232
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"
233
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
234
(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)
238
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
240
----->
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
241
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
246
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
248
- 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
250
- 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
253
Some points about the GRAMMAR of
predicate logic:
- abbreviated propositional functions by
themselves ARE grammatical:
Fx
- this isn't a sentence, but it is a
grammatical formula
- but quantifier symbols by themselves are
NOT grammatical:
(x)
- this symbolizes a universal quantifier
word such as "everything," and by
itself is not a grammatical formula
254
- quantifier symbols tacked onto predicate
letters without individual variables are
NOT grammatical:
(x)G
- quantifier symbols with individual
constants are NOT grammatical:
(a)Gx
KEEP IN MIND that in PREDICATE logic
capital letter abbreviations replace
predicates (like "is sunny") and are not
abbreviations for simple sentences
255
Quantifier scope
The SCOPE of a quantifier is the first
complete formula following the quantifier.
Keep in mind: abbreviated propositional functions are complete formulas
----->
examples:
(x)Fx
- Fx is within the scope of (x)
(y)Fy  Gy
- Fy is within the scope of (y),
but Gy isn't
(z)(Fz  Gz)
- the whole parenthetical
statement following (z) is
within the scope of (z)
- note: the scope of a quantifier is as
extensive as the scope of a tilde;
compare the above examples with:
~Fx
~Fy  Gy
~(Fz  Gz)
256
Bound and free variables
A BOUND variable is a variable that
falls within the scope of its own
quantifier.
----->
examples:
(x)Fx
- since the x in Fx is within the
scope of (x), this x is a bound
variable
(y)Fy  Gy
- since the y in Gy is not within
the scope of (y), this y isn't a
bound variable
A FREE variable is a variable that does
not fall within the scope of its own
quantifier.
----->
example:
(y)Fy  Gy
- since the y in Gy is not within
the scope of (y), it is a free
variable
257
----->
more examples:
(x)(Fx  Gy)
- the y in Gy is within the scope
of (x), but it is free because to
be bound it must fall within the
scope of its OWN quantifier,
that is, a quantifier with a y in it
(x)(y)(Fx  Gy)
- here both the x in Fx and the y
in Gy are bound
- note that the first complete
formula following (x) is
NOT (y)--because (y) is not a
complete formula
- the formula
(x)(y)(Fx  Gy) is read "for
all x and for all y, if x is an F
then y is a G"
258
Negated quantifiers
- we still haven't symbolized a sentence like:
Nothing is settled.
How do we do this???
We can do this two ways...
259
FIRST way:
Consider that to say:
Nothing is settled.
is just to say:
It is not the case that something is
settled.
And now we can symbolize this:
"It is not the case that" with ~
"something" with (x)
"is settled" with Sx
Put it together: ~(x)Sx
Read: "it is not the case that there
exists an x such that x is settled"
- here we have negated an existential
statement
260
But there's a second way to symbolize this
sentence.
SECOND way:
Consider that to say:
Nothing is settled.
is just to say:
Everything is not settled.
And now we can symbolize this:
"Everything" with (x)
"not" with ~
"is settled" with Sx
Put it together: (x)~Sx
Read: "for all x, it is not the case
that x is settled"
- note that we haven't negated a universal
statement, but rather the propositional
function Sx
261
The sentences:
It is not the case that something is settled.
Everything is not settled.
are equivalent. And we can generalize their
equivalence with the following
QUANTIFIER NEGATION EQUIVALENCE:
~(x)x  (x)~x
- here the symbol x stands for any
propositional function simple or
compound (just like p stands for any
statement simple or compound)
- this equivalence, along with three
other QN equivalences, will be
replacement rules in Unit 15
262
Quantifier Negation (QN) equivalences
(1)
~(x)x  (x)~x
-----> example:
(a) It is not the case that something is clear.
Symbolize:
"It is not the case that" with ~
"something" with (x)
"is clear" with Cx
~(x)Cx
(b) Everything is not clear.
Symbolize:
"Everything" with (x)
"not" with ~
"is clear" with Cx
(x)~Cx
NOTE: Both (a) and (b) are paraphrases of:
Nothing is clear.
NOTE: (a) is a negated existential
statement; (b) is a universal statement with
a negated propositional function
263
(2)
~(x)x  (x)~x
(a) It is not the case that everything is
funny.
Symbolize:
"It is not the case that" with ~
"everything" with (x)
"is funny" with Fx
~(x)Fx
(b) Something is not funny.
Symbolize:
"Something" with (x)
"not" with ~
"is funny" with Fx
(x)~Fx
NOTE: (a) is a negated universal
statement; (b) is an existential statement
with a negated propositional function
264
There are two more QN equivalences:
(3)
~(x)~x  (x)x
- this makes sense given the following two
equivalences:
(i) ~(x)~Fx  (x)~~Fx
- you get the formula on the right from the
formula on the left using QN (1):
~(x)x  (x)~x
Note: x stands for any propositional
function simple or compound (just like
p stands for any statement simple or
compound); in this example x stands
for ~Fx
(ii) (x)~~Fx  (x)Fx
- you get the formula on the right from the
formula on the left using DN
- result: ~(x)~Fx  (x)Fx
Note: QN equivalences are
REPLACEMENT RULES, just like DN
265
----->
example of ~(x)~x  (x)x:
(a) It's not the case that something does not
have a purpose.
Symbolize:
"It's not the case that" with ~
"something" with (x)
"not" with ~
"does have a purpose" with Px
~(x)~Px
(b) Everything has a purpose.
Symbolize:
"Everything" with (x)
"has a purpose" with Px
(x)Px
266
(4)
~(x)~x  (x)x
- this makes sense given the following two
equivalences:
(i) ~(x)~Fx  (x)~~Fx
- you get the formula on the right from the
formula on the left using QN (2):
~(x)x  (x)~x
Note: x stands for any propositional
function simple or compound (just like
p stands for any statement simple or
compound); in this example x stands
for ~Fx
(ii) (x)~~Fx  (x)Fx
- you get the formula on the right from the
formula on the left using DN
- result: ~(x)~Fx  (x)Fx
267
----->
example of ~(x)~x  (x)x:
(a) It is not the case that everything is not
clear.
Symbolize:
"It is not the case that" with ~
"everything" with (x)
"not" with ~
"is clear" with Cx
~(x)~Cx
(b) Something is clear
Symbolize:
"Something" with (x)
"is clear" with Cx
(x)Cx
268
Categorical sentences (also called categorical
propositions)
CATEGORICAL SENTENCES are sentences
where the subject expression and the
predicate expression both refer to classes (or
categories or groups) of things and the
sentence states an inclusion or exclusion
relation between these classes.
----->
example:
All whales are mammals.
269
So now we have three types of sentences in
predicate logic:
SINGULAR sentences
where the subject expressions are names, that
is, terms that refer to particular things or
people
----->
example:
Wisconsin is cold.
SIMPLE QUANTIFIED sentences
where the subject expressions don't refer, but
rather make an assertion with respect to how
many things a propositional function is true
of.
----->
examples:
Something is an animal.
Everything is made of quarks.
CATEGORICAL sentences
where the subject expressions are class terms,
that is, terms that refer to classes (or categories
or groups) of things
----->
examples:
All cats are animals.
Some animals are warm blooded.
270
Simple quantified sentences make an
assertion with respect to how many things OF
ANY CLASS (or category or group) a
propositional function is true of.
----->
examples:
Something is a mineral.
Everything has a purpose.
By contrast, categorical sentences make an
assertion with respect to how many things OF
A SPECIFIED CLASS (or category or group) a
propositional function is true of.
----->
examples:
Some minerals are valuable.
All tigers are ferocious.
- in the first example, the sentence asserts that
the propositional function x is a valuable
thing is true of SOME things of the MINERAL
CLASS
- in the second example, the sentence asserts
that the propositional function x is a ferocious
thing is true of ALL things of the TIGER
CLASS
271
Symbolization of categorical sentences in
predicate logic
BACKGROUND
- we represent class terms with propositional
functions, whether class terms are subject
expressions or predicate expressions
- since categorical sentences relate classes,
symbolizations of categorical sentences
relate propositional functions
- since symbolizations of categorical
sentences relate propositional functions,
they are COMPOUND QUANTIFIED
sentences
----->
examples:
(x)(Sx  Px)
(y)(Sy  Py)
- NOTE that in these examples, the
scopes of the quantifiers extend over
compound formulas
(REMEMBER: the scope of a
quantifier is the first complete formula
following the quantifier.)
272
There are four basic types of categorical
sentences:
UNIVERSAL AFFIRMATIVE (form A)
----->
example:
All politicians are honest.
form: All S are P
- the subject class term is "politicians"
- the predicate class term is "honest
people"
- "all" is a universal quantifier word
- the sentence asserts that the subject
class is TOTALLY INCLUDED in the
predicate class
273
----->
example:
All politicians are honest.
Now to symbolize this:
- represent the class terms with
propositional functions:
Px  x is a politician
Hx  x is an honest person
- symbolize "all" with (x)
- symbolize the total inclusion relation as
follows:
(x)(Px  Hx)
274
----->
example:
All politicians are honest.
We can represent what this sentence says
GRAPHICALLY with a Venn diagram:
politicians
honest people
275
PARTICULAR AFFIRMATIVE (form I)
----->
example:
Some politicians are honest.
Form: Some S are P
- the subject class term is "politicians"
- the predicate class term is "honest
people"
- "some" is an existential quantifier word
- the sentence asserts that the subject
class is PARTIALLY INCLUDED in the
predicate class
276
----->
example:
Some politicians are honest.
Now to symbolize this:
- represent the class terms with
propositional functions:
Px  x is a politician
Hx  x is an honest person
- symbolize "some" with (x)
- symbolize the partial inclusion relation
as follows:
(x)(Px  Hx)
277
----->
example:
Some politicians are honest.
The Venn diagram for this sentence is:
politicians
honest people
278
UNIVERSAL NEGATIVE (form E)
----->
example:
No politicians are honest.
Form: No S are P
- the subject class term is "politicians"
- the predicate class term is "honest
people"
- "no" is a quantifier word (like "nothing");
we'll interpret it as a universal quantifier
word
- the sentence asserts that the subject
class is TOTALLY EXCLUDED from
the predicate class
279
----->
example:
No politicians are honest.
Now to symbolize this:
- represent the class terms with
propositional functions:
Px  x is a politician
Hx  x is an honest person
- paraphrase "no" as "all. . .not"; so we'll
symbolize it with (x) and ~
- symbolize the total exclusion relation as
follows:
(x)(Px  ~Hx)
280
----->
example:
No politicians are honest.
The Venn diagram for this sentence is:
politicians
honest people
281
PARTICULAR NEGATIVE (form O)
----->
example:
Some politicians are not honest.
Form: Some S are not P
- the subject class term is "politicians"
- the predicate class term is "honest
people"
- "Some" is an existential quantifier word
- "not" is sentential operator
- the sentence asserts that the subject
class is PARTIALLY EXCLUDED from
the predicate class
282
----->
example:
Some politicians are not honest.
Now to symbolize this:
- represent the class terms with
propositional functions:
Px  x is a politician
Hx  x is an honest person
- symbolize "some" with (x) and "not"
with ~
- symbolize the partial exclusion relation
as follows:
(x)(Px  ~Hx)
283
----->
example:
Some politicians are not honest.
The Venn diagram for this sentence is:
politicians
honest people