Abstract
AbstractIn modern logic, various systems have been proposed extending classical Boolean logic & switching theory. Such logic frameworks include multiple-valued logic, probability logic, fuzzy logic, module logic, quantum logic and various other frameworks. Although these extensions have been applied to many applications in mathematics, in science and in engineering, all extensions to Boolean logic invalidates at least one of the six fundamental rules of Boolean logic shown in L1 to L6. We propose a new framework of logic, variant logic, extending Boolean logic whilst satisfying the six fundamental rules (L1–L6). By defining the Variant–Invariant behaviour of logical operations, this framework can be constructed using four types of general operators. Main results of the chapter are summarized in Theorems 8–10, respectively. To show significant differences between classical logic and new variant logic, invariant properties of this hierarchical organization are discussed. Simplest cases of one-variable conditions are illustrated. Variant logic can provide the necessary framework to support analysis and description of Cellular Automata, Fractal Theory, Chaos Theory and other systems dealing with complexity. Such applications of this framework will be explored in future papers.
Highlights
In modern logic, various systems have been proposed extending classical Boolean logic & switching theory
Classical logic identifies a class of formal logic that are characterized by a number of properties [1–17]
The five properties of classical logic (CL1–CL5) are listed as follows: CL1: Law of the excluded middle and double negative elimination CL2: Law of non-contradiction CL3: Monotonicity and idempotency of entailment CL4: Commutativity of conjunction CL5: De Morgan duality. Examples of such classical logic systems include works of philosophy and religion (Aristotle’s Organon; Nagarjuna’s tetralemma; and Avicenna’s temporal modal logic) as well as foundational logic systems such as reformulations by George Bool and Gottlob Frege [4–17]. These properties can be rewritten as simplified equations describing basic properties of a logic system using characteristics of the five classical properties
Summary
Classical logic identifies a class of formal logic that are characterized by a number of properties [1–17]. The five properties of classical logic (CL1–CL5) are listed as follows: CL1: Law of the excluded middle and double negative elimination CL2: Law of non-contradiction CL3: Monotonicity and idempotency of entailment CL4: Commutativity of conjunction CL5: De Morgan duality. Examples of such classical logic systems include works of philosophy and religion (Aristotle’s Organon; Nagarjuna’s tetralemma; and Avicenna’s temporal modal logic) as well as foundational logic systems such as reformulations by George Bool and Gottlob Frege [4–17]. If any logic system does not, they are categorized as non-canonical
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