In Simple and Compound Statements you split sentences into simple parts. Now we give those parts letters and the connectives symbols. We use lowercase letters $p$, $q$, $r$, $s$ for simple statements, and five connectives.
| Symbol | Name | Read as | Example |
|---|---|---|---|
| $\neg$ | negation | not | $\neg p$ |
| $\land$ | conjunction | and | $p \land q$ |
| $\lor$ | disjunction | or | $p \lor q$ |
| $\to$ | conditional | if … then | $p \to q$ |
| $\leftrightarrow$ | biconditional | if and only if | $p \leftrightarrow q$ |
Key idea
Why it works
Every formula is either a single statement letter or is built from smaller formulas by one connective. So $p$, $\neg p$, $p \land q$ and $\neg (p \land q) \to r$ are all formulas, and each of the last three was made by one final step. That recipe is what lets a computer, or you, check any formula piece by piece.
The meanings are fixed by truth tables, which you will use throughout Truth Values of the Connectives. Here is the one surprise to remember now: logical "or" is inclusive. $p \lor q$ is true when $p$ is true, when $q$ is true, and also when both are true.
Worked examples
Worked example: Symbolize one connective
- "Either … or" is disjunction, $\lor$.
- The first disjunct is $p$, the second $q$.
Answer: $p \lor q$
Worked example: Two words, one symbol
- "But" claims both parts, so it is $\land$.
- The second part denies $q$: $\neg q$.
- Join them.
Answer: $p \land \neg q$
Worked example: Name the parts
- The main connective is $\to$.
- Everything before it, $p \land q$, is the antecedent.
- Everything after it, $r$, is the consequent.
Answer: Antecedent $p \land q$, consequent $r$.
Common mistake