29 Mar Is Anything True or False

I like to use pictures in my posts. The more evocative they are, the better. Why? Because they may trigger brain activity in areas that may be otherwise untouched by my writing. In other posts, I speak of this brain activity as “patterns of activation” (Understanding, Vision, Enchanted, Flow). If we think in patterns (Tou 1974), then understanding and reasoning are not questions of true or false as much as of fitting a recognized pattern or not. Nevertheless, “true/false” logic has its place, and that’s what I want to talk about today.

George Boolos, Philosophy Professor at MIT tells us of a puzzle to wring the logic out of your mind. Raymond Smullyan apparently holds the title for “Hardest Logical Puzzle Ever.” The puzzle: Three gods A, B, and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for “yes” and “no” are “dam and “ja,” in some order. You do not know which word means which.” Please download the pdf. here or check out the Wikipedia article. If this exercise messes up your brain, you might try to reset with some Feng Shui.

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Section 5 # 28

Table of Context

Forms of Proof

We spoke of inference as one example of using logical form. inference may take many forms, any of which may be assigned a latin name.  The following forms of proof are used in logical reasoning to show that a conclusion is justifiable.

Although we seldom go through mathematically complete proofs as we learn and reason about the things we see and hear, the underlying processes described above generally reflect how we think about premises and conclusions. We may even think in these ways about walking and driving, though different people think in very different ways.

Translating how we think so a computer can imitate it requires us to develop formalisms or equations that can be implemented as computer instructions. The first three forms of proof above, plus the process of reduction, qualify as useful formalisms.

Shades of Gray

The citizenship syllogism in my prior post lends itself to dichotomous logic. Though there may be other ways of obtaining citizenship, and though the methods may differ from country to country, the rules can usually be answered with yes-or-no questions. Citizenship is largely black and white. But what about race? How do you know if a person is black or white? What rules can you apply to give you yes-or-no answers that could lead to a conclusion of one or the other?

I personally write “caucasian” on forms that are ridiculous enough to pose the question, even though the term caucasian implies a dichotomy (caucasian or not caucasian) that does not meaningfully exist. My ancestry is Czech, German, Irish, French, and who knows what else. My kids are more Scandinavian than anything else (much whiter than I with my olive complexion). My wife and I have many friends who are as much caucasian as they are black. In central Africa there are peoples like Hutus and Tutsis, both arguably black, whose differences are as long and deep as those between the Serbs and the Muslims. So what is the point of distinguishing race in such a complicated family tree as that of the human race?

For questions like race, dichotomous logic fails to capture the shades of gray that pervade so much of our knowledge of the way things (and people) are. In this post, and others in this section we will discuss the way humans reason in the absence of clear-cut distinctions of black and white. Look at the variety available in the gray scale. The examples here are just a fraction of all the shades of gray. Now think about the entire spectrum of colors, then the spectrum of light!

Multi-valued Logic

In many instances, two-valued logic is not expressive enough to capture the nature or interaction of things in the world. An entire domain in which two-valued logic falls short is probability theory. If a thing is absolutely true or false, it cannot be probably true or occasionally false. Probability theory builds predictive models on statistical constraints. For example, the probability that a rider on the bus has black hair may be calculated at 15% in Minneapolis, 25% in Rome, and 55% in New York, while the probable percentage of all people with black hair riding a bus in the world at any given time may exceed 80%.

Two-valued logic is not sufficient for expressing probability factors. The dichotomy of “has or has not black hair” can be represented with true/false logic, but even this is not as expressive as using multiple values to express multiple colors of hair and their relative statistical probabilities at a given place. Statistical probability is not the only application for multi-valued logic. The states of objects, such as the door at right, and states of knowledge and reasoning, such as confidence and belief levels, can also be expressed easily using scales of values.

Is this door partly open or partly closed? Is it possible for me to enter the room without an access code or a key or without turning the knob? is it possible to secure the door in this position to preclude the entry of anyone or anything greater than certain dimensions?

Perhaps, with computers, we are bet served by matching patterns and approximating confidence values. Perhaps we can define thresholds where common sense tells us whether a thing is or is not in a specific state. Perhaps dichotomous logic is enough to develop systems that are useful enough. I propose that we can have the best of all worlds in our automated systems, and that the elastic brain is the perfect model.

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