Logical Connectives: And Or Not If-Then

Five symbols handle almost all of logical reasoning.

• ∧ means "and" (conjunction)
• ∨ means "or" (disjunction)
• ¬ means "not" (negation)
• → means "if-then" (conditional)
• ↔ means "if and only if" (biconditional)

Each connective transforms truth values in a specific way. Learn how each one works, and you can analyze any argument's logical structure.

These aren't arbitrary symbols. They're precise definitions of what we mean when we say "and," "or," and "if-then." Natural language is fuzzy — these symbols make it exact.

Negation (¬)

Negation flips truth values.

If P is true, ¬P is false. If P is false, ¬P is true.

"Not raining" is true exactly when "raining" is false.

This is the simplest connective: one input, one output, flip the value.

Conjunction (∧)

Conjunction is true only when both parts are true.

P ∧ Q means "P and Q."

"It's raining and the ground is wet" requires both conditions.

One false part makes the whole conjunction false. Both must be true for the conjunction to be true.

Disjunction (∨)

Disjunction is true when at least one part is true.

P ∨ Q means "P or Q (or both)."

"It's raining or the sprinklers are on" — true if either condition holds.

This is inclusive or — true when both are true. The only way for the disjunction to be false is if both parts are false.

(Exclusive or — "one or the other but not both" — exists but is less common. We'd write P ⊕ Q.)

Conditional (→)

The conditional is where things get subtle.

P → Q means "if P, then Q."

A conditional is false only when the antecedent is true and the consequent is false.

The third and fourth rows surprise people. "If pigs fly, then I'm a billionaire" is true? Yes — in logic. The conditional only makes a claim about what happens when P is true. If P is false, the conditional is satisfied by default.

This is called vacuous truth. "All my unicorns are purple" is true — I have no unicorns, so the statement can't be violated.

Why the Conditional Works This Way

Think of P → Q as a promise: "If P happens, Q will happen."

When is the promise broken?

• P happens and Q happens: promise kept (T)
• P happens and Q doesn't: promise broken (F)
• P doesn't happen: promise wasn't tested, not broken (T)

The promise "if it rains, I'll bring an umbrella" isn't broken on sunny days. You didn't claim anything about sunny days.

Biconditional (↔)

The biconditional is true when both sides have the same truth value.

P ↔ Q means "P if and only if Q."

"You pass if and only if you score above 60" — passing requires exactly that, and scoring above 60 guarantees passing.

The biconditional is true when P and Q match — both true or both false. It's equivalent to (P → Q) ∧ (Q → P).

Precedence Rules

When multiple connectives appear, we need order of operations:

1. ¬ (negation) — highest precedence
2. ∧ (conjunction)
3. ∨ (disjunction)
4. → (conditional)
5. ↔ (biconditional) — lowest precedence

So ¬P ∧ Q means (¬P) ∧ Q, not ¬(P ∧ Q).

When in doubt, use parentheses. Clarity beats cleverness.

Translating English

Natural language to logic:

"Neither P nor Q" = ¬P ∧ ¬Q = ¬(P ∨ Q)

"P unless Q" = ¬Q → P (or equivalently, P ∨ Q)

"P only if Q" = P → Q

"P is necessary for Q" = Q → P (Q requires P)

"P is sufficient for Q" = P → Q (P guarantees Q)

The translations aren't always intuitive. "Only if" and "if" go in different directions.

Building Complexity

From these five connectives, we can express any truth function.

Want "exactly one of P, Q, R"? That's: (P ∧ ¬Q ∧ ¬R) ∨ (¬P ∧ Q ∧ ¬R) ∨ (¬P ∧ ¬Q ∧ R)

Want "at least two of P, Q, R"? That's: (P ∧ Q) ∨ (P ∧ R) ∨ (Q ∧ R)

The connectives are building blocks. Combine them to express any logical relationship between propositions.

The Complete Picture

Five connectives. That's the entire vocabulary of propositional logic.

Part 3 of the Logic series.

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