In this opening chapter, we will attempt to develop a logical basis for belief which allows for our limitations yet optimizes our potential. In subsequent chapters, we will show how this can be applied to several subjects in a comprehensive manner. This approach minimizes the likelihood of logical contradictions. An honest attempt at objectivity is central to this discussion. A person's beliefs should follow from a collection of facts coupled with logical analysis. In reality, most beliefs have emotional origins and are supported by one sided fact gathering and rationalizations. These emotions may be instinctual and/or conditioned responses developed from past experience. True objectivity is difficult, if not impossible.
The method of deduction derives conclusions from general statements. Typically, formal logic uses syllogisms which consist of a major premise, a minor premise, and a conclusion. It involves the proper use of syntax and the elimination of inconsistencies. The most common type of syllogism is the categorical (or Aristotelian) syllogism. It deals with categories of objects or ideas in the form of statements. An object or idea is either in a given category on not in that category. Furthermore, logical arguments are either valid or invalid. This is called the "law of the excluded middle." There is no allowance for intermediate degrees of validity. A typical textbook example is the following argument:
All men are mortal.This argument may be converted to a geometrical representation as follows:
Socrates is a man.
Therefore, Socrates is mortal.
The entire contents of the outer circle represents all things that are mortal. Within this circle is another circle which represents all men. The `men' circle is entirely within the "mortal" circle and thus represents the statement that all men are mortal. The littlest circle is "Socrates." It is entirely within the "men" circle and represents the statement that Socrates is a man.
Since the "men" circle is entirely within the "mortal" circle and the "Socrates" circle is entirely within the "men" circle, it can be seen that "Socrates" is also entirely within the "mortal" circle. Therefore, Socrates is mortal.
The above example would be considered a valid logical argument. Another twist on the above example is as follows:
All men are mortal.
Socrates is mortal.
Therefore, Socrates is a man.
If we apply our geometrical analysis to these statements, we might get Figure 2
As before, the "mortal" circle represents all things that are mortal. Thee "men" circle represents all men. We could place Socrates within the "men" circle as in Figure 1 and satisfy the statement that "Socrates is Mortal". But instead, we placed Socrates outside the "men" circle but still inside the "mortal" circle. This also satisfies the "Socrates is Mortal" statement. Thus it is possible that Socrates is mortal yet not a man. Since Socrates could be either inside or outside the "men" circle, we cannot conclude that Socrates is a man. Therefore, the above sequence of premises and conclusion is an example of an invalid argument in formal logic.
Other categorical syllogisms can be analyzed in this same way. Circles may be entirely within other circles or they may overlap or be outside, depending on the premises.
The preceding examples use circles of varying size. The larger circles represent categories that contain more elements than the smaller circles. If, in Figure 2, we found the men circle to be almost as big as the mortal things circle, it would imply that the great majority of mortal things are men. In this case, would it be illogical to beleve that Socrates is a man? Agreed, you still can not be sure - but it would seem to be more rational to believe Socrates is a man rather than he isn't.
These are the types of decisions we must make in everyday life. We often cannot wait or expend the effort to produce a perfectly valid argument to make a decision. After all, a decision not made is also a decision. Thus we can see that many situations force us to form our beliefs based on probabilities. Will it be warmer at noon tomorrow than it is tonight at midnight? Probably.
In short, the deductive method relies on generalizations. We can always form abstract manmade generalizations that, by definition, are true - but can absolute true generalizations be found in nature>? This requires inductive logic.
Induction is the method we use to create generalizations. Without valid generalizations, there can be no deductions. Unfortunately, induction has more weaknesses than deduction. Induction is mainly predicated on the idea that if we see an event occur many times under a certain set of conditions, this same event will occur if those conditions should arise again. There is no logical assurance that this method is infallible. Agreed, we often can build a good track record using this method but we always stand the chance of being wrong with our generalizations.
The use of categories is one type of generalization. The old philosophers could imagine things which are nicely grouped into categories. Their deductive logic followed from these generalizations. They were thinking abstractly. The question is, can perfect categorization be accomplished in the real world?
What is a coward?
What are trees?
What is red?
The above categories do not have the sharp outer line of the circles in our previous illustrations. The outer line is somewhat fuzzy. Where does the color red end and orange begin? Some categories will have fuzzier lines than others but there is always some fuzziness when you depart from the abstract. In order to reduce fuzziness, we often must define categories more exactly. This may require the use of measurements and mathematics. Better still, let's elminate categories altogether. The most primitive form of knowledge is the use of categories. We should define each object or idea by a set of measurable attributes. When this is done we often find that our categories are arbitrary. But the use of specific measurable attributes reduces the power of generalization and is therefore counter-productive when we are looking for universal truths.
The power and infallibility of the inductive basis for logical systems has been questioned in this century. Bertrand Russell spent much of his life attempting to build a logical system that would produce absolute "knowledge". Both he and Kurt Godel, probably this century's greatest mathematicians, found inconsistencies, paradoxes, and incompleteness in "logical" systems.
I wanted certainty in the kind of way in which people want religious faith . I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
- Bertrand Russell
Mathematics is our most powerfull logical tool. But even it does not necessarily create truth. As Bertrand Russell said, "two plus three equals five because we say it does." Mathematical relationships are abstractly defined rather than discovered. The happy coincidence is that many of these relationships can often be applied to nature and used to predict various phenomena with a high degree of accuracy.
What is the primary reason that ancient philosophers and even modern logicians have not been able to make an airtight case for logic? The deductive powers of logic are only as good as the inductive generalizations upon which they are based. Too much time has been spent on the use of primitive categories for the purpose of generalizaion. Abstract thought lulls people into thinking that categorization is possible. When we force ourselves to examine the real world, we find that categories are a product of the human mind - and not nature.
How can we improve our generalizations? There is an heirarchy of ways to describe things and ideas which provide greater accuracy:
CATEGORIZATION This is the method we have described above. An example would be room temperature. We might describe it as either warm or cold. The problem is that one person may describe it as warm whiie another may describe the same room as cold.
RANKING The method of ranking allows us to compare various room temperatures. Room A is "colder" then room B. Room C is warmer than room B. Room D is "warmer" than room B but "colder" than room C. We can now make a list of room temperatures as follows: A, B, D, C, with the temperature getting higher as we advance along the list.
RELATIVE SCALING The previous ranking example doesn't tell us anything about the distance between the room temperatures. This can be done by using a temperature scale such as our Fahrenheit scale. For example we can say room A is 64 deg; room B is 68 deg; room D is 74 deg; and room C is 75 deg. As a result, we have more information about the temperatures in terms of how far apart they are from each other.
ABSOLUTE SCALING Finally we have absolute scaling. Relative scaling gave us the distance between temperatures but did not tell us about the "ratio" or temperatures. Is 4 degrees Fahrenheit twice as warm as 2 degrees? Not really. Ratios provide additional information and are necessary for many scientific calculations. In order to get ratios, we must define absolute zero. This is done using the Rankine scale for temperatures. Zero on the Rankine scale means that there is no heat present. A temperaure of 4 degrees on this scale is twice as warm as 2 degrees. Zero degrees Rankine is -460 degrees Fahrenheit.
The following writings will assume that logic is very powerful but not infallible. In fact, some logic may require reflection on experiences rather than abstract thought. We believe that nature has a logical basis because of our observation that repeatable results are often obtained from a controlled set of initial conditions. The absolute accuracy of scientific laws are always being challenged. Newton's Laws, for example, were found to be in error under the unusual circumstances posed by Albert Einstein. Some of Einstein's laws are in turn being challenged by quantum mechanics. What leads us to believe we are making progress is that we have to reach further and deeper into the world of measurement to find these contradictions.
Knowledge, to be useful, must allow us to predict the future and explain the past. Logic is necessary for the determination of relationships and the analysis of facts. Without inductive logic we do not have knowledge but only a series of factual stimuli which have no context or meaning.
The use of probability in logic may not only be convenient, it may also be mandatory. When we are faced with fallibility, it is prudent to retain a certain degree of skepticism with respect to our beliefs. Probability is a man made concept which is used as a measure of our degree of uncertainty. Science has provided strong evidence for many relationships in nature. These relationships will produce a very high probability of belief in an educated person. Studies of many other types of relationships are not so well substantiated. These more tentative relationships should not be ignored but they should produce a lower level of certainty, Unfortunately, a weakness in classical logic is the tendency to categorize all knowledge into "true" or "false" statements. No allowance is made for uncertainty. As a result, many person's belief systems tend to be absolute and simple concepts are often extrapolated to an extreme position on many issues.
The existence of absolute truth seems to be supported by the fact that evolution produced humans whose primary survival advantage is intelligence. Without absolute truth, intelligence would be of little evolutionary value. If we assume the existence of absolute truth, it does not necessarily follow that we know what it is. Quite often when we say we "know" something, we assume the classical logic concept of the excluded middle. In other words, we either know it to be true or we know it to be false. This is a fallacy. Although we may assume that absolute truth exists, we cannot assume we have absolute knowledge of that truth.
Further considerations in a philosophical discussion are the matters of complexity and scope. It has been said that the human intellect can only consider the relationships between a few concepts at one time. This makes it much easier to think about a system consisting of a few general principles rather than a multitude of specific concepts. If truth requires the understanding of many concepts, it then becomes incomprehensible. On the other hand, as the subject matter expands beyond a very limited scope, it would seem that the probability that a few general principles will provide an accurate explanation of reality becomes more remote.
On the first level, this may be true. Many primitive generalizations are found to be weak. But as our degree of sophistication becomes greater, we may find that only a few simple generalizations are the basis for all the complexity we observe.
Let us introduce the concept of cause and effect. The belief in this concept has been the driving force for the progess of much recent science and technology. In fact, our every action is predicated on the idea that our actions have consequences. Other factors beyond our control may also play a part but these factors also produce effects or they wouldn't be important. But who can deny that belief in cause and effect has been the basis for much knowledge and/or predictabliity? It must also be admitted that progress in the future will depend on this basic concept. The mention of the future brings up another essential component to the complexity issue: TIME.If we combine cause and effect with time we get evolution. Very simple relationships can, through cause and effect over time, produce very complex systems. The more time, the more complex the systems can become. It is thus possible that very simple and general principles are all that is required to explain "everything". But all the individual connections in that chain of events may be impossible to determine. Does this make knowledge of the basic beginning principles of no value? As usual - time will tell.
The above concepts give a flavor of what type of thinking has gone into the discussions contained in the following chapters.