8.1 Propositional Logic, Truth Tables & Logical Connectives

8.1 Propositional Logic, Truth Tables & Logical Connectives

Key Exam Fact: Logic and Critical Thinking accounts for 9 questions (18%) of the 50-question Bocconi Admission Test. Questions within this domain test deductive validity rather than factual recall or subjective interpretation. Under the Bocconi scoring system (+1.00 for correct, 0.00 for skipped, -0.20 for incorrect), understanding Boolean truth values and symbolic connectives enables candidates to systematically eliminate flawed answer choices in under 60 seconds without relying on intuitive guesswork.

Formal logic provides the structural architecture for analytical reasoning. While reading comprehension passages evaluate your ability to synthesize rhetorical and contextual information, formal logic problems assess your command of deductive necessity: whether a conclusion must follow unconditionally from given premises. In the Bocconi test, these questions frequently appear disguised as corporate regulations, algorithmic compliance criteria, or economic policy statements.

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Propositions: Atomic vs. Compound Statements

At the core of propositional logic (also termed sentential logic) is the concept of a proposition (or statement).

Defining a Proposition

A proposition is a declarative sentence that is either strictly True ($T$) or strictly False ($F$), but never both simultaneously, and never indeterminate. Questions, commands, opinions, and paradoxes are not propositions.

• Proposition: "Bocconi University is located in Milan, Italy." (Truth value: $T$)
• Proposition: "Every real quadratic equation has two distinct real roots." (Truth value: $F$)
• Non-Proposition: "Submit the financial statement by noon." (Imperative command)
• Non-Proposition: "Is macroeconomics more challenging than microeconomics?" (Interrogative inquiry)
• Non-Proposition: "This statement is false." (Self-referential paradox)

Atomic vs. Compound Propositions

• Atomic Propositions: Basic statements containing no logical operators, typically denoted by lowercase or uppercase letters: $P, Q, R, S$.
• Compound Propositions: Expressions formed by joining one or more atomic propositions using logical connectives (operators). The truth value of a compound proposition depends entirely on the truth values of its constituent atomic propositions and the definitions of the connectives applied.

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The Five Essential Logical Connectives

To master Bocconi logic problems, you must know the exact operational definitions and symbolic representations of five logical connectives.

1. Negation (NOT: $\neg P$ or $\sim P$)

Negation reverses the truth value of a proposition. If $P$ is true, $\neg P$ is false; if $P$ is false, $\neg P$ is true.

• Natural Language: "It is not the case that $P$," "Not $P$," "$P$ fails to hold."

2. Conjunction (AND: $P \land Q$)

A conjunction asserts that both constituent propositions are true simultaneously. $P \land Q$ is True if and only if both $P$ and $Q$ are True. If either conjunct is false, the entire conjunction is false.

• Natural Language: "$P$ and $Q$," "$P$ but $Q$," "$P$ although $Q$," "$P$ as well as $Q$," "While $P$, also $Q$."
• Exam Trap: In formal logic, "but" and "although" have the identical logical truth conditions as "and." The contrastive tone in English does not alter the truth-functional conjunction.

3. Disjunction (Inclusive OR: $P \lor Q$ vs. Exclusive OR: $P \oplus Q$)

• Inclusive Disjunction ($P \lor Q$): True if $P$ is true, $Q$ is true, or both are true. It is false only when both $P$ and $Q$ are false.
• Exclusive Disjunction ($P \oplus Q$ or XOR): True if either $P$ is true or $Q$ is true, but not both. It is false if both are true or both are false.
• Critical Standard: In formal logic and throughout the Bocconi exam, the word "or" is ALWAYS inclusive ($P \lor Q$) unless the text explicitly states: "either $P$ or $Q$, but not both."

4. Material Implication / Conditional (IF-THEN: $P \rightarrow Q$)

The conditional proposition asserts that if the antecedent $P$ holds, the consequent $Q$ must hold.

• The Golden Truth Rule: $P \rightarrow Q$ is False in exactly ONE circumstance: when $P$ is True and $Q$ is False. In all other three scenarios, $P \rightarrow Q$ is True.
• The Vacuous Truth Principle: When the antecedent $P$ is False, the conditional $P \rightarrow Q$ is automatically True, regardless of whether $Q$ is True or False.
• Intuitive Analogy: Think of a contractual warranty: "If the device fails within 12 months ($P$), the manufacturer will issue a refund ($Q$)." If your device does not fail ($P$ is False), the manufacturer has not breached the warranty, whether they issue a goodwill discount or not. The contract is broken ($F$) only if the device fails ($P = T$) and no refund is provided ($Q = F$).

5. Biconditional (IF AND ONLY IF: $P \leftrightarrow Q$)

The biconditional asserts mutual implication: $(P \rightarrow Q) \land (Q \rightarrow P)$.

• $P \leftrightarrow Q$ is True when $P$ and $Q$ have identical truth values (both are $T$, or both are $F$).
• $P \leftrightarrow Q$ is False when their truth values diverge ($T$ and $F$, or $F$ and $T$).

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Master Truth Table of Core Connectives

The table below summarizes the truth values for all five primary operations across all four possible combinations of atomic values:

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Evaluating Compound Statements: Truth Tables with $2^n$ Rows

When an argument or problem statement contains $n$ distinct atomic variables, an exhaustive truth table requires exactly $2^n$ rows:

• 1 variable: $2^1 = 2$ rows ($T, F$)
• 2 variables: $2^2 = 4$ rows
• 3 variables: $2^3 = 8$ rows

Step-by-Step Truth Table Construction Example

Consider the compound statement: $(P \lor Q) \rightarrow (\neg P \land Q)$. Let us evaluate its truth value across all 4 rows:

1. Identify columns needed: $P, Q, \neg P, (P \lor Q), (\neg P \land Q)$, and the main conditional $[(P \lor Q) \rightarrow (\neg P \land Q)]$.
2. Compute each intermediate column systematically:

Because the final column contains both $T$ and $F$ entries, this compound statement is neither universally true nor universally false.

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Classifications: Tautology, Contradiction & Contingency

Bocconi exam items often ask whether a complex symbolic expression is logically valid, necessarily false, or contingent upon facts:

1. Tautology: A compound proposition that evaluates to True under every possible assignment of truth values to its atomic variables (symbolized by $\top$).
   • Canonical Example: The Law of Excluded Middle: $P \lor \neg P$.
   • Another Example: $[(P \rightarrow Q) \land P] \rightarrow Q$ (Modus Ponens in conditional form).
2. Contradiction (Self-Contradiction): A compound proposition that evaluates to False under every possible assignment of truth values (symbolized by $\bot$).
   • Canonical Example: The Principle of Non-Contradiction: $P \land \neg P$.
3. Contingency: A proposition that is True under at least one truth assignment and False under at least one truth assignment (such as our worked example above, which is true in rows 3 and 4, and false in rows 1 and 2).

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Fundamental Logical Equivalences & De Morgan's Laws

Two propositions $A$ and $B$ are logically equivalent (written $A \equiv B$) if and only if they possess identical truth values in every row of their truth tables. Recognizing equivalences allows you to transform difficult expressions into simple ones instantly.

De Morgan's Laws

De Morgan's Laws explain how negation distributes across conjunction and disjunction:

$\neg(P \land Q) \equiv \neg P \lor \neg Q$ $\neg(P \lor Q) \equiv \neg P \land \neg Q$

• In Words:
  • "It is not the case that both $P$ and $Q$ are true" means "Either $P$ is false, or $Q$ is false, or both are false."
  • "Neither $P$ nor $Q$ is true" means "$P$ is false AND $Q$ is false."
• Exam Application: If a Bocconi problem states: "It is false that the European Central Bank will raise interest rates and the euro will depreciate," De Morgan's Law translates this immediately to: "Either the ECB will not raise interest rates, or the euro will not depreciate (or neither will occur)."

The Disjunctive Form of Material Implication

A cornerstone equivalence frequently tested on the Bocconi exam is the identity between the conditional and disjunction:

$P \rightarrow Q \equiv \neg P \lor Q$

To prove this, inspect the truth tables:

• If $P$ is True, $\neg P$ is False, so $\neg P \lor Q$ depends entirely on $Q$. If $Q$ is True, it is True; if $Q$ is False, it is False. This matches $P \rightarrow Q$.
• If $P$ is False, $\neg P$ is True, making $\neg P \lor Q$ automatically True. This matches the vacuous truth of $P \rightarrow Q$.

The Negation of a Conditional Statement

What is the exact formal negation of "If $P$, then $Q$"?

$\neg(P \rightarrow Q) \equiv \neg(\neg P \lor Q) \equiv \neg(\neg P) \land \neg Q \equiv P \land \neg Q$

Critical Exam Warning: The negation of $P \rightarrow Q$ is NOT "If $P$, then not $Q$" or "If not $P$, then not $Q$"! The negation of a promise is the specific circumstance that breaks the promise: the antecedent occurs, but the consequent fails to occur ($P \land \neg Q$).

Other Useful Equivalences

• Double Negation: $\neg(\neg P) \equiv P$
• Commutativity: $P \land Q \equiv Q \land P$ and $P \lor Q \equiv Q \lor P$
• Distributivity:
  • $P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$
  • $P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$

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Translating Natural Language into Symbolic Notation

English phrasing contains idioms and grammatical nuances that must be converted into symbolic operators:

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Worked Bocconi Exam Scenario: Compound Statement Analysis

Problem: A corporate compliance rule at a multinational investment bank states: "It is not the case that a trader may execute a proprietary trade without compliance pre-clearance or while possessing material non-public information (MNPI)."

Let:

• $T$: A trader executes a proprietary trade.
• $C$: The trader has compliance pre-clearance.
• $M$: The trader possesses material non-public information.

Symbolic Formalization: The phrase "without compliance pre-clearance" is $\neg C$. The phrase "while possessing MNPI" is $M$. The condition describes forbidden execution: $\neg [T \land (\neg C \lor M)]$.

Deductive Simplification via De Morgan's Laws:

1. Apply De Morgan's Law to the outer brackets: $\neg T \lor \neg(\neg C \lor M)$
2. Apply De Morgan's Law to the inner disjunction: $\neg T \lor (\neg(\neg C) \land \neg M) \equiv \neg T \lor (C \land \neg M)$
3. Use the equivalence $\neg A \lor B \equiv A \rightarrow B$: $T \rightarrow (C \land \neg M)$

Conclusion: In plain language, the bank's rule is logically equivalent to: "If a trader executes a proprietary trade, then the trader must have compliance pre-clearance AND must NOT possess material non-public information."