Logical Connectives and Truth ACET Reviewer: Lesson and Practice

TEACHER ABI ACET LOGICAL REASONING

Logical Connectives and Truth

Evaluate compound statements built with AND, OR, NOT, IF–THEN, and IF AND ONLY IF.

One focused competency6 worked examples27 original questionsEasy to ACET-level

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Logical Connectives and Truth

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One connective changes the truth rule

Suppose these are the two simple statements:

P: The library is open.Q: The study hall is open.

P AND Q is true only when both places are open. P OR Q is normally true when at least one is open. NOT P reverses P’s truth value.

P ∧ Q = both
P ∨ Q = at least one
¬P = not P

ACET questions often use words, not symbols. Translate the sentence first; then apply the truth rule.

DO IT FAST

Use PARTS → CONNECTIVE → FAILURE CASE

1. PARTS: Separate the compound statement into P and Q.
2. CONNECTIVE: Identify and, or, not, if–then, or if and only if.
3. FAILURE CASE: Ask what exact combination makes the compound statement false.

AND fails if either part is false.
OR fails only if both parts are false.
P → Q fails only when P is true and Q is false.

WORKED EXAMPLES

Six forms you must recognize

1. AND

The pass is signed and dated.

If it is signed but not dated, the compound statement is false.

Both parts must be true.

2. Inclusive OR

Applicants may submit a passport or a national ID.

In standard logic, “or” allows either one or both unless the wording excludes both.

At least one is enough.

3. Exclusive OR

Choose either the morning session or the afternoon session, but not both.

Exactly one choice must be true.

“But not both” makes OR exclusive.

4. NOT and De Morgan

It is not true that the room is warm and bright.

At least one part fails: the room is not warm, or it is not bright, or both.

¬(P ∧ Q) = ¬P ∨ ¬Q

5. IF–THEN

If the alarm is armed, the indicator glows.

The rule is broken only by an armed alarm with no glow. An unarmed alarm does not violate the rule, whatever the indicator does.

P → Q is false only when P is true and Q is false.

6. IF AND ONLY IF

The door unlocks if and only if the correct code is entered.

The two parts must have the same truth value: both true or both false.

P ↔ Q = (P → Q) AND (Q → P)

GUIDED TRY

Find the failure case

If the permit is active, its barcode scans.The permit is active, but its barcode does not scan.Is the conditional rule true in this case?

1. The trigger P is true.
2. The required result Q is false.
3. This is the one combination that breaks P → Q.

No. A true trigger followed by a false result makes the conditional false.

COMMON TRAPS

Read the connective exactly

OR means exactly one: Standard OR includes the possibility of both.

AND survives one false part: It does not; both must be true.

Negating only one side: De Morgan changes the connective too.

False trigger makes implication false: A conditional is violated only by true P and false Q.

Biconditional is one-way: It requires both directions.

Ignoring scope: “Not both” differs from “neither.”

FIVE-FORM SKILL CHECK

Do you need Foundations or Core Practice?

CHOOSE YOUR PRACTICE

Build the skill in stages

Foundations

AND, OR, and NOT in plain language.

Core

Conditionals, biconditionals, and negation.

ACET Transfer

Nested statements and truth conditions.

FRESH MASTERY CHECK

Can you decide without the scaffolding?

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QUICK ANSWERS

Logical Connectives FAQ

Does OR allow both?

Yes, in standard logic. It becomes exclusive only when wording such as “but not both” or “exactly one” appears.

When is AND true?

Only when every joined statement is true.

How do I negate AND?

“Not both P and Q” becomes “not P or not Q.” At least one fails.

How do I negate OR?

“Not P or Q” as the negation of the whole disjunction becomes “not P and not Q.” Both must fail.

When is an implication false?

Only when its trigger is true but its promised result is false.

When is a biconditional true?

When both component statements have the same truth value.

RELATED LOGICAL REASONING COMPETENCIES

Continue the reasoning pathway

ConditionalsNecessary and sufficient conditionsSyllogismsValid deductionsDefinitionsCounterexamples

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