02 Jun Framing Formal Logic

Formal Logic

Formal logic often uses set theory. Set theory uses existential (an assertion that something applies to some members of a set) and universal (a statement that applies to all members in a set) quantifiers. Despite the utility and noncommittal correctness of existential quantifiers, set operations using existential quantifiers are weaker then those using universal quantifiers. The unfortunate problem in the domains of most serious AI applications is that universal quantifiers are generally inapplicable or simply wrong. They are so rigid that, when flexibility, a hallmark of good AI, is required, the universal assumption proves too brittle to handle reality.

A classic example is that of birds. To say birds fly is correct, but to say all birds fly is not correct because penguins do not. In classifying animals, a simple program might ask “DOES IT FLY?” and presume that an affirmative answer automatically places the animal in the BIRD category. Unfortunately, we know that some fish and small mammals fly as well so an assumption that all flying animals are birds fails to capture the variety in reality. Things that are true are not false, and things that are false are not true, but in nature and AI application domains, very few things fall neatly into such categories. Most things can better be expressed in terms of likelihood, belief, percentage or confidence.

Understanding Context Cross-Reference
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Section 7 #9

Table of Context

In Section 5 we discussed fuzzy logic. Perhaps one of the most fuzzy aspects of artificial intelligence systems can be expressed in a silly cliché: “Never say never.” Besides that, it is always extremely difficult to say “always.” Let’s take a look at cutting the brittle edges off universal statements with fuzzy techniques. But first, consider the types of knowledge that may always be fuzzy and never 100 percent certain:

There may be other types of knowledge, but this is a good set of ideas about differentiating knowledge for our purposes.

Fuzzy KR

Many different kinds of knowledge exist. The list in the blue ellipse below shows some types of knowledge. All knowledge in the human brain is fuzzy. Fuzzily is the only way we are capable of thinking of things. Fuzzy logic is the only kind of logic supported by the hardware of the brain. The yellow ellipses show different ways of expressing or modeling fuzziness. In automated systems we can use multi-valued logic to imitate the multiple levels of positive and negative impulses that bounce around the brain. Any of the fuzzy models named below can be used to process all kinds of knowledge.

I have spoken of constraints as a model for characterizing the limiting aspects that may influence an outcome. The more different constraints affecting the situation, the more complex the system has to be and the more important it is to incorporate an efficient scheme for fuzzy reasoning. This applies to all types of KR, not just rule-based systems. The application domain of medical diagnosis described in the discussion of rule-based KR is both instructional and accessible, but many good examples are available in other real-world domains as well.

As I explore different approaches for implementing fuzzy models for processing knowledge, I will try to show the strengths and weaknesses of each.

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