Logical Connectives — Mathematics Knowledge System

Overview

A logical connective is an operation that combines one or more propositions into a new proposition, whose truth value is determined entirely by the truth values of its inputs. The standard connectives are negation (¬, unary), conjunction (∧, 'and'), disjunction (∨, 'or', inclusive), the conditional (→, 'implies'), and the biconditional (↔, 'if and only if'); XOR and NAND/NOR are common derived or minimal alternatives. Every one of these connectives is truth-functional — its output depends only on the truth values of its arguments, never on their content or context — which is what makes truth tables a complete method for evaluating any propositional formula. A small set of connectives, such as {¬, ∧} or the single NAND connective, is functionally complete, meaning every possible truth function can be built from it alone.

Intuition

Connectives are the logical equivalent of arithmetic operators: just as + and × combine numbers into new numbers, ∧, ∨, ¬, →, and ↔ combine truth values into new truth values, following fixed rules with no ambiguity. The conditional p → q is the one that trips people up — it's defined to be true whenever p is false, regardless of q, which feels odd in English but is exactly what's needed for it to match statements like 'if it's a square, it's a rectangle': the statement isn't lying just because we're talking about a shape that isn't a square at all.

Formal Definition

Definition

The five standard connectives, defined by their truth conditions:

\neg p \text{ is true} \iff p \text{ is false}

Negation

p \land q \text{ is true} \iff p \text{ is true and } q \text{ is true}

Conjunction

p \lor q \text{ is true} \iff p \text{ is true or } q \text{ is true (or both)}

Disjunction (inclusive)

p \rightarrow q \text{ is false} \iff p \text{ is true and } q \text{ is false}

Conditional (only case it's false)

Worked Examples

p is false, so p → q is automatically true (a false antecedent makes any conditional vacuously true).
$p \rightarrow q = \text{T}$
p ∧ q requires both to be true; p is false, so the conjunction is false.
$p \land q = \text{F}$

Answer: p → q is true; p ∧ q is false.

Practice Problems

Difficulty 1/10

If p is true and q is false, what is the truth value of p ↔ q?

Difficulty 2/10

Express the exclusive or (XOR) of p and q, which is true exactly when p and q differ, using ∧, ∨, and ¬.

Difficulty 3/10

Show that the single connective NAND (written p ↑ q ≡ ¬(p ∧ q)) can express negation alone.

Quiz

Summary

Logical connectives (¬, ∧, ∨, →, ↔) are truth-functional operations: their output depends only on the truth values of the inputs.
The conditional p → q is true whenever p is false, which can feel counterintuitive but matches its intended logical role (vacuous truth).
A small set like {¬, ∧} or even the single NAND connective is functionally complete — sufficient to express every possible truth function.

References

WebsiteWikipedia — Logical connective