• Examples:
  • Mathematics: "If $x = 1$, then $x + 5 = 6$." This statement remains true whether or not $x$ actually equals $1$ at this moment. We are only asserting the relationship between the premise and the conclusion.
  • Weather: "If it is raining, then it is cloudy." This is a fact. If it is not raining, this fact is not invalidated; therefore, the statement remains true.
  • Absurdity: "If the moon is made of green cheese, then the lecturer is a pink elephant." Because the premise is false, the entire statement is logically true.
  • Political Promises: "If I win the election, I will lower taxes." If the politician loses, they have not broken their promise; hence, they haven't lied.

Equivalence / Biconditional ($\leftrightarrow$ or $\iff$)

• The "If and Only If" Structure: Represented as $P \iff Q$.
• Function: This is only true when both $P$ and $Q$ share the same truth value (both true or both false).
• Directionality Caution: Implication usually works in only one direction.
  • Example: "If it's raining, then it's cloudy" ($P \implies Q$) is true. However, the converse "If it's cloudy, then it's raining" ($Q \implies P$) is not necessarily true because it can be cloudy without rain.
  • Example: "If I break my arm, I am in pain." Being in pain ($Q$) does not automatically mean one has a broken arm ($P$).

Translating Arguments into Propositional Logic

To analyze the logic of an argument, sentences must be converted into variables that include a verb and possess clear truth values.

• Defining Variables (Example):
  • $P$: "I win the lottery."
  • $Q$: "I travel overseas."
  • $R$: "I buy a new car."
• Key Linguistic Connectives:
  • "But": Logically functions exactly like AND ($\land$).
  • "Therefore": Signifies a logical conclusion ($\implies$). Other synonyms include hence, in which case, so, thus, and consequently.
• Sentence Translation Exercise:
  1. Sentence 1: "If I win the lottery, I will travel overseas or buy a new car."
     • Translation: $P \implies (Q \oplus R)$
     • Note: Use brackets to show that the outcome of winning is the entire "either/or" scenario.
  2. Sentence 2: "I win the lottery but I do not travel overseas."
     • Translation: $P \land \neg Q$
  3. The Full Argument: "Therefore, I buy a new car."
     • The word "therefore" implies that everything stated previously leads to the final conclusion.
     • Logical Structure: $((P \implies (Q \oplus R)) \land (P \land \neg Q)) \implies R$