| Line 8: |
Line 8: |
| | ===Axioms=== | | ===Axioms=== |
| | | | |
| − | The axioms are just four in number, and they come in a couple of flavors: | + | The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math> |
| | | | |
| − | * The ''arithmetic initials'', ''I''<sub>1</sub> and ''I''<sub>2</sub>.
| + | {| align="center" border="0" cellpadding="10" cellspacing="0" |
| − | | + | | [[Image:PERS_Figure_01.jpg|500px]] || (1) |
| − | * The ''algebraic initials'', ''J''<sub>1</sub> and ''J''<sub>2</sub>.
| + | |- |
| − | | + | | [[Image:PERS_Figure_02.jpg|500px]] || (2) |
| − | o-----------------------------------------------------------o
| + | |- |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| + | | [[Image:PERS_Figure_03.jpg|500px]] || (3) |
| − | | ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| + | |- |
| − | | ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| + | | [[Image:PERS_Figure_04.jpg|500px]] || (4) |
| − | | ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| + | |} |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` |
| |
| − | | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| − | | Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` |
| |
| − | o-----------------------------------------------------------o
| |
| | | | |
| | Notice that all of the axioms in this set have the form of equations. This means that all of the inference steps licensed by them are fully reversible. In the proof annotation scheme used below, a double bar "=====" is used to mark this fact, but it may at times be left to the reader to decide which direction of axiom application is the one that is called for in a particular case. | | Notice that all of the axioms in this set have the form of equations. This means that all of the inference steps licensed by them are fully reversible. In the proof annotation scheme used below, a double bar "=====" is used to mark this fact, but it may at times be left to the reader to decide which direction of axiom application 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. 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. | + | 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.→ |
| | | | |
| | ===Frequently used theorems=== | | ===Frequently used theorems=== |