ONT Re: New List & Classification of Signs
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New Listers,
My immediate concern is to get to some concrete examples,
of a sort that will be simple enough for a beginning but
still embedded and filled out enough to help us keep in
mind the uses of logic in realistic applied situations.
Otherwise, we will be in danger of spinning off into
the anomie of syntactic rootlessness.
| NB. On this ascii transcription.
|
| !P! = Pi = Product, !S! = Sigma = Sum
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| `A` = All = Universal, `E` = Some = Existential
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| "c" = "contained as a subset in"
|
| I will not always quote a formula if it is clearly being mentioned, not used.
One of the differences between the way that Peirce -- like most practicing
math, stat, compsci folks still do today -- would have looked at a formula
like !S!_x F(x)G(x) and the way that most of us probably learned to regard
a formula like `E`x(Fx&Gx) is that the F and G are thought of as functions
F, G : X -> B from a suitable space X, the "universe of discourse", to the
space B of boolean values, usually written 0 and 1 or 'false' and 'true',
respectively. The first caution is not to get too excited about the kind
of "value" that is meant by the truth value 'true'. It is not as if this
little "gold star" were the be-all end-all of logical inquiry as coded
into a logical expression. It is merely an "indicator" of the objects
in the space X that one is concerned to point out by means of a sign.
Indeed, functions like F, G : X -> B are commonly called by either
of the names "indicator functions" or "characteristic functions"
in math and stat today. What they indicate or characterize is,
in the case of F, the subset of X such that F(x) = 1. This
is spoken "F-inverse of 1" and written "(F^(-1))(1)" and
various people call it the "antecedent", the "fiber",
the "level set", or the "pre-image under F of 1
in X". To make the notation prettier, one can
introduce the "fiber bars" "[| ... |]" and
write [|F|] = (F^(-1))(1) c X. This is
all just 1-dim notation for what you
are doing when you shade a region
of a venn diagram, nothing more.
Jon Awbrey
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