PHIL 12A // Answers, 25 January 2011
Julian Jonker
1
How much do you know?
Consider a firstorder language that has the names
Jesse
,
Kendall
,
Lou
,
Maria
and
Nate
, and the function symbols
leftof(x)
referring to the person to the left of
x (from your perspective), and
rightof(x)
referring to the person to the right of x
(from your perspective). Furthermore the language has the predicate
Taller(x,y)
,
meaning x is taller than y, and
x=y
, meaning x and y are the exact same person.
1. Formalize the following English sentences:
(a) The person standing to the left of the person standing to the left of Nate is
standing to the right of Maria.
leftof(leftof(Nate))=rightof(Maria)
(b) The person standing to the right of Kendall is shorter than Maria.
Taller(Maria, rightof(Kendall))
(c) Jesse is taller than Maria.
Taller(Jesse, Maria)
(d) Lou is shorter than Kendall.
Taller(Kendall, Lou)
2. Jesse, Kendall, Lou, Maria and Nate (and nobody else) are lined up from tallest
(on your left) to shortest (on your right). In what order are they lined up if the
above sentences are all true?
Jesse, Maria, Kendall, Lou, Nate
1
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2
Something slightly harder, if there’s time.
The following question is based on section 1.7 of Barwise and Etchemendy. The first
order language of arithmetic has two names,
0
and
1
, two binary relation symbols
=
and
<
, and two function symbols
+
and
×
. Terms are defined inductively as follows:
•
The names
0
and
1
are terms.
•
If
t
1
and
t
2
are terms, then so are
(
t
1
+
t
2
)
and
(
t
1
×
t
2
)
.
•
Nothing else is a term.
1. Show that there are infinitely many terms in the firstorder language of arithmetic
referring to the number one.
Since
0
and
1
are terms by the first clause of the definition,
(0 + 1
) is a
term by the second clause, and it refers to the number 1. But now, by the
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