o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

	



o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

	



o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` 



o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ---> Delete ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

	



o-----------------------------------------------------------o





 ` ` ` ` ` ` ` ` ` 



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.

	



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
C1.  Double negation theorem
The first theorem is known as the Double Negation Theorem (DNT).

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ---> Collect` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o


	





o-----------------------------------------------------------o



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o




The proof that follows it is derived from the one that was given by George Spencer Brown in his book Laws of Form and credited to two of his students, John Dawes and D.A. Utting.  This result is annotated as Consequence 1 (C1) or as Reflection in LOF.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> Reflect ` ` ` ` ` ` ` |
o-----------------------------------------------------------o


	


o-----------------------------------------------------------o
< I2. Unfold "(())" >=========o

o=============================
< J1. Insert "(a)" >==========o

o=============================
< J2. Distribute "((a))" >====o

o=============================
< J1. Delete "(a)" >==========o

o=============================
< J1. Insert "a" >============o

o=============================
< J2. Collect "a" >===========o

o=============================
< J1. Delete "((a))" >========o

o=============================
< I2. Refold "(())" >=========o

o=============================
< QED >=======================o


C2.  Generation theorem
One theorem of frequent use is the so-called Weed and Seed Theorem.  The proof is just an inductive exercise, so we can let it go till later, but it says that a label can be freely distributed or freely erased (retracted or withdrawn) anywhere in a subtree whose root is labeled with that label.  The second theorem on the list to be shown here amounts to the inductive base case for the Weed and Seed theorem.  It has the LOF name of Generation.

	

o-----------------------------------------------------------o

o=============================

	




o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` 



o-----------------------------------------------------------o




Here is a proof of the Generation Theorem.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> Regenerate` ` ` ` ` ` |
o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C1. Reflect "a(b)" >========o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< I2. Unfold "(())" >=========o

o=============================


 `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< J1. Insert "a" >============o

o=============================


 `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< J2. Collect "a" >===========o

o=============================


 `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C1. Reflect "a", "b" >======o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< QED >=======================o


C3.  Dominant form theorem
The third theorem to be proved here is one that GSB annotates as Integration, but it may also be regarded as a matter of Dominance or Recession among forms.

	

o-----------------------------------------------------------o

o=============================

	




o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` 



o-----------------------------------------------------------o




Here is a proof of the Dominant Form Theorem.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> Recess ` ` ` ` ` ` ` `|
o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C2. Regenerate "a" >========o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< J1. Delete "a" >============o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< QED >=======================o




If you scan the elementary steps that lead up to this point, you will notice two distinct qualities of the proofs so far:

	


	
 One brand of proof has that falling off a log and rolling downhill sort of quality that is earnestly to be wished for but seldom to be seen, at least, never so often as we'd wish.
	



	
 The other kind, more kith o' death than kind, has a quality strained past mercy, with a how in the heck did anybody ever think of that? sort of subtlety that all too unfortunately rules the roost whenever we begin to extend our practice to more and more compelling theories.
	




This is, to me, at least, a surprising observation, and though I have no grand conclusion to draw from it at the moment, it occurs to me that it might be a useful measure to keep in mind as we essay forth.

	

Exemplary proofs
With the meagre means afforded by the axioms and theorems given so far, it is already possible to prove a multitude of much more complex theorems.  A couple of all-time favorites are given next.

	

Peirce's law
(Peirce's law|Main article: Peirce's law)



This section presents a proof of Peirce's law, commonly written:

	


	* [[p ⇒ q] ⇒ p] ⇒ p
	
The first order of business is present the statement as it  appears in the so-called existential interpretation of Peirce's own logical graphs.  Here is the statement of Peirce's law, as rendered under the existential interpretation into (the topological dual forms of) Peirce's logical graphs:

	

o-----------------------------------------------------------o

o=============================

	

	

	

	

	






	

	


o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o




Finally, here's the promised proof of Peirce's law:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Collect >==============o

o==================================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Recess >===============o

o==================================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Refold >===============o

o==================================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Delete >===============o

o==================================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Refold >===============o

o==================================
< QED >==================o


Praeclarum theorema
Now to take up a more interesting example, here is the statement and a proof of the Praeclarum Theorema or Splendid Theorem of Leibniz.

	


	 If a is b and d is c, then ad will be bc.

	 


	 This is a fine theorem, which is proved in this way:

	 


	 a is b, therefore ad is bd (by what precedes),

	 


	 d is c, therefore bd is bc (again by what precedes),

	 


	 ad is bd, and bd is bc, therefore ad is bc.  Q.E.D.

	 


	 (Leibniz]], Logical Papers, p. 41).
	
o-----------------------------------------------------------o

o==================================

	



	

o-----------------------------------------------------------o


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o




And here's a neat proof of this nice theorem:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C1. Reflect "ad(bc)" >======o

o=============================


 ` 




 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Weed "a", "d" >=============o

o=============================


 ` 




 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C1. Reflect "b", "c" >======o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Weed "bc" >=================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< C3. Recess "abcd" >=========o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< I2. Refold "(())" >=========o

o=============================
< QED >=======================o


Formal extension: Cactus calculus
Let us now extend the CSP-GSB calculus in the following way:

	



The first extension is the reflective extension of logical graphs, or what may be described as the cactus language, after its principal graph-theoretic data structure.  It is generated by generalizing the negation operator "(—)" in a particular direction, treating "(—)" as the controlled, moderated, or reflective negation operator of order 1, and adding another such operator for each integer parameter greater than 1.  In sum, these operators are symbolized by bracketed argument lists of the following shapes:  "(—)", "(—, —)", "(—, —, —)", and so on, where the number of slots is the order of the reflective negation operator in question.

	



The formal rule of evaluation for a "k-lobe" or k-operator is:

	

o-----------------------------------------------------------o

o=============================

	




o-----------------------------------------------------------o

 ` ` ` ` ` >


` `=` `( )` ` |





The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


Case analysis-synthesis theorem
The task at hand is to lay out what I think of as the pontoon bridge between the model-theoretic and the proof-theoretic shores, and thus between their diverging perspectives on logical procedure, even if I can construct it at a point but so close to their common source that it may not seem like it's worth the candle.

	



The substance of this principle was known to Boole in the 1850's, tantamount to what we now call the boolean expansion of a propositional expression.  The only novelty here resides in a certain manner of presentation, in which we will prove the basic principle from the axioms given before.  One name for this rule is the Case Analysis-Synthesis Theorem (CAST).

	



I am going to revert to my customarily sloppy workshop manners and refer to propositions and proposition expressions on rough analogy with functions and function expressions, which implies that a proposition will be regarded as the chief formal object of discussion, enjoying many proposition expressions, formulas, or sentences that express it, but worst of all I will probably just go ahead and use any and all of these terms as loosely as I see fit, taking a bit of extra care only when I see the need.

	



Let Q be a proposition with an unspecified, but context-appropriate number of variables, say, none, or x, or x1, … xk, as the case may be.  (To be more precise, I should have said "sentence Q".)

	


	
 Strings and graphs sans labels are called bare.
	
 A bare terminal node, "o", is known as a stone.
	
 A bare terminal edge, "|", is known as a stick.
	




Let the replacement expression of the form Q[o/x] denote the proposition that results from Q by replacing every token of the variable x with a blank, that is to say, by erasing x.

	



Let the replacement expression of the form Q[|/x] denote the proposition that results from Q by replacing every token of the variable x with a stick stemming from the site of x.

	



In the case of a proposition Q, that is, an expression of it, not having a token of the designated variable x, let it be stipulated that Q[o/x] = Q = Q[|/x].

	



I think that I am at long last ready to state the following:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	



	

	


o-----------------------------------------------------------o



` ` ` ` ` ` |
/x] ` ` |



o-----------------------------------------------------------o

/x] (x) ) |




In order to think of tackling even the roughest sketch toward a proof of this theorem, we need to add a number of axioms and axiom schemata.  Because I abandoned proof-theoretic purity somewhere in the middle of grinding this calculus into computational form, I never got around to finding the most elegant and minimal, or anything near a complete set of axioms for the cactus language, so what I list here are just the slimmest rudiments of the hodge-podge of rules of thumb that I have found over time to be necessary and useful in most working settings.  Some of these special precepts are probably provable from genuine axioms, but I have yet to go looking for a more proper formulation.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |




o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> Merge ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

	



o-----------------------------------------------------------o

o-----------------------------------------------------------o



 ` ` ` ` ` ` ` ` /` ` ` ` ` ` ` ` 


 ` ` ` ` ` ` ` ` ` /` ` ` ` |


o-----------------------------------------------------------o

o-----------------------------------------------------------o

	



o-----------------------------------------------------------o

o-----------------------------------------------------------o

 1, the following equation holds:` ` ` ` ` ` ` ` ` >




To see why the Dispersion Rule holds, look at it this way:  If x is true, then the presence of x makes no difference on either side of the equation, but if x is false, then both sides of the equation are false.

	



Here is a proof sketch for the Case Analysis-Synthesis Theorem (CAST):

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> ` ` Disperse` ` ` ` ` |
o-----------------------------------------------------------o


	


o-----------------------------------------------------------o
< L1. Split " " >=============o

o=============================



` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



< L3. Disperse "Q" >==========o

o=============================



` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



< C1. Reflect "x" >===========o

o=============================



` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



< C2. Weed "x", "(x)" >=======o

o=============================



` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
/x] ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


< QES >=======================o




NB.  QES = "Quod Erat Sketchiendum".

	



Some of the jobs that the CAST can be usefully put to work on are proving propositional theorems and establishing equations between propositions.  Once again, let us turn to the example of Leibniz's Praeclarum Theorema as a way of illustrating how.

	

o-----------------------------------------------------------o

o=============================

	


o-----------------------------------------------------------o


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< CAST "a" >==================o

o=============================



 ` 
 ` ` ` ` ` ` ` ` ` 
 ` ` 


 ` ` ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 











< Domination >================o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` 


 ` ` ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 











< Cancellation >==============o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` 


 ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 











< Domination >================o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` 











< Cancellation >==============o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |










< CAST "d" >==================o

o=============================



 ` 
 ` ` ` ` ` ` ` 
/` ` 


 ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 












< Domination >================o

o=============================



 ` 
 ` ` ` ` ` ` ` 


 ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 












< Cancellation >==============o

o=============================



 ` 
 ` ` ` ` ` ` ` 


 ` ` ` ` ` ` ` `` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 












< Domination >================o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 












< Cancellation >==============o

o=============================


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |











< CAST "b" >==================o

o=============================


 ` ` ` ` 


 ` 
 ` ` ` ` ` ` ` 
 ` ` 


 ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 














< Cancellation >==============o

o=============================


 ` ` ` ` ` ` ` ` |

 ` ` 
 ` ` 


 ` ` ` ` ` ` ` ` /` ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 














< Domination >================o

o=============================


 ` ` 


 ` ` ` ` ` ` ` `` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 














< Cancellation >==============o

o=============================


 ` ` 


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |













< CAST "c" >==================o

o=============================


 ` ` 


 ` ` 
 ` ` 


 ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 
















< Cancellation >==============o

o=============================


 ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` ` ` ` ` 
















< Cancellation >==============o

o=============================
< QED >=======================o




What we have harvested is the succulent equivalent of a disjunctive normal form (DNF) for the proposition with which we started.  Remembering that a blank node is the graphical equivalent of a logical value true, we can read this brand of DNF in the following manner:

	

o-----------------------------------------------------------o

o=============================

	


o-----------------------------------------------------------o




That is tantamount to saying that the proposition being submitted for analysis is true in each case.  Ergo we are justly entitled to title it a Theorem.

	

Logic as sign transformation
We have been looking at various ways of transforming propositional expressions, expressed in the parallel formats of character strings and graphical structures, all the while preserving certain aspects of their "meaning" — and here I risk using that vaguest of all possible words, but only as a promissory note, hopefully to be cached out in a more meaningful species of currency as the discussion develops.

	



I cannot pretend to be acquainted with or to comprehend every form of intension that others might find of interest in a given form of expression, nor can I speak for every form of meaning that another might find in a given form of syntax.  The best that I can hope to do is to specify what my object is in using these expressions, and to say what aspects of their syntax are meant to serve this object, lending these properties the interest I have in preserving them as I put the expressions through the paces of their transformations.

	



On behalf of this object I have been spinning in the form of this thread a developing example base of propositional expressions, in the data structures of graphs and strings, along with many examples of step-wise transformations on these expressions that preserve something of significant logical import, something that might be referred to as their logical equivalence class (LEC), and that we could as well call the constraint information or the denotative object of the expression in view.

	



To focus still more, let us return to that Splendid Theorem noted by Leibniz, and let us look more carefully at the two distinct ways of transforming its initial expression that we just used to arrive at an equivalent expression, one that made its tautologous character or its theorematic nature as evident as it could be.

	



Just to remind you, here is the Splendid Theorem again:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	




o-----------------------------------------------------------o


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


o-----------------------------------------------------------o




The first way of transforming the expression that appears on the left hand side of the equation can be described as proof-theoretic in character.  That was given in Note 5.

	


	
 PERS 5. weblink
	




The other way of transforming the expression that appears on the left hand side of the equation can be described as model-theoretic in character.  That was given in Note 9.

	


	
 PERS 9. weblink
	




What we have here amounts to a couple of different styles of communicational conduct, or conductive communication, if you prefer, that is to say, two sequences of signs of the form e1, e2, …, en, each one beginning with a problematic expression and eventually ending with a clear expression of the appropriate logical equivalence class (LEC) to which each and every sign or expression in the sequence belongs.

	



Ordinarily, any orbit through a locus of signs can be taken to reflect an underlying sign-process, a case of semiosis.  So what we have here are two very special cases of semiosis, and what we might just find it useful to contemplate is how to characterize them as two species of a very general class.

	



We are starting to delve into some fairly picayune details of a particular sign system, non-trivial enough in its own right but still rather simple compared to the types of our ultimate interest, and though I believe that this exercise will be worth the effort in prospect of understanding more complicated sign systems, I feel that I ought to say a few words about the larger reasons for going through this work.

	



My broader interest lies in the theory of inquiry as a special application or a special case of the theory of signs.  Another name for the theory of inquiry is logic and another name for the theory of signs is semiotics.  So I might as well have said that I am interested in logic as a special application or a special case of semiotics.  But what sort of a special application?  What sort of a special case?  Well, I think of logic as formal semiotics — though, of course, I am not the first to have said such a thing — and by formal we say, in our etymological way, that logic is concerned with the form, indeed, with the animate beauty and the very life force of signs and sign actions.  Yes, perhaps that is far too Latin a way of understanding logic, but it's all I've got.

	



Now, if you think about these things just a little more, I know that you will find them just a little suspicious, for what besides logic would I use to do this theory of signs that I would apply to this theory of inquiry that I'm also calling logic?  But that is precisely one of the things signified by the word formal, for what I'd be required to use would have to be some brand of logic, that is, some sort of innate or inured skill at inquiry, but a style of logic that is casual, catch-as-catch-can, formative, incipient, inchoate, unformalized, a work in progress, partially built into our natural language and partially more primitive than our most artless language.  In so far as I use it more than mention it, mention it more than describe it, and describe it more than fully formalize it, then to that extent it must be consigned to the realm of unformalized and unreflective logic, where some say "there be oracles", but I don't know.

	



Still, one of the aims of formalizing what acts of reasoning that we can is to draw them into an arena where we can examine them more carefully, perhaps to get better at their performance than we can unreflectively, and thus to live, to formalize again another day.  Formalization is not the be-all end-all of human life, not by a long shot, but it has its uses on that behalf.

	



This looks like a good place to pause and take stock.  The question arises:  What is really going on here?  We have all these signs, but what is the object?

	



One object worth the candle is simply to study a non-trivial example of a syntactic system, simple in design but not entirely a toy, just to see how these systems tick.

	



More than that, we would like to understand how sign systems come to exist or can be placed in relation to object systems, in the likes of which we possess some compelling independent reason to take an interest.

	



What is the utility of setting up sets of strings and sets of graphs, and sorting them according to their semiotic equivalence class (SEC) based on this or that abstract notion of transformational equivalence?

	



Good questions.

	



I can but begin to address these questions in the present frame of work, but I can't hope to answer them in anything like a satisfactory fashion.  Nevertheless, I will not mind one bit if you keep them in mind as we go.

	

Analysis of contingent propositions
For all of the reasons mentioned above, and for the sake of a more compact illustration of the in and outs of a typical propositional equation reasoning system (PERS), let's now take up a much simpler example of a contingent proposition:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	

	

	

	

	





 ` 








For the sake of simplicity in discussing this example, I will revert to the existential interpretation (Ex) of logical graphs and their corresponding parse strings.

	



Under Ex the expression "(p (q))(p (r))" interprets as the vernacular expression "p implies q and p implies r", in symbols, {p ⇒ q} ∧ p ⇒ r, so this is the reading that we'll want to keep in mind for the present.

	



Where brevity is required, and it occasionally is, we may invoke the propositional expression "(p (q))(p (r))" under the name "f" by making use of the following definition:  "f = (p (q))(p (r))".

	



Since the expression "(p (q))(p (r))" involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact a couple of different ways to execute the picture.

	



Figure 1 indicates the points of the universe of discourse X for which the proposition f : X → B has the value 1 (= true).  In this paint by numbers style of picture, one simply paints over the cells of a generic template for the universe X, going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under f remain untinted, and let the cells that get the value 1 under f be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the paints, in other words, the 0, 1 in B, but in the pattern of regions that they indicate.  NB.  In this Ascii version, I use [```] for 0 and [^^^] for 1.

	

o-----------------------------------------------------------o




There are a number of standard ways in mathematics and statistics for talking about "the subset W of the domain X that gets painted with the value z by the indicator function f : X → B".  The subset
W ⊆ X is called by a variety of names in different settings, for example, the antecedent, the fiber, the inverse image, the level set, or the pre-image in X of z under f.  It is notated and defined as W = f–1(z).  Here, f–1 is called the converse relation or the inverse relation — it is not in general an inverse function — corresponding to the function f.  Whenever possible in simple examples, we use lower case letters for functions f : X → B, and its is sometimes useful to employ capital letters for subsets of X, if possible, in such a way that F is the fiber of 1 under f, in other words, F = f–1(1).

	



The easiest way to see the sense of the venn diagram is to notice that the expression "(p (q))", read as "p ⇒ q", can also be read as "not p without q".  Its assertion effectively excludes any tincture of truth from the region of P that lies outside the region Q.

	



Likewise for the expression "(p (r))", read as "p ⇒ r", and also readable as "not p without r".  Asserting it effectively excludes any tincture of truth from the region of P that lies outside the region R.

	



Figure 2 shows the other standard way of drawing a venn diagram for such a proposition.  In this punctured soap film style of picture — others may elect to give it the more dignified title of a logical quotient topology or some such thing — one goes on from the previous picture to collapse the fiber of 0 under X down to the point of vanishing utterly from the realm of active contemplation, thereby arriving at a degenre of picture like so:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	










` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` P ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `








 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ Q ^ ^ ^ ^ 
 ^ ^ ^ ^ R ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 










o-----------------------------------------------------------o
Figure 1.  Venn Diagram for (p (q))(p (r))


	









 ` ` ` ` ` ` ` ` 
 ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` 
 ` ` ` ` ` ` ` ` 

 ` ` ` ` Q ` ` ` 
 ` ` ` R ` ` ` ` 

 ` ` ` ` ` ` ` ` 
 ` ` ` ` ` ` ` ` 

 ` ` ` ` ` ` ` ` 
 ` ` ` ` ` ` ` ` 












This diagram indicates that the region where p is true is wholly contained in the region where both q and r are true.  Since only the regions that are painted true in the previous figure show up at all in this one, it is no longer necessary to distinguish the fiber of 1 under f by means of any stipple.

	



In sum, it is immediately obvious from the venn diagram that in drawing a representation of the propositional expression:

	


	 (p (q))(p (r)),
	
in other words,

	


	 [p ⇒ q] ∧ [p ⇒ r],
	
we are also looking at a picture of:

	


	 (p (q r)),
	
in other words,

	


	 p ⇒ [q ∧ r].
(	
Let us now examine the following propositional equation:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
Figure 2.  Venn Diagram for (p (q r))


	

	

	

	

	


o-----------------------------------------------------------o



` `
` ` ` ` ` ` ` |

` ` ` ` ` ` ` |


q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r)( ` ` ` ` >




There are three distinct ways that I can think of right off as to how we might go about formally proving or systematically checking the proposed equivalence, the evidence of whose truth we already have before us clearly enough, and in a visually intuitive form, from the venn diagrams that we examined above.

	



While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.

	



A sign relationL is a subset of a cartesian product O × S × I, where O, S, I are sets known as the object, sign, and interpretant sign domains, respectively.  One writes L ⊆ O × S × I, where the symbol "⊆"
indicates the subset relation, contained as a subset of.  Accordingly, a sign relation L consists of ordered triples of the form (o, s, i), where o, s, i belong to the domains O, S, I, respectively.

	



The syntactic domain of a sign relation L ⊆ O × S × I is just the set-theoretic union S ∪ I of its sign domain S and its interpretant domain I.

	



It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have S = I.

	



Elsewhere I have discussed examples of sign relations that consist of a finite set of triples of the form (o, s, i), where o, s, i are the object, sign, interpretant sign, respectively, of what is called the sign triple or the elementary sign relation (o, s, i).

	



We will be taking a bit of a jump up from the finite case now, since most of the examples of sign relations that interest us in logic will have S and I being infinite sets, and usually O will be infinite, too, in the long run, at least, although we will frequently work up to the infinite object domains by way of various series of finite approximations and gradual stages.

	



With that preamble behind us, let us turn to consider the case of semiosis, or sign transformation process, that is generated by our first proof of the propositional equation E1.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	 


o-----------------------------------------------------------o


` `




< Double Negation >===========o

o=============================


` `




` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Collection >================o

o=============================


` `




` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< Double Negation >===========o

o=============================



` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< QED >=======================o




For some reason I always think of this as the way that our DNA would prove it.

	



We are in the process of examining various proofs of the propositional equation "(p (q))(p (r)) = (p (q r))", and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations.

	



Here is a reminder of the equation in question:

	

o-----------------------------------------------------------o

o=============================

	


o-----------------------------------------------------------o



` `
` ` ` ` ` ` ` |

` ` ` ` ` ` ` |


q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r)( ` ` ` ` >




The second way of establishing the truth of this equation is one that I see, rather loosely, as model-theoretic, for no better reason than the sense of its ending with a pattern of expression, a variant of the disjunctive normal form (DNF), that is commonly recognized to be the form that one extracts from a truth table by pulling out the rows of the table that evaluate to true and constructing the disjunctive expression that sums up the senses of the corresponding interpretations.

	



In order to apply this model-theoretic method to an equation between a couple of contingent expressions, one must transform each expression into its associated DNF and then compare those to see if they are equal.  In the current setting, these DNF's may indeed end up as identical expressions, but it is possible, also, for them to turn out slightly off-kilter from each other, and so the ultimate comparison may not be absolutely immediate.  The explanation of this is that, for the sake of computational efficiency, it is useful to tailor the DNF that gets developed as the output of a DNF algorithm to the particular form of the propositional expression that is given as input.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` 




< CAST "p" >==================o

o=============================



 ` 
 ` 










< Domination >================o

o=============================


 ` 










< Cancellation >==============o

o=============================


 ` 










< CAST "q" >==================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` 
 ` 














< Cancellation >==============o

o=============================


 ` ` ` ` ` 














< Domination >================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |













< CAST "r" >==================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` 








 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









< Cancellation >==============o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









< DNF >=======================o




What we have harvested is the succulent equivalent of a disjunctive normal form (DNF) for the proposition with which we started.  Remembering that a blank node is the graphical equivalent of a logical value true, we can read this brand of DNF in the following manner:

	

o-----------------------------------------------------------o

o=============================

	


o-----------------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |










o-----------------------------------------------------------o




Sorry, the half-time show was cancelled by the censors.  But I'm guessing that the reader can probably finish off the second half of the proof with a few scribbles on paper faster than I can asciify it on my own, so at least there's that entertaiment to occupy the interval.

	



We are still in the middle of contemplating a particular example of a propositional equation, namely, "(p (q))(p (r)) = (p (q r))", and we are still considering the second of three formal methods that I intend to illustrate in the process of thrice-over establishing it.

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o



` `
` ` ` ` ` ` ` |

` ` ` ` ` ` ` |


q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r) ` ` ` ` >




I know that it must seem tedious, but I probably ought to go ahead and carry out the second half of this analogically model-theoretic strategy, just so that we will have the security of this concrete and shared experience on which to fall back at every later point in whatmay quickly become a rather abstruse discussion.  Here then is the rest of the necessary chain of equations:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< CAST "p" >==================o

o=============================



 ` ` `


 ` ` ` 






< Domination >================o

o=============================



 ` ` `` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` 






< Cancellation >==============o

o=============================



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





< CAST "q" >==================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` 


 ` ` ` 










< Domination >================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` 


 ` ` ` 










< Cancellation >==============o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` 










< CAST "r" >==================o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` 


 ` ` ` 




 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









< Cancellation >==============o

o=============================


 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |



 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









< DNF >=======================o




This is not only a logically equivalent DNF, but exactly the same DNF expression that we obtained before, so we have established the given equation "(p (q))(p (r)) = (p (q r))".

	



Incidentally, one may wish to note that this DNF expression quickly folds into the following form:

	

o-----------------------------------------------------------o

o=============================

	





This can be read to say "either p q r, or not p", which gives us yet another expression equivalent to the sentences "(p (q))(p (r))" and "(p (q r))".

	



Still another way of writing the same thing would be like so:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	







 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





In other words, "p is equivalent to p and q and r".

	



Let's pause to refresh ourselves with a few morsels of lemmas bread.  One lemma that I can see just far enough ahead to see our imminent need of is the principle that I canonize as the Emptiness Rule.  It says that a bare lobe expression like "(… , …)", with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression "( )" that Ex interprets as denoting the logical value false.  To depict the rule in graphical form, we have the continuing sequence of equations:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` ` ` ` ` /` ` ` ` ` /` ` ` ` ` ` ` ` ` ` ` ` |





Yet another rule that we'll need is the following:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o



 ` ` ` ` ` /` ` ` ` ` /` ` ` ` ` ` ` ` ` ` ` ` |





This one is easy enough to derive from rules that are already known, but call it the Indistinctness Rule just on behalf of ready reference and employment.

	



Finally, let me introduce a rule-of-thumb that is a bit more suited to routine computation, and that will serve to replace the indistinctness rule in many of the cases where we actually have to call on it.  This is actually just a special case of the evaluation rule listed above:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 `x_2` `... x_k` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









o-----------------------------------------------------------o




To continue with the beating of this still kicking horse that is known as the propositional equation "(p (q))(p (r)) = (p (q r))", let's now take up the third way that I mentioned for examining propositional equations, but I believe that you will be relieved to know that it is literally a third way only at the very outset, almost immediately breaking up according to whether one proceeds by way of the more "routine" model-theoretic path or else by way of the more "strategic" proof-theoretic path.  I think that I'll take the low road today.

	



Let's convert the equation between propositions:

	


	 (p (q))(p (r))  =  (p (q r)),
	
into the corresponding equational proposition:

	


	 (( (p (q))(p (r)) , (p (q r)) )).
	
If you're like me, you'd rather see it in pictures:

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o
 ----> ` ` `Spike` ` ` ` ` ` |
o-----------------------------------------------------------o


	

	

	


o-----------------------------------------------------------o


` `
` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` |






 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


p=>q] & [p=>rp=>q] & [p=>r` ` ` ` ` ` ` ` >




We may now interrogate the alleged equation for the third time, working by way of the case analysis-synthesis theorem (CAST).

	

o-----------------------------------------------------------o

o-----------------------------------------------------------o


	


o-----------------------------------------------------------o


 ` 
 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |






` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

< CAST "p" >=============o

o==================================



` `
` ` `
/ ` `


` ` `  / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `










< Domination >===========o

o==================================



` `
` ` ` / ` / ` ` / ` ` ` ` ` ` ` ` ` ` |

` ` `  / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `










< Cancellation >=========o

o==================================



` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `










< Emptiness >============o

o==================================



` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `










< Cancellation >=========o

o==================================



` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |









< CAST "q" >=============o

o==================================


` `r` ` `


` `
` ` `
` ` `


` ` `  / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `














< Domination >===========o

o==================================


` `r` ` `


` `
` ` `
` ` `


` ` `  / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `














< Cancellation >=========o

o==================================



` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` `  / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `














< Domination >===========o

o==================================



` ` `


` ` `  ` ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `














< Spike >================o

o==================================



` ` `


` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `














< Cancellation >=========o

o==================================



` ` `


` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |













< CAST "r" >=============o

o==================================


` ` `


` ` `
` ` `


` ` ` ` / ` ` `






` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `


















< Cancellation >=========o

o==================================


` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |





` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `


















< Emptiness & Spike >====o

o==================================


` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `


` ` ` ` ` ` ` `

` ` ` ` ` ` ` `

` ` ` ` ` ` ` `


















< Cancellation >=========o

o==================================
< QED >==================o




And that, of course, is the DNF of a theorem.

	

Proof as semiosis
We have been looking at several different ways of proving one particular example of a propositional equation, and along the way we have been exemplifying the species of sign transforming process that is commonly known as a proof, more specifically, an equational proof of the propositional equation at issue.

	



Let us now draw out these semiotic features of the business of proof and place them in relief.

	



Our syntactic domain S contains an infinite number of signs or expressions, which we may choose to view in either their text or their graphic forms, glossing over for now the many details of their parsicular correspondence.

	



Here are some of the expressions that we find salient enough to single out and confer an epithetic nickname on:

	


	 e0 = "( )"
	



	 e1 = " "
	



	 e2 = "(p (q))(p (r))"
	



	 e3 = "(p (q r))"
	



	 e4 = "(p q r, (p))"
	



	 e5 = "(( (p (q))(p (r)) , (p (q r)) ))"
	
Under Ex we have the following interpretations:

	


	 e0 expresses the logical constant "false"
	



	 e1 expresses the logical constant "true"
	



	 e2 says "not p without q, and not p without r"
	



	 e3 says "not p without q and r"
	



	 e4 says "p and q and r, or else not p"
	



	 e5 says that e_2 and e_3 say the same thing
	
We took up the Equation E1 that reads as follows:

	


	 (p (q))(p (r))  =  (p (q r)).
	
Each of our proofs is a finite sequence of signs, and thus, for a finite integer n, takes the form:

	


	 s1, s2, s3, …, sn.
	
Proof 1 proceeded by the straightforward approach, starting with e2 as s1 and ending with e3 as sn.  That is, it commenced from the sign "(p (q))(p (r))" and ended up at the sign "(p (q r))" by legal moves.

	



Proof 2 lit on by burning the candle at both ends, changing e2 into a normal form that reduced to e4, and changing e3 into a normal form that also reduced to e4, in this way tethering e2 and e3 to a common stake.  In more detail, one route went from "(p (q))(p (r))" to "(p q r, (p))", and another went from "(p (q r))" to "(p q r, (p))", thus equating the two points of departure.

	



Proof 3 took the path of reflection, expressing the met-equation between e2 and e3 in the naturalized equation e5, then taking e5 as s1 and exchanging it by dint of value preserving steps for e1 as sn.  Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))" to the blank expression that Ex recognizes as the value true.

	



Review:

	


	
 Pf 1.  PERS 14. weblink
	
 Pf 2a. PERS 15. weblink
	
 Pf 2b. PERS 16. weblink
	
 Pf 3.  PERS 18. weblink
	


Computation and inference as semiosis
Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks.  Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.

	



By way of a very preliminary orientation, let us consider the distinction between an information maintaining process (IMP) and an information reducing process (IRP).  To conform with prudent practice, let's make our first acquaintance with this difference in the medium of some concrete and simple examples.

	


	
 Example 1.  Modus Ponens
	



	 IRP Version:
	



	   p ⇒ q

	   p

	 ––––––––

	    q
	



	 IMP Version:
	



	   p ⇒ q

	   p

	 ========

	   p q
	
Let us examine these two types of inference in a little more detail.  A display of the form:

	


	   Expression1

	   Expression2

	 ––––––––––––

	   Expression3
	
is used to state a rule of inference (ROI).  The expressions above the line of inference are called premisses and the expression below the line is called a conclusion, (also outcome, result, or upshot).

	



If the ROI in question is succinct enough, one may write it in-line, as Premiss1, Premiss2 |– Conclusion, where the symbol "|–" is called the (proof-theoretic) turnstile.

	



Either way, one reads such a ROI in the following manner:  "From Expression1 and Expression2, infer Expression3".

	



Looking to our first Example, the ROI that is classically known as modus ponens says the following:  If one has that p implies q, and one has that p is true, then one has a way of putting it forward that q is true.

	



Modus ponens is an illative or implicational rule.  Passage through its turnstile incurs the toll of some information loss, and thus from a fact of q alone one cannot infer with any degree of certainty that p ⇒ q and p are the reasons why q happens to be true.

	



Further considerations along these lines may lead us to appreciate the difference between implicational rules of inference (IROI's) and equational rules of inference (EROI's), the latter indicated by an equational line of inference (ELOI) or 2-way turnstile "|–|".

	

Variations on a theme of transitivity
The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

	



In discussing this Example, it is convenient to observe that the implication relation that is ordinarily indicated by the propositional form x ⇒ y is equivalent to an order relation x ≤ y on boolean values, where 0 is taken to be less than 1.

	


	
 Example 2.  Transitivity
	



	 IRP Version:
	



	   p ≤ q

	   q ≤ r

	 ––––––––

	   p ≤ r
	



	 IMP Version:
	



	   p ≤ q

	   q ≤ r

	 ============

	   p ≤ q ≤ r
	
In stating the IMP analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a catenated syntax.  Thus, p ≤ q ≤ r means that p ≤ q and that q ≤ r.  The claim that this 3-adic relation holds among the 3 propositions p, q, r is a stronger claim — contains more information — than the claim that the 2-adic relation p ≤ r holds between the 2 propositions p and r.

	



To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.

	

Table 21.  Composite and Compiled Order Relations
o---------o------------o-----------------o----------------o-------------o




Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.

	

o-----------------------------------------------------------o




Among other things, these images make it visually obvious that the constraint on the three boolean variables p, q, r that we indicate by asserting either of the forms "(p (q))(q (r))" or "p ≤ q ≤ r" is one that implies a constraint on the two boolean variables p, r that we indicate by either of the forms "(p (r))" or "p ≤ r", but that it imposes additional constraints on these variables that are not captured by the illative conclusion.

	



One way to view a proposition f : Bk → B is to consider its fiber of truth, f−1(1) ⊆ Bk, and to regard it as a k-adic relation L ⊆ Bk.

	



By way of general definition, the fiber of a function f : X → Y at a given value y of its co-domain Y is the antecedent (pre-image or inverse image) of y under f.  This is a subset, possibly empty, of the domain X, notated as f−1(y) ⊆ X.

	



In particular, if f is a proposition f : X → B, then we think of f−1(y) as the subset of X that is indicated by the proposition f.  Whenever we assert a proposition f : X → B, we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse X.  Because the special case of the fiber of truth is used so often in logical contexts, we will sometimes use the notation [|f|] = f−1(1).

	



Using this panoply of notions and notations, we may treat the fiber of truth of each proposition f : B3 → B as if it were a relational data table of the shape {(p, q, r)} ⊆ B3, where the (p, q, r) are bit vectors indicated by the proposition f.

	



Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.

	

[| q_207 |]  =  [| p =< q |]
o---------o---------o---------o
< r |]
o---------o---------o---------o
< r |]
o---------o---------o---------o
< q =< r |]
o---------o---------o---------o




In the medium of these unassuming examples, we begin to see the activities of logical inference and methodical inquiry as information clarifying operations (ICO's).

	



First, we drew a distinction between information maintaining and information reducing processes and we noted the related distinction between equational and implicational inferences.  I will use the acronyms EROI and IROI, respectively, for the equational and implicational analogues of the various rules of inference.

	



For example, we considered the brands of information fusion that are involved in a couple of standard rules of inference, taken in both their equational and their illative variants.

	



In particular, let us assume that we begin from a state of uncertainty about the universe of discourse X = B3 that is standardly represented by a uniform distribution u : X → B such that u(x) = 1 for all x in X, in short, by the constant proposition 1 : X → B.  This amounts to the maximum entropy sign state (MESS).  As a measure of uncertainty, let us use either the multiplicative measure given by the cardinality of X, commonly notated as |X|, or else the additive measure given by log |X|.  In this frame we have |X| = 8 and log |X| = 3, to wit, 3 bits of doubt.

	



Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.

	


	
 Transitive Law (IROI)
	



	    p ≤ q

	    q ≤ r

	 ––––––––

	    p ≤ r
	
By itself, the information p ≤ q would reduce our uncertainty from log 8 bits to log 6 bits.

	



By itself, the information q ≤ r would reduce our uncertainty from log 8 bits to log 6 bits.

	



By itself, the information p ≤ r would reduce our uncertainty from log 8 bits to log 6 bits.

	



In this situation, the application of the IROI for transitivity to the information p ≤ q and the information q ≤ r to get the information p ≤ r does not increase the measure of information beyond what any one of the three propositions has independently of the other two.  In a sense, then, this IROI operates only to move the information around without changing its measure in the slightest bit.

	


	
 Transitive Law (EROI)
	



	   p ≤ q

	   q ≤ r

	 ============

	   p ≤ q ≤ r
	
The contents and the measures of information that are associated with the propositions p ≤ q and q ≤ r are the same as before.

	



On its own, the information p ≤ q ≤ r would reduce our uncertainty from log(8) = 3 bits to log(4) = 2 bits, a reduction of 1 bit.

	



These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it.  In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.

	



For ease of reference during the rest of this discussion, let us refer to the propositional form f : B3 → B such that f(p, q, r) = q139(p, q, r) = (p (q))(q (r)) as the syllogism mapping, written as syll : B3 → B, and let us refer to the fiber syll−1(1) ⊆ B3 as the syllogism relation, written as Syll ⊆ B3.  Table 25-a shows Syll as a relational dataset.

	

Table 25-a.  Syllogism Relation
o---------o---------o---------o




One of the first questions that we might ask about a 3-adic relation, in this case Syll, is whether it is determined by its 2-adic projections.  I will illustrate what this means in the present case.

	



Table 25-b repeats the relation Syll in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or planar projections of Syll in the next three columns.  For instance, Syllpq is the 2-adic projection of Syll on the pq plane that is arrived at by deleting the r column and counting each 2-tuple that results just one time.  Likewise, Syllpr is obtained by deleting the q column and Syllqr is derived by deleting the p column, ignoring whatever duplicate pairs may result.  The final row of the right three columns gives the propositions of the form f : B2 → B that indicate the 2-adic relations that result from these projections.

	

Table 25-b.  Dyadic Projections of the Syllogism Relation
o-------------o-------------o-------------o-------------o




Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus.  What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.

	



One will find that some relations can be reconstructed from various types of derivatives and projections, others cannot.  The reconstuctible relations are said to be reducible to the types of reductive data in question, while the others are said to be irreducible with respect to those means.

	



The analogies between derivation, differentiation, implication, projection, and others sorts of information reducing operation will undergo extensive development in the remainder and sequel of the present discussion.

	



We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.

	



The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules q139, q175, q187, q207.

	



The function q139 : B3 → B and its fiber [|q139|] ⊆ B3 appeared to be key to many structures in this setting, and so I singled them out under the new names of syll : B3 → B and Syll ⊆ B3, respectively.

	



Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.

	



But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so I want next to lay out some visual schemata that are designed to facilitate that.

	



Figure 28-a shows the familiar picture of a boolean 3-cube, wherein the points of B3 are coordinated as bit strings of length three.  Looking at the functions f : B3 → B and the relations L ⊆ B3 on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation L = [|f|] and which points are out of it.  Bowing to common convention, we may use the color "1" for points that are "in" a given relation and the color "0" for points that are "out" of that same relation.  However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

	

o-------------------------------------------------o




Table 28-b shows the 3-adic relation Syll ⊆ B3 again, and Figure 28-c shows it plotted on a 3-cube template.

	

Table 28-b.  Syll c B^3
o-----------------------o




We return once more to the plane projections of Syll ⊆ B3.

	

Table 29-a.  Syll c B^3
o-----------------------o




In showing the 2-adic projections of a 3-adic relation L ⊆ B3, I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.

	



Figure 29-c shows Syll and its three 2-adic projections:

	

o-------------------------------------------------o




We now compute the tacit extensions of the 2-adic projections of Syll, alias q139, and this makes manifest its relationship to the other functions and fibers, namely, q175, q187, q207.

	

Table 30-a.  Syll c B^3
o-----------------------o




The reader may wish to contemplate Figure 31 and use it to verify the following two facts:

	


	 Syll = TE(Syll12) ∩ TE(Syll23)
	



	 Syll13 = Syll12 ο Syll23
	
o-------------------------------------------------o




I don't know about you, but I am still puzzled by all of thus stuff, that is to say, by the entanglements of composition and projection and their relationship to the information processing properties of logical inference rules.  What I lack is a single picture that could show me all of the pieces and make the pattern of their informational relationships clear.

	



In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

	



At first or maybe second sight, the relationships seem easy enough to write out.  Figure 32 shows how the various logical expressions are related to each other:  The expressions "(p (q))" and "(q (r))" are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression "(p (q)) (q (r))" that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference "(p (r))" by the most conventional implication.

	

o-------------------o ` ` ` ` o-------------------o




Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the expressive aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.

	



In the way of many experiments, the word expressive does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective.  Aside from coining a word like ennotative, some other options would be connotative, hermeneutic, semiotic, syntactic — each of which works in some range of interpretation but fails in others.  Trial 2.  Let's try formulaic.

	



Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith.  Thus we engage the denotative semantics or the model theory of these extremely simple programs that we call propositions.

	



Table 33 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference.  In it I use a number of abbreviated notations:

	


	
 I use the forms X:Y:Z and x:y:z as alternative notations for the cartesian product X × Y × Z and the tuple (x, y, z), respectively.
	
 In situations where we have products like X:Y:Z with X = Y = Z = B, and relations like L ⊆ X:Y, M ⊆ X:Z, N ⊆ Y:Z, I will use forms like L ⊆ B:B:~, M ⊆ B:~:B, N ⊆ ~:B:B to remind us that we are considering particular ways of situating L, M, N within the product space X:Y:Z.
	


o-------------------o ` ` ` ` o-------------------o




A piece of syntax like "(p (q))" or "p ⇒ q" is an abstract description, and abstraction is a process that loses information about the objects described.  So when we go to reverse the abstraction, as we do when we look for models of that description, there is a degree of indefiniteness that comes into play.

	



For example, the proposition (p (q)) is typically assigned the functional type B2 → B, but that is only its canonical or its minimal abstract type.  No sooner do we use it in a context that invokes additional variables, as we do when we next consider the proposition (q (r)), than its type is tacitly adjusted to fit the new context, for instance, acquiring the extended type B3 → B.  This is one of those things that most people eventually learn to do without blinking an eye, that is to say, unreflectively, and this is precisely what makes the same facility so much trouble to implement properly in computational form.

	



Both the fibering operation, that takes us from the function (p (q)) to the relation [| (p (q)) |], and the tacit extension operation, that takes us from the relation [| (p (q)) |] ⊆ B:B to the relation [|q207|] ⊆ B:B:B have this same character of abstraction-undoing or modelling operations that require us to re-interpret the same pieces of syntax under different types.  This accounts for a large part of the apparent ambiguities.

	



Up till now I've concentrated mostly on the abstract types of domains and propositions, things like Bk and Bk → B, respectively.  This is a little like trying to do physics all in dimensionless quantities without keeping track of the qualitative physical units.  So much abstraction has its obvious limits, not to mention its hidden dangers.

	



To remedy this situation I will start to introduce the concrete types of domains and propositions, once again as they pertain to our current collection of examples.

	



We have been using the lower case letters p, q, r for the basic propositions of abstract type B3 → B and the upper case letters P, Q, R for the basic regions of the universe of discourse where p, q, r hold true, respectively.

	



The set of signs X = {"p", "q", "r"} is the alphabet for the universe of discourse that is notated as X • = [X] = [p, q, r], already getting sloppy about quotation marks to single out the signs.

	



The universe X • is composed of two different spaces of objects.  The first is the space of positions X = áp, q, rñ = {<p, q, r>}.  The second is the space of propositions X↑ = (X → B).

	



Let us take make the following definitions:

	


	 P‡ = Xp = {(p), p},

	 Q‡ = Xq = {(q), q},

	 R‡ = Xr = {(r), r}.
	
These are three sets of two abstract signs each, altogether staking out the qualitative dimensions of the universe of discourse X •.

	



Given this framework, the concrete type of the space X is P‡ × Q‡ × R‡ ≈B3 and the concrete type of each proposition in X↑ = (X → B) is P‡ × Q‡ × R‡ → B.  Given the length of the type markers, we will often omit the cartesian product symbols and write just P‡ Q‡ R‡.

	



An abstract reference to a point of X is a triple in B3.  A concrete reference to a point of X is a conjunction of signs from the dimensions P‡, Q‡, R‡, picking exactly one sign from each dimension.

	



To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 35 translates the contents of Figure 33 into the new language.

	

o-------------------o ` ` ` ` o-------------------o


References

	
 Leibniz, G.W. (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in Parkinson, G.H.R. (ed.), Leibniz : Logical Papers, Oxford University Press, London, UK, 1966.  (Cf. Gerhardt, 7, p. 223).
	



	
 Peirce, C.S., Bibliography.
	



	
 Peirce, C.S. (1931–1935, 1958), Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA.  Cited as CP volume.paragraph.
	



	
 Peirce, C.S. (1981–), Writings of Charles S. Peirce:  A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN.  Cited as CE volume, page.
	



	
 Peirce, C.S. (1885), "On the Algebra of Logic:  A Contribution to the Philosophy of Notation", American Journal of Mathematics 7 (1885), 180–202.  Reprinted as CP 3.359–403 and CE 5, 162–190.
	



	
 Peirce, C.S. (c. 1886), "Qualitative Logic", MS 736.  Published as pp. 101–115 in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy, Mouton, The Hague.
	



	
 Peirce, C.S. (1886 a), "Qualitative Logic", MS 582.  Published as pp. 323–371 in Writings of Charles S. Peirce:  A Chronological Edition, Volume 5, 1884–1886, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
	



	
 Peirce, C.S. (1886 b), "The Logic of Relatives: Qualitative and Quantitative", MS 584.  Published as pp. 372–378 in Writings of Charles S. Peirce:  A Chronological Edition, Volume 5, 1884–1886, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
	



	
 Spencer Brown, George (1969), Laws of Form, George Allen and Unwin, London, UK.
	


See also
Related essays and projects

	
 Futures Of Logical Graphs
	
 Information = Comprehension × Extension
	
 Inquiry Driven Systems
	
 Introduction to Inquiry Driven Systems
	
 Peirce’s Logic Of Information
	
 Semiotic Theory Of Information
	
Related concepts and topics

	
 Ampheck
	
 Boolean algebra
	
 Boolean domain
	
 Boolean function
	
 Boolean logic
	
 Boolean-valued function
	
 Conceptual graph
	
 Entitative graph
	
 Existential graph
	
 Graph
	
 Graph theory
	
 Laws of Form
	
 Logical graph
	
 Logical matrix
	
 Logical NAND
	
 Logical NNOR
	
 Minimal negation operator
	
 Peirce's law
	
 Peirce’s logic of information
	
 Propositional calculus
	
 Semiotic information theory
	
 Truth table
	
 Zeroth order logic
	


External links

	
 Laws of Form Web Site.
	
 Spencer-Brown's talks at Esalen 1973 — Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types".
	
 Louis H. Kauffman — Box Algebra, Boundary Mathematics, Logic, and Laws of Form.
	


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(last updated by Jon Awbrey, 6:54pm EDT - Sat, Apr 07 2007)
		
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o==================================

	



	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	

	



	

	

	

	

	

	

	

	

	

	



	

	

	

	

	

	

	

	

	

 L_2 ` ` ` `
 L_4 ` ` ` ` ` `

 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

 Binary` ` `
 Cactus` ` ` ` `

o---------o------------o-----------------o----------------o-------------o
 ` ` ` ` `p : 1 1 1 1 0 0 0 0 
 ` ` ` ` ` ` |
 ` ` ` ` `q : 1 1 0 0 1 1 0 0 
 ` ` ` ` ` ` |
 ` ` ` ` `r : 1 0 1 0 1 0 1 0 
 ` ` ` ` ` ` |
o---------o------------o-----------------o----------------o-------------o
 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

 q_11001111 
 (p` `(q)) ` ` `

 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

 q_10111011 
 ` ` `(q ` (r)) 

 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

 q_10101111 
 (p` ` ` ` (r)) 

 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

 q_10001011 
 (p (q))(q (r)) 

 ` ` ` ` ` `
 ` ` ` ` ` ` ` `

o---------o------------o-----------------o----------------o-------------o


	










` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` P ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `








 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ Q ^ ^ ^ ^ 
 ^ ^ ^ ^ R ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 










o-----------------------------------------------------------o
q_207.  (p (q))

	



o-----------------------------------------------------------o









^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ P ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^








 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` Q ` ` ` ` 
 ^ ^ ^ ^ R ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 










o-----------------------------------------------------------o
q_187.  (q (r))

	



o-----------------------------------------------------------o









` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` P ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `








 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ^ ^ ^ Q ^ ^ ^ ^ 
 ^ ^ ^ ^ R ^ ^ ^ 

 ^ ^ ^ ^ ^ ^ ^ ^ 
 ^ ^ ^ ^ ^ ^ ^ ^ 










o-----------------------------------------------------------o
q_175.  (p (r))

	



o-----------------------------------------------------------o









` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` P ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` ` ` ` ` ` ` ` `








 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 

 ` ` ` Q ` ` ` ` 
 ^ ^ ^ ^ R ^ ^ ^ 

 ` ` ` ` ` ` ` ` 
 ^ ^ ^ ^ ^ ^ ^ ^ 










o-----------------------------------------------------------o
q_139.  (p (q))(q (r))


	

` ` q ` `

o---------o---------o---------o






o-----------------------------o

	



[| q_187 |]  =  [| q =
` ` q ` `

o---------o---------o---------o






o-----------------------------o

	



[| q_175 |]  =  [| p =
` ` q ` `

o---------o---------o---------o






o-----------------------------o

	



[| q_139 |] = [| p =
` ` q ` `

o---------o---------o---------o




o-----------------------------o


	

	

	

	

	

	

	

` ` q ` `

o---------o---------o---------o




o-----------------------------o


	

 ` Syll_pq ` 
 ` Syll_qr ` |
o-------------o-------------o-------------o-------------o
 ` ` 00 ` ` `
 ` ` 00 ` ` `|
 ` ` 00 ` ` `
 ` ` 01 ` ` `|
 ` ` 01 ` ` `
 ` ` 11 ` ` `|
 ` ` 11 ` ` `
 ` ` 11 ` ` `|
o-------------o-------------o-------------o-------------o
 ` (p (q)) ` 
 ` (q (r)) ` |
o-------------o-------------o-------------o-------------o


	  



 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

 ` ` /  ` ` /

`` `/` `` `/`

`  / ` `  / `

` `` ` ` `/` `

` /  ` ` /  `

`/` `` `/` ``

/ ` `  / ` ` 


` ` ` / ` ` ` ` ` ` ` ` |
` ` `/` ` ` ` ` ` ` ` ` |
` ` / ` ` ` ` ` ` ` ` ` |
` `/` ` ` ` ` ` ` ` ` ` |
` / ` ` ` ` ` ` ` ` ` ` |
`/` ` ` ` ` ` ` ` ` ` ` |
/ ` ` ` ` ` ` ` ` ` ` ` |


o-------------------------------------------------o
Figure 28-a.  Boolean 3-Cube B^3


	


o-----------------------o




o-----------------------o

	



o-------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`  / ` `  / `

` `` ` ` `/` `

` /  ` ` /  `

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/ ` `  / ` ` 


` ` ` / ` ` ` ` ` ` ` ` |
` ` `/` ` ` ` ` ` ` ` ` |
` ` / ` ` ` ` ` ` ` ` ` |
` `/` ` ` ` ` ` ` ` ` ` |
` / ` ` ` ` ` ` ` ` ` ` |
`/` ` ` ` ` ` ` ` ` ` ` |
/ ` ` ` ` ` ` ` ` ` ` ` |


o-------------------------------------------------o
Figure 28-c.  Triadic Relation Syll c B^3


	


o-----------------------o




o-----------------------o

	



Table 29-b.  Dyadic Projections of Syll
o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o
 
 

 
 

 
 

o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o


	



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`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`` `/` `` `/`

`  / ` `  / `

` `` ` ` `/` `

` /  ` ` /  `

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/ ` `  / ` ` 


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` ` ` |
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` ` ` |
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` ` ` ` ` ` ` ` ` `

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` ` ` ` 1.1 ` ` ` `

` ` ` ` /  ` ` ` `

` ` ` `/` `` ` ` `

` ` ` / ` `  ` ` `















o-------------------------------------------------o
Figure 29-c.  Syll c B^3 and its Dyadic Projections


	


o-----------------------o




o-----------------------o

	



Table 30-b.  Dyadic Projections of Syll
o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o
 
 

 
 

 
 

o-----------o o-----------o o-----------o
 
 

o-----------o o-----------o o-----------o

	



Table 30-c.  Tacit Extensions of Projections of Syll
o---------------o o---------------o o---------------o
 
 

o---------------o o---------------o o---------------o
 
 

o---------------o o---------------o o---------------o
 
 

 
 

 
 

 
 

 
 

 
 

o---------------o o---------------o o---------------o
 (p (q)) 
 
 (p (r)) 
 
 (q (r)) 

o---------------o o---------------o o---------------o
 `q_207` 
 
 `q_175` 
 
 `q_187` 

o---------------o o---------------o o---------------o

	



o-------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`  / ` `  / `

` `` ` ` `/` `

` /  ` ` /  `

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/ ` `  / ` ` 


` ` ` / ` ` ` ` ` ` ` ` |
` ` `/` ` ` ` ` ` ` ` ` |
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`` ` ` ` ` `` `

`  ` ` ` ` `  `

` `` ` ` ` ` ``

` `  ` ` ` ` ` 

` ` `` ` ` ` ` 000 ` ` ` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |


o-------------------------------------------------o
Figure 30-d.  Tacit Extension TE_12_3 (Syll_12)

	



o-------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`  / ` `  / `

` `` ` ` `/` `

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/ ` `  / ` ` 


` ` ` / ` ` ` ` ` ` ` ` |
` ` `/` ` ` ` ` ` ` ` ` |
` ` / ` ` ` ` ` ` ` ` ` |
` `/` ` ` ` ` ` ` ` ` ` |
` / ` ` ` ` ` ` ` ` ` ` |
`/` ` ` ` ` ` ` ` ` ` ` |
/ ` ` ` ` ` ` ` ` ` ` ` |





















o-------------------------------------------------o
Figure 30-e.  Tacit Extension TE_13_2 (Syll_13)

	



o-------------------------------------------------o


 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`  / ` `  / `

` `` ` ` `/` `

` /  ` ` /  `

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/ ` `  / ` ` 


` ` ` / ` ` ` ` ` ` ` ` |
` ` `/` ` ` ` ` .11 ` ` |
` ` / ` ` ` ` ` /

` `/` ` ` ` ` `/`

` / ` ` ` ` ` / `

`/` ` ` ` ` `/` `

/ ` ` ` ` ` / ` `

` ` ` |
` ` ` |

` ` ` / ` ` ` |
` ` `/` ` ` ` |
` ` / ` ` ` ` |
` `/` ` ` ` ` |
` / ` ` ` ` ` |
`/` ` ` ` ` ` |
/ ` ` ` ` ` ` |


o-------------------------------------------------o
Figure 30-f.  Tacit Extension TE_23_1 (Syll_23)


	

	

	

	


 ` ` ` ` ` ` ` ` ` ` ` |
`` ` ` ` ` ` ` ` ` ` ` |
`  ` ` ` ` ` ` ` ` ` ` |
` `` ` ` ` ` ` ` ` ` ` |
` `  ` ` ` ` ` ` ` ` ` |
` ` `` ` ` ` ` ` ` ` ` |
` ` `  ` ` ` ` ` ` ` ` |

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`` `/` `` `/`

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` `` ` ` `/` `

` /  ` ` /  `

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/ ` `  / ` ` 


` ` ` /  ` ` `  ` ` ` |
` ` `/` `` ` ` `*` ` ` |
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` ` ` |
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` ` ` |
`  ` ` ` ` /  `
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` `  ` ` / ` ` 
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` ` `  / ` ` ` /  ` ` ` ` / ` ` 


` ` ` / ` `  ` ` `

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` / ` ` /  ` `  `

`/` ` `/` `` ` ``

/ ` ` / ` `  ` ` 















o-------------------------------------------------o
Figure 31.  Syll = TE(Syll_12) |^| TE(Syll_23)


	

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` ` ` ` ` `` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` `
` ` ` ` ` `  ` ` ` Conjunction ` ` ` / ` ` ` ` ` `
` ` ` ` ` ` `` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` `
` ` ` ` ` ` ` v ` ` ` ` ` ` ` ` ` ` v ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
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` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` q ` r ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` o ` o ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` | ` | ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` p o ` o q ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ``/` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` @ ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
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` ` ` ` ` ` ` `| `(p (q)) (q (r))` |` ` ` ` ` ` ` `
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` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` Implication ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `
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` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` r ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` o ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` | ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` p o ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` | ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` @ ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` `(p (r))` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` q_175 ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
Figure 32.  Expressive Aspects of Transitive Inference


	

	

	

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

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o-------------------o ` ` ` ` o-------------------o
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o-------------------o ` ` ` ` o-------------------o
 q_207 
 ` ` ` ` 
 q_187 

o----o---------o----o ` ` ` ` o----o---------o----o
` ` `^` ` ` ` `  ` ` ` ` ` ` ` ` / ` ` ` ` `^` ` `
` ` `|` ` ` ` ` ``Intersection `/` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `  ` ` ` ` ` ` / ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `v` ` ` ` ` `v` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:0:0 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:0:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:1:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 1:1:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `Syll c B:B:B ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `[| q_139 |]` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` Projection` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `0:0` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `0:1` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `1:1` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| `Syll_13 c B:~:B` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` [| (p (r)) |] ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o----o---------o----o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `^` ` ` ` ` `^` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` / ` ` ` ` ` `  ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `/` Composition `` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` / ` ` ` ` ` ` ` `  ` ` ` ` `|` ` `
o----o---------o----o ` ` ` ` o----o---------o----o
 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

o-------------------o ` ` ` ` o-------------------o
 ` ` ` ` 

o-------------------o ` ` ` ` o-------------------o
 (p (q)) 
 ` ` ` ` 
 (q (r)) 

o---------o---------o ` ` ` ` o---------o---------o

	



Figure 33.  Denotative Aspects of Transitive Inference


	  

	 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

 ` ` ` ` 

o-------------------o ` ` ` ` o-------------------o
 ` ` ` ` 

o-------------------o ` ` ` ` o-------------------o
 q_207 
 ` ` ` ` 
 q_187 

o----o---------o----o ` ` ` ` o----o---------o----o
` ` `^` ` ` ` `  ` ` ` ` ` ` ` ` / ` ` ` ` `^` ` `
` ` `|` ` ` ` ` ``Intersection `/` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `  ` ` ` ` ` ` / ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `v` ` ` ` ` `v` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p)(q)(r) ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p)(q)`r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p) q `r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `p` q `r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| `Syll c P‡ Q‡ R‡` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `[| q_139 |]` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` Projection` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `(p) (r)` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `(p)` r ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` p ` r ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| `Syll_13 c P‡ R‡` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` [| (p (r)) |] ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o----o---------o----o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `^` ` ` ` ` `^` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` / ` ` ` ` ` `  ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `/` Composition `` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` / ` ` ` ` ` ` ` `  ` ` ` ` `|` ` `
o----o---------o----o ` ` ` ` o----o---------o----o
 ` ` ` ` 

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o-------------------o ` ` ` ` o-------------------o
 ` ` ` ` 

o-------------------o ` ` ` ` o-------------------o
 (p (q)) 
 ` ` ` ` 
 (q (r)) 

o---------o---------o ` ` ` ` o---------o---------o

	



Figure 35.  Denotative Aspects of Transitive Inference