Mathematical Logic_160506_01

© Art Traynor 2011
Mathematics
Definition
Mathematics
Wiki: “ Mathematics ”
1564 – 1642
Galileo Galilei
Grand Duchy of Tuscany
( Duchy of Florence )
City of Pisa
Mathematics – A Language
“ The universe cannot be read until we have learned the language and
become familiar with the characters in which it is written. It is written
in mathematical language…without which means it is humanly
impossible to comprehend a single word.
Without these, one is wandering about in a dark labyrinth. ”

© Art Traynor 2011
Mathematics
Definition
Algebra – A Mathematical Grammar
Mathematics
A formalized system ( a language ) for the transmission of
information encoded by number
Algebra
A system of construction by which
mathematical expressions are well-formed
Expression
Symbol Operation Relation
Designate expression
elements or Operands
( Terms / Monomials )
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical Structure
between operands
represented by a well-formed
Expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( Laws of Composition - LOC’s )
may structure a Relation
1. Identifies the explanans
by non-tautological
correspondences
Definition
2. Isolates the explanans
as a proper subset from
its constituent
correspondences
3. Terminology
a. Maximal parsimony
b. Maximal syntactic
generality
4. Examples
a. Trivial
b. Superficial
Mathematics
Wiki: “ Polynomial ”
Wiki: “ Degree of a Polynomial ”

© Art Traynor 2011
Mathematics
Disciplines
Algebra
One of the disciplines within the field of Mathematics
Mathematics
Others are Arithmetic, Geometry,
Number Theory, & Analysis

The study of expressions of symbols ( sets ) and the
well-formed rules by which they might be manipulated
to preserve validity .

Algebra
Elementary Algebra
Abstract Algebra
A class of Structure defined by the object Set and
its Operations ( or Laws of Composition – LOC’s )

Linear Algebra
Mathematics

© Art Traynor 2011
Mathematics
Definitions
Expression
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical structure
between operands represented
by a well-formed expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( LOC’s ) may structure a Relation
Expression – A Mathematical Sentence
Proposition
A declarative expression
asserting a fact, the truth
value of which can be
ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Mathematics
Predicate
A Proposition admitting the
substitution of variables
O’Leary, Section 2.1,
Pg. 41
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial
An Expression composed of Constants ( Coefficients ) and Variables ( Unknowns) with
an LOC’s of Addition, Subtraction, Multiplication and Non-Negative Exponentiation

© Art Traynor 2011
Mathematics
Definitions
Expression
Transformations capable of
rendering an expression
into a relation
A mathematical structure between operands represented
by a well-formed expression
Proposition
the truth value of which can
be ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Equation
A formula stating an
equivalency class relation
Inequality
A formula stating a relation
among operand cardinalities
Function
A Relation between a Set of inputs and a Set of permissible
outputs whereby each input is assigned to exactly one output
Univariate: an equation containing
only one variable
( e.g. Unary )
Multivariate: an equation containing
more than one variable
( e.g. n-ary )
Mathematics
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial

© Art Traynor 2011
Mathematics
Definitions
Expression
Proposition Formula
VariablesConstants
Operands ( Terms )
Equation
A formula stating an
equivalency class relation
Linear Equation
An equation in which each term is either
a constant or the product of a constant
and (a) variable[s] of the first degree
Mathematics
Polynomial

© Art Traynor 2011
Mathematics
Expression
Mathematical Expression
A representational precursive discrete composition to a
Mathematical Statement or Proposition ( e.g. Equation )
consisting of :

Operands / Terms
Expression
A well-formed symbolic
representation of Operands
( Terms or Monomials ) ,
of discrete arity, upon which one
or more Operations ( LOC’s ) may
structure a Relation
Mathematics
n Scalar Constants ( i.e. Coefficients )
n Variables or Unknowns
The Cardinality of which is referred to as the Arity of the Expression
Constituent representational Symbols composed of :
Algebra
Laws of Composition ( LOC’s )
Governs the partition of the Expression
into well-formed Operands or Terms
( the Cardinality of which is a multiple of Monomials )

© Art Traynor 2011
Mathematics
Arity
Arity
Expression
The enumeration of discrete symbolic elements ( Variables )
comprising a Mathematical Expression
is defined as its Arity

The Arity of an Expression can be represented by
a non-negative integer index variable ( ℤ + or ℕ ),
conventionally “ n ”

A Constant ( Airty n = 0 , index ℕ )or Nullary
represents a term that accepts no Argument

A Unary expresses an Airty n = 1
A relation can not be defined for
Expressions of Arity less than
two: n < 2
A Binary expresses Airty n = 2
All expressions possessing Airty n > 1 are n-ary, Multary, Multiary, or Polyadic
VariablesConstants
Operands
Expression
Polynomial

© Art Traynor 2011
Mathematics
Expression
Arity
Operand
 Arithmetic : a + b = c
The distinct elements of an Expression
by which the structuring Laws of Composition ( LOC’s )
partition the Expression into discrete Monomial Terms
 “ a ” and “ b ” are Operands
 The number of Variables of an Expression is known as its Arity
n Nullary = no Variables ( a Scalar Constant )
n Unary = one Variable
n Binary = two Variables
n Ternary = three Variables…etc.
VariablesConstants
Operands
Expression
Polynomial
n “ c ” represents a Solution ( i.e. the Sum of the Expression )
Arity is canonically
delineated by a Latin
Distributive Number,
ending in the suffix “ –ary ”

© Art Traynor 2011
Mathematics
Arity
Arity ( Cardinality of Expression Variables )
Expression
A relation can not be defined for
Expressions of Arity less than
two: n < 2
Nullary
Unary
n = 0
n = 1
Binary n = 2
Ternary n = 3
1-ary
2-ary
3-ary
Quaternary n = 4 4-ary
Quinary n = 5 5-ary
Senary n = 6 6-ary
Septenary n = 7 7-ary
Octary n = 8 8-ary
Nonary n = 9 9-ary
n-ary
VariablesConstants
Operands
Expression
Polynomial
0-ary

© Art Traynor 2011
Mathematics
Operand
Parity – Property of Operands
Parity
n is even if ∃ k n = 2k
n is odd if ∃ k n = 2k+1
Even ↔ Even
Integer Parity
Same Parity
Even ↮ Odd Opposite Parity
|:
|:

© Art Traynor 2011
Mathematics
Polynomial
Expression
A well-formed symbolic
representation of operands, of
discrete arity, upon which one
or more operations can
structure a Relation
Expression
Polynomial Expression
A Mathematical Expression ,
the Terms ( Operands ) of which are a compound composition of :
Polynomial
Constants – referred to as Coefficients
Variables – also referred to as Unknowns
And structured by the Polynomial Structure Criteria ( PSC )
arithmetic Laws of Composition ( LOC’s ) including :
Addition / Subtraction
Multiplication / Non-Negative Exponentiation
LOC ( Pn ) = { + , – , x bn ∀ n ≥ 0 }
An excluded equation by
Polynomial Structure Criteria ( PSC )
Σ an xi
n
i = 0
P( x ) = an xn + an – 1 xn – 1 +…+ ak+1 xk+1 + ak xk +…+ a1 x1 + a0 x0
Variable
Coefficient
Polynomial Term
From the Greek Poly meaning many,
and the Latin Nomen for name





© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Degree of a Polynomial
Polynomial
The Degree of a Polynomial Expression ( PE ) is supplied by that
of its Terms ( Operands ) featuring the greatest Exponentiation
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P = Variable Cardinality & Variable Product
Exponent Summation
& Term Cardinality
Arity
Latin “ Distributive ” Number
suffix of “ – ary ”
Degree
Latin “ Ordinal ” Number
suffix of “ – ic ”
Latin “ Distributive ” Number
suffix of “ – nomial ”
0 =
1 =
2 =
3 =
Nullary
Unary
Binary
Tenary
Constant
Linear
Quadratic
Cubic
Monomial
Binomial
Trinomial
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation

© Art Traynor 2011
Mathematics
Degree
Polynomial
Nullary
Unary
p = 0
p = 1 Linear
Binaryp = 2 Quadratic
Ternaryp = 3 Cubic
1-ary
2-ary
3-ary
Quaternaryp = 4 Quartic4-ary
Quinaryp = 5 5-ary
Senaryp = 6 6-ary
Septenaryp = 7 7-ary
Octaryp = 8 8-ary
Nonaryp = 9 9-ary
“ n ”-ary
Arity Degree
Monomial
Binomial
Trinomial
Quadranomial
Terms
Constant
Quintic
P
Septic
Octic
Nonic
Decic
Sextic
aka: Heptic
aka: Hexic

© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Polynomial
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation
For a PE with multivariate term(s) ,
the Degree of the PE is supplied by
that Term featuring the greatest summation
of individual Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
Unary Quadratic Monomial
ai xi
1 yi
1P( x , y ) =
Binary Quadratic Monomial
Univariate
Bivariate

© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Polynomial
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
ai xi
1 yi
1P( x , y ) = Binary Quadratic Monomial
ai xi
1 yi
1zi
1P( x , y , z ) = Ternary Cubic Monomial
Univariate
Bivariate
Trivariate
Multivariate

© Art Traynor 2011
Mathematics
Quadratic
Expression
Polynomial
Quadratic Polynomial
Polynomial
A Unary or greater Polynomial
composed of at least one Term and :
Degree precisely equal to two
Quadratic ai xi
n ∀ n = 2
 ai xi
n yj
m ∀ n , m n + m = 2|:
Etymology
From the Latin “ quadrātum ” or “ square ” referring
specifically to the four sides of the geometric figure
Wiki: “ Quadratic Function ”
Arity ≥ 1
 ai xi
n ± ai + 1 xi + 1
n ∀ n = 2
Binary Quadratic Monomial
Unary Quadratic Binomial
 ai xi
n yj
m ± ai + 1 xi + 1
n ∀ n + m = 2 Binary Quadratic Binomial

© Art Traynor 2011
Mathematics
Equation
Equation
Expression
An Equation is a statement or Proposition
( aka Formula ) purporting to express
an equivalency relation between two Expressions :

Expression
Proposition
asserting a fact whose truth
value can be ascertained
Equation
A symbolic formula, in the form of a
proposition, expressing an equality relationship
Formula
A concise symbolic
expression positing a
relationship between
quantities
VariablesConstants
Operands
Symbols
Operations
The Equation is composed of
Operand terms and one or more
discrete Transformations ( Operations )
which can render the statement true
( i.e. a Solution )
Polynomial

© Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
 Free Variable: A symbol within an expression specifying where
a substitution may be made
Contrasted with a Bound Variable
which can only assume a specific
value or range of values
 Solution: A value when substituted for a free variable which
renders an equation true
Analogous to independent &
dependent variables
Unique Solution: only one solution
can render the equation true
(quantified by $! )
General Solution: constants are
undetermined
value-specified (bound?)
Unique Solution
Particular Solution
General Solution
Solution Set
n A family (set) of all solutions –
can be represented by a parameter (i.e. parametric representation)
 Equivalent Equations: Two (or more) systems of equations sharing
the same solution set
Section 1.1, (Pg. 3)
Any of which could include a Trivial Solution

© Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
 Solution: A value when substituted for a free variable which
renders an equation true
Unique Solution: only one solution
can render the equation true
(quantified by $! )
undetermined
value-specified (bound?)
Solution Set
n For some function f with parameter c such that
f(xi , xi+1 ,…xn – 1 , xn ) = c
the family (set) of all solutions is defined to include
all members of the inverse image set such that
f(x) = c ↔ f -1(c) = x
f -1(c) = {(ai , ai+1 ,…an-1 , an ) Ti· Ti+1 ·…· Tn-1· Tn |f(ai , ai+1 ,…an-1 , an ) = c }
where Ti· Ti+1 ·…· Tn-1· Tn is the domain of the function f
o f -1(c) = { }, or Ø empty set ( no solution exists )
o f -1(c) = 1, exactly one solution exists ( Unique Solution, Singleton)
o f -1(c) = { cn } , a finite set of solutions exist
o f -1(c) = {∞ } , an infinite set of solutions exists
Inconsistent
Consistent
Section 1.1,
(Pg. 5)

© Art Traynor 2011
Mathematics
Linear Equation
Linear Equation
Equation
An Equation consisting of:
Operands that are either
Any Variables are restricted to the First Order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
n Constant(s) or
n A product of Constant(s) and
one or more Variable(s)
The Linear character of the Equation derives from the
geometry of its graph which is a line in the R2 plane

As a Relation the Arity of a Linear Equation must be
at least two, or n ≥ 2 , or a Binomial or greater Polynomial

Polynomial

© Art Traynor 2011
Mathematics
Equation
Linear Equation
Linear Equation
 An equation in which each term is either a constant or the product
of a constant and (a) variable[s] of the first order
Term ai represents a Coefficient
b = Σi= 1
n
ai xi = ai xi + ai+1 xi+1…+ an – 1 xn – 1 + an xn
Equation of a Line in n-variables
 A linear equation in “ n ” variables, xi + xi+1 …+ xn-1 + xn
has the form:
n Coefficients are distributed over a defined field
( e.g. ℕ , ℤ , ℚ , ℝ , ℂ )
Term xi represents a Variable ( e.g. x, y, z )
n Term a1 is defined as the Leading Coefficient
n Term x1 is defined as the Leading Variable
Coefficient = a multiplicative factor
(scalar) of fixed value (constant)

© Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form ( Polynomial )
 Ax + By = C
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b
Blitzer, Section 3.2, (Pg. 226)
Test for Linearity
 A Linear Equation can be expressed in Standard Form
As a species of Polynomial , a Linear Equation
can be expressed in Standard Form
 Every Variable term must be of precise order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
Polynomial

© Art Traynor 2011
Mathematics
Logic
Taxonomy
Logic
A system of inquiry addressed to gauging the validity of Arguments
Wiki: “ Logic ”
Informal Logic
That branch of Logic addressed to considerations of
Natural Language Arguments
Formal Logic
That branch of Logic addressed to considerations of Inference
Restricted to consideration
of the Form ( as cont-
rasted with the Content )
of an Argument
Formal
Symbolic Logic
That branch of Formal Logic whereby the Inferential Validity of Arguments
are reckoned by a representational scheme employing Symbolic Substitution

Mathematical Logicnn
An extension of Symbolic Logic into specific consideration of
Models, Proof, Sets, and Recursion

© Art Traynor 2011
Mathematics
Logic
Argument
Logic
Wiki: “ Logic ”
Informal Logic
Formal Logic
of an Argument
Formal
Symbolic Logic
Mathematical Logicnn
Argument
A prefatory composition of Propositions ( i.e. Declarative Statements ) ,
or Premises , the Truth Value of which can be reckoned ,
and by which the application of Inference will permit a Conclusion to be gauged
Deductive
Inductive
Asserts the Truth of the Conclusion is a Logical Consequence of the Premises
Wiki: “ Argument ”
Asserts the Truth of the Conclusion is a Probable Consequence of the Premises

© Art Traynor 2011
Mathematics
Logic
Symbolic Logic
Logic
Wiki: “ Logic ”
Informal Logic
That branch of Logic addressed to considerations of
Natural Language Arguments
Formal Logic
That branch of Logic addressed to considerations of Inference
of an Argument
Formal
Symbolic Logic

Propositional Logicnn
Predicate Logicnn

© Art Traynor 2011
Mathematics
Symbolic Logic
A Declarative Statement ( which declares a fact)
that is either True or False but not both
No variables
Propositional Logic
For which a Truth Value
can be ascertained
Rosen, Section 1.1, Pg. 2
Formal Logic
Symbolic Logic

Propositionnn
Logic
o Propositional Calculus ( Pro-Calc )
Wiki: “ Propositional Calculus ”
Akin to a Propositional Algebra , Pro-Calc supplies the Laws
of Composition ( LOC’s ) governing the structure of well-
formed Propositions , specifying the Logical Connectives by
which Compound Propositions may be constructed .

© Art Traynor 2011
Mathematics
Propositional Logic
A Declarative Statement ( which declares a fact)
that is either True or False but not both
No variables
Proposition
can be ascertained
Formal Logic
Symbolic Logic

Propositionnn
Logic
o Propositional Calculus ( Pro-Calc )
Wiki: “ Propositional Calculus ”
Propositional Variable ( Pro-Var )
The assignment of a symbolic representation
to a well-formed Proposition
P ≔ It is raining
O’Leary, Section 1.2, Pg. 14
Q ≔ I have my umbrella
R ≔ I am struck by lightning
Note carefully that though these Propositions
are represented symbolically , they are not
“ Variables ” in the sense of an “ Unknown ”,
in that they can only assume one value, akin
to a Constant

© Art Traynor 2011
Mathematics
Propositional Logic
Propositional Calculus
Propositional Calculus ( Pro-Calc ) Wiki: “ Propositional Calculus ”
Akin to a Propositional Algebra,
P-Calc supplies the Laws of Composition ( LOC’s )
governing the structure of well-formed Compound Propositions,
specifying the Logical Connectives by which
Compound Propositions may be constructed.
Logical Connectives ( LC’s )
LC’sPro-Var
Proposition
Supply the LOC’s governing the structure of
well-formed Compound Propositions.
Unary : Negation
Binary :
Conjunctionnn
Disjunctionnn
Material Implicationnn
Biconditionalnn
ascertained
Proposition
Compound Proposition
Wiki: “ Logical Connectives ”
Connectives
Conditionals
Propositional Variables ( Pro-Var )
Implicitly quantified by only
a single value , ( i.e.
exhibiting a Unary
Predicate ) or with a DoD
of only one possible
element, akin to a
Constant
Propositional Variable

© Art Traynor 2011
Mathematics
Logical Connectives
Propositional Calculus ( Pro-Calc ) Wiki: “ Propositional Calculus ”
LC’sPro-Var
Proposition
Unary
ascertained
Proposition
Negation: ¬ p
p ¬ p
T F
F T
The Negation Operator
not “ p ”, it is not the case that “ p ”,
the opposite of the truth value of “ p ”
Let P ≔ It is raining
¬ P = It is not raining

© Art Traynor 2011
Mathematics
Propositional Calculus ( P-Calc ) Wiki: “ Propositional Calculus ”
LC’sPro-Var
Proposition
Binary Compound Proposition
Conjunction ( Connective )nn
Let p and q be propositions
The conjunction of p and q, is p q
p q is true when both p and q are true, false otherwise
And…
However…
alternative
formulations
Plus…
Yet…
But…
Logical Connectives
For a conjunction to be
True both propositions
must be True
Falsity “infects” a conjunction
The And is Near
The Conjunction Operator
Let P ≔ It is raining
Let Q ≔ I have my umbrella
P Q = It is raining and I have my umbrella

© Art Traynor 2011
Mathematics
Propositional Logic
LC’sPro-Var
Proposition
Binary
For a conjunction to be
True both propositions
must be True
Wiki: “ Logical Connectives ”Conjunction ( Connective )nn
p q
T T
T F
The Conjunction Operator
The conjunction of p and q, is p q
p q is true when both p and q are true, false otherwise
p q
F T
F F
T
F
F
F
Falsity “infects” a conjunction
P Q = It is raining and I have my umbrella
P Q = It is raining plus I have my umbrella
The And is Near
P Q = It is raining but I have my umbrella
P Q = It is raining yet I have my umbrella
P Q = It is raining however I have my umbrella

© Art Traynor 2011
Mathematics
Propositional Logic
LC’sPro-Var
Proposition
Binary
For a inclusive disjunction
to be True either
propositions can be True
Inclusive Disjunction ( Connective )nn
The inclusive disjunction of p and q, is p q
p q is false when both p and q are false, true otherwise
At least one proposition must
be True
By mathematical convention
disjunction is interpreted as
inclusive unless noted
otherwise ( UNO )
p q
T T
T F
The Inclusive Disjunction Operatorp q
F T
F F
T
T
T
F
p or q : An inclusive Or

© Art Traynor 2011
Mathematics
 Logical Operators: applied to simple propositions create compound propositions
 Connectives: Exclusive Disjunction
p q
T T
T F
The Exclusive Disjunction Operator
The exclusive disjunction of p and q, is p q
p q is true when exactly one of p and q are true, false otherwise
p q
F T
F F
F
T
T
F
p or q : An exclusive Or
For an exclusive disjunction to be True only one propositions can be True
At least one proposition must be True
e.g.: Soup or Salad?
Proposition
A declarative sentence (a sentence which declares a fact)
that is either True or False
but not both
No variables
can be ascertained
Propositional Logic
Operators

© Art Traynor 2011
Mathematics
Conditionals
Conditional Statements
If p and q are propositions,
then the conditional statement p  q
is the proposition
“ if p, then q ”
p  q is false when p is true and q is false, true otherwise
p q
T T
T F
The Conditional Statement Operatorp  q
F T
F F
T
F
T
T
if p then q
For an exclusive disjunction to be True only one propositions can be True
At least one proposition must be True
“ p ” is the hypothesis, antecedent, or premise
“ q ” is the conclusion, consequent
p  q aka an implication
p is sufficient for q
q if p
q when p
a necessary condition for p is q
a sufficient condition for q is p
q is necessary for p
q whenever p
q only if p
p implies q
q follows from p
q unless ¬ p
alternative formulations
Note: The Consequent “controls” the Truth Value of the Conditional Statement
Propositional Logic

© Art Traynor 2011
Mathematics
Conditional Statement Variants
then the conditional statement p  q
is the proposition
“ if p, then q ”
 Converse: q  p if q then p
 Contrapositive: ¬ q  ¬ p if not q then not p
 Inverse: ¬ p  ¬ q if not p then not q
p q
T T
T F
p  q
F T
F F
T
F
T
T
¬p  ¬q
T
T
F
T
¬p ¬q
F F
F T
T F
T T
q  p
T
T
F
T
¬q  ¬p
T
F
T
T
Inverse Converse ContrapositiveConditional
Conditionals
Propositional Logic

© Art Traynor 2011
Mathematics
Biconditional Statement
then the biconditional statement p  q
is the proposition
“ p if and only if q ”
p  q is true when p and q have the same truth value, false otherwise
p  q aka a bi-implication
p q
T T
T F
p  q
F T
F F
T
F
T
T
q  p
T
T
F
T
( p q ) ( q p )
T
F
F
T
Converse ConjunctionConditional
Conditional/Converse
p  q
T
F
F
T
Biconditional
p is necessary and sufficient for q
p iff q
if p then q and conversely
alternative formulations
p  q aka a statement which implicitly includes its converse
Biconditionals
Propositional Logic

© Art Traynor 2011
Mathematics
Precedence of Logical Operators
¬

Precedence


1
2
3
4
Operation
Negation
Conjunction
Disjunction
Conditionality
Operator
 Biconditionality5
Operator Precedence
Propositional Logic

© Art Traynor 2011
Mathematics
Sequence Collection Notation
 Disjunction: n
pj = p1  p2 … pn
j =1
 Conjunction: n
pj = p1  p2  … pnj =1
Example: Sudoku Puzzle
 Each cell: p( i, j, n ) for row = i, column = j, value = n
 Every row contains every number (1-9):   p ( i, j, n )
i =1
9
n =1
9 9
j =1
 Every column contains every number (1-9):   p ( i, j, n )
j =1
9
n =1
9 9
i =1
 Each of nine 3 x 3 blocks contain every number:    p ( 3r + i, 3s + j, n )
r =0
2
s =0
2 9
n =1
3 3
j =1i =1
Sequence Collection
Propositional Logic

© Art Traynor 2011
Mathematics
Logic Gates
p ¬p
Inverter
p
q
p q
OR gate
p
q
p q
AND gate
p
q
r
¬q
¬r
p ¬q
( p ¬q ) ¬r
Logic Gates
Propositional Logic

© Art Traynor 2011
Mathematics
Propositional / Logical Equivalence
 Tautology : a compound proposition which is True notwithstanding the truth values of any of
its constituent propositional variables
 Contradiction : a compound proposition which is False notwithstanding the truth values of any
of its constituent propositional variables
 Contingency : a compound proposition which is neither a Tautology nor a Contradiction
p ¬ p
T F
F T
p ¬ p
T
T
p ¬ p
F
F
Tautology Contradiction
 Symbology : logical equivalence is denoted by either p ≡ p or p p
De Morgan’s Laws
¬( p  q ) ≡ ¬ p ¬ q
¬( p  q ) ≡ ¬ p ¬ q
p  q ≡ ¬ p q
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
p  T ≡ p
Identity
p F ≡ p
p T ≡ T
Domination
p F ≡ F
p ¬ p ≡ T
Negation
p ¬ p ≡ F
p  T ≡ p
Idempotent
p F ≡ p
¬ ( ¬ p ) ≡ p Double Negation
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
Commutative
p q ≡ q p
Distributive
¬ ( p  q ) ≡ ¬ p ¬ q
De Morgan’s Law
Absorption
Associative
p q ≡ q p
( p q ) r ≡ p ( q r )
( p q ) r ≡ p ( q r )
p ( q r ) ≡ ( p q ) ( p r )
p ( q r ) ≡ ( p q ) ( p r )
¬ ( p  q ) ≡ ¬ p ¬ q
p  ( p q ) ≡ p
p ( p q ) ≡ p
Only DM’s requires negations
Changes Order of Operations
as per “PEM-DAS”, Parentheses
are the principal or first operation
Re-Orders Terms
Does Not Change
Order of Operations – PEM-DAS
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
for Conditional Statements
p  q ≡ ¬ p  q
p  q ≡ ¬ q  ¬ p Recall that the consequent “controls”
the truth value of the conditional
p  q ≡ ¬ p  q
p  q ≡ ¬ ( p  ¬ q )
¬ ( p  q ) ≡ p  ¬ q
Notice how the ALL require negations
For Implications (aka Conditionals)
The “If” Term “P” is the Antecedent
The “Then” Terrm is the Consequent
Lay, Section 1, Pg. 4
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
for Conditional Statements
( p  q )  ( p  r ) ≡ p  ( q  r )
( p  r )  ( q  r ) ≡ ( p  q )  r
Recall that the consequent “controls”
the truth value of the conditional
Note that none of these expressions
use negation (among conditional
statement logical equivalences)
( p  q )  ( p  r ) ≡ p  ( q  r )
( p  r )  ( q  r ) ≡ ( p  q )  r
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
for Bi-Conditional Statements
p  q ≡ ( p  q )  ( q  p )
p  q ≡ ¬ p  ¬ q
p  q ≡ ( p q )  ( ¬ p  ¬ q )
¬ ( p  q ) ≡ p  ¬ q
Logical Equivalence
Propositional Logic

© Art Traynor 2011
Mathematics
Propositional Satisfiability
Satisfiable : A compound proposition is satisfiable if
there is an assignment of truth values to its variables
that makes it True
Unsatisfiable : A compound proposition for which
no assignment of truth values to its variables
can result in a True statement
i.e.: iff its negation is true for all variable truth value assignments
(negation results in a tautology)
i.e.: there exists a solution
Satisfiability
Propositional Logic

© Art Traynor 2011
Mathematics
Argument
A sequence of statements
comprised of premises
and a single conclusion
A sequence of propositions
whose validity can said to follow
from the truth of the statements
Fallacy
Incorrect reasoning
leading to
an invalid argument
Valid if the truth of all premises
implies a true conclusion
Argument Form
A sequence of compound propositions
comprised of propositional variables
A sequence of propositions
Validity of the Argument
follows from its form
whose validity for any substitution of statements
is true for its conclusion if all its premises are true
Definitions
Propositional Logic

© Art Traynor 2011
Mathematics
Symbolic Logic
A Conditional Statement
encoding a Relation , capable of Quantification ,
which can be evaluated to a Truth Value
determined by one or more Variables Includes “ Unknowns ”
Predicate Logic
can be ascertained
– i.e. evaluates to a
Boolean Value
Formal Logic
Symbolic Logic

Predicatenn
Logic
Ф ( x ) → Ф is determined by “ x ”
Wiki: “ Predicate
(mathematical logic )”
o A Predicate may be rendered into a well-formed Proposition by
substituting values into the Variable(s) and evaluating the Expression
Ф ( x ) → Ф ≔ “ x ” is a philosopher
Ф ( Socrates ) → “ Socrates ” is a philosopher
x = “ Socrates ”

© Art Traynor 2011
Mathematics
Predicate Logic
Includes “ Unknowns ”
Predicate
can be ascertained
Boolean Value
A Conditional Statement encoding a Relation , capable of Quantification ,
which can be evaluated to a Truth Value determined by one or more Variables
Predicate
Ф ( x ) → Ф is determined by “ x ” O’Leary, Section 2.1, Pg. 41
Wiki: “ Predicate
A Predicate may be rendered into a well-formed Proposition by
Propositional Transformation
Functional Analogue
Considered as a Function ,
the Predicate may be regarded as a Map ,
and the Independent Variable as the Argument of that Function
Ф ( Socrates ) → “ Socrates ” is a philosopher
x = “ Socrates ”

© Art Traynor 2011
Mathematics
Predicate Logic
Predicate
can be ascertained
Boolean Value
Predicate
Wiki: “ Predicate
Functional Analogue
Considered as a Function ,
the Predicate may be regarded as a Map ,
and the Independent Variable as the Argument of that Function

© Art Traynor 2011
Mathematics
Predicate Logic
Predicate
can be ascertained
Boolean Value
A Conditional Statement encoding a Relation , capable of Quantification ,
which can be evaluated to a Truth Value determined by one or more Variables
Predicate
Ф ( x ) → Ф is determined by “ x ” O’Leary, Section 2.1, Pg. 41
Wiki: “ Predicate
Ф ( Socrates ) → Ф ≔ “ Socrates ” is a philosopher
S = { x | Ф( x ) }
Solution Set
Employing Set Builder Notation ( SBN ) we can state the Solution to the
transformed Predicate as a Solution Set

( Logical ) Predicate
Argument
:
Conditional Separator

© Art Traynor 2011
Mathematics
Predicate Logic
Predicate Calculus
A Conditional Statement encoding a Relation ,
capable of Quantification ,
determined by one or more Variables
Predicate
Predicate Calculus ( Pred-Calc )
Akin to a Predicate Algebra ,
Pred-Calc supplies the Laws of Composition ( LOC’s )
governing the structure of well-formed Predicates ,
which when Quantified as a Domain, form a Proposition ,
the Truth Value of Which can be ascertained .
ArgumentPredicate
Quantification
Wiki: “ First-order logic ”
The interval of values over which a Quantification is defined
Proposition
Though represented
symbolically , Pro-Vars are
not “ Variables ” in the
sense of an “ Unknown ”,
as they can assume only
one value, akin to a
Constant
Domain of Discourse ( DoD )
Assumption of Non-Empty DoDnn
It is assumed by convention that the DoD for any
Quantification is non-empty
Trivial Quantification

© Art Traynor 2011
Mathematics
Predicate Logic
Predicate Calculus
Predicate
ArgumentPredicate
Quantification
The interval of values over which a Quantification is defined
Proposition
Domain of Discourse ( DoD )
Assumption of Non-Empty DoDnn
It is assumed by convention that the DoD for any
Quantified is non-empty
Proof: Rosen, Section 1.4, Pg. 43
for x = Ø → Ф is always “ false ”
Ф is determined by “ x ”
Ф cannot be rendered True
S = { x | Ф( x ) }:
∴ x = Ø represents a trivial Quantification

© Art Traynor 2011
Mathematics
Predicate Logic
Predicate Calculus
A Conditional Statement encoding a Relation ,
capable of Quantification ,
determined by one or more Variables
Predicate
which when Quantified by a Domain of Discourse ( DoD ),
form a Proposition , the Truth Value of Which can be ascertained .
ArgumentPredicate
Quantification
A Predicate Quantified by a singular fixed value
( i.e. a Constant or Singleton ) is degenerate ,
with only a single argument populating its DoD ,
akin to a simple Proposition
Proposition
Though represented
Constant
Superficial Quantification

© Art Traynor 2011
Mathematics
Predicate Calculus
Quantification
Predicate
which when Quantified as a Domain, form a Proposition ,
the Truth Value of Which can be ascertained .
ArgumentPredicate
Quantification
Proposition
Though represented
Constant
if Socrates is the only philosopher , Ф reduces to a simple Proposition

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Though represented
Constant
if Socrates is the only philosopher , Ф reduces to a simple Proposition
Existential Quantificationnn
The Superficial case , degenerate quantification of at least one DoD
element to satisfy the Predicate, represents the simplest form of
Existential Quantification , denoted by the symbol “ ∃ ”
meaning “ there exists ( at least one ) ” element of the DoD
for which the proposition can be rendered true
Predicate Calculus
Quantification

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Though represented
Constant
if Socrates is the only philosopher ,
Ф reduces to a simple Proposition
Existential Quantificationnn
Uniqueness Quantificationnn
Alternatively, the Superficial case, degenerate quantification can be
narrowed further to exclude Domain membership to all but one element
populating the Predicate Argument, denoted by the symbol “ ∃! ”
Predicate Calculus
Quantification
The Superficial case , degenerate quantification of at least one DoD
element to satisfy the Predicate, represents the simplest form of
Existential Quantification , denoted by the symbol “ ∃ ”
meaning “ there exists ( at least one ) ” element of the DoD

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Wiki: “ Quantifier (logic) ”
Proposition
exhibiting a Unary
element, akin to a
Constant
Propositional VariablePredicate Quantificationnn
Supplied with a Quantification the Predicate Variable(s) are both:
Quantification
A construct which specifies the Cardinality
of a Domain of Discourse over which a Predicate
may be satisfied ( i.e. rendered true )
o Free Variables : To assume any value defined by the DoD
o Bound : To the Quantifier
Predicate Calculus
Quantification
Existential Quantification ( ExQ )nn
There exists at least one DoD element to satisfy the Predicate ,
denoted by the symbol “ ∃ ” , for which the proposition can be rendered true

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Predicate Quantificationnn
Quantification
Predicate Calculus
Quantification
There exists at least one DoD element
to satisfy the Predicate , denoted by the symbol “ ∃ ” ,
There is…
For at least one…
alternative
formulations
There exists…
For some…
There is/exists at least one…
o A Field or some interval of a Field must always be
specified for the DoD when declaring ExQ
( e.g.: N , Z , Q , R , C ) inclusion in which is
denoted by the symbol “  ” which assigns the Relation
“ is an element of ” to members of the DoD Set
S = { x | Ф( x ) } → ∃ x ℝ → S = { x | ∃ x ℝ , Ф( x ) }: :

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
There exists at least one DoD element
to satisfy the Predicate , denoted by the symbol “ ∃ ” ,
There is…
For at least one…
alternative
formulations
There exists…
For some…
There is/exists at least one…
o Disjunctive ExQ Logical Equivalence
Rosen, Section 1.4, Pg. 43For DoD elements D = { xi , xi+1 ,… , xn – 1 , xn }
an ExQ such as ∃ x Ф ( x ) is logically equivalent to
the compound disjunction:
S = { Ф ( xi ) ∨ Ф ( xi+1 ) ∨… ∨ Ф ( xn – 1 ) ∨ Ф ( xn ) }
As the disjunction can be rendered True by a True
evaluation of any of its elements

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
exhibiting a Unary
element, akin to a
Constant
Quantification
Predicate Calculus
Quantification
Universal Quantification ( UnQ )nn
For DoD elements D = { xi , xi+1 ,… , xn – 1 , xn }
a Quantification to include all elements satisfying Ф ( xn )
would necessarily yield a compound conjunction:
S = { Ф ( xi ) ∧ Ф ( xi+1 ) ∧… ∧ Ф ( xn – 1 ) ∧ Ф ( xn ) }
However this formulation will only work for a DoD defined
over “ countable ” fields such as N , Z , and Q
Over “ uncountable ” Fields such as R and C there is no means to
enumerate the infinite conjuncts, thus a Quantification construct
encompassing all elements of such a Field or Set interval connoting “
for all ” elements, denoted “ ∀ ” captures Universal Quantification
Wiki: “ Quantifier (logic) ”

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
S = { x | Ф( x ) } → ∀ x ℝ → S = { x | ∀ x ℝ , Ф( x ) }: :
S = { Ф ( xi ) ∧ Ф ( xi+1 ) ∧… ∧ Ф ( xn – 1 ) ∧ Ф ( xn ) }
For all…
For arbitrary…
alternative
formulations
For every…
Given any…
All of…
For any…
For each…

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
S = { x | Ф( x ) } → ∀ x ℝ → S = { x | ∀ x ℝ , Ф( x ) }: :
S = { Ф ( xi ) ∧ Ф ( xi+1 ) ∧… ∧ Ф ( xn – 1 ) ∧ Ф ( xn ) }
For all…
For arbitrary…
alternative
formulations
For every…
Given any…
All of…
For any…
For each…
UnQ implies the functional equivalent of an Injection ( over fields R and C )
o Functional Analogue
UnQ implies the functional equivalent of a Bijection ( over fields N , Z , and Q )
[AT] This is only my
speculation

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
O’Leary, Section 2.4, Pg. 62Example:
( ∀ x ℤ ) Ф ( x ) ⊢ Ф ( a )
For a D “ a ” is an arbitrary element
of the DoD “ D ”
∴ any element x = a will satisfy ФϘ ( x )
RHS logically follows from LHS
The validity of this proposition
may only hold when “ a ” is
a constant??
1
2
3
( ∀ x D ) [ Ф ( x ) ⇒ Ϙ( x ) ] ⊢ Ф ( a ) ⇒ Ϙ( a )
( ∀ x D ) [ Ф ( x ) ∨ ( ∀ y D ) Ϙ( y ) ] ⊢ Ф ( a ) ∨ ( ∀ y D ) Ϙ( y )
( ∀ x D ) ( ∀ y D ) [ Ϙ( x ) ∨ Ф( y ) ] ⊢ ( ∀ y D ) [ Ϙ( x ) ∨ Ф( y ) ]

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
o Arbitrary Universally Quantified DoD Element
An element satisfying a Universally Quantified Predicate is
understood to be Arbitrary if it is capable of representing
randomly selected element of the DoD ( i.e. without restriction )
o Particular Universally Quantified DoD Element
understood to be Particular if it imparts some greater structure to
the DoD than an inclusion Relation ( i.e. with restriction )

© Art Traynor 2011
Mathematics
Predicate
ArgumentPredicate
Quantification
Proposition
Quantification
Predicate Calculus
Quantification
o Arbitrary Universally Quantified DoD Element
understood to be Arbitrary if it is capable of representing
randomly selected element of the DoD ( i.e. without restriction )

© Art Traynor 2011
Mathematics
Logic
Propositional Logic
A Declarative Sentence (a sentence which declares a fact)
that is either True or False
but not both
No variables
Predicate Logic
A Symbolic Formal composition
the elements of which Instantiate Variables
that can quantified
 Statement: “ x is greater than three ” x > 3
Subject: the variable
Predicate: a property that the Subject can exhibit
 Once a value has been assigned to the Variable, it has a Truth Value
Symbolic Logic
can be ascertained
can be ascertained

© Art Traynor 2011
Mathematics
Symbolic Logic
Predicate Logic
Predicate Logic
A symbolic formal system
whose formulae contain variables
that can quantified
 Statement: “ x is greater than three ” x > 3
Subject: the variable
Predicate: a property that the subject can possess
 Propositional Function: P(x) can be substituted for the
predicate to form a propositional function
 P(x) is the value of the propositional function P at x
 Once a value has been assigned to the variable, it has a Truth Value Whose truth value can be
ascertained
 P ( x1 , x2 ,…, xn ) is the value of the propositional function,
or the n-place or n-ary predicate, at the n-tuple ( x1 , x2 ,…, xn )

© Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Rules of Inference
p
p  q
q
Modus Ponens( p ( p  q ) )  q
¬ q
p  q
¬ p
Modus Tollens( ¬ q ( p  q ) )  ¬ p
The lines of the premises equate
to an “and”
Recall that the consequent
“controls” the truth value of the
conditional
Because the consequent “controls”
the truth value of the conditional, it
“drives” the negation back through
to the antecedent
q  r
p  r
Hypothetical
Syllogism
( ( p  q ) ( q  r ) )  ( p  q )
p  q
¬ p
q
Dysjunctive
Syllogism
( ( p q ) ¬ p )  q
p q
MP “operates” on the antecedent
MT “operates” on the (negated)
consequent

© Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Rules of Inference
p
p q
Additionp  ( p  q )
Simplification( p  q )  p
q
p  q
( ( p )  ( q ) )  ( p  q )
p
¬ p r
q r
( ( p q ) ( ¬ p r ) )  ( q r )
p q
p  q
p
Conjunction
Resolution
Could just as well be “ q ” by the
associative principle?

© Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Rules of Inference - Resolution
¬ p r
q r
( ( p q ) ( ¬ p r ) )  ( q r )
p q
Resolution
Resolvent: the final disjunction in the resolution rule of inference
 A Tautology
 Basis for the Disjunctive Syllogism rule of inference
Let q = r , then: ( p q ) ( ¬ p r )  q
Let r = F , then: ( p q ) ( ¬ p )  q Which is the Disjunctive Syllogism,
Rule of Inference
 Possible in propositional logic to construct proofs using resolution
as the only rule of inference
 Hypotheses and Conclusion replaced by clauses composed
exclusively of disjunction and negation of variables

© Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Fallacies
 Affirming the Conclusion (ATC)
An impostor of Modus Ponens: ( p ( p  q ) )  q
Recall that the consequent
“controls” the truth value of the
conditional
MP “operates” on the antecedentOf the form: ( q ( p  q ) )  p
 Not a tautology because its truth value is F when P is F and q is T
 It attempts to errantly infer something about the antecedent from the
consequent, unlike MP which correctly implies something about the
antecedent given the consequent
 Denying the Hypothesis (DTH)
An impostor of Modus Tollens: ( ¬ q ( p  q ) )  ¬ p
Of the form: ( ¬ p ( p  q ) )  ¬ q
 Not a tautology because its truth value is F when P is F and q is T
 It attempts to errantly infer something about the lack of an antecedent
from the lack of a consequent, unlike MT which correctly implies
something about the lack of an antecedent given the lack of a
consequent
e.g.: “So no ‘p’…’so what…’”

© Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Quantified Statements
"x P(x )
P( c )
Universal Instantiation
P( c ) for an arbitrary “ c ”
"x P(x )
Universal Generalization
$ x P( x )
P( c ) for some element “ c ”
Existential Instantiation
P( c ) for some element “ c ”
$ x P( x )
Existential Generalization

Mathematical Logic_160506_01

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Mathematical Logic_160506_01