Logical Module

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). The true statement can be also signified as '1' and the false statement as '0.'

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

To create the above table, first, select ( Data, Format ), set the number of decimal points to zero, and click on the Format command.

Next, select ( Structures, Tautological ) and, using the rotating button, set the number of arguments to 2. Click the Accept button.
Note, that the number of possible outcomes of all possible combinations of true and false statements, n, is given by the equation n = 2^k

where k denotes the number of statements. For three statements, the n = 2³ = 8, as shown in the table below.

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 generated as

and summarized as shown in the table below.

CONJUNCTION

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.

The above table was created by selecting ( Structures, Tautological ), setting the Number of Arguments to 2, clicking of Accept command,

selecting the the Logical Module, marking the p and q Arguments, clicking on the Conjunction command

and clicking the Apply command. Analogous steps can be used for generation of other logical functions.

IMPLICATION 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.

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 (Select ( Structures, Implicational ) )

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 the implication scale.

Consider a data matrix, consisting of all possible response patterns to a set of three variables (Select (Structures, Tautological )) and set

the Number of Arguments to 3

Click on the Logical module, select the p and q Arguments, and click on the Implication function and the Apply button.

Next, select the q and r Arguments, and click the Implication function and the Apply button.

Finally, select the p->q and q->r Arguments

and click on the Conjunction function and on the Apply command.

Select the p->q &q->r Argument and click on the Rectify command.

Delete the last three columns of the above table. and select ( Operations, Add Variables, Select All ), Name the Sum G and click the Accept command.

Data on the Vector display will now contain the triangular pattern of the p q r variables and their corresponding Guttman scale 0 1 2 3.