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04 / 18 · Make the structure visible

Not, and, or

Translate simple combinations without importing everyday ambiguity.

Builds on The missing bridge

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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

  1. 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.
  2. If only the PDF was submitted, ¬(P ∧ Q) is true, but ¬P ∧ ¬Q is false.
  3. Unknown is not the same as false. The table lists hypothetical assignments; it does not decide which real upload occurred.

Your turn

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Practice 1Not checked

Complete the table for the three connectives.

Your answer
T = true · F = false
PQ¬PP ∧ QP ∨ Q
TT
TF
FT
FF
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.

Practice 2Not checked

P is true and Q is false. Which is true?

Your answer
Read solution · does not award completion

¬(P ∧ Q)

'Not both' is weaker than 'neither'. This distinction matters in rules and eligibility conditions.

Practice 3Not checked

A form says 'choose exactly one: public or private'. Which expression matches?

Your answer
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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