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LAFS. Note 8 (02)
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The definition of a "sign" as a thing involved in a
specific kind of triadic relation, a "sign relation",
is one that Peirce gives pursuant to his definition of
logic as "formal semiotic", that is, as a formal theory
of signs. (04)
Further details on how the definition of a sign relation
bears on the conception of logic as formal semiotic can
be found by examining the following two variant texts: (05)
 On the Definition of Logic [Version 1]

 Logic will here be defined as ''formal semiotic''.
 A definition of a sign will be given which no more
 refers to human thought than does the definition of
 a line as the place which a particle occupies, part
 by part, during a lapse of time. Namely, a sign is
 something, ''A'', which brings something, ''B'', its
 ''interpretant'' sign determined or created by it, into
 the same sort of correspondence with something, ''C'',
 its ''object'', as that in which itself stands to ''C''.
 It is from this definition, together with a definition
 of "formal", that I deduce mathematically the principles
 of logic. I also make a historical review of all the
 definitions and conceptions of logic, and show, not
 merely that my definition is no novelty, but that my
 nonpsychological conception of logic has ''virtually''
 been quite generally held, though not generally
 recognized. (C.S. Peirce, NEM 4, 2021).

 On the Definition of Logic [Version 2]

 Logic is ''formal semiotic''. A sign is something, ''A'', which
 brings something, ''B'', its ''interpretant'' sign, determined
 or created by it, into the same sort of correspondence (or
 a lower implied sort) with something, 'C', its 'object',
 as that in which itself stands to 'C'. This definition
 no more involves any reference to human thought than
 does the definition of a line as the place within
 which a particle lies during a lapse of time.
 It is from this definition that I deduce the
 principles of logic by mathematical reasoning,
 and by mathematical reasoning that, I aver,
 will support criticism of Weierstrassian
 severity, and that is perfectly evident.
 The word "formal" in the definition is
 also defined. (C.S. Peirce, NEM 4, 54).

 ''The New Elements of Mathematics'', Volume 4,
 Carolyn Eisele (ed.), Mouton, The Hague, 1976. (06)
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