Lecture Notes: Logical Equivalences in Propositional Logic

Key Concepts

• Logical Equivalence:
  • Two propositions, R and S, are logically equivalent if they have the same truth value regardless of the truth values of any other propositions they depend on.
  • The importance of logical equivalences is in simplifying expressions and solving logical problems.
  • Example given: P implies Q is logically equivalent to not P or Q.

Checking Logical Equivalence

• Use truth tables to determine logical equivalence:
  • List all possible truth values for the propositions (e.g., P and Q).
  • Analyze to see if the two propositions agree in their truth values for all combinations.
  • Example: P implies Q is only false if P is true and Q is false.

Laws of Propositional Logic

Similarities with Boolean Algebra

• Commutative Law:
  • P and Q is equivalent to Q and P
  • P or Q is equivalent to Q or P
  • Truth table does not depend on the order of P and Q.
• Idempotent Laws:
  • P and P is equivalent to P.
  • P or P is equivalent to P.
• Associative Laws:
  • Order of operations in intersections does not change the outcome.
  • Example using Venn diagrams to illustrate: P and (Q and R) equals (P and Q) and R.

Proving Equivalences

• Use truth tables or connect them with laws of Boolean logic.
• Conditional Logic:
  • P implies Q and Q implies P are equivalent to P if and only if Q.
  • To prove P if and only if Q, both the direct and reverse implications must be shown.

Conclusion

• Exercises will help solidify understanding by practicing these laws and proving logical equivalences.
• Conditional symbol and its role in propositional logic discussed.

Next Steps

• Complete exercises provided to reinforce understanding of logical equivalences and their proofs.