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I will begin with one of the gamma cuts. I call it the broken cut. I scribe it thus
[diagram "it rains" in broken cut]
This does not assert that it does not rain. It only asserts that the alpha and beta rules do not compel me to admit that it rains, or what comes to the same thing, a person altogether ignorant except that he was well versed in logic so far as it is embodied in the alpha and beta parts of existential graphs, would not know that it rained.
The rules of this cut are very similar to
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those of the alpha cut.
Rules of the Broken Cut
Rule 1. In a broken cut already on the sheet of assertion any graph may be inserted.
Rule 2. An evenly enclosed alpha cut may be half erased so as to convert it into a broken cut, and an oddly enclosed broken cut may be filled up to make an alpha cut. Whether the enclosures are by alpha or broken cuts is indifferent.
Consequently [diagram: g enclosed in broken enclosed in alpha cut] will mean that the graph g is beta-necessarily true. By Rule 2, this is converted into [diagram] which is equivalent to g
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the simple assertion of g. By the same rule [diagram] is transformable into [diagram]. which means that the beta rules do not make g false. That is g is beta-possible.
So if we start from [diagram] which denies the last figure and thus asserts that it is beta-impossible that g should be true, Rule 2 gives [diagram] equivalent to [diagram] the simple denial of g
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And from this we get again [diagram]
It must be remembered that possibility and necessity are relative to the state of information.
Of a certain graph g let us suppose that I am in such a state of information that it may be true and may be false; that is I can scribe on the sheet of assertion [diagrams] Now I learn that it is true. This gives me a right to scribe on the sheet [diagrams] But now relative to this new state of information, [diagram] ceases to be true; and