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Revision as of 00:40, 19 October 2008
, 00:40, 19 October 2008→Formal development: add explanation of EN & EX
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One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows:
Peirce introduced these formal equations at a level of abstraction that is one step higher than their customary interpretations as propositional calculi, which two readings he called the ''Entitative'' and the ''Existential'' interpretations, here referred to as ''En'' and ''Ex'', respectively. The early CSP, as in his essay on "Qualitative Logic", and also GSB, emphasized the ''En'' interpretation, while the later CSP developed mostly the ''Ex'' interpretation. When it comes down to this very primitive level of formal structure, it is important to note the significance of the circumstance that this formal system is a ''very abstract calculus'' (VAC), devoid of meaning in the usual logical sense.→
{| align="center" border="0" cellpadding="10"
|
<math>\begin{array}{ccccc}
I_1 & : &
\operatorname{true}\ \operatorname{or}\ \operatorname{true} & = &
\operatorname{true} \\
I_2 & : &
\operatorname{not}\ \operatorname{true}\ & = &
\operatorname{false} \\
J_1 & : &
a\ \operatorname{or}\ \operatorname{not}\ a & = &
\operatorname{true} \\
J_2 & : &
(a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = &
a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\
\end{array}</math>
|}
Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows:
{| align="center" border="0" cellpadding="10"
|
<math>\begin{array}{ccccc}
I_1 & : &
\operatorname{false}\ \operatorname{and}\ \operatorname{false} & = &
\operatorname{false} \\
I_2 & : &
\operatorname{not}\ \operatorname{false} & = &
\operatorname{true} \\
J_1 & : &
a\ \operatorname{and}\ \operatorname{not}\ a & = &
\operatorname{false} \\
J_2 & : &
(a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = &
a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\
\end{array}</math>
|}
All of the axioms in this set have the form of equations. This means that all of the inference licensed by them are reversible. The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{~~~~~~}}</math> to mark this fact, but it will often be left to the reader to decide which of the two possible ways of applying the axiom is the one that is called for in a particular case.
Peirce introduced these formal equations at a level of abstraction that is one step higher than their customary interpretations as propositional calculi, which two readings he called the ''Entitative'' and the ''Existential'' interpretations, here referred to as "EN" and "EX", respectively. The early CSP, as in his essay on "Qualitative Logic", and also GSB, emphasized the EN interpretation, while the later CSP developed mostly the EX interpretation.
===Frequently used theorems===
===Frequently used theorems===
