intro

Connectives & Well-Formed Formulas

Atomic sentences, the five truth-functional connectives, and the rules for building formulas.

Formal logic starts by stripping arguments down to their form. We represent simple declarative sentences with capital letters — $P$ , $Q$ , $R$ — called atomic sentences, and we glue them together with a small set of connectives. Each connective is truth-functional: the truth value of the whole is fixed entirely by the truth values of its parts.

Type a formula below (or use the symbol buttons) and read its truth value under every assignment.

Type symbols or ASCII: ~ ! → ¬, & → ∧, | → ∨, -> → →, <-> → ↔.

P	Q	¬(P ∧ Q)
T	T	F
T	F	T
F	T	T
F	F	T

This formula is a contingent — true under some assignments, false under others.

The five connectives

Connective	Here	Teller’s primer	Read as	True exactly when
Negation	$\lnot P$	$\sim P$	not $P$	$P$ is false
Conjunction	$P \land Q$	$P \mathbin{\&} Q$	$P$ and $Q$	both are true
Disjunction	$P \lor Q$	$P \lor Q$	$P$ or $Q$	at least one is true
Conditional	$P \rightarrow Q$	$P \supset Q$	if $P$ then $Q$	$P$ is false or $Q$ is true
Biconditional	$P \leftrightarrow Q$	$P \equiv Q$	$P$ if and only if $Q$	both sides match

The disjunction is inclusive ( $P \lor Q$ is true when both are true). The conditional is material: $P \rightarrow Q$ only fails in the single case where $P$ is true and $Q$ is false — so a false antecedent makes the whole conditional true (“vacuously true”).

Well-formed formulas

Not every string of symbols is legal. A well-formed formula (WFF) is built by these rules:

every atomic sentence ( $P$ , $Q$ , …) is a WFF;
if $X$ is a WFF, so is $\lnot X$ ;
if $X$ and $Y$ are WFFs, so are $(X \land Y)$ , $(X \lor Y)$ , $(X \rightarrow Y)$ , and $(X \leftrightarrow Y)$ .

Parentheses prevent ambiguity. To cut down on clutter we drop outer parentheses and use a precedence order — $\lnot$ binds tightest, then $\land$ , then $\lor$ , then $\rightarrow$ , then $\leftrightarrow$ — so $\lnot P \land Q \rightarrow R$ means $((\lnot P \land Q) \rightarrow R)$ .

The main connective

Every WFF has exactly one main connective — the last one applied as you build it up. It determines what kind of sentence you have: $\lnot(P \land Q)$ is a negation, while $\lnot P \land Q$ is a conjunction. Misreading the main connective is the most common beginner mistake, and the visualizer above makes the difference concrete — compare the two formulas and watch their columns differ.

Takeaways

Atomic sentences plus five truth-functional connectives ( $\lnot, \land, \lor, \rightarrow, \leftrightarrow$ ) generate every formula of sentence logic.
A connective is truth-functional: the whole’s value depends only on the parts’ values.
The conditional $P \rightarrow Q$ is false only when $P$ is true and $Q$ is false.
Precedence and the main connective decide how a formula is read.

References

Paul Teller, A Modern Formal Logic Primer, Prentice Hall (1989) — free at tellerprimer.ucdavis.edu. This track follows the primer’s curriculum; the explanations, examples, and exercises here are original.