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
  T	F
  F	T

  • 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:

  α	β	(α ∧ β)
  T	T	T
  T	F	F
  F	T	F
  F	F	F

  • Has 4 rows due to two component propositions.

Disjunction (∨)

• Definition: A disjunction is true if at least one disjunct is true.
• Truth Table:

  α	β	(α ∨ β)
  T	T	T
  T	F	T
  F	T	T
  F	F	F

  • 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:
  

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


• Truth Table:
  

• 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.