Boolean Logic: Implications, Biconditionals, and Truth Tables (Recap)

Attendance, Communication, and Focus

• Attendance and being “on the ball” are stressed as real-world expectations (what happens if you miss meetings in professional settings).
  • Class announcements about Friday’s meeting were made in three different ways; still some students didn’t show.
  • The instructor asks you to show up, rather than rely on last-minute reminders; in the real world this reliability matters.
• Reachability and identification in class systems
  • If you’re not in the class Discord channel, the instructor can’t find you; you may have created an account with a confusing name or multiple accounts.
  • Action item: consolidate identity on class platforms so communications reach you.
• Phone policy and attention
  • Put phones away; even when face down, nearby phones can reduce attention and learning efficacy.
  • The instructor cites a recent article showing phones near you can impair attention; exception only if you have a specific reason to use the phone.
• Why these requests matter
  • The instructor clarifies: this is not about meanness; the goal is to help you become consistent (second-nature) so you reliably check email, messages, etc.
  • Life happens; if you miss something, acknowledge and try to be on the road as much as possible.
• Recap and preparation mindset
  • The course emphasizes familiarity with logical connectives and ongoing practice with Boolean logic as a tool to interpret information in the world.
  • The aim is to prepare for exams through understanding, not cramming; the emphasis is on preparation, not on excuses.

Core Concepts in Boolean Logic (Propositions and Connectives)

• Propositions and truth values
  • A proposition is a statement with a truth value: true (T) or false (F).
  • The instructor encourages you to generate your own propositions (p, q, etc.) to practice evaluating truth values.
• Boolean logic as a tool for evaluating real-world claims
  • Use truth values to assess statements people make in media and everyday life.
  • Example focus: evaluate claims from online sources or TV by constructing logical arguments.
• Basic connectives introduced
  • Implication: P \to Q
  • Biconditional: P \leftrightarrow Q
  • Negation: \neg P (NOT P)
  • Converse: (flip the antecedent and consequent of an implication)
  • Inverse: (negate both antecedent and consequent)
  • Contrapositive: (negate both and flip the antecedent and consequent)
• How to think about these connectives
  • Sometimes it helps to use real-world analogies (e.g., studying leading to passing; grass being wet leading to the sprinkler being on) to grasp implication structures.
  • The key is to understand how each form behaves across all possible truth values of P and Q.

Biconditional (P ⇔ Q) and Its Meaning

• Definition and intuition
  • A biconditional expresses a very strong, bidirectional relationship: both directions must hold.
  • It is equivalent to the conjunction of two implications: (P \to Q) \land (Q \to P)
• How it’s represented
  • Commonly written as P \leftrightarrow Q or described as two implications occurring together.
• Logical requirement
  • For P \leftrightarrow Q to be true, both P and Q must share the same truth value (both true or both false).
• Example used in class (egg and twins)
  • If an egg divides, we get identical twins. If we have identical twins, that implies the egg divided. This captures the idea that both implications are true for the biconditional to hold.
  • Discussed truth outcomes when one side is true and the other false, and why that breaks the biconditional.
• Practical interpretation
  • A biconditional can be used to diagnose consistency in a system: if you observe one side, the other must be true as well, and vice versa.
• Notion of logical equivalence in this context
  • The biconditional is true exactly when the two implications are both true (TT) or both false (FF) in the underlying pair of propositions.

Truth Tables and How to Use Them

• Why truth tables help
  • They force you to consider all possible combinations of truth values for P and Q and see how the connective evaluates.
• Basic implication truth table (P → Q)
  • The only false case is when P is true and Q is false.
  • Table (P, Q, P → Q):
    egin{array}{c|c|c}
    P & Q & P \to Q \\hline
    T & T & T \\hline
    T & F & F \\hline
    F & T & T \\hline
    F & F & T \
    \end{array}
• Biconditional truth table (P ⇔ Q)
  • True when P and Q have the same truth value; false when they differ.
  • Table (P, Q, P ⇔ Q):
    \begin{array}{c|c|c}
    P & Q & P\leftrightarrow Q \\hline
    T & T & T \\hline
    T & F & F \\hline
    F & T & F \\hline
    F & F & T \
    \end{array}
• Contrapositive and equivalence
  • contrapositive of P → Q: \neg Q \to \neg P
  • This is logically equivalent to the original implication: P \to Q \equiv (\neg Q) \to (\neg P)
• Using negations to simplify thinking
  • Sometimes it’s easier to negate both sides and flip, then fill in the truth table step by step.

The Four Types of Implications (Common Forms)

• Straight implication: P → Q
  • True in all cases except P = T, Q = F.
• Converse: Q → P
  • Not logically equivalent to P → Q in general; has its own truth table and can be true even when P → Q is false.
• Inverse: ¬P → ¬Q
  • Does not have the same truth as P → Q in general; can differ in truth values.
• Contrapositive: ¬Q → ¬P
  • Logically equivalent to P → Q; if one is true, the other is true, and if one is false, the other is false.
• Why this matters
  • Some problems are easier to reason about in contrapositive form or by using negations.
  • For debugging or troubleshooting (e.g., code or processes), contrapositive can be a powerful tool since one counterexample can disprove the entire implication when using contrapositive equivalence.

Worked Examples: Sprinkler, Grass, Attendance, and Studying

• Sprinkler example (classic):
  • Let P = “the sprinkler is on” and Q = “the grass is wet.”
  • Base implication: P → Q (If the sprinkler is on, the grass will be wet.)
  • Converse: Q → P (If the grass is wet, the sprinkler must be on.)
  • Inverse: ¬P → ¬Q (If the sprinkler is not on, the grass is not wet.)
  • Contrapositive: ¬Q → ¬P (If the grass is not wet, the sprinkler is not on.)
• Realistic readings and counterexamples
  • Case 1: P is true, Q is true → P → Q is true (the expected behavior).
  • Case 2: P is true, Q is false → P → Q is false (the sprinkler on but grass not wet is a breakdown).
  • Case 3: P is false, Q is true → P → Q is true (grass could be wet due to other reasons, e.g., rain).
  • Case 4: P is false, Q is false → P → Q is true (vacuous truth).
• Practical interpretation of the four forms in this example
  • Converse (Q → P) examines whether wet grass guarantees a sprinkler on; often not true in real life (rain, dew, etc.).
  • Inverse (¬P → ¬Q) considers what happens if the sprinkler is off; the grass not being wet doesn’t necessarily mean the sprinkler is off (other causes for wet grass).
  • Contrapositive (¬Q → ¬P) is logically equivalent to the original implication and often useful for reasoning about what would happen if the grass is not wet.
• Attendance and performance example (class policy)
  • Let P = “you attend the class” and Q = “you pass the course.”
  • Implication discussion: If you attend, you will pass (P → Q) under the policy; also discuss the converse, inverse, contrapositive in the context of class policy.
  • Real-world constraint: there are cases where you attend but still do not pass due to other factors; conversely, you might pass without attendance if there are other compensating factors.
• Analyzing with truth-values and policy realities
  • The instructor uses these examples to show that real-world statements might not always mirror ideal logical implications because additional factors exist (e.g., other requirements, policy specifics).
  • The truth-table approach helps isolate when the implication holds, and when it does not, under different scenarios.

Step-by-Step Problem-Solving Strategy for Logic (as emphasized in lecture)

• Build from simple to complex
  • Start with a basic implication P → Q; determine truth in all four cases.
  • If needed, construct the contrapositive or converse to gain additional insight.
• Use negations first when constructing truth tables
  • Especially helpful for contrapositive and inverse forms: first determine ¬P and ¬Q, then evaluate the implication structures.
• Keep notation clear
  • In-class practice sometimes uses L (left) and R (right) to denote the two sides of a conditional; avoid swapping with other symbols that imply different relations.
  • If using other letters (e, i, etc.), ensure you maintain consistency with the rest of the table.
• Compare equivalence and implication carefully
  • Two statements can produce the same truth table outputs under some formations, but you must show the steps to prove logical equivalence formally, not just claim.
• Use real-world analogies to aid memory
  • Athlete analogy: if I practice, I’ll be on the team; if I’m on the team, I’ll win a game; use such analogies to remember how implications and converses function.
• Exam strategy highlighted by instructor
  • Do not rely on memorized “laws” alone; be prepared to show your reasoning steps (Morgan steps) and prove the laws with explicit truth-table or logical derivations.
  • Expect to prove laws and demonstrate understanding rather than giving brief conclusions.

Practice Insights and Exam Readiness

• Prove laws with steps, not only statements of truth
  • On exams, you should be able to justify why a given form is valid or invalid by constructing truth tables or logical derivations.
• Understand that contrapositive is especially powerful
  • Because of logical equivalence, a counterexample to the original implication will also counter the contrapositive, and vice versa.
• Build your own examples to test understanding
  • Create personal analogies (e.g., studying leading to passing; attendance affecting the grade) to internalize the forms.
• Remember real-world relevance
  • Logical forms help evaluate public statements, media claims, and decisions by examining the structure of the argument, not just the surface content.

Final Reminders and Takeaways

• Slides and resources
  • The instructor will post slides in Blackboard for continued practice.
• The goal of the course
  • To develop a solid, practice-driven understanding of logic that can be applied both on exams and in real-world critical thinking.
• Open Q&A approach
  • Students were encouraged to ask questions; if unclear, use analogies or ask for more examples (e.g., via ChatGPT for practice, but use it as a learning aid, not a solution source for exams).
• Keep engagement and focus
  • The class emphasizes staying engaged, avoiding distractions, and keeping up with communications, as these are essential to building the skills needed for the exam and for real-world problem solving.

Summary of Key Formulas and Notation (LaTeX)

• Implication: P \to Q
• Biconditional: P \leftrightarrow Q
• Negation: \neg P
• Converse: Q \to P
• Inverse: \neg P \to \neg Q
• Contrapositive: \neg Q \to \neg P
• Truth-table reminder for P → Q:
  egin{array}{c|c|c}
  P & Q & P \to Q \\hline
  T & T & T \\hline
  T & F & F \\hline
  F & T & T \\hline
  F & F & T \
  \end{array}
• Truth-table reminder for P ⇔ Q:
  egin{array}{c|c|c}
  P & Q & P \leftrightarrow Q \\hline
  T & T & T \\hline
  T & F & F \\hline
  F & T & F \\hline
  F & F & T \
  \end{array}
• Equivalence of P → Q and its contrapositive:
  P \to Q \equiv \neg Q \to \neg P
• Common intuition: a biconditional holds iff both P → Q and Q → P hold, i.e., both directions are true.