04 / 18 · Make the structure visible
Not, and, or
Translate simple combinations without importing everyday ambiguity.
Builds on The missing bridge
Go to practice ↓A question to keep in mind
If you may submit a PDF or a text file, does 'or' forbid submitting both?
In classical two-valued logic each statement is assigned true (T) or false (F). Letters such as P and Q stand for whole statements. Negation, written ¬P, reverses P's truth value. P ∧ Q is true only when both are true.
Disjunction P ∨ Q uses inclusive 'or': at least one is true, possibly both. Exclusive 'or' requires one but not both, which can be written (P ∨ Q) ∧ ¬(P ∧ Q). Real instructions may intend either reading; clarify them before translating.
Parentheses determine scope. ¬(P ∧ Q) says they are not both true; it does not say they are both false. We deliberately write parentheses around compound parts instead of relying on a reader to guess the grouping.
Work through an example
- Let P mean 'a PDF was submitted' and Q mean 'a text file was submitted'. If both were submitted, P ∨ Q is true and P ∧ Q is true.
- If only the PDF was submitted, ¬(P ∧ Q) is true, but ¬P ∧ ¬Q is false.
- Unknown is not the same as false. The table lists hypothetical assignments; it does not decide which real upload occurred.
Your turn
0 / 3Complete the table for the three connectives.
Read solution · does not award completion
- P=T, Q=T →
¬P: F;P ∧ Q: T;P ∨ Q: T - P=T, Q=F →
¬P: F;P ∧ Q: F;P ∨ Q: T - P=F, Q=T →
¬P: T;P ∧ Q: F;P ∨ Q: T - P=F, Q=F →
¬P: T;P ∧ Q: F;P ∨ Q: F
Negation flips P. Conjunction needs two T values. Inclusive disjunction fails only at F,F.
P is true and Q is false. Which is true?
Read solution · does not award completion
¬(P ∧ Q)
'Not both' is weaker than 'neither'. This distinction matters in rules and eligibility conditions.
A form says 'choose exactly one: public or private'. Which expression matches?
Read solution · does not award completion
(P ∨ Q) ∧ ¬(P ∧ Q)
Translate the actual requirement, not just the word 'or'.
Bring it back to your own work
Find a rule containing 'and' or 'or'. Write two interpretations and a case on which they differ.
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