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generally don™t capitalize them. For example, we might use max as an individ-
ual constant to denote a particular person, named Max, or 1 as an individual
constant to denote a particular number, the number one. In either case, they
would basically work exactly the way names work in English. Our blocks



19
20 / Atomic Sentences


language takes the letters a through f plus n1, n2 , . . . as its names.
The main di¬erence between names in English and the individual constants
of fol is that we require the latter to refer to exactly one object. Obviously,
names in fol
the name Max in English can be used to refer to many di¬erent people, and
might even be used twice in a single sentence to refer to two di¬erent people.
Such wayward behavior is frowned upon in fol.
There are also names in English that do not refer to any actually existing
object. For example Pegasus, Zeus, and Santa Claus are perfectly ¬ne names
in English; they just fail to refer to anything or anybody. We don™t allow such
names in fol.1 What we do allow, though, is for one object to have more than
one name; thus the individual constants matthew and max might both refer
to the same individual. We also allow for nameless objects, objects that have
no name at all.

Remember

In fol,

—¦ Every individual constant must name an (actually existing) object.

—¦ No individual constant can name more than one object.

—¦ An object can have more than one name, or no name at all.




Section 1.2
Predicate symbols

Predicate symbols are symbols used to express some property of objects or
some relation between objects. Because of this, they are also sometimes called
predicate or relation
symbols relation symbols. As in English, predicates are expressions that, when com-
bined with names, form atomic sentences. But they don™t correspond exactly
to the predicates of English grammar.
Consider the English sentence Max likes Claire. In English grammar, this
is analyzed as a subject-predicate sentence. It consists of the subject Max
followed by the predicate likes Claire. In fol, by contrast, we view this as
a claim involving two “logical subjects,” the names Max and Claire, and a
logical subjects
1 There is, however, a variant of ¬rst-order logic called free logic in which this assumption
is relaxed. In free logic, there can be individual constants without referents. This yields a
language more appropriate for mythology and ¬ction.




Chapter 1
Predicate symbols / 21



predicate, likes, that expresses a relation between the referents of the names.
Thus, atomic sentences of fol often have two or more logical subjects, and the
predicate is, so to speak, whatever is left. The logical subjects are called the
“arguments” of the predicate. In this case, the predicate is said to be binary, arguments of a
predicate
since it takes two arguments.
In English, some predicates have optional arguments. Thus you can say
Claire gave, Claire gave Scru¬y, or Claire gave Scru¬y to Max. Here the
predicate gave is taking one, two, and three arguments, respectively. But in
fol, each predicate has a ¬xed number of arguments, a ¬xed arity as it is arity of a predicate
called. This is a number that tells you how many individual constants the
predicate symbol needs in order to form a sentence. The term “arity” comes
from the fact that predicates taking one argument are called unary, those
taking two are binary, those taking three are ternary, and so forth.
If the arity of a predicate symbol Pred is 1, then Pred will be used to
express some property of objects, and so will require exactly one argument (a
name) to make a claim. For example, we might use the unary predicate symbol
Home to express the property of being at home. We could then combine this
with the name max to get the expression Home(max), which expresses the
claim that Max is at home.
If the arity of Pred is 2, then Pred will be used to represent a relation
between two objects. Thus, we might use the expression Taller(claire, max) to
express a claim about Max and Claire, the claim that Claire is taller than
Max. In fol, we can have predicate symbols of any arity. However, in the
blocks language used in Tarski™s World we restrict ourselves to predicates
with arities 1, 2, and 3. Here we list the predicates of that language, this time
with their arity.

Arity 1: Cube, Tet, Dodec, Small, Medium, Large

Arity 2: Smaller, Larger, LeftOf, RightOf, BackOf, FrontOf, SameSize, Same-
Shape, SameRow, SameCol, Adjoins, =

Arity 3: Between

Tarski™s World assigns each of these predicates a ¬xed interpretation, one
reasonably consistent with the corresponding English verb phrase. For exam-
ple, Cube corresponds to is a cube, BackOf corresponds to is in back of, and
so forth. You can get the hang of them by working through the ¬rst set of
exercises given below. To help you learn exactly what the predicates mean,
Table 1.1 lists atomic sentences that use these predicates, together with their
interpretations.
In English, predicates are sometimes vague. It is often unclear whether vagueness




Section 1.2
22 / Atomic Sentences



Table 1.1: Blocks language predicates.

Atomic
Sentence Interpretation
Tet(a) a is a tetrahedron
Cube(a) a is a cube
Dodec(a) a is a dodecahedron
Small(a) a is small
Medium(a) a is medium
Large(a) a is large
SameSize(a, b) a is the same size as b
SameShape(a, b) a is the same shape as b
Larger(a, b) a is larger than b
Smaller(a, b) a is smaller than b
SameCol(a, b) a is in the same column as b
SameRow(a, b) a is in the same row as b
a and b are located on adjacent (but
Adjoins(a, b)
not diagonally) squares
a is located nearer to the left edge of
LeftOf(a, b)
the grid than b
a is located nearer to the right edge
RightOf(a, b)
of the grid than b
a is located nearer to the front of the
FrontOf(a, b)
grid than b
a is located nearer to the back of the
BackOf(a, b)
grid than b
a, b and c are in the same row, col-
Between(a, b, c) umn, or diagonal, and a is between b
and c



an individual has the property in question or not. For example, Claire, who
is sixteen, is young. She will not be young when she is 96. But there is no
determinate age at which a person stops being young: it is a gradual sort of
thing. Fol, however, assumes that every predicate is interpreted by a deter-
minate property or relation. By a determinate property, we mean a property
determinate property
for which, given any object, there is a de¬nite fact of the matter whether or
not the object has the property.
This is one of the reasons we say that the blocks language predicates are



Chapter 1
Atomic sentences / 23



somewhat consistent with the corresponding English predicates. Unlike the
English predicates, they are given very precise interpretations, interpretations
that are suggested by, but not necessarily identical with, the meanings of the
corresponding English phrases. The case where the discrepancy is probably
the greatest is between Between and is between.

Remember

In fol,

—¦ Every predicate symbol comes with a single, ¬xed “arity,” a number
that tells you how many names it needs to form an atomic sentence.

—¦ Every predicate is interpreted by a determinate property or relation
of the same arity as the predicate.




Section 1.3
Atomic sentences
In fol, the simplest kinds of claims are those made with a single predicate
and the appropriate number of individual constants. A sentence formed by a
predicate followed by the right number of names is called an atomic sentence. atomic sentence
For example Taller(claire, max) and Cube(a) are atomic sentences, provided
the names and predicate symbols in question are part of the vocabulary of
our language. In the case of the identity symbol, we put the two required
names on either side of the predicate, as in a = b. This is called “in¬x” no- in¬x vs. pre¬x notation
tation, since the predicate symbol = appears in between its two arguments.
With the other predicates we use “pre¬x” notation: the predicate precedes
the arguments.
The order of the names in an atomic sentence is quite important. Just
as Claire is taller than Max means something di¬erent from Max is taller
than Claire, so too Taller(claire, max) means something completely di¬erent
than Taller(max, claire). We have set things up in our blocks language so that
the order of the arguments of the predicates is like that in English. Thus
LeftOf(b, c) means more or less the same thing as the English sentence b is
left of c, and Between(b, c, d) means roughly the same as the English b is
between c and d.
Predicates and names designate properties and objects, respectively. What



Section 1.3
24 / Atomic Sentences


makes sentences special is that they make claims (or express propositions).
claims
A claim is something that is either true or false; which of these it is we call
its truth value. Thus Taller(claire, max) expresses a claim whose truth value is
truth value
true, while Taller(max, claire) expresses a claim whose truth value is false.
(You probably didn™t know that, but now you do.) Given our assumption
that predicates express determinate properties and that names denote de¬nite
individuals, it follows that each atomic sentence of fol must express a claim
that is either true or false.


You .try. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . it

1. It is time to try your hand using Tarski™s World. In this exercise, you
will use Tarski™s World to become familiar with the interpretations of the
atomic sentences of the blocks language. Before starting, though, you need
to learn how to launch Tarski™s World and perform some basic operations.
Read the appropriate sections of the user™s manual describing Tarski™s
World before going on.

2. Launch Tarski™s World and open the ¬les called Wittgenstein™s World and
Wittgenstein™s Sentences. You will ¬nd these in the folder TW Exercises. In
these ¬les, you will see a blocks world and a list of atomic sentences. (We
have added comments to some of the sentences. Comments are prefaced
by a semicolon (“;”), which tells Tarski™s World to ignore the rest of the
line.)

3. Move through the sentences using the arrow keys on your keyboard, men-
tally assessing the truth value of each sentence in the given world. Use
the Verify button to check your assessments. (Since the sentences are all
atomic sentences the Game button will not be helpful.) If you are sur-
prised by any of the evaluations, try to ¬gure out how your interpretation
of the predicate di¬ers from the correct interpretation.

4. Next change Wittgenstein™s World in many di¬erent ways, seeing what hap-
pens to the truth of the various sentences. The main point of this is to
help you ¬gure out how Tarski™s World interprets the various predicates.
For example, what does BackOf(d, c) mean? Do two things have to be in
the same column for one to be in back of the other?

5. Play around as much as you need until you are sure you understand the
meanings of the atomic sentences in this ¬le. For example, in the original
world none of the sentences using Adjoins comes out true. You should try



Chapter 1
Atomic sentences / 25


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