5. On Definitions

5.

 Definitions

definiendum: that which is defined (noted to the left).

 definiens: that which it is defined as meaning (to the right with =)

“p É q . = . ~p v q  Df.”

Three primitive ideas have been introduced above which are not “defined” but only descriptively described. Their primitiveness is only relative to our expansion of logical connection and is not absolute; though of course such an exposition gains in importance according to the simplicity of its primitive ideas.

 These ideas are symbolized by “~p” and “p v q” and by “├” prefixed to a proposition.

Three definitions have been introduced:

p . q . = . ~(~p v ~ q)  Df,

p . É q .= . ~ p v q  Df,

p ≡ q . = . p É q . q ) p  Df.

Primitive propositions: some propositions must be taken without
proof, since all inference proceeds from propositions previously
asserted. These, like the primitive ideas are to some extent a matter
of arbitrary choice. Though, as in the previous case, a logical
system grows in importance according as the primitive propositions
are few and simple.
 
It will be found that owing to the weakness of the imagination in
dealing with simple abstract ideas no very great stress can be laid
upon their obviousness.
 
They are obvious to the instructed mind, but then so are many
propositions which cannot be quite true, as being disproved by their
contradictory consequences.
 
The following are the primitive propositions employed in the
calculus of propositions. The letters “Pp: stand for “primitive
proposition.”

(1)     Anything implied by a true premise is true  Pp.

This is the rule which justifies inference.

(2)     ├ : p v p . É . p  Pp.

i.e. if p or p is true, then p is true.

   (3)  ├ : q  . É . q v p  Pp.

i.e. if q is true then p or q is true.

   (4)   ├ : p v q . É . q v p  Pp.

i.e. if p or q is true, then q or p is true.

   (5)   ├ : p v É q v r) . É . q v (p v r)  Pp

i.e. if either p is true or “q or r” is true then either q is true or “p or r” is true.

   (6)   ├ :. q É r . É : p v q . É . p É r  Pp’

i.e. if q implies r, then “p or q” implies “p or r.”

   (7)   “The axiom of identification of real variables.”

When we have separately asserted two different functions of x where x is undetermined it is often important to know whether we can identify the x in one assertion with the x in the other. i.e. if     φ(x,y,z…) is a constituent in one assertion, and φ(x,u,v…) is a constituent in another.

*3·03, *1·7, and *1·72 (which is the statement of this axiom).

Some simple propositions:

The law of the excluded middle: ├ . p v ~p.  (*2·11)

The law of contradiction: ├ . ~(p . ~p).  (*3·24)

The law of double negation: ├ . p ≡ ~(~p).   (*4·13)


The principle of transposition: this principle has various forms, namely:

       (*4·1)   ├ : p É q .≡ . ~q É ~p,

       (*4·11) ├ : p ≡ q . ≡ . ~p ≡ ~q,

       (*4·14) ├ :. p . q . É . r : ≡ : p . ~r . ) . ~q. as well as other variations of these.

The handsome gent in the photo at the top of this post is a Tlingit shaman circa 1900. His culture was already in flux from half a century of contact with European culture. Centuries old realities were fast changing by concepts and technologies that brought about changes that were unexpected, and affected each person in individual ways. Cultures neither rise nor decline by broad strokes, it is the way that persons perceive, understand and communicate large and small changes among themselves that makes present and future dimensional realities

The base idea of this work is not specifically to be always correct in all assertions, inferences or propositions; more important is to be sure that that these are not in error. Perhaps these seem like the same thing, but they are heads and tails on the same coin. One may choose to see a coin as a prospective gumball,  or take a more existential view that there are two  dimensional-aspects whose differences are recognizable and definable as significant fine distinctions that are useful or not as each person ultimately decides  for herself.