13. Formal Implication and Formal Equivalence

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.



Formal implication and formal equivalence  

When an implication, say φx . É . y x, is said to hold always, i.e. when (x) : φx .  É  . yx, we shall say that φx formally implies yx; and propositions of the form "(x): φx . É . yx" will be said to state formal implications. In the usual instances of implication, such as "'Socrates is a man' implies 'Socrates is mortal,"' we have a proposition of the form "φx . É . yx " in a case in which "(x): φx . É . yx" is true. In such a case, we feel the implication is a particular case of a formal implication. Thus it has come about that implications which are not particular cases of formal implications have not been regarded as implications at all. There is also a practical ground for the neglect of such implications, for, speaking generally, they can only be known when it is already known either that their hypothesis is false or that their conclusion is true; and in neither of these cases do they serve to make us know the conclusion, since in the first case the conclusion need not be true, and in the second it is known already. Thus such implications do not serve the purpose for which implications are chiefly useful, namely that of making us know, by deduction, conclusions of which we were previously ignorant. Formal implications, on the contrary, do serve this purpose, owing to the psychological fact that we often know         "(x) : φx . É . yx" and yy, in cases where by (which follows from these premises) cannot easily be known directly.   

These reasons, though they do not warrant the complete neglect of implications that are not instances of formal implications, are reasons which make formal implication very important. A formal implication states that, for all possible values of x, if the hypothesis φx is true, the conclusion yx is true. Since       "φx . É . yx" will always be true when φx is false, it is only the values of x that make φx true that are important in a formal implication; what is effectively stated is that, for all these values, yx is true. Thus propositions of the form "all a is b," "no a is b" state formal implications, since the first (as appears by what has just been said) states

(x) : x is an a . É . x is a b,

while the second states

(x) : x is an a . É . x is not a b.

And any formal implication "(x): φx . É . yx" may be interpreted as: "All values of x which satisfy* φx satisfy yx," while the formal implication                           "(x) : φx . É . ~yx" may be interpreted as: "No values of x which satisfy φx satisfy yx."

We have similarly for "some a is b " the formula

($x) . x is an a . x is a b,

and for "some a is not b" the formula

($x). x is an a . x is not a b.

Two functions φx, yx are called formally equivalent when each always implies the other, i.e. when

(x) : φx º . yx,

and a proposition of this form is called a formal equivalence. In virtue of what was said about truth-values, if φx and yx are formally equivalent, either may replace the other in any truth-function. Hence for all the purposes of mathematics, or of the present work, φz^ may replace yz^ or vice versa in any proposition with which we shall be concerned. Now to say that φx and yx are formally equivalent is the same thing as to say that φz^ and yz^ have the same extension, i.e. that any value of x which satisfies either satisfies the other. Thus whenever a constant function occurs in our work, the truth-value of the proposition in which it occurs depends only upon the extension of the function. A proposition containing a function φz^ and having this property (i.e. that its truth-value depends only upon the extension of φz^) will be called an extensional function of φz^. Thus the functions of functions with which we shall be specially concerned will all be extensional functions of functions.

What has just been said explains the connection (noted above) between the fact that the functions of propositions with which mathematics is especially concerned are all truth-functions and the fact that mathematics is concerned with extensions rather than intensions.  

* A value of x is said to satisfy φx or yx when φx is true for that value of x.

Walt Kelly’s Songs of the Pogo© Respective copyright/trademark holders.

When contemplating a complex problem of any sort, the answers are very often unknowable by empirical means. This conundrum however is not an insurmountable obstacle to finding a solution to any such problem by means of tools including formal implication and formal equivalence.

These same tools are paramount to the Indian’s perceptual toolbox. Instead of expecting a new object, creature, person or idea to fit into her collective experience, she will seek implications and equivalences to reach some means to bring the unknown into a level of common knowledge.

We use the tools of implication and equivalence to find meaning in many ways, including in this beautiful song by Jackson Browne and performed with David Lindley, with vocals by renowned Spanish songbird Luz Casal.

Script: Bob Haney  Pencils and inks: John Rosenberger

Superman™ Wonder Woman™ © Respective copyright/trademark holders.

hanBLOGlaka

Logical Philosophy In American

Indian thought and Perception

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

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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.

4. Use of Dots

Fletcher Hanks’ Stardust

© Respective copyright holders.

The Use of Dots –

1. To bracket off propositions.

2. To indicate the logical product of two propositions.

The general principle is that a large number of dots indicate an outside bracket and a smaller number of dots indicate an inside bracket.


Group I      consists of dots adjoining a sign of implication (É), of equivalence (≡), of disjunction (v), or of equality by definition (= Df).


Group II     consists of dots following brackets indicative of an apparent variable such as (x), or (x,y), or (Эx), or (Эx,y), or [(ίx) (φx)] or analogous expressions.


Group III      consists of dots which stand between propositions in order to indicate a logical product. The scope of any collection of dots extends backwards or forwards beyond any smaller number of dots or equal number of dots from a group of less force until we reach the end of the proposition or a greater number of dots or an equal number of dots belonging to a group of equal or superior force.

 Dots indicating a logical product have a scope which works backwards and forwards. Other dots work only away from adjacent signs of disjunction, implication or equivalence, or forward from others in Group II.

“├ : p v q . É . q v q”

shows what is asserted is the whole of what follows the assertion sign.

“├ :. p É q . É : q É r . É . p É r”

means “if p implies q, then if q implies r, p implies r.”



“p É q . É . q É r : É . p É r”

will mean “if ‘p implies q’ implies ‘q implies r’ then

p implies r.”  (This is in general untrue.)

“p É q . q É r . É . p É r”

will mean “p implies q; and if q implies r then

p implies r.”

In this formula the first dot indicates a logical product, hence the second dot extends back to the beginning of the proposition.

“p É q : q É r . É . p É r”

will mean “p implies q; and if q implies r, then p implies r.” (this in general is not true). Here the two dots indicate a logical product, since the two dots do not occur anywhere else, the scope of these two dots extend backwards to the beginning of the proposition, and forwards to the end.

“p v q . É :. p . v . q É r : É p v r”

will mean “if either p or q is true, then if either p or ‘q implies r’ is true, it follows that either p or r is true.” If this is to be asserted, we must put four dots before the assertion sign thus:

“┌ :: p . v . q . É :. p . v . q É r : É . p v r.”

To assert an equivalent proposition to the phrase: “if either p or q is true, and either p or ‘p implies r’ is true, and either p or ‘q implies r’ is true,” we write:

“├ :. p v q : p . v . q É r : É . p v r.”

here the first pair of dots indicates a logical product, while the second pair does not. Thus the second pair of dots passes over the first pair, and back until we reach the three dots after the assertion sign.

In reading a proposition, the dots should be noticed first, as they show its structure.

In a proposition containing, several signs if implication, or equivalence, the one with the greatest number of dots before or after is the principle one: everything that goes before this one is stated by the proposition to imply or be equivalent to everything that comes after it.

 

 

Script, Pencils and Inks: Russ Manning  © Respective copyright/trademark holders.

The dots system and other symbols are carefully thought out to compress the language of propositional equations to as short a form as possible. In fact it seems the entire work is a response to the basic problem of writing about or speaking about very complex subjects, that of the severe limitation of the amount of time and attention the reader or listener is willing or able to devote to taking it in. As the nomenclature is absorbed and practiced very complex expressions may be absorbed and evaluated instantly, or practically so, without the need for the time consuming demands of conventional written language or speech.

Also by this system error manifests itself clearly, so one is not taxed by having to decide whether or not to believe the material presented.

                     Artist Unknown: Little Dot© Respective copyright/trademark holders.