6. Tautology and Absorption

Script:  Phil Evans  Pencils and Inks: Jesse Marsh   

© Respective copyright/trademark holders.

6.

The law of tautology, in two forms:

       (*4·24)  ├ : p . ≡ : p . p,

       (*4·25)  ├ : p . ≡ . p v p.

i.e. “p is true” is equivalent to “p is true and p is true,” as well as to “p is true or p is true,” from a formal point of view, it is through the law of tautology and its consequences that the algebra of logic is chiefly distinguished from ordinary algebra.

 

The law of absorption:

       (*4·71)  ├ :. p É q . ≡ : p . ≡ . p . q .

This shows the factor q is absorbed by the factor p, if p implies q. This principle allows us to replace an implication (p ) q) by an equivalence

(p . ≡ . p . q .) whenever it is convenient to do so.

An analogous and very important principle is the following:

       (*4·73)  ├ :. q . É : p . ≡ . p . q .

Logical addition and multiplication of propositions obey the associative and communicative laws, and the distributive laws in two forms, namely:

       (*4·4)   ├ :. p . q v r . ≡ : p . q v p . r,

       (*4·41)  ├ :. p . v . q . r : ≡ : p v q . p v r.

The second of these distinguishes the relations of logical addition and multiplication from of those of mathematical addition and multiplication.

Script: Bob Powell   Pencils and Inks: Bob Powell  © Respective copyright/trademark holders.

Tautology is not welcome generally in language or standard written communication; but it is a building block of thought and the determining of fine distinctions. Tautology is the repetition of an idea, especially in other words than fit the present context without imparting more clear or forceful emphasis. This is detrimental to speech or journalism in that is wastes time and energy in communicating abstract ideas to others and most often impedes our purpose.

However in the field of logic and logical philosophy under present examination, tautology represents a compound propositional form in which all instances are true or all instances are false. This is useful in our thought process and in propositional notation which we can see, consider and share without the lengthy exposition that destroys its value by inertia.

Absorption is a process that allows us to, within set guidelines, recognize and convert the nature of more speculative propositions into ones that are more provable and acceptable. These are propositional equations that resemble mathematic equations, but do not function in the same manner. This dissonance causes mathematicians to want to change terms and structures in these introductory chapters, but this is a big mistake. Any changes or simplifications in the metaphysical construct here affects and corrupts perception and understanding of the actual mathematical content in the body of Principia Mathematica.

Script: Robert Schaefer; Eric Freiwald  Pencils and Inks: Russ Manning  © Respective copyright/trademark holders.