Lecture Notes: Logic, Implications, and Truth Tables

Attendance, conduct, and logistics

• Instructor emphasizes attendance tracking and caps off no-shows; if you’ve shown up at least once, you’re considered present for the channel cadence.

• New rule: headphones off in class; earpieces discouraged due to attention issues (example given of a prior incident where a jack came off and distracts the class).

• Message: you’re here for your future, not the instructor or parents; four years of hard work can lead to a better forty-year life; alternatively, poor effort risks limited job prospects (examples include working at Walmart or as a cashier).

• Professionalism is stressed: avoid profanity in the classroom/around the building; maintain control and act professionally; your work ethic ties directly to your future prospects.

• Class culture: collaboration is encouraged through formal teams and informal study groups; you can learn from instructors, tutors, and peers; you may form study groups to survive/finish college.

• Tutoring announcements:

  • Schedule tutoring for computational thinking and Elements of Computing; use Twitter for announcements, follow the rules, and treat tutoring as ongoing support.

  • Scheduling a tutoring appointment is date-specific, not recurring; plan ahead to build a regular schedule, or adapt week-to-week if your work schedule is irregular.

  • If you have other courses requiring tutoring, also schedule those.

• Friday recitation update: no recitation this Friday; continue practicing truth tables and logic on your own; assignment will involve coming up with your own logical propositions and validating them with truth tables.

• Study group practical tip: even if you know the material, run it by tutors to confirm understanding; collaboration helps reveal gaps in understanding.

• Reminder about class logistics: a Discord channel is in the works; use real names; the instructor is creating the roster and roster channel; expect updates soon.

• Instructor’s personal note: false starts and last-minute issues can be avoided with routines and pre-class prep; routine building is encouraged.

• Overall goal: build a strong foundation in logic to prepare for future programming, problem solving, and rigorous reasoning.

What we’re covering today: implications, truth tables, and varieties

• Four (plus one “on steroids”) ways to think about implications; today and next time, we’ll cover three more forms and a stronger bidirectional form.

• Implication basics: if-then statements drive logic, programming, and everyday reasoning; a proposition can be true or false.

• Truth tables are the exhaustive way to capture all the truth possibilities for logical statements.

• Key vocabulary:

  • Negation:

  • Conjunction:

  • Disjunction:

  • Exclusive OR (XOR):

  • Implication:

  • Converse:

  • Biconditional:

  • Logical equivalence (when two statements always share the same truth value):

• Real-world framing helps memorize: the instructor uses concrete examples (food, timing, weather-like scenarios) to reason about logic.

• Safety note: this material emphasizes careful thinking and avoiding quick coding mistakes (a humorous anecdote about a missing semicolon causing debugging delays).

Core connectives and their meanings

• Negation:
  eg p means “not p.” If p is true, ¬p is false and vice versa.

• Conjunction: p \,\land\, q is true only when both p and q are true.

• Disjunction: p \,\lor\, q is true when at least one of p or q is true; inclusive or allows both to be true.

• Exclusive or (XOR): p \oplus q is true when exactly one of p or q is true (one true, the other false).

• Common intuition examples:

  • XOR example: choosing one side (mashed potatoes or fries) at a restaurant; you can pick one but not necessarily both unless you pay extra.

  • Regular or: you can have one or both sides; both being true still satisfies the condition.

• Recap mental model: read and interpret these operators clearly; the instructor warns that understanding can be tricky and can require slow, careful thinking.

Implication (if A then B) and its truth table

• Basic form: p \rightarrow q means: if p is true, then q should be true.

• Evidence-based analysis with a sprinkler example:

  • A: "If you turn on the sprinkler, the grass will get wet." (p: sprinkler on; q: grass wet)

  • Case 1: sprinkler on (p = True) and grass wet (q = True) → statement supported; implication true.

  • Case 2: sprinkler on (p = True) and grass not wet (q = False) → implication false; evidence contradicts the promise.

  • Case 3: sprinkler not on (p = False) and grass wet (q = True) → implication true (vacuous truth, since antecedent is false).

  • Case 4: sprinkler not on (p = False) and grass not wet (q = False) → implication true (vacuous truth).

• Practical takeaway: an implication can be true even when the antecedent is false or when evidence contradicts the consequent, as long as the antecedent is false or the implication holds with the given evidence.

• Truth table for p \rightarrow q:

  • p q | p → q

  • T T | T

  • T F | F

  • F T | T

  • F F | T

• Converse of implication: q \rightarrow p flips the direction; sometimes easier to reason about in practice.

• Note on “innocent until proven guilty” in logic: you don’t have evidence against the implication unless you have a counterexample (p true and q false).

The converse and the biconditional (implication on steroids)

• Converse: the statement flipped, q \rightarrow p.

• Biconditional: p \leftrightarrow q, meaning both directions hold: (p → q) ∧ (q → p).

• Truth table intuition for biconditional:

  • It is true only when both implications hold, i.e., when p and q have the same truth value (both true or both false).

  • Rows where both true: true; both false: true. Mixed truth values: false.

• Example: pregnancy twins (single egg divides → identical twins) and its converse (identical twins → egg divided).

  • This pair forms a biconditional: E ↔ I, where E = egg divides, I = identical twins.

  • Truth table analysis shows the biconditional is true exactly when both directions hold and is false when one direction holds but the other does not.

• Summary: biconditional is an “implication in both directions” and is true only when both directions are true; it encodes a strong, bidirectional relationship.

Working with biconditionals and practical reasoning

• The lecture emphasizes drawing truth tables to verify biconditionals rather than relying on intuition alone.

• A failure of either direction invalidates the biconditional; if you can show one direction fails, the biconditional is false.

• Real-world debugging analogy: a missing semicolon or a subtle logic error can turn a correct-looking implication into a faulty program; careful logical analysis helps prevent such errors.

• Converse vs. biconditional distinction:

  • Converse (q → p) is only one-way; it may or may not be true independently.

  • Biconditional (p ↔ q) requires both p → q and q → p to be true simultaneously.

Logical equivalence and De Morgan-type relations

• Logical equivalence: two statements have identical truth values across all possible assignments.

• Example task: show that p \land q is logically equivalent to \neg(\neg p \lor \neg q) (De Morgan-related equivalence for conjunction).

• Truth-table-based proof approach:

  • Build truth tables for both sides across all combinations of p and q (TT, TF, FT, FF).

  • Show the output columns match exactly for all four combinations.

• Practical takeaway: truth tables are the formal method to prove logical equivalences; when asked to prove equivalence on an exam, you must present a truth table rather than a verbal argument.

• Worked outline (brief):

  • p q | p ∧ q | ¬p | ¬q | ¬p ∨ ¬q | ¬(¬p ∨ ¬q)

  • TT: T | F | F | F | F | T

  • TF: F | F | T | T | T | F

  • FT: F | T | F | F | F | F

  • FF: F | T | T | T | T | F

  • The columns p ∧ q and ¬(¬p ∨ ¬q) match for all rows, establishing equivalence.

Examples to solidify understanding

• Additional intuitive examples for implication and converse:

  • Example 1: “If you eat a lot of food, you’ll gain weight.” One direction: weight gain does not prove you ate a lot (could be other factors); the converse is also not guaranteed; use evidence to validate the rule, and consider exceptions.

  • Example 2: “If you plant a seed, you grow a garden.” One-directional causal link that, in typical cases, requires the seed to be planted; the contrapositive and converse can be explored for deeper understanding.

• Polish idiom example used to illustrate equivalence/affordances of reasoning about causes and effects; cautions against over-simplified moral claims (stability of cause and effect in real-world scenarios).

Study strategies and exam readiness

• Key takeaways:

  • Truth tables are your best tool for validating logical statements and equivalences.

  • Practice reading and translating logical expressions into truth tables and back into plain English.

  • Practice constructing and evaluating: p ∧ q, p ∨ q, ¬p, p → q, q → p, and p ↔ q across all possible assignments.

  • Build personal examples that connect to your interests (sports, cars, music, arts) to reinforce implications and rule-based thinking.

  • Regular collaboration (study groups, tutoring) helps uncover gaps and provides multiple perspectives.

• Exam-ready protocol:

  • When asked to prove logical equivalence, present a truth table and explain why the rows align.

  • When asked to analyze an implication, discuss cases with direct evidence, vacuous cases, and counterexamples.

  • Distinguish between implication, converse, and biconditional; know which are true in which contexts.

Recap: what to remember for the next session

• Four core ways to think about implications (plus the biconditional):

  • p → q (implication) and q → p (converse)

  • (p → q) ∧ (q → p) (biconditional) = p ↔ q

  • p ↔ q is true only when p and q have the same truth value (both true or both false).

  • Equivalence with logical negation: p ∧ q ≡ ¬(¬p ∨ ¬q).

• Prepare to dive deeper into logical equivalence and related laws next time, including more advanced forms of implication.

Final reminders and next steps

• No class Friday; recitation canceled; continue practicing truth tables on your own.

• Monday: continue with more implications and equivalence, and start exploring equivalence of logical propositions in more detail.

• If you have questions, bring concrete examples (from your life or coursework) to illustrate how these implications operate in real problems.

• Stay engaged, mute/unmute as needed, and leverage tutoring as a regular habit to reinforce understanding.