Lecture 3 - PHIL222: Truth Tables and Logical Propositions

Preliminaries

  • Syntax of Propositional Logic (PL):

    • A well-formed formula (wff) represents a meaningful expression in propositional logic.

  • Semantics of PL:

    • Relates wffs to truth and falsity (truth values).
    • Bivalence Assumption: Every proposition is either true (T) or false (F), not both.

  • Truth-functional Connectives:

    • The truth or falsity of a compound proposition depends on its component propositions, based on the connective used.

Truth Tables

Negation (¬)

  • Truth Table:
    α¬α
    TF
    FT
    • If α is true, ¬α (negation of α) is false, and vice versa.

Conjunction (∧)

  • Definition: A conjunction is true only if both conjuncts are true.
  • Truth Table:
    αβ(α ∧ β)
    TTT
    TFF
    FTF
    FFF
    • Has 4 rows due to two component propositions.

Disjunction (∨)

  • Definition: A disjunction is true if at least one disjunct is true.
  • Truth Table:
    αβ(α ∨ β)
    TTT
    TFT
    FTT
    FFF
    • Is inclusive; true if at least one is true.

Conditional (→)


  • Definition: A conditional is false only if the antecedent is true and the consequent is false.

  • Truth Table:

αβ(α → β)
TTT
TFF
FTT
FFT

Biconditional (↔)


  • Definition: A biconditional is true if both propositions have the same truth value.

  • Truth Table:

αβ(α ↔ β)
TTT
TFF
FTF
FFT

Truth Values of Complex Propositions

  • Example: Evaluate (¬P ∧ (Q ∨ R)) given P=T, Q=F, R=F.

    1. Compute ¬P: T -> F
    2. Compute (Q ∨ R): F ∨ F -> F
    3. Final computation: F ∧ F = F
    • Thus, (¬P ∧ (Q ∨ R)) is false.

  • Another Example: Evaluate ((A ↔ B) → (¬A ∨ B)) given A=F, B=F.

    1. A ↔ B: F ↔ F = T
    2. ¬A: F -> T
    3. ¬A ∨ B: T ∨ F = T
    4. Final computation: T → T = T

Building Truth Tables

  • General Approach: List all possible combinations of truth values for component propositions to fill out the truth table.
  • Number of Rows: Each component doubles the number of rows:
    • 1 component: 2^1 = 2 rows
    • 2 components: 2^2 = 4 rows
    • 3 components: 2^3 = 8 rows
    • n components: 2^n rows

Complex Proposition Example

  • Setup: Create a truth table for ((P → Q) ∧ (P ∧ ¬Q)). Enter all combinations of truth values for P and Q, then compute each segment step-by-step.

Conclusion

  • Truth tables are fundamental in assessing the truth values of complex propositions in propositional logic, allowing logic operators to be understood and analyzed systematically.

  • The process applies universally regardless of the structure of the proposition, reinforcing the consistency of logical analysis.