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).

 

 

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

 

 

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 summarized as shown in the tables below.

 

 

 

 

 

 

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.

 

                                                                    

 

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:

 

                                                                     

 

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

 

                                                                                          

 

 

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.

 

                                                                     

 

 

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.

                                                                    

 

 

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.

 

 

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.