# Elements of Propositional Calculus by keralaguest

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```									         Cruise Scientific   Visual Statistics Studio Measurement and Scaling

Krus, D. J. (2005) Elements of propositional calculus. Journal of Visual Statistics (December,
2005)

Elements of Propositional Calculus
COMBINATORY LOGIC FUNCTIONS - BUILDING BLOCKS OF MATHEMATICAL
LOGIC - PREDICATE (PROPOSITIONAL) CALCULUS

Relationships among properties of attributes and entities of concrete or abstract
phenomena can be described on several levels. On one level these properties can be
described algebraic formulae and operated upon by algebraic algorithms. On another
level, these properties can be visualized by the graphs of the analytic geometry.
However, underlying these and other levels are the relationships defined in terms of
propositional calculus of the formal logic.

ARGUMENTS OF LOGICAL FUNCTIONS
A statement p can be either true (T) or false (F).

p
T
F

The true statement can be also signified as '1' and the false statement as '0.' In this
case it is more natural to rewrite the above table in the reverse order, to preserve the
natural order of numbers. Two statements can be both true, both false, and either one
can be true while the other is false. Designating one statement as p and the other as q,
this can be schematized as
p       q
0       0
0       1
1       0
1       1

The number of possible outcomes of all possible combinations of true and false
statements, n, is given by the equation n = 2k where k denotes the number of
statements. For three statements, the n = 23 = 8, as shown in the table below.

p q r
0       0 0
0       0 1
0       1 0
0       1 1
1       0 0
1       0 1
1       1 0
1       1 1

For four statements, n = 24 = 16, etc. To construct a plenum of all possible true-false
response patterns, first, write half of the n for the first determinant as 0, half as 1. For an
example of 3 statements (n = 8), write four zeroes and four ones, as [ 0 0 0 0 1 1 1 1].
Next, construct the second column by halving the number of zeroes in the first column.
This number is the repeat-alternate factor for the second vector, written as an alternating
series of zeroes and ones: [ 0 0 1 1 0 0 1 1 ]. Finally, write the last column as alternating
zeroes and ones [ 0 1 0 1 0 1 0 1 ]. The above table illustrates these operations.

LOGICAL FUNCTIONS
The main functions of the propositional calculus can be summarized as shown in the
tables below.

p                               p                    p       p
p    p0q p               p p      q    p                   p   p   q       p       p|    p1q
q                                                                              
&   q          q                    q          q        q       q   q
q                          q            q
00    0       0     0   0   0       0    0     0      1    1     1    1     1     1    1      1
01    0       0     0   0   1       1    1     1      0    0     0    0     1     1    1      1
10    0       0     1   1   0       0    1     1      0    0     1    1     0     0    1      1
11    0       1     0   1   0       1    0     1      0    1     0    1     0     1    0      1

p 0 0     1     1
q 0 1     0     1
0 0     0     0   p is never q        p0q                                 Contradiction         never
0 0     0     1     p and q           p&q        Conjunction       AND                                     Conjun
0 0     1     0   p and not q         pq                                     Assertion          p!
0 0     1     1     p exists            p                                     Existence          p
0 1     0     0  q and not p          pq                                     Assertion          q!
0 1     0     1     q exists            q                                     Existence          q
0 1     1     0      p or q            pq       Disjunction       XOR                                   Disjuncti
0 1     1     1 either p or q         pq        Disjunction       OR                                    Disjunct
1 0     0     0 neither p nor q       pq          Denial                                                  Negate
1 0     0     1       p is q          pq        Equivalence        EQ                                    Negated
1 0     1     0       not q             q                                     Negated q          q
1 0     1     1   q implies p         pq        Implication                  Negated q !       pq
1 1     0     0      not p              p                                     Negated p          p
1 1     0     1   p implies q         pq     Implication                     Negated p !       pq
1 1     1     0  not p and q           p|q   Disassociation        NAND                                 Negated Co
1 1     1     1  p is always q        p1q                                     Tautology     always

CONJUNCTION

CONJUNCTION AND
The statement
Last night we saw Venus and Mars

is a conjunction, a compound statement formed by 'and' between two statements, called
conjuncts. The symbol for conjunction is the ampersand (&) and if p and q are any two
statements, their conjunction is written as p & q. A conjunction is true if both its conjuncts
are true and is false otherwise, as shown in the following table.

p q     p&q                               p               q      p&q
0   0     0                                                          0
0   1     0       Last night we saw                      Mars        0
1   0     0                            Venus                         0
1   1     1                            Venus and         Mars        1

NEGATED CONJUNCTION NAND

Last night we did not see Venus and Mars

The negation of conjunction, NAND, (also called the Scheffer stroke), has the truth
values as shown below:

p q     p|q                                     p               q       p|q
0   0     1                                                              1
0   1     1      Last night we did not see                      Mars     1
1   0     1                                    Venus                     1
1   1     0                                    Venus and        Mars     0

DISJUNCTION

JOINT DENIAL
When two statements are combined by inserting the word 'or' between them, the
resulting compound statement is called disjunction and the two statements thus
combined are called disjuncts.

NOR
Last nigh neither Venus nor Mars were visible.
Negated Disjunction

While the NAND, is the negated conjunction, the joint denial is the negated disjunction,
with the truth values as

p q       p|q                 neither      p     nor    q     p|q
0   0     1                                                   1
0   1     0      Last night                            Mars   0     we
1   0     0                             Venus                 0
1   1     0                             Venus          Mars   0

The disjunction can be either inclusive or exclusive.

INCLUSIVE DISJUNCTION
OR
The inclusive disjunction is symbolized as 'V' and the exclusive disjunction is
symbolized as an inverted A. In Latin the inclusive or, vel, one, or the other, or both is
differentiated from the exclusive or, aut, introducing a second alternative which positively
excludes the first. An example of the inclusive or is

'In theatro comediae vel tragediae aguntur.'

in theater, comedies or tragedies are played, as shown in the following table.
p q     pq                           p                  q         pq
0   0     0                                                         0
0   1     1       In theatro                     tragediae          1    agun
1   0     1                    comediae                             1
1   1     1                    comediae vel tragediae               1

EXCLUSIVE DISJUNCTION
XOR
An example of the exclusive or is

'In bellum vinceris aut vincis.'

In war win or be enslaved (bound), as shown in the following table.

p q     pq                       p       aut     q          pq
0   0     0                                                  0
0   1     1       In bellum                     vincis       1
1   0     1                    vinceris                      1
1   1     0                    vinceris         vincis       0

IMPLICATION
The p q argument of the implication function consists of the antecedent 'if' and the
consequent 'then.' The implication is false when the antecedent is true and the
consequent is false. The truth-values of the implication function are shown in the
following table.
p q     pq
0   0     1
0   1     1
1   0      0
1   1     1

The sentence 'If you will touch the hot stove then you will burn your finger' is an
implication, also called a conditional. A conditional does not assert that the antecedent
or the consequent is true or false. It pertains to the truth or falsity of the relationship
between these statements. Consider a conditional statement 'If Phobos is the satellite of
Venus then I am Tycho de Brahe.' Both statements are false, but the conditional is true.

Consider another example. 'If gold is placed in aqua regia then it dissolves.' Aqua regia
is a mixture of nitric and hydrochloric acids that dissolves gold or platinum. Observation
of gold dissolving in aqua regia (argument 1 1) lends credence to the above conditional
statement.
Not placing the gold into aqua regia and gold not dissolving (argument 0 0) does not
disprove the truth-value of this conditional.
To establish the falsehood of an inductive conditional we must establish the falsehood of
the consequent. The falsity of the conditional 'If gold is placed to agua regia then it
dissolves' could be proved only if the gold is actually placed in aqua regia and it does not
dissolve. Thus, only a conditional with a true antecedent and false consequent is false.
Not placing the gold into aqua regia and observing it to dissolve (argument 0 1) does not
disprove the truth-value of the sentence 'If gold is placed to agua regia then it dissolves.'
This particular configuration of the truth-values of the conditional explains the dictum that
correlation does not imply causation. Scrutiny of the above set diagram can also explain
why a better dictum in this respect is that 'correlation is necessary, but not sufficient
condition of causality.’ There are several ways how to express the relationship of
implication, given in table below.

if p then q
all p is q   p implies q               p causes q
q if p       when p then q             p is sufficient for q
p only if q those who are p are q q is necessary for p
Of all arguments, the F T argument of implication bears most relevance to social studies,
as fallacious reasoning about causes of social events has often this form.

ASSERTION

The sentence 'Pistachio ice cream is better than the vanilla ice cream' can be
decomposed into two statements:

Agree Disagree

p I like pistachio ice cream
q   I like vanilla ice cream

ASSERTING P

The p > q logical function asserting that pistachio ice cream is better tasting than the
vanilla ice cream is

p q     pq                   p             q        pq
0   0     0                                           0
0   1     0      I like                  vanilla      0    ice cream.
1   0     1               pistachio                   1
1   1     0               pistachio      vanilla      0

ASSERTING Q
The p < q logical function asserting that the vanilla ice cream is better tasting than the
pistachio ice cream is
p q        pq                     p          q       pq
0     0      0                                         0
0     1      1         I like               vanilla    1    ice cream.
1     0      0                  pistachio              0
1     1      0                  pistachio   vanilla    0

RECTIFYING VARIABLES
To the category with all values true (or none value true) contains the contradictory and
tautological logical functions. The tautological functions often serve as rectifying
variables. Thus, e.g., for the logical function of implication the plenum of the two
propositions p and q can be rectified by the tautological function as

p q           p  q ( p  q)
0     0         1          0 0
0     1         1          01
1     0         0
1     1         1          11

and written as

0       0
0           0 0
1 
  1 0
 p q 
1       0     
          1 1 
   
1       1

FOUR-FOLD POINT TABLES
In the second group, there are four logical functions with a singular truth value and the
four-fold table of responses, typical of the chi-square analysis, than can be constructed
by using these four singular - value logical functions as
p q         pq               pq                  pq          p&q
0   0        1                     0                0            0
0   1        0                     1                0            0
1   0        0                     0                1            0
1   1        0                     0                0            1

In a condensed form as

p q             &
0   0        1       0       0   0
0       1       0   0
0   1                             
1   0        0       0       1   0
                     
1   1        0       0       0   1

or, alternatively, as

q                                                             q

1            pq                   p&q                        1   Assertion of q    Conjun

0            pq                   pq                        0   Disassociation    Asserti
0                        1            p                    0                1

An instance of the four-fold point table of frequencies or proportions, such as

q                                                         q

1            25            30                             1   .238       .286

0            15                35                         0   .143       .333
0                1            p                  0          1     p

can be interpreted in terms of subjects’ responses rating the palatability of the pistachio
and rum raisin ice creams.
As another example consider the book by Harris I am OK you are OK. It describes four
types of interpersonal behavior. Substituting p for I and q for you, these styles can be
expressed as p & q , p  q , p  q , and p  q

q

You are better        I am OK
1
than me            you are OK

We are both         I am better
0
no good             than you
0                   1            p

The p & q style ' I am OK you are OK' describes healthy relationships with both parties
having positive self-esteem. The other styles are typical of maladaptive behavior.

IMPLICATORY SCALES
Among the group with three true values, the implication function is often of interest.

Consider the following questions from a questionnaire designed to measure a person's
attitude toward some ethnic or religious group.

A   They should be denied entry visa to our country
B   I would not like to live in the same neighborhood as them
C I would object if my daughter wanted to marry one of them

Assume that answers to these questions are agree (1) and disagree (0) a pattern of
responses on a scale of animosity against that particular group, in its idealized form,
should look as
A B C
S1   0   0 0
S2    0   0 1
       
S3    0   1 1
       
S4    1   1 1

The scale [0 1 2 3], associated with the above data, would correctly classify the subjects
with respect to their attitudes toward such a group. Note that within such a scale
response patterns such as 'They should be denied entry visa to our country' (Agree) and
'I would object if my daughter wanted to marry one of them.' (Disagree) do not make
sense. These considerations were guiding Louis Guttman to define implication scales as
prototypes of homogenous scales.

Data matrices congruent with logical functions of implication, when rearranged in either
ascending or descending order show a characteristic triangular shape with zeroes
clustering in one order and ones in the opposite corner. Such an arrangement of data is
also known as the Guttman or implicational scale. Consider a data matrix, consisting of
all possible response patterns to a set of three variables, shown below, as analyzed by
conjunction of logical implication functions.

p q r      p  q q  r ( p  q ) & (q  r )  ( p q r )
0   0 0       1        1             1            0    0    0
0   0 1       1        1             1            0    0    1
0   1 0       1        0             0
0   1 1       1        1             1            0    1    1
1   0 0       0        1             0
1   0 1       0        1             0
1   1 0       1        0             0
1   1 1       1        1             1            1    1    1

The rectified data matrix hen can be written as
0       0 0
0       0 1
           
0       1 1
           
1       1 1

For the above instance of the variables p, q, and r, the process of rectification can be
detailed as

p q r          ( p, q, r )
0   0 0 1           0    0   0
0   0 1       1     0    0   1
0   1 0 0
0   1 1       1     0    1   1
1   0 0 0
1   0 1       0
1   1 0 0
1   1 1 1           1    1   1

i.e., rows in the above matrix corresponding to the false (0) values of the logical function
( p  q) & (q  r ) [1 1 0 1 0 0 0 1] are deleted.

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