Lecture 1 - Propositions and Propositional Logic

Propositions

• Definition: A proposition is a declarative statement that is either true or false. It cannot be a question, command, or exclamation.

• Naming: Propositions are labeled with letters such as p, q, and r.

  • Example assignments:

  • p = “donkeys can fly”

  • q = “2+2 = 4”

  • r = “It will not rain tomorrow”

• In this world (example truth values):

  • p is false

  • q is true

  • r is unknown

• Non-propositions: Questions, commands, and exclamations are not propositions.

  • Examples: “How are you doing?”, “Close the door.”, “Wow!”

• Significance: Propositions form the basic units for building more complex statements using logical connectives.

Logical Connectives (Binary and Unary)

• Conjunction (AND): p \land q

  • Read: “p and q.”

  • Truth condition: p \land q is true iff both p and q are true; false otherwise.

  • Truth table for when (p, q) is p ∧ q:

    

  • (T, T) → T

  • (T, F) → F

  • (F, T) → F

  • (F, F) → F

  • Example: “donkeys can fly and the sky is blue.”

• Disjunction (Inclusive OR): p \lor q

  • Read: “p or q.”

  • Truth condition: p \lor q is true if at least one of p or q is true.

  • False only when both are false.

  • Truth table for when (p, q) is p ∨ q:

    

  • (T, T) → T

  • (T, F) → T

  • (F, T) → T

  • (F, F) → F

  • Example: “donkeys can fly or the sky is blue.”

• Exclusive OR (XOR): p \oplus q

  • Read: “p XOR q” (true when exactly one of p or q is true).

  • Truth condition: p \oplus q is true iff exactly one of p or q is true.

  • False when both true or both false.

  • Truth table for when (p, q) is p ⊕ q:

    

  • (T, T) → F

  • (T, F) → T

  • (F, T) → T

  • (F, F) → F

  • Note: XOR is often used for scenarios where one, and only one, of the conditions holds.

• Negation (NOT): \neg p

  • Read: “not p.”

  • Truth condition: flips the truth value of p.

  • Truth table (p) → ¬p:

  • p = T → ¬p = F

  • p = F → ¬p = T

  • Example: If p is “the sky is blue,” then \neg p is “the sky is not blue.”

• Conditional (IF-THEN): p \to q

  • Read: “if p then q.”

  • Interpretation: p is the hypothesis; q is the conclusion.

  • Truth condition: p \to q is false only when p is true and q is false; true otherwise.

  • Truth table for when (p, q) is p → q:

    

  • (T, T) → T

  • (T, F) → F

  • (F, T) → T

  • (F, F) → T

  • Note: If the hypothesis p is false, the conditional is vacuously true.

Conditional Example

• Example: m → p where:

  • m: You mow my lawn

  • p: I pay you $20

  • Interpretation: “If you mow my lawn, I will pay you $20.”

• Truth table for when (m, p) m → p:

  

  • (T, T) → T

  • (T, F) → F

  • (F, T) → T

  • (F, F) → T

• Practical note: This demonstrates how a promise or obligation might be modeled with a conditional.

Conditional Examples

• Example statements:

  • “If donkeys can fly, then the sky is blue.”

  • “If the sky is blue, then donkeys can fly.”

• These two conditionals illustrate that a conditional’s truth value depends on p and q, and that the converse of a conditional is not necessarily true.

Expressing p → q in Alternative Forms

• p → q can be expressed as:

  • If p, then q

  • p implies q

  • p is sufficient for q

  • q is necessary for p

  • Equivalent form: p \to q \equiv \neg p \lor q

• These verbal/structural equivalents help connect natural language with formal logic.

Biconditional (IF AND ONLY IF)

• Biconditional: p \leftrightarrow q

  • Read: “p iff q” or “p if and only if q.”

  • Meaning: p and q have the same truth value.

  • Equivalences: p \leftrightarrow q \equiv (p \to q) \land (q \to p)

  • Truth table for when (p, q) is p ↔ q:

    

  • (T, T) → T

  • (T, F) → F

  • (F, T) → F

  • (F, F) → T

• Note: The biconditional is true exactly when p and q agree in truth value.

Operator Precedence and Expression Clarity

• Precedence (from highest to lowest):

  • ¬ (NOT)

  • ∧ (AND)

  • ∨ (OR)

  • → (IMPLICATION)

  • ↔ (BICONDITIONAL)

• Practical advice: Use parentheses to make the intended order of operations explicit.

  • Example: To express ¬p ∨ q ∧ ¬q, use parentheses according to the precedence rules: since ∧ binds before ∨, write as ¬p ∨ (q ∧ ¬q). If you mean (¬p ∨ q) ∧ ¬q, write as (¬p ∨ q) ∧ ¬q.

Example: Truth Table for p ↔ (¬p ∧ q)

• Consider the proposition: p \leftrightarrow (\neg p \land q)

• Assumed interpretation: \neg p \land q means “not p and q.”

• Columns to consider: p, q, ¬p, ¬p ∧ q, p ↔ (¬p ∧ q)

• Computation:

  • If p = T, q = T:

  • ¬p = F; ¬p ∧ q = F; p ↔ (¬p ∧ q) = T ↔ F = F

  • If p = T, q = F:

  • ¬p = F; ¬p ∧ q = F; p ↔ (¬p ∧ q) = T ↔ F = F

  • If p = F, q = T:

  • ¬p = T; ¬p ∧ q = T; p ↔ (¬p ∧ q) = F ↔ T = F

  • If p = F, q = F:

  • ¬p = T; ¬p ∧ q = F; p ↔ (¬p ∧ q) = F ↔ F = T

• Result: The formula is true only in the (p, q) = (F, F) case; false in the other three cases.

• Summary truth values:

  • (p, q) = (T, T) → F

  • (p, q) = (T, F) → F

  • (p, q) = (F, T) → F

  • (p, q) = (F, F) → T

Example: Truth Table for p ↔ (¬p ∧ q) (Alternative Row Representation)

• For quick reference, the rows can be listed as:

  • p = T, q = T: p ↔ (¬p ∧ q) = F

  • p = T, q = F: p ↔ (¬p ∧ q) = F

  • p = F, q = T: p ↔ (¬p ∧ q) = F

  • p = F, q = F: p ↔ (¬p ∧ q) = T

Translating English Statements into Logic

• Define base propositions:

  • p: The weather is bad.

  • q: The trip is cancelled.

• Translate the following sentences:
  1) The weather is good. → \neg p
  2) The weather is bad, but the trip is not canceled. → p \land \neg q
  3) The weather was good and the trip was not canceled. → \neg p \land \neg q
  4) If the weather is good, the trip will not be canceled. → \neg p \to \neg q

• Notes:

  • The negation corresponds to the opposite of the stated proposition.

  • The conjunction represents “and” between two statements.

  • The conditional captures dependency of one statement on another, with the usual truth-functional interpretation.

Real-World Relevance and Implications

• Foundational role: Propositional logic underpins boolean logic used in computer science, digital circuits, and formal reasoning.

• Practical use: Truth tables provide a simple, tabular method for evaluating the outcomes of compound statements.

• Philosophical angle: The idea of a statement being true, false, or unknown touches on the semantics of truth and the limits of bivalent logic; real-world statements may be incomplete or context-dependent.

• Ethical/practical considerations: Clear formalization of rules (e.g., in contracts or programming) helps prevent misinterpretation and ambiguity.

Summary of Key Symbols and Equivalences

• Propositions: p, q, r, \dots

• Conjunction: p \land q

• Disjunction (inclusive): p \lor q

• Exclusive OR: p \oplus q

• Negation: \neg p

• Conditional: p \to q

• Biconditional: p \leftrightarrow q

• Equivalences: p \to q \equiv \neg p \lor q and p \leftrightarrow q \equiv (p \to q) \land (q \to p)

• Precedence (highest to lowest): \neg, \land, \lor, \to, \leftrightarrow

• Practice tip: Use parentheses to make evaluation order explicit.