Notes on Negations and Variations of Conditional Statements

Negations and De Morgan's Laws

• De Morgan's laws give the negations of conjunctions and disjunctions:

  • $\neg(P \land Q) \equiv (\neg P) \lor (\neg Q)$
  • $\neg(P \lor Q) \equiv (\neg P) \land (\neg Q)$

• Intuition: negation flips the sign upside down when dealing with conjunctions or disjunctions.

• Quick takeaway: negation of a conjunction becomes a disjunction of negations; negation of a disjunction becomes a conjunction of negations.

• Example 1 (conjunction): "I got an A in math and in anthropology".

  • Let $P$ = "I got an A in math" and $Q$ = "I got an A in anthropology".
  • The negation is $\neg(P \land Q) \equiv (\neg P) \lor (\neg Q)$, which reads: "I didn't get an A in math or I didn't get an A in anthropology."
  • Why: for a conjunction to be true, both $P$ and $Q$ must be true. If either fails (at least one is false), the conjunction is false, so the negation is true.

• Example 2 (disjunction): "I'm going out or I'm going to stay in and study".

  • Let $P$ = "I'm going out" and $Q$ = "I'm going to stay in and study".
  • The negation is $\neg(P \lor Q) \equiv (\neg P) \land (\neg Q)$, which reads: "I'm not going out and I'm not going to stay in and study."\
  • Why: for a disjunction to be true, at least one of $P$ or $Q$ must be true. To make the disjunction false, negate both parts.
  • This illustrates the second De Morgan's law: the negation of an OR becomes an AND of negations.

Negation of Conditional Statements

• Original conditional example: "If I study hard, then I will get an A."

  • Let $P$ = "I study hard" and $Q$ = "I will get an A".
  • Symbolically: $P \rightarrow Q.$

• Common but incorrect negations people often guess:

  • "If I study hard, I will not get an A." (not the correct negation)
  • "If I don't study hard, I will not get an A." (this is the inverse, not the negation of the original)

• Correct negation:

  • $\neg(P \rightarrow Q) \equiv P \land \neg Q$, which reads: "I will study hard and I will not get an A."
  • Important note: the negation of a conditional is not itself a conditional; it is a conjunction of the antecedent with the negation of the consequent.

• Summary formula:

  • $\neg(P \rightarrow Q) \equiv P \land \neg Q$

Variations of Conditional Statements: Converse, Inverse, Contrapositive

• Start with the same original: $P \rightarrow Q$ where

  • $P$ = "I study hard" and $Q$ = "I will get an A".

• Converse: swap the order of the implication

  • Symbol: $Q \rightarrow P$
  • Verbal: "If I get an A, then I study hard."

• Inverse: negate both parts but keep the order

  • Symbol: $\neg P \rightarrow \neg Q$
  • Verbal: "If I don't study hard, then I won't get an A."

• Contrapositive: negate both parts and flip the order

  • Symbol: $\neg Q \rightarrow \neg P$
  • Verbal: "If I didn't get an A, then I did not study hard."

• Key point about logical equivalence:

  • The contrapositive is logically equivalent to the original statement: $P \rightarrow Q \equiv \neg Q \rightarrow \neg P$
  • The converse and the inverse are logically equivalent to each other, but not to the original: $Q \rightarrow P \not\equiv P \rightarrow Q$ and $\neg P \rightarrow \neg Q \equiv Q \rightarrow P$ (but both differ from the original).

Concrete Example: Bowl Game and Road Trip

• Setup:

  • Let $P$ = "We go to a bowl game this year".
  • Let $Q$ = "I am making a road trip".
  • Original: $P \rightarrow Q$ (If we go to a bowl game this year, then I am making a road trip.)

• Variations:

  • Converse: $Q \rightarrow P$, verbally: "If I am making a road trip, then we are going to a bowl game this year."
  • Inverse: $\neg P \rightarrow \neg Q$, verbally: "If we do not go to a bowl game this year, then I am not making a road trip."
  • Contrapositive: $\neg Q \rightarrow \neg P$, verbally: "If I am not making a road trip, then we are not going to a bowl game this year."

• Truth-table check (summary in row form for the four possible truth assignments of P and Q):

  • Row (P, Q) = (T, T):
  • $P \rightarrow Q$ = True
  • $Q \rightarrow P$ = True
  • $\neg P \rightarrow \neg Q$ = True
  • $\neg Q \rightarrow \neg P$ = True
  • Row (P, Q) = (T, F):
  • $P \rightarrow Q$ = False
  • $Q \rightarrow P$ = True
  • $\neg P \rightarrow \neg Q$ = True
  • $\neg Q \rightarrow \neg P$ = False
  • Row (P, Q) = (F, T):
  • $P \rightarrow Q$ = True
  • $Q \rightarrow P$ = False
  • $\neg P \rightarrow \neg Q$ = False
  • $\neg Q \rightarrow \neg P$ = True
  • Row (P, Q) = (F, F):
  • $P \rightarrow Q$ = True
  • $Q \rightarrow P$ = True
  • $\neg P \rightarrow \neg Q$ = True
  • $\neg Q \rightarrow \neg P$ = True

• Conclusions from the truth table:

  • The contrapositive (\neg Q -> \neg P) is equivalent to the original (P -> Q).
  • The converse (Q -> P) and the inverse (\neg P -> \neg Q) are equivalent to each other, but not to the original.

Practical Takeaways and Critical Thinking

• When analyzing statements, be careful with negations:

  • For conjunctions: negate each part and switch AND to OR. As a rule: $\neg(P \land Q) \equiv (\neg P) \lor (\neg Q)$.
  • For disjunctions: negate each part and switch OR to AND. As a rule: $\neg(P \lor Q) \equiv (\neg P) \land (\neg Q)$.

• For conditionals:

  • The negation is not another conditional; it is a conjunction: $\neg(P \rightarrow Q) \equiv P \land \neg Q$.
  • Variations (converse, inverse, contrapositive) behave differently in terms of logical equivalence.
  • The contrapositive is the only variation equivalent to the original; the converse and inverse form an equivalent pair to each other but not to the original.

• This material helps prevent being misled by superficially similar statements and supports critical listening and reasoning in everyday discourse and in exams.

• Quick reference formulas (LaTeX):

  • De Morgan's laws:
  • $\neg(P \land Q) \equiv (\neg P) \lor (\neg Q)$
  • $\neg(P \lor Q) \equiv (\neg P) \land (\neg Q)$
  • Negation of conditional:
  • $\neg(P \rightarrow Q) \equiv P \land \neg Q$
  • Variations:
  • Converse: $Q \rightarrow P$
  • Inverse: $\neg P \rightarrow \neg Q$
  • Contrapositive: $\neg Q \rightarrow \neg P$
  • Equivalences:
  • $P \rightarrow Q \equiv \neg Q \rightarrow \neg P$ (contrapositive)
  • Converse and Inverse are equivalent to each other but not to the original