This paper, first released on May 23rd 2018 is comprised of several discoveries. First the paper gives a way to represent propositional logical operators via matrices called correspondence matrices (CM’s). Furthermore that logical expressions can be computed by way of a CM operators action on the operands, the operands being similar are similar to the notion of wave-functions as seen in quantum mechanics. What is found is that these correspondence matrices are linear operators and the notion of ‘measurement’ can be extended to the theory of computation of of propositional expressions.
The representation of logical operators as matrices allow for algebraic operations to be performed on logical expressions. Not only can logical expressions be computed from the rotation and transposition of correspondence matrices, CM’s also give the ability to quotient one logical expression by another! The ability to operate on CM operators themselves give rise to a very important discovery in which negation of a logical expression to be computed by the negation of its associated CM! This ability is possible as there are sixteen 2 by 2 CM’s possible which give every CM a unique compliment. This ability for operator compliments allows for the proof of an extraordinary result – that similarly formatted logical expressions of two variables can be combined over another operator solely by operating on the each of the expressions respective CM’s! The ability for composition of logical expressions also gives a constraint as to possible decomposition’s any logical expression can be decomposed to.
The paper describes computations of more than two logical variables, what is found is that we not only can operate on the logical operators but that this can be done before even valuating logical variables themselves! This work shows that we can both compute any finite variable logical expression by a finite length computation involving two by two CM’s as well as creating a higher dimensional CM which itself is an operator and can be combined element wise with other CM’s to perform the overall computation! This allows us to apply a measurement on any expression with a finite number of logical variables. The paper also showed that CM’s are a special case of their more general logical matrices (LM’s) and that similar properties and computational abilities exist for them as well.
The image above gives a way to tensor together smaller dimensional logical matrices in order to create a larger LM. Near the bottom of the image it is shown how to measure the created LM by bra-ket operands consisting of the logical state-vectors by which the LM was created.