Logical Connectives
Negation, and, or, if-then, and if-and-only-if — the truth-table machinery behind every proof, plus the contrapositive and De Morgan's laws.
The idea
Logical connectives are the precise forms of the words that join statements together. A statement is a sentence that is either true or false, with no third option, and from statements $P$ and $Q$ the connectives build five compounds: the negation $\lnot P$ ("not $P
quot;), the conjunction $P \land Q$ ("$P$ and $Qquot;), the disjunction $P \lor Q$ ("$P$ or $Qquot;), the implication $P \Rightarrow Q$ ("if $P$ then $Qquot;), and the biconditional $P \Leftrightarrow Q$ ("$P$ if and only if $Qquot;).A table defines each connective by listing the truth value of the compound for every combination of truth values of its parts, and that table is the whole of its meaning. A connective sees only truth values: $P \land Q$ depends on whether $P$ is true and whether $Q$ is true, never on what either statement is about.
The tables and everyday usage do not always agree. Where they part company, the table wins — it is the definition, and English was only the motivation for it.
Ways to work on it
- Walkthrough. Build up the five connectives one truth table at a time, ending with the contrapositive equivalence and the biconditional.
- Practice. Find converses and contrapositives, evaluate connectives, and apply De Morgan's laws.
- Hardest. Negate a compound statement and decide which related implications are logically equivalent.
Not sure where to start? Take the ten-question placement test.