12,080
edits
Changes
Directory talk:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 16:03, 27 April 2009
, 16:03, 27 April 2009→Commentary Work Area
==Place For Discussion==
==Place For Discussion==
==Commentary Work Area==
==Work Area==
===Commentary Note 12.2===
The logic of terms is something of a lost art these days, when the current thinking in logic tends to treat the complete proposition as the quantum of discourse, ''ne plus infra''. With absolute terms, or monadic relatives, and the simpler operations on dyadic relatives, the necessary translations between propositions and terms are obvious enough, but now that we've reached the threshold of higher adic relatives and operations as complex as exponentiation, it is useful to stop and consider the links between these two languages.
The term ''exponentiation'' is more generally used in mathematics for operations that involve taking a base to a power, and is slightly preferable to ''involution'' since the latter is used for different concepts in different contexts. Operations analogous to taking powers are widespread throughout mathematics and Peirce frequently makes use of them in a number of important applications, for example, in his theory of information. But that's another story.
The ''function space'' <math>Y^X,\!</math> where <math>X\!</math> and <math>Y\!</math> are sets, is the set of all functions from <math>X\!</math> to <math>Y.\!</math> An alternative notation for <math>Y^X\!</math> is <math>(X \to Y).</math> Thus we have the following equivalents:
{| align="center" cellspacing="6" width="90%"
| <math>\begin{matrix}Y^X & = & (X \to Y) & = & \{ f : X \to Y \}\end{matrix}</math>
|}
If <math>X\!</math> and <math>Y\!</math> have cardinalities <math>|X|\!</math> and <math>|Y|,\!</math> respectively, then the function space <math>Y^X\!</math> has a cardinality given by the following equation:
{| align="center" cellspacing="6" width="90%"
| <math>\begin{matrix}|Y^X| & = & |Y|^{|X|}\end{matrix}</math>
|}
In the special case where <math>Y = \mathbb{B} = \{ 0, 1 \},</math> the function space <math>\mathbb{B}^X</math> is the set of functions <math>\{ f : X \to \mathbb{B} \}.</math> If the elements <math>0, 1 \in \mathbb{B}</math> are interpreted as the logical values <math>\operatorname{false}, \operatorname{true},</math> respectively, then a function of the type <math>X \to \mathbb{B}</math> may be interpreted as a ''proposition'' about the elements in <math>X.\!</math>
===Old Commentary Notes===
<pre>
<pre>