The symbolic analysis of ordinary compound statements covered in the first three sections of
Chapter 1 is called statement calculus or propositional calculus. This analysis involves the use
of simple statements joined by the NOT, OR, AND, IMPLICATION, and BICONDITIONAL
connectives.
This type of analysis defines the form of logical argument, but cannot be used to determine
the validity of an argument.
Here’s an example
All snowboarders wear big pants
Erika is a snowboarder
 Erika wears big pants
This seems to follow but nothing in conditional logic allows us to determine the validity of the
statements.
We need to separate the statements into parts: subjects and predicates. We need to look
carefully at words like “all” and “some”.
We need to analyze the predicates of these statements which describe their subjects.
This analysis of predicates and quantified statements is called predicate calculus.
What are predicates, predicate variables, propositional functions (open sentences) domains,
and truth set?
We can arrive at the predicates of this statement by removing nouns.
Aloma is a student at Cabrillo College.
P = is a student at Cabrillo College
Q = is a student at
Both P and Q are predicate symbols.
P(x) = " x is a student at Cabrillo College "
( x = Aloma)
Q(x,y) = " x is a student at y "
(y = Cabrillo College)
x and y are predicate variables. When values are substituted for these variables, a statement
results.
P(x) and Q(x,y) are propositional functions (open sentences)
, R, Q are domains; they represent the set of all integers, real numbers, and rational numbers
respectively.
R is a set of all points on a line, and is referred to as continuous mathematics.
Z is a set of points set at intervals one unit apart on a line.
These points are called discrete; discrete mathematics ……
Notation to watch out for
P(x)  Q(x)
means every element in the truth set of P(x) is in the truth set of Q(x)
P(x)  Q(x)
means P(x) and Q(x) have identical truth sets
Example:
The domain of x is +, the set of positive integers
P(x)
x is a factor of 8
Q(x)
x is a factor of 4
R(x)
x ≤ 4 and x  3
List truth sets:
P(x)
{1,2,4,8}
Q(x)
{1,2,4}
R(x)
{1,2,4}
Use the symbols  and  to indicate the relationships among the three predicates.
Q(x)  P(x)
every elements in the truth set of Q(x) is in P(x)
Q(x)  R(x)
truth sets are identical
When concrete values are substituted for predicate variables, a statement results. There are two
ways to change predicates into statements: assign specific values to all their variables, or utilize
quantifiers.
QUANTIFIERS:  and 
(Charles S. Peirce/Gottlob Frege, late 19th C)
 = "for all" and is called the universal quantifier.
All human being are mortal =  human beings x, x is mortal.
Universal Statement = x, in D, Q(x) where Q(x) is the predicate and D is the domain of x.
You only need one counterexample to prove it false.



 = "there exists" and is called the existential quantifier.
There is a student in Math 23 =  a person s such that s is a student in Math 23.
Existential Statement = x, in D, Q(x) where Q(x) is the predicate and D is the domain of x.
You only need one example to prove it true.
What do you need to test? Domain & statement
Negations of Quantified Statements
The negation of a universal statement ("all are") is logically equivalent to an existential
statement ("some are not")
statement: All computer software is costly to develop.
Negation: Some computer software is not costly to develop.
The negation of an existential statement ("some are") is logically equivalent to a universal
statement ("all are not")
statement: Some ski instructors are Olympians.
Negation: All ski instructors are not Olympians.
MULTIPLE QUANTIFIERS
Negations of Multiple Quantified Statements
Statement:  people x,  a person y such that x likes y
Its Negation:
 a person x such that ~ ( a person y such that x likes y)
people y, x does not like y
Statement:  a person x such that  people y, x likes y
Its Negation:
 people x, ~ ( people y, x likes y)
 a person y such that x does not like y
This idea is similar to DeMorgan’s law with and & or statements.
Universal statements are generalizations of and statements.
Existential statements are generalization of or statements.
Universal Conditional Statement = x, if P(x) then Q(x)
Contrapositive
Converse
Inverse
Necessary & sufficient conditions
(Q(x) P(x)) & (P(x) Q(x))
ARGUMENTS WITH QUANTIFIED STATEMENTS
Universal instantiation = if a property is true for everything in a domain, then it is true of any
particular thing in the domain.
All snowboarders wear big pants
Erika is a snowboarder
 Erika wears big pants
Universal instantiation is a fundamental tool of deductive reasoning.
UNIVERSAL MODUS PONENS
DEMORGAN'S LAWS
UNIVERSAL MODUS TOLLENS
To say that an argument form is valid means the following: No matter what particular predicates
are substituted for the predicate symbols in its premises, if the resulting premise statements are all
true, then the conclusion is also true.
An argument is called valid if and only if its form is valid.
Converse Error: create an argument that illustrates this
Guilt by association
Inverse Error: create an argument that illustrates this
Verify the following ERROR using a diagram. (Leibnez)
All sharks are mammals.
(major premise)
For all sharks, if s is a shark then s is a mammal.
Rover is a mammal.
(minor premise)
Therefore, Rover is a shark.
(Conclusion)
A VARIATION ON CONVERSE ERROR CAN BE USEFUL.
 Process of Elimination
 In the case of an expert system, the closer
the necessary (q  p) get to the sufficient (p  q), the more certain the diagnosis. This is
called abduction by A.I. researchers.