Download 8.1 Symbols and Translation


Predicate Logic combines the distinctive
features of syllogistic and propositional logic.
◦ The fundamental component in predicate logic is
the predicate, which is always symbolized with
upper case letters.
◦ Symbolic predicates may be used to translate three
kinds of statements:
 Singular Statements: Socrates is dead.
 Universal Statements: All S are P or No S are P .
 Particular Statements: Some S are P or Some S are not
P.

Translating singular statements requires
lower case letters called individual constants
and the logical operators of propositional
logic.
◦ “Socrates is mortal” is symbolically translated as
“Ms.”
◦ “If Paris is beautiful, then André told the truth,”
becomes Bp  Ta.

Translating universal statements requires
universal quantifiers, individual variables, and
the horseshoe operator:
◦ “All S are P” is translated as x (Sx  Px), meaning,
for any x, if x is an S, then x is a P.
◦ “No S are P” is translated as x (Sx  ~Px), meaning,
for any x, if x is an S, then x is not a P.
◦ The Venn diagrams responding to these statements
are:

Translating particular statements requires
existential quantifiers, individual variables,
and the dot operator.
◦ “Some S are P” is translated as (Ǝx) (Sx • Px) or
“There exists such an x such that x is an S and x is
a P.”
◦ “Some S are not P” is translated as (Ǝx) (Sx • ~Px) or
“There exists such an x such that x is an S and x is
not a P.”
◦ The Venn diagrams responding to these statements
are:

For both universal and particular statements,
we need to distinguish between free and
bound variables.
◦ Free variables are not bound by any quantifier.
Bound variables are.

Four rules govern the removal and
introduction of quantifiers:
◦ Universal Instantiation (UI): This allows universal
quantifiers to be removed. It requires using either a
constant or a variable.
◦ Universal Generalization (UG): This allows universal
quantifiers to be introduced. Universal
Generalization cannot be performed if the instantial
letter is a constant.
◦ Existential Generalization allows existential
quantifiers to be introduced. There are no
restrictions on Existential Generalization.
◦ Existential Instantiation allows for existential
quantifiers to be removed. Existential instantiation
requires that the existential name must be a new
name that does not occur on the line that indicates
the conclusion to be derived. This name must be
new and many not occur earlier in the proof.

Common Misapplications:
◦ All four of these rules of inference require that the
rules be applied only to whole lines in the proof.

As long as negation signs or ~s precede
quantifiers, statements cannot be
instantiated; thus, conclusions cannot be
derived. The Quantifier Negation Rule allows
us to remove the negation signs.
The Quantifier Negation Rule (QN) is expressed as:
(x) Fx :: ~(Ǝx) ~Fx
~(x) Fx :: (Ǝx) ~Fx
(Ǝx) Fx :: ~(x) ~Fx
~(Ǝx) Fx :: (x) ~Fx

One type of quantifier can be replaced by the
other type if and only if it appears
immediately before and after the new
quantifier:
1. Tilde operators that were originally present are
deleted.
2. Tilde operators that were not originally present
are inserted.

Many arguments with conclusions that are
either difficult or impossible to derive by the
conventional method can be handled with
ease in using either conditional or indirect
proof.
1. (x) (Hx  Ix)
/ (Ǝx)Hx  (Ǝx) Ix
2. (Ǝx) Hx
ACP
3. Ha
2, EI
4. Ha  Ia
I, UI
5. Ia
3, 4, MP
6. (Ǝx) Ix
5, EG
7. (Ǝx)Hx  (Ǝx) Ix
2-6, CP

UG must not be used within the scope of an
indented sequence if the instantial variable y
occurs free (unbound by any quantifier) in the
first line of that sequence. The following
defective proof shows why:
1. (x)Rx  (x)Sx
/ (x) (Rx  Sx)
2. Rx
ACP
3. x(Rx)
2, UG (invalid)
4. x(Sx)
1, 3, MP
5. Ia
4, UI
6. Rx  Sx
2-5, CP
7. (x) (Rx  Sx)
6, UG

If Rx means “x is a rabbit,” and Sx means “x is
a snake,” then the premise translates as, “If
everything in the universe is a rabbit,
everything in the universe is snake.”
◦ The statement is true because the antecedent is
false: not everything in the universe is a rabbit.
However, the conclusion is false because it asserts
that all rabbits are snakes. Thus, the argument is
invalid.

The Counterexample Method consists of
finding a substitution instance of a given
invalid argument for or a given invalid
symbolized argument that has true premises
and a false conclusion:
◦ Some animals are not mammals.
◦ All cats are mammals.
◦ Therefore some cats are not mammals.
(Ǝx) (Ax • ~Bx)
(x) (Cx  Bx)
/ (Ǝx) (Cx • ~Ax)

The Finite Universe Method can be used to
establish the validity of an invalid argument
expressed in terms of a single variable.
◦ Conditional Logical Equivalence:
We shrink the universe from billions of things to one
thing, named Abigail.
“Everything in the universe is perfect” is equivalent to
“Abigail is perfect.”
“Something in the universe is perfect” is also equivalent
to “Abigail is perfect” because “Abigail” is “something” as
well as “everything.”
(x)Px :: Pa
(Ǝx)Px :: Pa

The method for proving an argument invalid
consists in translating the premises and
conclusion into singular statements, then testing
the result with an indirect truth table.
◦ A universe of one is tried. If it is possible for the
premises to be true and the conclusion false in this
universe, the argument is immediately identified as
invalid.
◦ If a contradiction results from this assumption, a
universe of two is tried. If, in this second universe, it is
possible for the premises to be true and the conclusion
false, the argument is invalid.
◦ If not, a universe of three is tried, and so on.


Monadic (one-place) predicates assign
attributes to individual things while Relational
(N-place) Predicates establish connections
between or among individuals.
Translating Relational Statements:
◦ The Willis Tower is taller than the Empire State
Building = Twe.
◦ However, as in monadic predicates, Tew = The
Empire State Building is taller than the Willis Tower
(which is not true).

Overlapping Quantifiers:
◦ When two quantifiers of the same sort appear
adjacent to each other, the order does not make a
difference.
◦ When different quantifiers appear adjacent to one
another the order makes a difference, even when
the statement function is nonrelational.

Using the Rules of Inference: The quantifier
negation rule is applied in basically the same
way as with single quantifiers. As the tilde
operator is moved past a quantifier, the
quantifier in question is switched to its
correlative.
1. (Ǝx)(Ǝy)Pxy
2. (Ǝy) Pay
1, EI
3. Pab
2, EI
4. (Ǝx)Pxb
3, EG
5. (Ǝy)(Ǝx)Pxy
4, EG
◦ The CQ rule is used in the same way.
◦ UI, EG and EI are all used in the same way.
◦ UG must not be used if the instantial variable y is
free in any preceding line obtained by EI.

Translating and Negating Identity Relations:
◦ “The only friend I have is Elizabeth. Elizabeth is not
Nancy. Nancy is a Canadian. Therefore, there is a
Canadian who is my friend.” This relationship
involves the following signs: “=” and “”
◦ Simple Identity Statements:
 Whoopi Goldberg is Caryn Johnson: w = c
 Whoopi Goldberg is not Roman Polanski: w  r.
◦ “Only,” “The Only,” and “No…Except”:
 Only Nolan Ryan has struck out 5,000 batters: Sn • (x)
(Sx  x = n)
 The only opera written by Beethoven is Fidelio. Of • Bf
• (x) [Ox • Bx)  x = f]
 No nation except Australia is a continent: Na • Ca • (x)
[(Nx • Cx)]  x = a]
◦ “All Except”
 All painters except Jackson Pollack make sense: Pj •
~Mj • (x) [(Px • x j)  Mx]
◦ Superlatives:
 The largest planet is Jupiter: Pj • (x)[(Px • x j)  Ljx]
◦ Numerical Statements:
 There is at most one god: (x) (y) [Gx • Gy)  x = y]
◦ Definite Descriptions:
 The author of Middlemarch was a Victorian freethinker:
(Ǝx) [Wxm • (y)(Wym  y = x) • Vx • Fx]
◦ Using the Rules of Inference:
◦ Special Rules, known collectively as Id govern the
identity relation:
 Rule 1: a=a
 Rule 2: a=b is logically equivalent to b=a
 Rule 3: If a=b and b=c then a=c