Practice in Logic and Extra Logic Homework 1
B.H. Partee, March 12, 2007 p.1
“Practice in Logic” and “Extra Logic Homework 1”.
This homework is not required; it is just for those of you for whom logic is new, to give you some
extra practice. Answers are provided. So try the problems, look at the answers to check yourself,
and if you have remaining questions, ask me. The two sets are quite similar.
1. Practice in 1st-Order Predicate Logic
Translate the following into 1st-order predicate logic (see Lecture 1). Add “=” to the grammar of
predicate logic (see the way it’s added in Montague’s Intensional logic, lecture 2.)
1.
Mary loves everyone.
[assuming D contains only humans]
2.
Mary loves everyone. [assuming D contains both humans and non-humans, so we
need to be explicit about ‘everyone’ as ‘every person’]
3.
No one talks. [assume D contains only humans unless specified otherwise.]
4.
Everyone loves himself.
5.
Everyone loves everyone.
6.
Everyone loves everyone except himself. (= Everyone loves everyone else.) (This
one needs “=”. See note on problem 14 on the next page.)
7.
Every student smiles.
8.
Every student except George smiles.
9.
Everyone walks or talks.
10.
Every student walks or talks.
11.
Every student who walks talks.
12.
Every student who loves Mary is happy.
13.
Every boy who loves Mary hates every boy who Mary loves.
14.
Every boy who loves Mary hates every other boy who Mary loves.
(So if John loves Mary and Mary loves John, sentence 13 requires that John
hates himself, but sentence 14 doesn’t require that.)
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Practice in Logic and Extra Logic Homework 1
B.H. Partee, March 12, 2007 p.2
Extra Logic Homework #1. (not to hand in)
Use the notation and the predicates and constants given in the handout in which we did the
syntax and semantics of predicate logic (Lecture 1). Add additional constants as needed.
Translate the following English sentences into first-order predicate calculus. If any of the
English sentences are ambiguous with respect to their ‘logical structure’, give two
different translations that capture their two interpretations. (You are free to assume that D
is a domain of humans, so that quantifier expressions like “No one”, “everyone”, etc., can
be represented without the need for an additional predicate person.)
Feel free to send me notes, comments, and questions. There is not always a single right
answer, and the meanings of the English sentences are not always clear-cut, e.g. when to
interpret “if” as “if and only if”. If, after checking the answer sheet, you have questions
about the interpretation of the sentence, or about whether what you wrote in your formula
corresponds to how you in fact interpreted the sentence, ask me.
1. Everyone loves Mary.
2. John does not love anyone. (Not ambiguous, but there are two equivalent and equally
good formulas for it, one involving negation and the existential quantifier, the other
involving negation and the universal quantifier. Give both.)
3. Everyone who sees Mary loves Mary.
4. Everyone loves someone. (Ambiguous)
5. Someone loves everyone. (Ambiguous)
6. Someone walks and talks.
7. Someone walks and someone talks.
8. Everyone who walks is calm.
9. No one who runs walks. (Not ambiguous, but same note as for number 2.)
10. Everyone who Mary loves loves someone who is happy.
11. If anyone cheats, he suffers.
12. If anyone cheats, everyone suffers.
13. Anyone who loves everyone loves himself.
14. Mary loves everyone except John. (For this one, you need to add the two-place
predicate of identity, “=”. Think of “everyone except John” as “everyone who is not
identical to John”.)
15. Redo the translations of sentences 1, 4, 6, and 7, making use of the predicate person,
as we would have to do if the domain D contains not only humans but cats, robots, and
other entities.
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