[Inquiry] Re: Differential Logic -- Series B

[Jon Awbrey]

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DLOG.  Note B14

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Table 5 sums up the facts of the physical situation at equilibrium.
If we let B = {note, rest} = {moving, steady} = {charged, resting},
or whatever candidates you pick for the 2-membered set in question,
the Table shows a function f : B x B -> B, where f[x, y] = (x , y).

Table 5.  Dynamics of (x , y)
o---------o---------o---------o
|    x    |    y    | (x , y) |
o=========o=========o=========o
| resting | resting | charged |
o---------o---------o---------o
| resting | charged | resting |
o---------o---------o---------o
| charged | resting | resting |
o---------o---------o---------o
| charged | charged | charged |
o---------o---------o---------o

There are two ways that this physical function
might be taken to represent a logical function:

1.  If we make the identifications:
    
    charged  =  true   (= indicated),
    
    resting  =  false  (= otherwise),
    
    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical equivalence",
    or just plain "equality":

    Table 6.  Equality Function
    o---------o---------o---------o
    | x       | y       | (x , y) |
    o=========o=========o=========o
    | false   | false   | true    |
    o---------o---------o---------o
    | false   | true    | false   |
    o---------o---------o---------o
    | true    | false   | false   |
    o---------o---------o---------o
    | true    | true    | true    |
    o---------o---------o---------o

2.  If we make the identifications:
    
    resting  =  true   (= indicated),
    
    charged  =  false  (= otherwise),
    
    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical difference",
    or "exclusive disjunction":

    Table 7.  Difference Function
    o---------o---------o---------o
    | x       | y       | (x , y) |
    o=========o=========o=========o
    | true    | true    | false   |
    o---------o---------o---------o
    | true    | false   | true    |
    o---------o---------o---------o
    | false   | true    | true    |
    o---------o---------o---------o
    | false   | false   | false   |
    o---------o---------o---------o

Although the syntax of the cactus language modifies the
syntax of Peirce's graphical formalisms to some extent,
the first interpretation corresponds to what he called
the "entitative graphs" and the second interpretation
corresponds to what he called the "existential graphs".
In working through the present example, I have chosen
the existential interpretation of cactus expressions,
and so the form "(jets , sharks)" is interpreted as
saying that everything in the universe of discourse
is either a Jet or a Shark, but never both at once.

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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