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The sentence:
All whales are mammals.
is what's called a CATEGORICAL
SENTENCE; we'll focus on this kind of
sentence when we get to Unit 12 (in Symbolic
Logic II). The first kind of sentence that we'll
focus on in predicate logic is the singular
sentence:
A SINGULAR SENTENCE is a sentence
that contains name(s) and predicate(s).
----->
for example:
Los Angeles is sunny.
- a NAME is an expression that picks out a
particular thing or person, for example:
Russia, the Concorde, Joe Torre; names
sometimes are not capitalized, for example:
e. e. cummings
- a PREDICATE of a singular sentence says
something about a named thing or person
- in our example, the predicate is:
"is sunny"
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SYMBOLIZING simple singular sentences in
predicate logic
Singular sentences can be simple or
compound. Our example:
Los Angeles is sunny.
is a simple sentence. To symbolize this
sentence:
(1) represent the predicate with a
PROPOSITIONAL FUNCTION; this is just a
technical term for what we get if we put an
individual variable in place of the name in a
singular sentence
- an INDIVIDUAL VARIABLE is a
variable that can be replaced by a name;
individual variables are written as lower
case letters: x, y, z
- for our example sentence, the
propositional function is:
x is sunny
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(2) ABBREVIATE the propositional function
by replacing the predicate with a capital letter
(this is called a predicate letter)
----->
example:
x is sunny
- replace "is sunny" with "S", and get:
Sx
(NOTICE that the predicate letter goes
BEFORE the individual variable)
235
REMEMBER, we're symbolizing the
singular sentence:
Los Angeles is sunny.
so we still have to symbolize "Los Angeles"
somehow.
(3) Complete symbolization of the sentence by
ABBREVIATING the name with an individual
constant.
- an INDIVIDUAL CONSTANT
abbreviates a name; individual constants
are written as lower case letters a-w
- we'll abbreviate "Los Angeles" with
"l" and get:
Sl
(NOTICE that the predicate letter goes
BEFORE the individual constant)
236
SUMMARY
In symbolizing:
Los Angeles is sunny
(1) represent the predicate with a propositional
function:
x is sunny
- notice this is a sentence form
(2) abbreviate the propositional function by
replacing predicate with a predicate letter:
Sx
- notice this is still a sentence form
(3) complete symbolization by abbreviating
the name with an individual constant:
Sl
- this is an INSTANCE of the
SENTENCE FORMS in (1) and (2)
237
MORE examples:
(a) The Herald is a newspaper.
(b) Tokyo is populous.
(c) Paris is beautiful.
(d) Bill Gates is wealthy.
(1) propositional functions for these:
(a) x is a newspaper.
(b) y is populous.
(c) x is beautiful.
(d) z is wealthy.
(2) abbreviations for the propositional
functions:
(a) Nx
(b) Py
(c) Bx
(d) Wz
(3) complete symbolization:
(a) Nh
(b) Pt
(c) Bp
(d) Wb
- the symbolizations in (3) are
INSTANCES of the SENTENCE FORMS
in (1) and (2)
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SYMBOLIZING compound singular sentences
in predicate logic
All of the examples of singular sentences
so far have been simple sentences.
(REMEMBER: a simple sentence is one
that does not contain any other
sentence as a component--for example,
dictionary entries used for
symbolization in sen-tential logic are
always simple sentences)
But singular sentences can be compound
sentences.
Symbolizing a compound sentence in
predicate logic is like symbolizing a
compound sentence in sentential logic,
only the simple sentence components are
represented differently:
- in sentential logic, the simple
sentence components are represented
with capital letter abbreviations: A, B,
C, etc.
- in predicate logic, the simple sentence
components are represented with
instances of propositional functions:
Nh, Pt, Bp, Wb, etc.
239
----->
example:
Jill and Stephanie both like to take the
bus to work.
(Paraphrase: Jill likes to take the bus to
work and Stephanie likes to take the
bus to work.)
- set up dictionary:
j = Jill
s = Stephanie
Lx  x likes to take the bus to work
- so we get:
Lj  Ls
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----->
example:
Jim wins the lottery only if he has a
ticket.
- set up dictionary:
j = Jim
Wx  x wins the lottery
Tx  x has a ticket
- so we get:
Wj  Tj
----->
another example:
Jim wins the lottery if he has a ticket.
- set up dictionary:
j = Jim
Wx  x wins the lottery
Tx  x has a ticket
- so we get:
Tj  Wj
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Apply DeM to: ~A  ~B
Apply DeM to: ~~A  ~B
Apply DeM to: ~~A  ~~B
Apply DeM to: ~(A  B)
Apply DeM to: ~(~A  B)
Apply DeM to: ~(~A  ~B)
Apply DeM to: ~~(A  B)
242
Apply CE to: ~A v B
Apply CE to: ~~A v B
Apply CE to: ~~A v ~B
243
Quantifiers
In the following sentences:
SOMETHING is wrong.
EVERYTHING is right.
NOTHING is here.
- the capitalized words are examples of
QUANTIFIER words
- these words are similar to names in
that they function grammatically as the
subjects of these sentences
- but these words are different from
names in that they don't refer:
"something" and "everything" don't
refer to particular things or people;
obviously "nothing" doesn't refer
244
There are two kinds of quantifier words in
English:
1. UNIVERSAL quantifier words: "all,"
"each," "every"
- these words are used to make what are
called UNIVERSAL STATEMENTS
----->
for example:
Everything is right.
- the predicate of this sentence is: "is
right"; as with singular sentences, we
represent this predicate with a
propositional function:
x is right
- quantifier words assert how many things
a propositional function is true of;
universal quantifier words indicate a
propositional function is true of all things
245
2. EXISTENTIAL quantifier words: "some," "a
few," "most"
- these words are used to make what are
called EXISTENTIAL STATEMENTS
----->
for example:
Something is wrong.
- represent the predicate with a
propositional function:
x is wrong
- again, quantifier words assert how many
things a propositional function is true of;
existential quantifier words indicate a
propositional function is true of
some--meaning at least one--thing
- NOTE that in predicate logic, the
word "some" has a technical meaning:
at least one
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Symbolization of simple quantified sentences
- universal and existential statements are
called QUANTIFIED (or general) sentences
- to symbolize SIMPLE quantified sentences:
- FIRST symbolize the quantifier word:
We symbolize UNIVERSAL quantifier
words with an individual variable in
parentheses:
(x)
- this is read "for all x"
We symbolize EXISTENTIAL quantifier
words with a backwards "E" along with an
individual variable in parentheses:
(x)
- this is read "there exists an x such
that"
247
----->
example:
Everything is right.
- in this example, "everything" is
symbolized as:
(x)
- SECOND symbolize the predicate:
- symbolization of predicates of
quantified sentences is done the same
way as symbolization of predicates of
singular sentences
----->
for our example:
Everything is right.
- we represent the predicate with a
propositional function:
x is right
- we abbreviate this propositional
function:
Rx  x is right
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- THIRD, put symbolization of the
quantifier word and predicate together
- for our example, we take the
abbreviation of the propositional
function and tack on a universal
quantifier symbol and get:
(x)Rx
- this is read "for all x, x is right"
249
----->
another example:
Something is wrong.
- FIRST, symbolize the quantifier word:
- in this example, "something" is
symbolized with as:
(x)
- SECOND symbolize the predicate:
- we represent the predicate with a
propositional function:
x is wrong
- we abbreviate this propositional
function:
Wx  x is wrong
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- THIRD, put symbolization of the
quantifier word and predicate together
- for our example, we take the
abbreviation of the propositional
function and tack on an existential
quantifier symbol and get:
(x)Wx
- this is read "there exists an x such that
x is wrong"
251
An important thing to notice:
Propositional functions are not sentences;
for example:
Rx (the abbreviation for "x is rich")
is not a sentence; propositional functions
only represent PREDICATES.
252
But now you've learned two ways to turn a
propositional function INTO a sentence:
FIRST way (Unit 10):
Replace individual variable with a
name:
Rb (where b = Bill Gates)
SECOND way (Unit 11):
Quantify over individual variable:
(x)Rx