Logical connective: meaning and examples

A logical connective is an operator that joins propositions into a compound whose truth value is fixed by the truth values of its parts: AND, OR, NOT, the conditional ->, the biconditional <-> and XOR are the usual ones.

Connectives are the propositional-logic name for Boolean operators, and each is defined by a truth table: AND is true only when both parts are, OR when at least one is, the conditional fails only when its antecedent is true and its consequent false, and the biconditional holds when both sides agree. NOT is the one unary connective. Because every connective is truth-functional, a compound's value on any row is computed from the inside out, which is what the engine's evaluator does.

The artefact on this page uses three connectives at once: (p -> q) AND (q XOR r). The engine's reading separates them: it holds when, if p is true then q is true, and exactly one of q and r is true. The 8-row table reads true on rows 1, 2 and 6. The satisfiable check holds with the witness p = FALSE, q = FALSE, r = TRUE, where the conditional is true because its antecedent is false; the tautology check is violated with the counterexample p = FALSE, q = FALSE, r = FALSE, where the XOR fails.

Connectives are the first thing a logic course teaches and the source of most translation errors, because natural language has more of them than logic does (“unless”, “but”, “only if”, “neither”) and each must be mapped to one of the few truth functions. In programming they are the Boolean operators of the language; in hardware, the gates. The term is confused with a conjunction in grammar and with a relation such as equality, which connects values rather than propositions.

Example

f := (p -> q) AND (q XOR r) · output

Exact checks

Always true (tautology)? No Counterexample: p = F, q = F, r = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: p = F, q = F, r = T · Exhaustively checked and solver cross-checked · 2 rows

Can be true? Yes Example assignment: p = F, q = F, r = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (8 rows)

Logical connective: meaning and examples
#	p	q	r	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	FALSE
5	TRUE	FALSE	TRUE	FALSE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	FALSE

Plain-language reading

Beginnerf is true exactly when both of the following hold:
1. if p is true, then q is true
2. exactly one of q and r is true (not both)
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.
- An “if … then” condition only constrains the case where its “if” part holds: when the “if” part is false, the condition is satisfied whatever the “then” part is. It does not say the reverse.
- XOR (exclusive or) is true when exactly one of its two parts is true.

Intermediatef is true exactly when both of the following hold:
1. if p is true, then q is true
2. exactly one of q and r is true (not both)
In every other case, f is false.

Advancedf is true exactly when both of the following hold [conjunction ∧]:
1. if p is true, then q is true [material implication →]
2. exactly one of q and r is true (not both) [exclusive disjunction ⊕]
In every other case, f is false.
Formal notation: (p → q) ∧ (q ⊕ r)

The common slip

Treating “or” as exclusive because the sentence sounds like a choice. The connective OR is inclusive; XOR is the exclusive one, and the two differ exactly on the row where both parts are true. The comparison page on XOR versus OR shows that row.

Questions

How many connectives does logic need?

Two suffice, for example NOT and AND, and one does if it is NAND or NOR. The others are kept because they match how people speak: a conditional is clearer than its equivalent NOT p OR q, and the engine's elimination laws translate between them.

Practice

Logic exercises for philosophy courses, with answers

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.