Why Truth Tables Matter

Remember playing a board game where you consider all possible moves before deciding? Truth tables do something similar for logical statements. They list every possible combination of truth values for your basic propositions (like "It's raining" or "I have an umbrella") and show whether a larger, compound statement (like "It's raining AND I have an umbrella") is true or false in each scenario.

Building on What We've Learned

In Part 2, we introduced three connectives: - Negation: "not" ($\mathrm{\neg }$)
- Conjunction: "and" ($\wedge$)
- Disjunction: "or" ($\vee$, used inclusively)

Now, we'll systematically show how each connective behaves in every possible scenario of truth and falsehood. This methodical approach helps us avoid mistakes (like confusing exclusive "or" with inclusive "or").

Truth tables will make these rules crystal clear by enumerating all possibilities.

Truth Table for Negation ($\mathrm{\neg }$)

Consider a single proposition $p$. It can be either true (T) or false (F). A simple table for $p$ and its negation $\mathrm{\neg }p$ is:

Reading the rows: 1. If $p$ is true, then $\mathrm{\neg }p$ is false.
2. If $p$ is false, then $\mathrm{\neg }p$ is true.

Truth Table for Conjunction ($p\wedge q$)

When we deal with two propositions $p$ and $q$, each can be either T or F. This means there are four possible combinations of truth values. The truth table for $p\wedge q$ is:

Interpretation: $p\wedge q$ is true only in the first row, where both $p$ and $q$ are true. In every other case, at least one of $p$ or $q$ is false, making $p\wedge q$ false.

Truth Table for Disjunction ($p\vee q$)

Again, consider two propositions $p$ and $q$. The truth table for the inclusive "or" ($\vee$) is:

Interpretation: $p\vee q$ is false only in the last row (when both $p$ and $q$ are false). In all other cases (at least one true), $p\vee q$ is true.

Example:
- $p$ = "I have a discount coupon."
- $q$ = "I have enough cash."

$p\vee q$ = "Either I have a discount coupon or I have enough cash (or both)."
- If I have at least one of those, the statement is true. Only if I have neither (no coupon and not enough cash) is the statement false.

Compound Truth Tables

We can create truth tables for more complex expressions that combine multiple connectives. For example, let's analyze $\mathrm{\neg }p\vee (p\wedge q)$.

Truth Tables in Decision-Making and Analysis

Truth tables have practical uses beyond homework exercises. They ensure no scenario is overlooked:

Common Misconceptions

• "The logical 'or' must exclude both being true."
  Reality: By default, logical "or" is inclusive; both can be true. Exclusive or is a special case we have to specify separately.

• "Truth tables are only for math classes, not real life."
  Reality: Truth tables (or the systematic thinking behind them) clarify any situation with clear yes/no conditions, from planning events to troubleshooting electronics.

• "If there are many propositions, truth tables are impossible to use."
  Reality: It's true that truth tables grow quickly (2^n rows for n propositions). For large $n$, other techniques or software are used. But for small numbers of propositions, truth tables are a straightforward and reliable tool.

• "A truth table can be done arbitrarily; there's no standard method."
  Reality: There is a standard method: list all combinations of truth values for the basic propositions, then compute stepwise. This ensures consistency and that no case is missed.

Exercises

3.1. Construct a truth table for exclusive or (XOR) of two propositions $p$ and $q$. Hint: XOR is true when exactly one of $p$ or $q$ is true (and false otherwise). Verify that your final column matches the definition (true for one true input, false for both true or both false).

3.2. Build a truth table for $(p\wedge \mathrm{\neg }q)\to \mathrm{\neg }r$. How many rows will it have? (List them all.) Which rows make the implication false?

3.3. Using a truth table, determine if the expression $p\vee (q\wedge \mathrm{\neg }q)$ is a tautology, contradiction, or contingency (see Part 8 for definitions of these terms, if needed).

3.4. Write the conditions for a simple household alarm system and then represent them in a truth table. For example: "$a$ = motion sensor detects movement, $b$ = door sensor detects door open. Alarm sounds if $a\vee b$."

3.5. (Challenge) If a compound statement has 3 distinct propositions, how many rows will a complete truth table have? If it has 4 distinct propositions? Explain the pattern.

Next: In Part 4: If-Then Statements, we'll focus on the conditional connective ($\to$), learn its truth table, and see why it's defined the way it is.

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