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Understanding Propositional Equivalences: Truth Tables and Logical Laws

This section explores propositional equivalences, examining how equivalent propositions possess the same truth values and can be interchanged in logical statements. It details methods for determining logical equivalence, including the use of truth tables and substitution proofs with established logical laws. Examples illustrate tautologies, contradictions, and contingencies while emphasizing the importance of logical relations such as DeMorgan’s Laws and the commutative and associative properties. Understanding these concepts enhances logical reasoning and critical thinking skills.

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Understanding Propositional Equivalences: Truth Tables and Logical Laws

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  1. Section 1.2 Propositional Equivalences

  2. Equivalent Propositions • Have the same truth table • Can be used interchangeably • For example, exclusive or and the negation of biconditional are equivalent propositions: p q p  q p  q  (p  q) T T F T F T F T F T F T T F T F F F T F

  3. Equivalent propositions • Logical equivalence is denoted with the symbol  • If p  q is true, then p  q

  4. Tautology • A compound proposition that is always true, regardless of the truth values that appear in it • For example, p p is a tautology: p p p p T F T F T T

  5. Contradiction • A compound proposition that is always false • For example, p  p is a contradiction: p p p  p T F F F T F

  6. Tautology vs. Contradiction • The negation of a tautology is a contradiction, and the negation of a contradiction is a tautology • Contingency: a compound proposition that is neither a tautology nor a contradiction

  7. Determining Logical Equivalence • Method 1: use truth table • Method 2: use proof by substitution - requires knowledge of logical equivalencies of portions of compound propositions

  8. Method 1 example Show that p  q  p  q p q p p  q q p  q T T F F F F T F F T T T F T T T F T F F T F T F

  9. Method 1 example Show that (p  q)  p  q p q p  q (p  q) p q p  q T T T F F F F T F F T F T T F T F T T F T F F F T T T T

  10. Method 1 example Show that p  (q  r)  (p  q)  (p  r) p q r qr p(qr) pq pr (pq)(pr) T T T T T T T T T T F T T T F T T F T T T F T T T F F F F F F F F T T T F F F F F T F T F F F F F F T T F F F F F F F F F F F F

  11. The limits of truth tables • The previous slide illustrates how truth tables become cumbersome when several propositions are involved • For a compound proposition containing N propositions, the truth table would require 2N rows

  12. Method 2: using equivalences • There are many proven equivalences that can be used to prove further equivalences • Some of the most important and useful of these are found in Tables 5, 6 and 7 on page 24 of your text, as well as on the next several slides

  13. Identity Laws p  T  p p  F  p In other words, if p is ANDed with another proposition known to be true, or ORed with another proposition known to be false, the truth value of the compound proposition will be the truth value of p

  14. Domination Laws p  T  T p  F  F A compound proposition will always be true if it is composed of any proposition p ORed with any proposition known to be true. Conversely, a compound proposition will always be false if it is composed of any proposition p ANDed with a proposition known to be false

  15. Idempotent Laws p  p  p p  p  p A compound proposition composed of any proposition p combined with itself via conjunction or disjunction will have the truth value of p

  16. Double negation (p)  p The negation of a negation is … well, not a negation

  17. Commutative Laws p  q  q  p p  q  q  p Ordering doesn’t matter in conjunction and disjunction (just like addition and multiplication)

  18. Associative Laws (p  q)  r  p  (q  r) (p  q)  r  p  (q  r) Grouping doesn’t affect outcome when the same operation is involved - this is true for compound propositions composed of 3, 4, 1000 or N propositions

  19. Distributive Laws p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) OR distributes across AND; AND distributes across OR

  20. DeMorgan’s Laws (p  q)  p  q (p  q)  p  q The NOT of p AND q is NOT p OR NOT q; the NOT of p OR q is NOT p AND NOT q Like Association, DeMorgan’s Laws apply to N propositions in a compound proposition

  21. Two Laws with No Name p  p  T p  p  F A proposition ORed with its negation is always true; a proposition ANDed with its negation is always false

  22. A Very Useful (but nameless) Law (p  q)  (p  q) The implication “if p, then q” is logically equivalent to NOT p ORed with q

  23. Method 2: Proof by Substitution • Uses known laws of equivalences to prove new equivalences • A compound proposition is gradually transformed, through substitution of known equivalences, into a proveable form

  24. Example 1: Show that(p  q)  p is a tautology 1. Since (p  q)  (p  q), change compound proposition to: (p  q)  p 2. Applying DeMorgan’s first law, which states: (p  q)  p  q, change compound proposition to: p  q  p 3. Applying commutative law: p  p  q 4. Since p  p  T, we have T  q 5. And finally, by Domination, any proposition ORed with true must be true - so the compound proposition is a tautology

  25. Example 2: Show thatp  q and p  q are logically equivalent 1. Start with definition of biconditional: p  q  p  q  q p; then the 2 expressions become: (p  q)  (q  p) and (p  q)  (q  p) 2. Since p  q  p  q, change expressions to: ((p)  q)  (q  p) and (p  q)  ((q)  p); same as: (p  q)  (q  p) and (p  q)  (q  p) 3. Reordering terms, by commutation, we get: (p  q)  (p  q) and (p  q)  (p  q) Since the two expressions are now identical, they are clearly equivalent.

  26. Section 1.2 Propositional Equivalences - ends -

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