MS 455-456 (1903) - Lowell Lecture II

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[scroll with small blot ??]

or even

[scroll with infinitesimal blot ??] This suggests that the relation which the cut asserts between the universe of discourse and what is scribed within it is simply that what is scribed within is false of the universe of discourse.

Then we may interpret

as meaning “It is false that it rains and that a pear is not ripe.” But we have already seen that this is precisely the whole meaning of the conditional de inesse; namely that it is false that the antecedent is true while the consequent is

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false. Thus, that which the cut asserts is precisely that that which is on its bottom is not, as a whole, true.

This agrees with the fact that if there is nothing on the bottom except the inner cut, its patch may be asserted. Thus

since a blank merely asserts known truth means, "if the truth is true then a pear is ripe," or it is not false that a pear is ripe. So in the following

or if it rains, then if it blows, a pear is ripe. The two cuts with nothing between can be taken away without altering the

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fact expressed If it both rains and blows a pear is ripe.

[Go to Vol 2 p 40 (MS 456)]

We now have three tules of necessary inference. First, from any premiss A, we can necessarily conclude B, if and only if we know that it is not the case that A is true while B is false.

We may, therefore, rub out everything on the board, because we know that the blank sheet of assertion asserts nothing false. Hence if two graphs are written one may erase either of them, because each has the same signification as if it stood alone, and either standing alone might be erased.

But now if any graph, G, could if it were scribed on the sheet of assertion be transformed into another graph H without fear of introducing falsity

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So much of the system of existential graphs as I have thus far described I call the alpha part of the system.

There are certain ways in which graphs scribed on the sheet of assertion can be modified without any possibility of changing a true graph into a false one. Such modifications, I call permissible transformations. In particular those which can be proved by the principles of the alpha part of the system to be permissible, that is, never to be capable of changing a true graph into a false one are called alpha permissible transformations. The alpha part of the system establishes three kinds of signs besides the graphs themselves. The first sign consists in writing scribing two graphs

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together on the sheet. As long as we recognize a blank as a graph, the writing scribing of a single graph is a case of scribing two graphs together, of which a blank is one. The second sign is the scroll. These two are the only indispensible signs; but we recognize as a third sign the filling up of an area with a blot.

I just defined a permissible transformation as one which can not produce a ?? never change a true graph to a false one one not true. Very good. But why should we regard this statement as constituting a definition of a permissible transformation rather than as constituting a definition of truth? Why might not truth be defined as that which we can assert with impunity? If the idea of penalties for breaking rules is more familiar to us than the idea of truth it might not be a bad definition.

In like manner we might define the three signs of the alpha part of existential graphs by means of permissible what is permitted. Since every act of definition involves two propositions,— they may be variously stated, but there will always be two,— one for example stating that if a word is used a certain interpretation is justified, the other stating that in given cases the use of the word will be legitimate,— if, I say we remember this double character of definition, we shall see that the definitions of the three signs will give six propositions. These six are as follows: 1st, If two graphs are together on the sheet of assertion, either may be cancelled or erased. 2nd, If either of two graphs might be written scribed, both may be scribed together. [written here by Peirce: "Turn over." and under that: "Transpose second & third Rules"]

The second third rule is that a cut with nothing within is a replica of the pseudograph and may be scribed on the sheet of assertion with enclosed in a cut on the sheet of assertion, while two coincident cuts may be removed.

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