Introduction to Symbolic Logic - PowerPoint by S5I8Y6

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									 Introduction
      to
Symbolic Logic
  San Diego Math Circle
    David W. Brown
        Why learn about logic?
   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,
                       B = “Bad weather is ahead.”
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,
                                B = “Bad weather is ahead.”
Sailor take warning.

If the sky is red at night,
then
Good weather is ahead
and
If the sky is red at morning,
then
Bad weather is ahead.
  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,
                                B = “Bad weather is ahead.”
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 the sky is red at morning,
then
Bad weather is ahead.
  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,
                                B = “Bad weather is ahead.”
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
Bad weather is ahead.
  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,
                                B = “Bad weather is ahead.”
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
Bad weather is ahead.
 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
  consequents to be addressed.
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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