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Mathematics and Logic

At this stage, it may be useful to take another look at logic. Let’s use a well-known example: “All men are mortal. A is a man. Therefore, A is mortal.” The conclusion seems beyond dispute. In fact, it is hard to imagine how it could not be true. But, what does true mean in this context? Suppose that we discover an apparent man B who appears to be immortal because he has substantially outlived any prior example of a man. There are two clear possibilities. Obviously, B may eventually turn out to be mortal—that is, B may die. Alternatively, we may conclude that B is not actually a man. Indeed, it may be that mortality is part of the definition of “man.” In other words, if B is not mortal, then he is not a “man” (he may be an “angel” or a “god”). But, when worded this way, the logic is not very satisfying.

There are other possibilities. We might find an ambiguity in the meaning of mortal or, at least, in the empirical observation of mortality (such as the length of life). Another possibility is in the meaning of the word “all.” If it has a probabilistic meaning, then occasional counter-examples are compatible.

In any event, we would hope that our logical argument could go beyond merely asserting, just in different words, a definition that we have adopted. To have some meat in the theory, we would want the definition of man to be in some sense independent of the assertion that man is mortal. Clearly, it cannot be completely independent. There could be aspects of man, defined without explicit reference to mortality, that would still mean that man was in fact necessarily mortal.

Example: One plus one

Take a simple arithmetical example. One plus one equals two. How could it be otherwise? If the proposition means no more than if you have one item and obtain another item, then you have two items; then it seems necessarily true—that is, true by reason of the meanings of the terms. By that, I do not mean just a matter of mere definition, but as a result of the structure of the proposition.

To illustrate, are there situations where one plus one does not “equal” two?

If the word “equals” in the question is limited to its definition in logic or mathematics, then the answer must be “no” because of how the term “equals” is defined. In contrast, if “equals” is taken to mean “result in” or “yield,” then there are some other interesting possible answers. For example, one male plus one female could turn out to result in three (or four or five, etc.). Or, one particle of matter plus one particle of anti-matter supposedly results in nothing or zero (ignoring the energy that assuredly would be released). One would say that these examples are fundamentally different from the original arithmetical proposition. If so, the difference is because the statements consist of more than arithmetic—they are statements that incorporate facts about the physical world and relationships in that world.

So, the arithmetical proposition is a statement about certain characteristics of things (closely connected to the concept of counting). But it abstracts from or ignores other characteristics (such as the potential for biological, chemical or physical interactions). The logical proposition is clear and unambiguous, but trivial. The possibilities in real world applications are regularly much more nuanced and complex.

Example: Proof by contradiction?

As a final example, take the assertion that two propositions that contradict each other cannot both be true. A corollary is the idea that the proposition that the demonstration that a statement logically leads to a contradiction constitutes a “proof” that the statement is not true (or vice versa). Are these concepts empirical, that is, do they reflect things that are true (or untrue) about the physical world?

Clearly, the propositions seem to be integral to the notion of rational thought and logic and, probably, inherent in the human mental processes. For example, Professor Arkes states that “[t]he law of contradiction expresses a necessary truth, and all efforts to refute it will fall into the embarrassment of self-contradiction.” First Things, p.51. This “necessary truth,” like the similar concept of causality (discussed above), may be “necessary” in terms of the functioning of the human mind, but is it necessarily “true”?⁹ It may be that we are incapable of conceiving of two contradictory propositions both being true, but our inability to conceive does not control the actual physical relationships of the world.

In fact, the concept of inconsistency itself seems necessarily to implicate human theories or models. We might say that we have never observed in nature an entity that is simultaneously dead and alive, as evidence that nature does not tolerate inconsistent or contradictory states. But, that claim requires definitions of “dead” and “alive”. One could imagine plausible definitions that overlap, allowing the alleged inconsistency to exist in the external world (say a definition of death that turns on brain activity and a definition of life that is based upon cell reproduction). If the definitions are structured to be mutually exclusive, then no entity could satisfy both—by definition. Take a different type of example. We recognize that a person could simultaneously be both “short” and “tall”, since those characteristics are relative and we could have two different points of references.

I note that scientists often use inconsistencies as a form of proof. A typical “proof” using this type of argument would be as follows: Proposition A is either true or false. Suppose that I can deduce (using deduction) that if A is false, then both B and not-B are true. Since that conclusion presents a contradiction that cannot be accepted, I conclude that A is true. Actually, the logic tells us only that A is not false, since that is the assumption that led to the contradiction. The conclusion that A is, therefore, true depends upon the initial assertion or assumption that A is either true of false. If A could be something else, then the conclusion that A is true is not established.