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Adam I. Gerard
ISU
NIU
CS MS TBD

Dimensional and Hyper-Dimensional Logics #2

Regarding cryptographic applications of Dimensional and Hyper-Dimensional Logics.

Summary

So, I introduced the concept of a "dimensional logic" as a way of abstracting general logical notions like sets of truth conditions. This is already familiar in modal logic where truth-values and various world conditions mutually determine the truth of many modal sentences.

This line of thinking leads to natural orderings between various systems of logic (at once harkening back to Tarski's object-language/meta-language distinction and topological treatments of logic while also offering alternatives to both). We arrive at a picture of logics overlaid (usually topological or geometric notions) and the more interesting idea that propositions can be freely computed within or between one or more logics (so-called "mixed-order propositions").

The idea is not about quantification over various kinds of predicates (e.g. - second and higher order logics).

Cryptographic Notions

So, a mixed-order proposition might be assembled from one or more languages (formal logics). The computation of its truth value would be given via a mixed-order truth computation.

Furthermore, the proposition might be purely inferential - indicating a space of logical truths but not expressly stating some proposition, declarative fact, or command.

  1. So, the proposition is at least dually encrypted (using the conjunction of components from multiple languages whose natures, themselves, might be generally unknown).
  2. The proposition may express some conjunction of facts that mutually entail a set of other, desired, propositions.
  3. So, even if the proposition is decrypted, it's nature and intended message are unknown. It's unknown what it's component languages are, what orders of logics are being used to compute its truth, and how inferences are decided on it by multiple propositions.

Example

To demonstrate this consider the following proposition P:

  1. ◔◒◭◭◶▶▶

Suppose it's a combination of two very primitive logics (that lack most properties characteristic of a logic like deduction, soundness, etc.):

Language A:

i. With lexicon: ◔◒◭⤳

ii. With axiom rule: ◭⤳ becomes ◒◭ (translation rather than inference per say)

Language B:

i. With lexicon: ◶▶▶⤳

ii. With axiom rule: ▶⤳ becomes ▶◶ (translation rather than inference per say)

Suppose further that each symbol of the proposition P is rotated by one to the right (the cipher).

  1. ◒◭⤳⤳▶▶⤳

Now, lastly suppose that the Mixed-Order proposition computation is:

i. A is True (in A) and B is True (in B) entails that A and B is True (in C).

ii. So, upon decrypting P to reveal the plaintext proposition we must locate the relevant two languages the expressions are a compound of.

iii. Then, we must derive the correct inferences (here, translations) of each to arrive at the resultant message.

  1. ◒◒◭⤳▶▶◶

Suppose there's padding we can also remove:

  1. .◒◭..▶◶

This yields the following true compound conjunction ◒◭ and ▶◶ (in language C).

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