1
It must assert something that can be judged.
"7 is a prime number."
proposition ✓
true
"10 is an odd number."
proposition ✓
false, but still a proposition
"Study hard!"
not a proposition ✗
a command asserts nothing
"n − 4 = 10"
not a proposition ✗
an open sentence: true or false only once n is fixed
Being false does not disqualify a sentence. Being unjudgeable does. Commands,
questions, exclamations and opinions with no fact behind them are all out.
2
Five operators build every compound statement.
¬p
not p
negation
p ∧ q
p and q
conjunction
p ∨ q
p or q
disjunction
p → q
if p then q
conditional
p ↔ q
p if and only if q
biconditional
order
¬ then ∧ then ∨ then → then ↔
brackets override
3
Translating means naming the simple parts first.
p: the jeepney is full
one simple proposition
q: we will walk
the other
Keep the negatives out of the letters. Setting p to "the jeepney is NOT full" makes every
later step harder to read, because ¬p then means "the jeepney is full".
4
"Or" in logic includes both.
p ∨ q, p true only
true
at least one holds
p ∨ q, both true
true
still counts
p ∨ q, neither
false
the only false case
Everyday "or" often means one or the other, not both
"Tea or coffee?" expects one answer. Logic's ∨ is the inclusive one: it is satisfied
by either, and by both. When a problem means strictly one, it has to say so.
5
Variations: negating "and", negating "or", negating "no".
Negating "and": "It is sunny and it is warm."
p ∧ q
p: it is sunny, q: it is warm
¬(p ∧ q) ≡ ¬p ∨ ¬q
one failure is enough
"It is not sunny or it is not warm."
the "and" turns into "or"
To break an "and", only one part needs to fail. "It is not sunny and not warm" claims far more than the negation does.
Negating "or": "Ana will sing or Ana will dance."
p ∨ q
p: Ana will sing, q: Ana will dance
¬(p ∨ q) ≡ ¬p ∧ ¬q
both must fail
"Ana will not sing and Ana will not dance."
the "or" turns into "and"
An inclusive "or" is false only when neither part holds, so its negation says neither.
Negating "no": "No jeepney is late."
"no" claims zero late jeepneys
the original
negation: at least one jeepney is late
one is enough to break it
This is the "some" case run backwards: the negation of "none" is "at least one", not "all".
6
Negating "all" gives "at least one is not".
"All students passed"
negation: at least one did not
NOT "no student passed"
"Some tricycles are electric"
negation: no tricycle is electric
"some" means at least one
"x < −2"
negation: x ≥ −2
the "or equal" comes back
"Manila is the capital of the Philippines" is the proposition
To break "all students passed" you need only one student who did not. Claiming "no
student passed" is a far stronger statement, and it is not what the original denies
— this is the single most common mistake in the whole topic.
On the exam
Periodical exams ask you to pick out the propositions, name or translate the operators, and write a negation in words; entrance tests concentrate on negating "all", "some" and "none", and on the inclusive "or". Name the simple parts first, then ask what single case would break the statement. For a 10-second check, the original and your negation must never be true together and never be false together.
Remember the flow
Can it be judged true or false? · Name the simple parts · Join with the right operator · Negate "all" to "at least one is not"
An open sentence with a free variable is not a proposition until the variable is fixed.
