Re: Sowa and the Meaning of Equivalence Relation. Was Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[John F. Sowa]

Jerry LRC, 
In logic, the statement "p is equivalent to
q" is a synonym for "p iff q", which is a synonym for
"(if p then q) and (if q then p)".  That is the relation that
Peirce intended when he said that the scroll is equivalent to a nest of
two negations.
In mathematics, there is an open-ended variety of
equivalence relations.  For more information, see my tutorial: 
http://jfsowa.com/logic/math.htm  where I state the following
definition:
An equivalence relation is reflexive, symmetric,
and transitive.  The archetype of an equivalence relation is equality:
it is reflexive, because x=x; it is symmetric,
because x=y implies y=x; and it is transitive,
because x=y and y=z imply x=z.
As another example, born under the same sign of the zodiac is
an equivalence relation over the set of all people.
Whenever an equivalence relation is defined over a set,
it divides the set into equivalence classes.
The zodiac relation divides the set of all human beings into 12
equivalence classes that have the traditional labels Aries, Taurus, ...,
Pisces.
That tutorial has been downloaded over 200,000 times.  You
might find it useful.
John
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Sowa and the Meaning of Equivalence Relation. Was Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[Jerry LR Chandler]

List:

> On Feb 10, 2021, at 7:44 PM, Jon Alan Schmidt  
> wrote:
> 
> On Tue, Feb 9, 2021 at 11:25 PM John F. Sowa  > wrote:
>  and Jerry LRC,
> 
> JFS> In mathematics and logic, equivalence means freely interchangeable in 
> all contexts without any change in meaning.
> 
> JLRC> Really?
> 
> Yes, indeed.  As Casey Stengel used to say, "Ya could look it up." 
> 
> John
> 

The question of interchangeability of meaning of mathematical symbols is remote 
from the simplistic view you present, John.

In standard mathematics, that is, the usual mathematics of analysis and applied 
mathematics of physical analysis,
including such graphic structures as categories, topological spaces, Hilbert 
spaces, etc, 
the term equivalence relations is used to distinguish one form of mathematical 
structure from another.
The constraints used to separate the meanings of equivalence relations can be 
interpreted as a triad,
firstness (or transitivity), secondness (or symmetry) and thirdness (or 
reflexivity).


One very simple usage of the notion of “equivalence relations” is to structure 
the ordering of mathematical spaces such as (x,y,z), by inclusivity, that is, 
x is contained in y and
 y is contained in z and therefore
x is contained in z. 

With these antecedent definitions, one can order the meanings of the structural 
graphs for various spaces.
For example, a Hilbert space is logically contained in a metric space and a 
metric space is contained in a general topological space.

These distinctions are critical for interpreting the relationships between CSP 
usage of “topology” / graph theory
and the usage of Hilbert spaces in modern chemical graph theory.   (That is, 
perplex number theory of electrical relations.)
Note that the “handedness” of biomolecules are NOT symmetric as a consequence 
of the existential graphic patterns of internal electrical relations of 
chemical atoms in these compositions.

If you are not aware of these advanced mathematical concepts, to quote Casey 
Stengel, you could look it up.

But, I will save you the time and effort and suggest that the term 
“equivalence” as you used it in the sentence:

> JFS> In mathematics and logic, equivalence means freely interchangeable in 
> all contexts without any change in meaning.
> 

does not have a triadic meaning and does not imply an “equivalence relation”. 

Cheers

Jerry 

Research Professor
Krasnow Institute of Advanced Study

Headwater House
Little Falls, MN




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Re: Sowa and the Meaning of Equivalence Relation. Was Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[Auke van Breemen]

Jerry,

I think you did hit the weak spot: 

> But, I will save you the time and effort and suggest that the term 
> “equivalence” as you used it in the sentence:
> 
> 
> > > 
> > > > > 
> > > JFS> In mathematics and logic, equivalence means freely 
> > > interchangeable in all contexts without any change in meaning.
> > > 
> > > > > 
> > > does not have a triadic meaning and does not imply an 
> > “equivalence relation”. 
> 


I only would add 'real' to your remark. For logical space it is freely 
interchangable as Jon acknowledges.


Best,

Auke

> Cheers
> 
> Jerry 
> 
> Research Professor
> Krasnow Institute of Advanced Study
> 
> Headwater House
> Little Falls, MN
> 
> 
> 
> 
> 


 

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