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Ladies and Gentlemen:
At the end of the last lecture I gave a rule for positively ascertaining whether or not a graph is alpha-possible. In case it be so, the rule further furnishes a description of the states of the universe to which its possibility is limited. If I had time to set before you the process of reasoning train of thought which determined the formulation of that rule, it would afford an instructive simple example of methodical reasoning. Unfortunately, there is not time for any such illustration in this course, far less can the scientific methodeutic be touched upon, that great third of logic that teaches the principles upon which reasonings are to be arranged into systematic developments
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of theory. I will only say that in formulating that rule, I considered that of the six fundamental permissible operations, two are non-reversible, namely erasure and insertion. Where That is where there can either of these can be performed the reverse operation cannot be performed. I further considered that three of the six fundamental permissible operations complicate the graph, while the other three simplify it. Namely, insertion, iteration, and the placing of marking introduction of a double cut, complicate the graph, while erasure, deiteration, and the removal of a double cut simplify it. It is necessary for the purpose to modify the non-reversible operations so as to render them reversible; and it fortunately happens
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that this can be done without erasure, the only one of them that is simplicative, being rendered complicative. In the next place, since the result aimed at must be of extreme simplicity, it is necessary to place restrictions upon the employment of the complicative operations. To this end, each of these was studied and the precise ways in which it could further the general purpose of the rule was ascertained; and thus in place of the complicative operations were substituted a set of operations conducive to the purpose.
The process of the rule is, as I said, probably not the most facile. But since the system of existential graphs is not intended as a calculus
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or general instrument tool of reasoning, but is only meant for use in the study of logic, where it quite infrequently happens that it is necessary to ascertain whether or not any very complicated graph is alpha-possible or not, the waste of time due to this rule not being the simplest possible, is not of much importance.
The rule prescribes a routine calling for the exercize of so little intelligence that a machines might be devised to perform it. When we pass to the problem of beta possibility that is of formulating a rule for positively ascertaining whether a graph with ligatures is logically possible or not, it would be quite easy to