The Author considers what is the deference between a “cognitive logic” as used in everyday life and a “mathematical logic” as used in meta-mathematics? A key difference is their assumptions on whether their knowledge and resources are sufficient to solve the problems they face. On this aspect, the author distinguish three types of reasoning systems:
1. Pure-axiomatic systems
As the author says pure-axiomatic systems are designed under the assumption that both knowledge and resources are sufficient. A typical example is the notion of “formal system” suggested by Hilbert (and many others), in which all answers are deduced from a set of axioms by a deterministic algorithm. The axioms and answers get their meaning by being mapped into a concrete domain using model-theoretical semantics. Such a system is built on the idea of sufficient knowledge and resources, because all relevant knowledge is assumed to be fully embedded in the axioms, and because questions have no time constraints, as long as they are answered in finite time. If a question requires information beyond the scope of the axioms, it is not the system’s fault but the questioner’s, so no attempt is made to allow the system to improve its capacities and to adapt to its environment.
2. Semi-axiomatic systems
These systems are designed under the assumption that knowledge and resources are insufficient in some, but not all, aspects. Consequently, adaptation is necessary. Most current non-classical logics fall into this category. For example, non-monotonic logics draw tentative conclusions (such as “Tweety can fly”) from defaults (such as “Birds normally can fly”) and facts (such as “Tweety is a bird”), and revise such conclusions when new facts (such as “Tweety is a penguin”) arrive. However, in these systems, defaults and facts are usually unchangeable, and time pressure is not taken into account. Fuzzy logic treats categorical membership as a matter of degree, but does not accurately explain where the degree come from.
3. Non axiomatic systems
Pei Wang introduces the main assumptions of the non axiomatic reasoning system which absorbs non axiomatic logic that is used within his project named Non Axiomatic Reasonic System (NARS). This cognitive logic is different from a mathematical logic firstly in its assumption on the sufficiency of knowledge and resources. Because of this difference, the formal language, semantic theory, and inference rules of these two types of logic are different too, and so are the memory structure and control mechanism when
they are implemented in a computer system.
The paper is a good perspective to consider why logic is a basic and useful tool for legal interpretations. The problems that arise in use of logic systems while interpreting laws result from fact we do not use a cognitive logic but more or less axiomatic systems. However the way we think is the only one and its based on cognitive skills that are modeled within the Pei Wang project.