In one of the matheology threads I asked ZG if he would be interested in certain geometric ideas I have formulated. This thread is a presentation of those ideas.
It will have repetition of statements made elsewhere.
The title of the initial post is meant to convey the idea that certain fundamental relations of propositional semantics can be modeled geometrically. There had been a time when I would have made foundational claims. There comes a time when one learns that such things are always a matter of belief. Enough said. It is not about belief.
This post has, as its primary purpose, the presentation of a projective geometry and its corresponding affine geometry. There are actually two projective geometries presented. The geometry corresponding to the main topic has labels corresponding to the connectives of propositional logic. The second geometry is the original source of the labeling using natural numbers. The basis of this construction is what is called a difference set.
serves as an an example from the literature of the underlying construction involved.
What is relevant to this particular form of presentation is the idea that a single label is used for both points and lines.
This is the "math trick" used to relate the relations of propositional logic to extensions of propositional logic in a "geometric" manner. If one numerically orders the lines of the second geometry, one will have a presentation similar to that in the link.
The 16-point affine geometry is labeled with names corresponding to propositional connectives. But, it has 20 lines. Since the lines have the same names as the points, the system of 20 lines incorporates the possibility of extension.
Consider, for example, extension by modal operators. Using a unary negation, one would have
, ~, <>, <>~
, ~, ~~, ~
But without using unary negation, the extension introduces a quadruple with specific relations.
For a variable of quantificational logic one has,
Ex, Ex~, ~Ex~, ~Ex
~Ax~, ~Ax, Ax, Ax~
So, the 20 lines of the affine geometry extend the 16 labels by 4. In this sense, it makes possible the notion of extension interpretable relative to geometric structure.
There is another part of the "trick" that will be discussed later. It involves a substitution in the labels. The substitution will be discussed here, but the "trick" also involves relations in a 20-element ortholattice with a 16-element Boolean sub-block.
In the presentation of labels, these 4 extension symbols are given Aristotelian labels. There is
ALL and NO
In addition, there is
SOME and OTHER
Technically, one would not see 'OTHER' in a syllogistic argument. The mnemonic here is that OTHER is what is left after SOME is chosen as the fixed species.
Relative to the affine structure, these four labels are located on the line at infinity of the projective structure.
Relative to the affine structure, the remaining label on the line at infinity of the projective structure is
An early motivating principle of this work had been to develop an understanding of logical structure without a unary negation. It seemed natural to use this label to complete the system of labels.
The second part of the "math trick" is to replace the label NTRU corresponding to an affine point with the label NTRU in the set of line labels. This is necessary by virtue of the construction.
One could compare this with Frege's use of self-contradictory descriptions to ground his quantificational logic or Kant's use of self-contradictory descriptions to ground the modal nature of his epistemological investigations. What I mean here is that something more is involved with the extension from propositional logic to its extensions. Given the necessary relations of the labeling, the substitution mentioned above may be given an interpretation along some such lines.
But, what is important at this point is merely that a namespace is given having certain relations determined by geometric structure.
Here are the two geometries. I shall add explanations to the thread a little at a time.