# Introduction to Symbolic Logic - PowerPoint by S5I8Y6

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```									 Introduction
to
Symbolic Logic
San Diego Math Circle
David W. Brown
   Sudoku                 Law
   Minesweeper            Contracts

   Mathematics            Debate
   Science                Philosophy
   Engineering
   Computers              Politics?
   Medical diagnosis      Policy?
Why study symbolic logic?
   Symbolic logic excels in separating
the formal structure of logical
relationships from the material
content of the statements being
related.
   With the material content of
statements out of the way, we can
discover how to “calculate” the
properties of logical relationships.
Goals
Understand:             No time now for:

•   Truth value         •   Boolean Algebra
•   Propositions        •   Sets
•   Connectives         •   Predicates
•   Truth Tables        •   Quantifiers
•   Theorems            •   Venn Diagrams
•   Inference           •   Logic gates
•   Fallacy             •   Etc.
What is Truth?
   A philosophical, not mathematical matter

   Mathematically, “true” and “false” are
simply the two mutually exclusive values
that can be taken by a well-formed
formula.

   True:   1, T, +, up,   on, Republican

   False: 0, F, -, down, off, Democrat
What is a
“Well-formed formula” ?
A “well-formed formula” is any construction
that has definite truth value, whether that
value is “true” or “false”.

Synonymous alternative terms:
•   Statement
•   Proposition
•   Sentence
•   Wff         ( Well-formed formula )
What is an atomic formula?
Not something from science fiction or a cold
war thriller …

Propositions may be compound, constructed
from simpler propositions.

An “atomic formula” or “atomic sentence”,
etc. is one that cannot be decomposed
into simpler wffs.
Terminology
   The symbols such as p or q used to
represent statements, whether
atomic or compound, may be called
variously
• Statement letters
• Literals
• Variables (provided the context is clear;
in predicate logic variables are used
differently)
Language

   We speak in natural language or informal
language, which is imprecise, flexible, and
expressive.

   Logic requires formal language, which is
precise, rigid, and constructive.

   Logic can connect natural language
statements, but only if those statements
have definite truth value.
Natural vs. Formal Language
Natural language       Formal language
Why?                   Why?

   Two’s company.        Today is Saturday.
   Three’s a crowd.      All men are created
   I love you.            equal.
   OMG!                  True or false
   Time flies.           2x2=5
   I voted.              1+1=1
   Obama rules!          McCain lost.
Translating Natural Language into
Formal Language

Roses are red,
Violets are blue,
Sugar is sweet,
and so are you.
Translating Natural Language into
Formal Language

Roses are red,      R = “Roses are red”
Violets are blue,   V = “Violets are blue”
Sugar is sweet,     S = “Sugar is sweet”
and so are you.     Y = “You are sweet”
Translating Natural Language into
Formal Language

Roses are red,         R = “Roses are red”
Violets are blue,      V = “Violets are blue”
Sugar is sweet,        S = “Sugar is sweet”
and so are you.        Y = “You are sweet”

Roses are red and
Violets are blue and
Sugar is sweet and
You are sweet.
Translating Natural Language into
Formal Language

Roses are red,         R = “Roses are red”
Violets are blue,      V = “Violets are blue”
Sugar is sweet,        S = “Sugar is sweet”
and so are you.        Y = “You are sweet”

Roses are red and      R and V and S and Y
Violets are blue and
Sugar is sweet and
You are sweet.
Translating Natural Language into
Formal Language

Roses are red,         R = “Roses are red”
Violets are blue,      V = “Violets are blue”
Sugar is sweet,        S = “Sugar is sweet”
and so are you.        Y = “You are sweet”

Roses are red and      R and V and S and Y
Violets are blue and
 = “and”
Sugar is sweet and
You are sweet.
Translating Natural Language into
Formal Language

Roses are red,         R = “Roses are red”
Violets are blue,      V = “Violets are blue”
Sugar is sweet,        S = “Sugar is sweet”
and so are you.        Y = “You are sweet”

Roses are red and      R and V and S and Y
Violets are blue and
 = “and”
Sugar is sweet and
You are sweet.
RVSY
Translating Natural Language into
Formal Language

Red sky at night,
Sailor’s delight,
Red sky at morning,
Sailor take warning.
Translating Natural Language into
Formal Language

Red sky at night,      N = “The sky is red at night.”
Sailor’s delight,      G = “Good weather is ahead.”
M = “The sky is red at morning”
Red sky at morning,
Sailor take warning.
Translating Natural Language into
Formal Language

Red sky at night,               N = “The sky is red at night.”
Sailor’s delight,               G = “Good weather is ahead.”
M = “The sky is red at morning”
Red sky at morning,
Sailor take warning.

If the sky is red at night,
then
and
If the sky is red at morning,
then
Translating Natural Language into
Formal Language

Red sky at night,               N = “The sky is red at night.”
Sailor’s delight,               G = “Good weather is ahead.”
M = “The sky is red at morning”
Red sky at morning,
Sailor take warning.

If the sky is red at night,     (if N then G) and (if M then B)
then
and
If the sky is red at morning,
then
Translating Natural Language into
Formal Language

Red sky at night,               N = “The sky is red at night.”
Sailor’s delight,               G = “Good weather is ahead.”
M = “The sky is red at morning”
Red sky at morning,
Sailor take warning.

If the sky is red at night,     (if N then G) and (if M then B)
then
Good weather is ahead            = “and”     → = “if … then”
and
If the sky is red at morning,
then
Translating Natural Language into
Formal Language

Red sky at night,               N = “The sky is red at night.”
Sailor’s delight,               G = “Good weather is ahead.”
M = “The sky is red at morning”
Red sky at morning,
Sailor take warning.

If the sky is red at night,     (if N then G) and (if M then B)
then
Good weather is ahead            = “and”     → = “if … then”
and
If the sky is red at morning,   (N → G)  (M → B)
then
Translating Natural Language into
Formal Language

Two wrongs don’t           W1    = “Act1 is wrong.”
make a right.              W2    = “Act2 is wrong.”
R     = “Act3 is right.”

It is not true that a      not ( W1 and W2 equivalent R )
wrong act and another
~ = “not”     = “and”
wrong act are equivalent   ↔ = “equivalent”
to a right one.
~( W1  W2 ↔ R )
Translating Natural Language into
Formal Language

All work and no play       W = “Jack always works”
makes Jack a dull boy.     P = “Jack sometimes plays”
D = “Jack is a dull boy”

If Jack always works and   If W and not P, then D.
it is not true that Jack
~ = “not”     = “and”
sometimes plays, then
→ = “if … then”
Jack is a dull boy.
( W  ~P ) → D
The Propositional Calculus
By:
• carefully translating natural language
statements into formal propositions,
• replacing statements with literals, and
• replacing relational language with symbols
representing precisely-defined logical
operations,

it becomes possible to calculate the truth value of
complex statements; i.e., to reduce
argumentation and proof to calculation.
Functions,
Truth Tables,
and Connectives
uh …

not quite
Functions in Logic

The concepts of domain and range familiar
from algebra apply to functions in logic as
well.

In propositional logic, however, the
elements of the domain and range are
truth values “T” and “F” rather than
numbers.

This greatly limits the possibilities.
Functions
x       f(x) = x2
Algebraic   0          0

f(x)
Functions   1          1
2          4
…          …
X

Logical         p       g(p)

Functions       T      T or F

F      T or F          T       F
Simple Functions in Algebra
We often denote functions in algebra “f(x)”,

But some functions are so simple that we don’t use
function notation for them. For example,

The negation function n(x)=-x we write as “-x”.

The identity function i(x)=x we write as “x”.

Constant functions c(x)=c we write as “c”.
Unary Functions in Logic
Unary functions in logic are functions of a
single variable much like the functions f(x)
of algebra except for the fact that the
“variables” represent propositions and the
domain and range are extremely small:

Domain = { T, F } ,   Range    { T, F }

This means that there can exist only 4 unary
functions in logic.
The four unary functions
   Identity              That these four are
• g(p) = p              the only ones that
   Negation                can exist can be
• g(p) = ~p (or ¬p)     seen using an
organizing tool
   Constant True
called a truth table
• g(p) = T
that explicitly
   Constant False          tabulates all
• g(p) = F              possibilities.
Truth Table of the Identity Function

Domain             Range

p                  p

T

F
Truth Table of the Identity Function

Domain             Range

p                  p

T                  T

F
Truth Table of the Identity Function

Domain             Range

p                  p

T                  T

F                  F
Truth Table of the Negation
Function

Domain            Range

p                 ~p

T

F
Truth Table of the Negation
Function

Domain            Range

p                 ~p

T                 F

F
Truth Table of the Negation
Function

Domain            Range

p                 ~p

T                 F

F                 T
Truth Table of the Constant-True
Function

Domain             Range

p                 T

T

F
Truth Table of the Constant-True
Function

Domain             Range

p                 T

T                 T

F
Truth Table of the Constant-True
Function

Domain             Range

p                 T

T                 T

F                 T
Truth Table of the Constant-False
Function

Domain            Range

p                 F

T

F
Truth Table of the Constant-False
Function

Domain            Range

p                 F

T                 F

F
Truth Table of the Constant-False
Function

Domain            Range

p                 F

T                 F

F                 F
Binary Functions
In the same sense that a unary
function is a function of only one
variable, a binary function is a
function of two variables. For
example:

Algebra:         f(x,y) = x2 + y2

Logic:           g(p,q) = p  q
Domain and Range for Binary
Functions
The domain of a binary      On the other hand, the
function is a Cartesian     range of a binary
product of unary            function still contains
domains; i.e., the set      just the simple truth
of all possible ordered     values T and F
pairs of truth values:      because the output of
a function is a single
D = { T, F }  { T, F }       value.

= { (T,T), (T,F),           R    { T, F }
(F,T), (F,F) }
How many binary functions are
there?
Domain   Range
That is, how many
p    q   g(p,q)      truth tables?

T    T

T    F

F    T

F    F
How many binary functions are
there?
Domain   Range
That is, how many
p    q   g(p,q)      truth tables?

T    T   T or F

T    F   T or F

F    T   T or F

F    F   T or F
How many binary functions are
there?
Domain   Range
That is, how many
p    q   g(p,q)       truth tables?

T    T   T or F   2 x 2 x 2 x 2 = 16

T    F   T or F
Thus, there exist
exactly 16 binary
F    T   T or F     functions in
propositional logic.
F    F   T or F
From logical functions to
connectives
It is convenient to notate logical functions g(p,q)
using relational symbols “?” called “connectives”,
so that “p ? q” means the same thing as g(p,q).

We already do this in algebra using relational
symbols such as “=”, “>”, etc. that can be
viewed as functions from  to {T,F}. For
example

2=2 is “true”         1 < 5 is “true”
2=1 is “false”        e > π is “false”
The 16 binary functions and their
notation as connectives
g(p,q):                               (negations)

=    T        “true”             =   F
=    p   q   “or”               =   p↓q
=    p   ←q   “only if”          =   p↚q
=    p        “left identity”    =   ¬p, ~p
=    p   →q   “implies”          =   p↛q
=    q        “right identity”   =   ¬q, ~q
=    p   ↔q   “equivalent”       =   pq
=    p   q   “and”              =   p↑q
Aren’t 16 connectives overkill?
sort of

8 connectives are negations of the other 8.

3 of the remaining 8 (T, p, and q) are not
particularly handy as “binary” relations.

4 of the remaining 5 are the real workhorses:

 , , →, ↔
(Technically … one connective is enough if it is the right one …  or .)
AND
pq
Conjunction – “AND”

p    q   pq
   The domain consists
of the four possible
T    T             ordered pairs of truth
values (p,q).
T    F
   The range consists of
the truth values taken
F    T
by the proposition
p  q for each of these
F    F             ordered pairs.
Conjunction – “AND”

p    q   pq
   The domain consists
of the four possible
T    T    T        ordered pairs of truth
values (p,q).
T    F
   The range consists of
the truth values taken
F    T
by the proposition
p  q for each of these
F    F             ordered pairs.
Conjunction – “AND”

p    q   pq
   The domain consists
of the four possible
T    T    T        ordered pairs of truth
values (p,q).
T    F    F
   The range consists of
the truth values taken
F    T
by the proposition
p  q for each of these
F    F             ordered pairs.
Conjunction – “AND”

p    q   pq
   The domain consists
of the four possible
T    T    T        ordered pairs of truth
values (p,q).
T    F    F
   The range consists of
the truth values taken
F    T    F
by the proposition
p  q for each of these
F    F             ordered pairs.
Conjunction – “AND”
p and q are called
p    q   pq
“conjuncts”
T    T    T     “AND” is true only
when both
T    F    F      conjuncts are true

F    T    F    “AND” is false when
at least one
conjunct is false
F    F    F
OR
pq
Disjunction – “OR”

p   q   pq
   The domain consists
of the four possible
T   T             ordered pairs of truth
values (p,q).
T   F
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F             ordered pairs.
Disjunction – “OR”

p   q   pq
   The domain consists
of the four possible
T   T    T        ordered pairs of truth
values (p,q).
T   F
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F             ordered pairs.
Disjunction – “OR”

p   q   pq
   The domain consists
of the four possible
T   T    T        ordered pairs of truth
values (p,q).
T   F    T
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F             ordered pairs.
Disjunction – “OR”

p   q   pq
   The domain consists
of the four possible
T   T    T        ordered pairs of truth
values (p,q).
T   F    T
   The range consists of
the truth values taken
F   T    T
by the proposition
p  q for each of these
F   F             ordered pairs.
Disjunction – “OR”
p and q are called
p   q   pq          “disjuncts”

T   T    T      “OR” is true when at
least one disjunct is
true
T   F    T
“OR” is false only when
both disjuncts are
F   T    T
false

F   F    F     Note that p and q may
be true simultaneously
Equivalence
p↔q
Biconditional - Equivalence

p   q   p↔q
   The domain consists
of the four possible
T   T              ordered pairs of truth
values (p,q).
T   F
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F              ordered pairs.
Biconditional - Equivalence

p   q   p↔q
   The domain consists
of the four possible
T   T     T        ordered pairs of truth
values (p,q).
T   F
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F              ordered pairs.
Biconditional - Equivalence

p   q   p↔q
   The domain consists
of the four possible
T   T     T        ordered pairs of truth
values (p,q).
T   F     F
   The range consists of
the truth values taken
F   T
by the proposition
p  q for each of these
F   F              ordered pairs.
Biconditional - Equivalence

p   q   p↔q
   The domain consists
of the four possible
T   T     T        ordered pairs of truth
values (p,q).
T   F     F
   The range consists of
the truth values taken
F   T     F
by the proposition
p  q for each of these
F   F              ordered pairs.
Biconditional - Equivalence
p ↔ q is true when p
p   q   p↔q      and q are both true,
or both false
T   T     T    p ↔ q is false when
exactly one of {p,q} is
false, or when exactly
T   F     F      one of {p,q} is true

F   T     F    Also:
“if and only if”
F   F     T      “iff”
“biconditional sentence”
Implication
p→q
The Asymmetry of Implication
   The truth values of conjunction, dysjunction, and
equivalence do not depend on the order in which
p and q appear; i.e., these connectives are
symmetric.
   Implication (p → q) is asymmetric – order
matters.
• p is the antecedent     (hypothesis, premise, sufficient condition)
• q is the consequent     (thesis, conclusion, necessary condition)

• the arrow symbolizes the “flow of truth” from p to q, or
the ability to conclude the truth of q from the truth of p.

   Some aspects of this asymmetry can be
challenging to understand at first.
Conditional – “Implies”

p     q   p→q
Part of what we mean by
implication is that the
T     T    T       truth of antecedent
assures the truth of
the consequent.
T     F

Thus, the implication
F     T
must be true for the
case (T,T).
F     F
Conditional – “Implies”
However, if p is true and
p     q   p→q      q is false, then the
truth of the
antecedent does NOT
T     T    T       assure the truth of the
consequent.
T     F    F     This is not what we
mean by implication,
so the implication
F     T            must be false.
Thus, the implication
F     F            must be false for the
case (T,F).
Truth Values of Implication

   What should the truth value of implication
be when the antecedent is false? That is,
what should be the truth value of F → q ?

   Rather than making an argument
regarding what the remaining truth values
of implication should be, we will consider
what these values must be by explicitly
examining all the possibilities.
What are the possible truth tables
to represent implication?
p     q     ?      p     q     ?
T     T     T      T     T     T
T     F     F      T     F     F
F     T     ?      F     T     ?
F     F     ?      F     F     ?

p     q     ?      p     q     ?
T     T     T      T     T     T
T     F     F      T     F     F
F     T     ?      F     T     ?
F     F     ?      F     F     ?
What are the possible truth tables
to represent implication?
p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     F      F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     F      F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     F      F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T            F     T
F     F            F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q            p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     T      F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q   p↔q      p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     T      F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q   p↔q      p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T
F     F     T      F     F
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q   p↔q      p     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     T      F     F     T
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q   p↔q      p     q   p→q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     T      F     F     T
What are the possible truth tables
to represent implication?
p     q    pq     p     q     q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     F      F     F     F

p     q   p↔q      p     q   p→q
T     T     T      T     T     T
T     F     F      T     F     F
F     T     F      F     T     T
F     F     T      F     F     T
Powerful Conclusion
   Of the four possible ways to finish the
truth table for implication, three are
already “taken” (pq, p↔q, q) and none of
these three “work” anyway: Two are
symmetric (pq, p↔q), the third is
independent of the antecedent (q), and
none of them have the meaning of
implication.

   We necessarily conclude that there is only
one binary function that can express the
meaning of implication.
Conditional – “Implies”

p     q   p→q    “Implies” is false only
when the antecedent
T     T    T       p is true and the
consequent q is false

T     F    F     Also:

F     T    T       “if p then q”
“implication”
“conditional sentence”
F     F    T       “material implication”
Vacuous Truth
   A principle is involved here that may be
unfamiliar from common experience but is
essential in mathematics – that from a false
premise one can “prove” anything; that is, that
the implication F → q true independent of q.

   Truth that follows by implication from a false
premise is sometimes called “vacuous truth”.

   However, there is no functional difference in logic
between “truth” and “vacuous truth”;

T is T
(and you can quote me on that!)
Understanding Implication
The properties of implication can appear to
allow us to arrive at absurd conclusions;
for example, the implications

• “If the moon is made of green cheese, then
Neil Armstrong walked on green cheese”,
and

• “If 1=0, then Pluto is officially a planet”

are both true, even though all of the
premises and conclusions are false and
each implication seems absurd.
Understanding Implication
The appearance of absurdity derives not from the formal
language of logic but from our natural language and
the presumptions implicit in it.
In natural language, we tend to use “if … then” in
different ways in different contexts without careful
discrimination.
Often, we presume there to exist some meaningful real-
world context that relates the antecedent and
consequent, and often there is a presumption of
causality involved; i.e., a presumption the antecedant
causes the consequent.
However, real-world context or causality has nothing to
do with material implication, which relates truth
values and nothing more.
Understanding Implication
There exist extensions of propositional logic that
introduce connectives permitting possible
relatedness between antecedents and
Such extended logics generally include propositional
logic as a subset, and thus include the material
implication among their connectives, but they
also permit generalizations of it that allow some
of the more “natural” expectations about
implication to be formalized.
“Vacuous truth” does not go away; material
implication continues to “mean” exactly what it
means in propositional logic.
Understanding Implication
redux

Meaning:
p    q   p→q

T    T    T    ← It is true that T → T

T    F    F    ← It is false that T → F

F    T    T
← It is true that F → anything
F    F    T
(Get used to it!)
Converse
Implication
p←q
Converse – “Only if”

p    q   p←q
“Only if” is false only
when the antecedent
T    T    T      p is false and the
consequent q is true
T    F    T
Also:

F    T    F
“if q then p”
“converse implication”
F    F    T
Other Connectives
   NAND   “”    Because negation is simple
to notate, these are not so
important in basic logic;
• ~ (p  q)
however …

   NOR    “”
NAND and NOR are
important in computer
• ~ (p  q)     science

   XOR    “”    Natural language “or” is
sometimes “OR”,
• ~ (p ↔ q)     sometimes “XOR”
Order of Precedence
   1st ~p           negation
   2nd       p   q     conjunction (and)
   3rd       p   q     disjunction (or)
   4th       p   →q     implication
   5th       p   ↔q     equivalence

Random example:

{ [ (~p)  (~q) ]  (p → r) } ↔ (p → s)

~p  ~q  (p → r) ↔ p → s
Four forms of Implication
From any two statements p and q, there are four related
implications that can be formed:

p→q                < Converses >
q→p
implication                                   converse

Note that none of
Inverses ↕            these are negations    ↕ Inverses

~p → ~q              < Converses >
~q → ~p
inverse                                   contrapositive

Diagonals are equivalent:

( p → q ) ↔ ( ~q → ~p )
( q → p ) ↔ ( ~p → ~q )
Consolidated Truth Table for
Conditionals
p       q      p→q          q→p         ~p → ~q       ~q → ~p
implication   converse      inverse    contrapositive

T       T         T            T            T             T

T       F         F            T            T             F

F       T         T            F            F             T

F       F         T            T            T             T

(Note that none of these four is a negation of any other.)
Consolidated Truth Table for
Conditionals
p       q      p→q          q→p         ~p → ~q       ~q → ~p
implication   converse      inverse    contrapositive

T       T         T            T            T             T

T       F         F            T            T             F

F       T         T            F            F             T

F       F         T            T            T             T

(Note that none of these four is a negation of any other.)
Consolidated Truth Table for
Conditionals
p       q      p→q          q→p         ~p → ~q       ~q → ~p
implication   converse      inverse    contrapositive

T       T         T            T            T             T

T       F         F            T            T             F

F       T         T            F            F             T

F       F         T            T            T             T

(Note that none of these four is a negation of any other.)
Consolidated Truth Table for
Conditionals
p       q      p→q          q→p         ~p → ~q       ~q → ~p
implication   converse      inverse    contrapositive

T       T         T            T            T             T

T       F         F            T            T             F

F       T         T            F            F             T

F       F         T            T            T             T

(Note that none of these four is a negation of any other.)
Consolidated Truth Table for
Conditionals
p       q      p→q          q→p         ~p → ~q       ~q → ~p
implication   converse      inverse    contrapositive

T       T         T            T            T             T

T       F         F            T            T             F

F       T         T            F            F             T

F       F         T            T            T             T

(Note that none of these four is a negation of any other.)
Consolidated Truth Table for
Conditionals
p   q    p→q          q→p        ~p → ~q     ~q → ~p
implication   converse    inverse   contrapositive

T   T       T            T          T            T

T   F       F            T          T            F

F   T       T            F          F            T

F   F       T            T          T            T

( p → q ) ↔ ( ~q → ~p )
Consolidated Truth Table for
Conditionals
p   q    p→q          q→p        ~p → ~q     ~q → ~p
implication   converse    inverse   contrapositive

T   T       T            T          T            T

T   F       F            T          T            F

F   T       T            F          F            T

F   F       T            T          T            T

( q → p ) ↔ ( ~p → ~q )
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
iff
p   q    p→q          p←q         (p → q)  (p ← q)      p↔q
implication   converse         If and only if   Equivalence

T T         T            T                  T               T

T   F       F            T                  F               F

F   T       T            F                  F               F

F   F       T            T                  T               T

(p → q)  (p ← q) ↔ (p ↔ q)
“Normal Forms” of
Implication and Equivalence
   Implication and equivalence both can be reduced
to useful expressions containing only the
connectives ~, , and . (Check the truth tables!)

(p → q) ↔ (~p  q)
(p ↔ q) ↔ (p  q)  (~p  ~q)

   RHSs are in “disjunctive normal form”.
   Any expression of propositional logic can
transformed into “disjunctive normal form” and
also “conjunctive normal form”.
   Any two propositions sharing the same normal
form are equivalent.
End of Part 1

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