A doctor says: 'The patient has disease A or disease B.' Tests confirm the patient has disease A. Based on logical OR alone, what can the doctor conclude about disease B?
ADisease B is ruled out — once one disjunct is confirmed, the other is excluded
BDisease B cannot be ruled out — logical OR allows both conditions to be true simultaneously
CDisease B is ruled out — 'or' in a medical context always means exactly one condition
DNothing can be concluded about disease B without more clinical information
Logical OR (P ∨ Q) is inclusive: it is true when at least one disjunct is true, including when both are true. P ∨ Q is false only when both P and Q are false. Confirming P makes P ∨ Q true, but says nothing about whether Q is true or false. This diverges from everyday English 'or,' which often implies exclusivity. In medical and scientific reasoning, treating OR as exclusive leads to missed diagnoses — both conditions can coexist.
Question 2 Multiple Choice
The statement 'Not all politicians are corrupt' is equivalent to which of the following?
ANo politicians are corrupt
BAll politicians are not corrupt
CAt least one politician is not corrupt
DMost politicians are not corrupt
Negation applies to the quantifier 'all,' not to 'corrupt.' 'Not all X are Y' means 'there exists at least one X that is not Y' — the negation of a universal claim is an existential claim. This is entirely compatible with most politicians being corrupt. 'No politicians are corrupt' (option A) and 'All politicians are not corrupt' (option B) both assert universal non-corruption, which is a much stronger claim. Option D adds a quantitative judgment ('most') not implied by the original statement.
Question 3 True / False
The statement 'P AND Q' is false whenever P is false, regardless of Q's truth value.
TTrue
FFalse
Answer: True
True. Conjunction (P ∧ Q) requires both parts to be true. If P is false, the compound sentence is false no matter what Q is — there is no truth value of Q that rescues a false P. This can be verified in the truth table: both rows where P = F yield P ∧ Q = F. This 'unanimous agreement' requirement is what makes AND a strong claim.
Question 4 True / False
Logical OR works the same way as everyday English 'or' — exactly one of the two options is expected to be true for the statement to be true.
TTrue
FFalse
Answer: False
False. Logical OR is inclusive: P ∨ Q is true when at least one of P or Q is true, including when both are true. It is false only when both are false. Everyday English 'or' is often exclusive (implying exactly one alternative), but this meaning is not built into logical OR. The distinction matters: 'you can have cake or ice cream' in everyday speech often implies not both, but in formal logic, a disjunction is satisfied even when both options hold.
Question 5 Short Answer
A programmer writes the condition: 'The file is valid if it contains a number OR a letter.' A user submits a file containing both a number and a letter. Using the logical definition of OR, is the file valid? Explain why.
Think about your answer, then reveal below.
Model answer: Yes, the file is valid. Logical OR (disjunction) is true when at least one condition holds, and it is also true when both hold. Since the file contains a number AND a letter, both disjuncts are true — but that is more than sufficient for the OR to be satisfied. The file would only be invalid if it contained neither a number nor a letter.
This is the practical consequence of inclusive OR. If the programmer intended exclusive OR (exactly one but not both), they would need to add an explicit exclusion condition. Many bugs in software arise from programmers who assume OR is exclusive when the language's logical OR is inclusive. The truth table entry for 'T OR T' is T — both conditions being satisfied is always sufficient.