Propositional Logic: Translation, Truth Tables, and Logical Equivalence

Tutorial Enrollment:
- Every student must be enrolled in a tutorial to have their assignments marked.
- The self-enrollment period has ended; students not currently in a tutorial must see the instructor at the end of class for manual enrollment.
Upcoming Assessments:
- Assignment 1: This is due on Monday of next week.
- Quiz 1: This is also due on Monday of next week.
- Quiz Policy: Students are permitted as many attempts as they like for the quiz to encourage initial engagement and mastery.
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- Help Desk: Operates from 1:00 PM to 2:00 PM every weekday during teaching weeks.
- Consultation: The instructor maintains dedicated consultation hours.
- Q&A Forum: A new question and answer forum has been established for students to post questions, provide answers, and share useful resources like videos.

Lexical Variations of "If-Then":
- The standard logical "if-then" or implication relationship can be expressed in numerous ways in the English language.
- Examples of synonyms or markers for implication include:
  - "Thus"
  - "Therefore"
  - "Implies"
  - "So"
  - "Then"
- The instructor challenged students to identify unique or less common ways of expressing this logical connection beyond the standard terms.

The Problem: Analyze a complex argument to determine its logical validity using tools from week one.
Defining Components (Propositions): To translate a paragraph into propositional logic, identifying the basic true/false components is necessary.
- $P$ : Interest rates are high.
- $Q$ : The exchange rate is low.
- $R$ : House prices will rise.
Translating the Argument:
- Sentence 1: "If interest rates are not high, house prices will rise." translated as $\neg P \rightarrow R$ .
- Sentence 2: "If interest rates are high, the exchange rate is low." translated as $P \rightarrow Q$ .
- Logical Conjunction of Premises: The full stop between these sentences functions as a logical "and" ( $\land$ ), combining the premises.
- Conclusion: "If the exchange rate is not low, then house prices will rise." translated as $\neg Q \rightarrow R$ .
- The Complete Well-Formed Formula (WFF): The "therefore" indicates the conjunction of all previous premises implies the final conclusion.
  - $((\neg P \rightarrow R) \land (P \rightarrow Q)) \rightarrow (\neg Q \rightarrow R)$

Determining Table Size:
- The number of rows in a truth table is determined by the formula $2^{n}$ , where $n$ is the number of variables.
- For 3 variables ( $P, Q, R$ ), there are $2^3 = 8$ rows.
- For 4 variables, there are $2^4 = 16$ rows.
- The number of rows grows exponentially as variables increases.
Ordering Rows:
- Rows can be ordered by treating False as $0$ and True as $1$ , then listing them in order as if they were binary numbers.
The Implication Operator ( $\rightarrow$ ):
- The implication $A \rightarrow B$ is only False when the antecedent ( $A$ ) is True and the consequent ( $B$ ) is False.
- $T \rightarrow T$ is True.
- $T \rightarrow F$ is False.
- $F \rightarrow T$ is True.
- $F \rightarrow F$ is True.
Operator Precedence:
- The logical "and" ( $\land$ ) is considered stronger than the implication ( $\rightarrow$ ). Therefore, in the absence of explicit brackets, the AND operation is performed first.

Valid (Tautology): An argument where the final column of the truth table is True for every possible valuation.
- In the provided exercise, the formula $((\neg P \rightarrow R) \land (P \rightarrow Q)) \rightarrow (\neg Q \rightarrow R)$ resulted in a final column of all True values, making it a valid argument.
Unsatisfiable (Contradiction): An argument where the final column is False for every possible valuation.
Satisfiable: An argument that is True for at least one valuation (at least one line in the truth table).
Models:
- A "model" is a specific evaluation of inputs (e.g., $P = F, Q = T, R = F$ ) that results in the overall formula being True.
- In a tautology, every possible input is a model.

Definition: Two formulas are logically equivalent if they produce the exact same final column in a truth table.
Contraposition:
- The statement $P \rightarrow Q$ is logically equivalent to $\neg Q \rightarrow \neg P$ .
- Example: "If it is raining, then there are clouds in the sky" is logically equivalent to "If there are no clouds in the sky, then it is not raining."
Biconditional Statements ( $\leftrightarrow$ ):
- The symbol for "if and only if" signifies that two statements are equivalent.
- $P \leftrightarrow Q$ is logically equivalent to the conjunction of implications in both directions: $(P \rightarrow Q) \land (Q \rightarrow P)$ .
- In mathematics, proving a biconditional requires proving both directions.
Mathematical Reversibility:
- Not all mathematical arguments are reversible.
- Rounding Example: $x = 1.35 \rightarrow \text{floor}(x) = 1$ . However, $\text{floor}(x) = 1$ does not imply $x = 1.35$ . This is not a biconditional statement.
- Squaring Example: $x = -2 \rightarrow x^2 = 4$ . However, $x^2 = 4 \rightarrow x = 2 \text{ or } x = -2$ . The reverse is not a single value, so it is not strictly reversible.

Question on Enrollment: A student noted they were not enrolled in a tutorial.
- Response: The instructor confirmed that they must be manually enrolled at the end of class to ensure their assignments can be marked.
Question on Punctuation Labels: A student asked what a "full stop" (period) means in logical translation.
- Response: Usually, a full stop acts as a conjunction ( $\land$ /AND). However, if followed by words like "thus" or "therefore," it acts as an implication ( $\rightarrow$ ) connecting the preceding information to the conclusion.
Question on Operator Order: A student asked about the order of operations for logic symbols.
- Response: AND is stronger than implies. Usually, brackets are provided in assessments to avoid ambiguity, but the rule is to work from the inner brackets outward and left to right within the same precedence level.
Question on Logic Outcomes: A student asked for clarification on the difference between valid and satisfiable.
- Response: If a table has a mixture of True and False values in the final column, it is satisfiable but not valid.