We now begin the formal development of Objectivist philosophy. We will start with its base in reality, reason, and logic.
The content and direction of these essays reflect my own thinking. None of this material contradicts Ayn Rand in any of her fundamentals. I hope these discussions allow you to better understand and apply her insights.
Logic has a very bad reputation in some circles as a technical discipline unrelated to human life. However, once you understand that logic is just the formalization of meaning, you will see how ill-deserved this reputation is. Logic is often considered cold, unfeeling and `inhuman', yet nothing is more fundamentally human than meaning -- only man can understand. Furthermore, only logic and reason can point the way to the deepest and truest expression of man's humanity.
Therefore, a word of caution and a plea for patience to those new to trying to reason everything through from fundamentals. You may not have thought it possible or you may not have thought it worthwhile. Be assured, it is both.
Much may seem so obvious that you could be tempted to say, ``so
what?''
You may feel that nothing deep or meaningful can evolve from
such transparent foundations or that the whole process is too far
removed from your `practical' daily concerns. Please reserve judgment.
If you are familiar with mathematics, you already know what dense, intricate, and practical structures can be assembled from a few basic axioms. You also know that time invested in understanding the fundamentals is always time well spent.
Even those things that should be the most clear and most evident have been rejected, distorted, or misrepresented for ulterior motives or by willful blindness -- right down to the validity of reason itself or the existence of reality.
Let me now apologize for breaking an earlier promise. I had intended to start with the absolute fundamentals, but I find it necessary to proceed somewhat out of sequence.
The foundation of philosophy is metaphysics, the study of what is the nature of reality. The next rung in the hierarchy of philosophy is epistemology, the study of what knowledge is and how it is obtained. If I were to start with metaphysics, I would face the problem of having no way to convince you that what I am asserting is valid knowledge, as we would have no agreed upon standards for making that determination.
Since I must communicate with you, I must first establish some rules for doing so, even though an understanding of the nature of reality grounds the derivation of these rules. I will need to use the principles of logic to communicate to you the understanding of reality on which the rules are themselves based. Once that is done, you will see that the nature of reality is fundamental and actually does lead to the rules.
In my own mind, an understanding of the nature of reality is my primary as it is Ayn Rand's. We derive logic from it. You will be able to understand the reasons for the rules of logic once you understand the nature of reality. Most see no reason to dispute the rules of logic, and, in truth, this is because there is none. Some do, and I suspect that their understanding of the nature of reality is at fault; they find it meaningful to require a `reason' to `trust reason'.
What would it take to shake the foundations of logic? It would take an impossible universe. If the nature of the universe were such that it could change totally every second, if this is even conceivable, the laws of logic would not be applicable. Fortunately, we do not live in such a universe, we could not, there would be no `we'.
This type of universe is not possible and cannot be given meaning precisely because it contradicts reality and therefore logic. What meaning could the term `second' have in such a universe?
A logical fallacy is committed when we attempt to transfer a word that has meaning in one context to another context where it has none. This logical error Ayn Rand termed ``the fallacy of the stolen concept.'' In our universe the concept of second is meaningful, but in a totally chaotic universe time would have no meaning. Concepts do not exist in isolation. They develop meaning in a specific context and that context is part of the meaning.
Suppose someone asks you to prove the validity of the laws of logic. They would be evading the knowledge that proof is a concept that can only have meaning within an established logical framework. A common method of attacking reason is to ignore the hierarchical structure of concepts. This type of error is extremely common in ethical reasoning and will be discussed in greater depth later.
Formal logical systems are built from axioms. By definition, an axiom cannot be proven within the system in which it is accepted as axiomatic. If it could be, you would not need it as an axiom. One proves with axioms, one does not prove axioms; one must start somewhere.
Note that this does not mean that there must be no way to validate one's axioms. Proof is only one method of validation. For example, if I tell you the sky is blue, you could verify my claim by looking at the sky. You would not need to construct a logical proof. Proof is a very complex and powerful way of validating a concept, but it is not the only way. Proof derives new knowledge from old knowledge -- there must be some starting knowledge to begin a proof.
Objectivism defines logic as the process of non-contradictory identification. Ayn Rand's three laws of logic were first enunciated by Aristotle, whom she admired immensely for this achievement. However, it was Ayn Rand herself who first appreciated their full force; they ground all knowledge. She was also the first person to make these laws the basis for a consistent philosophy.
Ayn Rand phrased the first law of logic, as ``A is A.'' For this statement, she received much criticism. Her critics felt that nothing meaningful was being said and that nothing useful could be derived from it. They were wrong on both counts, as you will soon see in the analysis following all three laws.
This first law, often called the law of identity, means that to exist, even as an idea, is to be something and that to be something is to be something specific. A thing is itself, it is what it is. If you prefer, it means that before you can talk about something, you must know what it is that you are talking about.
When we state that ``A is A'', we must first assert ``A''. The fundamental unit of thought is clear identity, the content of ``A'', whether as an existent entity in metaphysics or as a specific concept in epistemology.
This law is the most fundamental law of logic because it is the only one that deals directly with meaning. For a statement to be subject to any further logical manipulations, even by the second and third laws, it must first mean something. Failure to adhere to this requirement can lead to logical absurdities.
Let us examine the concept `axiom'. An axiom (or premise) is ordinarily taken to mean a self-evident truth basic to any further reasoning. It is a statement, not derivable from other statements, but prior to them.
Yet, this is not necessarily its use in a specific formal axiom system used to prove a set of theorems. It is quite possible, in fact very common, that for example, axioms A, B, and C together imply theorems D and E. While, had we accepted D and E as axioms, we could have derived A, B, and C. This time A, B, and C are theorems and D and E are the axioms.
In other words, you may have a consistent reasoned structure in which A, B, C, D, and E all hold, yet the choice of axiomatic base is arbitrary. Where does this leave the naive concept of a ``fundamental axiom?''
Just because you have shown that X can lead to Y, you are not yet entitled to proclaim X more fundamental than, or prior to, Y. Such a pronouncement must await a meta-axiomatic analysis of the meaning of the concepts involved. Meaning and reality are the final arbiters; truly fundamental axioms must be grounded in reality.
The second law of logic is the law of non-contradiction. As Nathaniel Branden worded it, this law states that an entity cannot both have a specific attribute and not have that same attribute at the same time and in the same respect. As Rand phrased it, something cannot be both A and non-A. The most precise phrasing of this law is as follows:
If we understand the meaning of a particular entity and a particular property and we know precisely both what it would mean for the entity to have the property and what it would mean for the entity not to have the property, then the entity cannot both have and not have the property.
The third law is the law of the excluded middle. It states that once a specific entity and a specific attribute are well defined and understood, either the entity possesses that attribute or it does not, at a specific time and in a specific respect. In other words, something is either A or non-A.
Though phrased in terms of single entities, the second and third laws each denote a relationship between two concepts. Even though one of the concepts is the negation of the other, each must be understood in its own right to be meaningful. Before these laws can be applied, each of the two concepts being related must be independently understood, that is, have meaning. As we shall see, though you may clearly understand what it means for a particular statement to be true, it does not automatically follow that you understand what it means for it to be false.
There is also a sense in which the second and third laws can be considered to be implied by the law of identity. For a concept to have clear meaning, for example, it cannot both possess and not possess the same well-defined characteristic.
Each of these laws has been challenged on grounds that show clearly that the challenger does not understand them. Without an agreement on these laws, it would not even be possible to attack them.
To attack the law of identity, it must be admitted that the law of identity is what it is, and that the arguments presented against it exist and are what they are.
To attack the law of non-contradiction it must be admitted that either the law is true or it is false. It must be agreed that it cannot be both true and false. The law's attackers must argue that their arguments, and yours, can not be both true and false.
To attack the law of the excluded middle, it must be admitted that the law is either true or false. It must be agreed that all arguments presented are either true or false, valid or invalid.
In short, these three rules are preconditions of rational discourse. No one can argue against them rationally. At most someone can claim that they are not useful. We will later see, from an analysis of the nature of reality, just how useful they are.
However, application of these rules must proceed with care.
One cannot use the law of the excluded middle to say, for example, that either a number is positive or it is negative, unless either ``positive'' means exactly not-negative or ``negative'' means exactly not-positive. What is zero? What we might say is that either a number is positive or it is not positive.
Yet, even if someone redefines positive to include zero, his job of identifying what he means is not yet finished. The law of identity requires that the identification be precise.
Thus, should someone say that either a number is positive or it is not positive, he still does not speak precisely. To what category does a complex number belong? In this case, the failure to specify precisely the meaning of the term `number' causes an error in reasoning, or more precisely, a loss of meaning.
We see here that the law of identity must be used to check applications of the other laws for validity. That check consists of making sure that the meanings of all terms used are clear and exact. Rand's critics were very wrong in denouncing the law of identity as useless or tautologic.
As a further example, analyze the claim that either there are a million consecutive sevens somewhere in the decimal expansion of pi or there are not. A careless use of the law of the excluded middle would accept this claim as obviously valid. It is not. It violates the law of identity and the law of identity precedes the law of the excluded middle.
What does it mean to say that there are not a million sevens in the decimal expansion of pi? If we assume, as is believed, that there is no other way of determining the digits of pi than calculating them individually and sequentially, there is no way, even in principle, to establish that there are not a million consecutive sevens anywhere in the expansion.
How could you go about proving it? How far would you go? No proof would ever be constructible; you would have to check all of the digits. Any attempted proof would require the completion of an infinite number of steps, and this concept is not meaningful.
If anyone ever found a million sevens, the practical issue would be settled. But until that happens, if ever, the application of the law of the excluded middle is not permitted. At least one of the supposed alternatives has no clear meaning.
In other words, even though we understand what it means for there to be a million consecutive sevens somewhere in the decimal expansion of pi, it does not follow that we understand what it means for there not to be such a sequence! And if you do not understand a concept, how can you hope to use it?
We may, however, say that either you can produce a proof that there are a million consecutive sevens in the decimal expansion of pi or you cannot produce such a proof. This is meaningful. The proof would be the demonstration of these digits beginning at a specified point. It would not mean anything to say that one could not ever produce such a proof.
Again, what we cannot say is that either there are a million sevens or there are not; this is not well defined. It is equivalent to stating that when the issue is resolved, ultimately either you can demonstrate a million consecutive sevens or prove that there are not. There is, however, no meaning to proving that there is not such a sequence of sevens when such a proof must entail an infinite number of steps.
This issue arises in many other areas of mathematics that deal with infinities. Unless one is always aware of the meaning of what one is talking about, one cannot indiscriminately apply the other laws of logic. This was discovered to the horror of those attempting to construct the foundations of mathematics on formal logical principles alone, without retaining meaning every step of the way.
Just as mathematicians have found it necessary to employ meta-mathematical reasoning and analysis if they wish to talk about mathematics, so must philosophers, hoping to derive valid conclusions from logic, have a set of tools, a `meta-logic' if you will, to talk about logic.
In both cases, the `meta' tool is meaning. Meaning grounded in the nature of reality -- in the deepest identity of the entity being considered. This is not a prescription for pronouncing ``A is A'' as some magic wand to navigate difficult shoals. It is a demand that you devote the deepest thought to that which you hope to use as a `premise' and to the reasoning to follow. There are no shortcuts. These concepts will be essential to our discussion of metaphysics, the nature of reality.
Even the law of non-contradiction itself is not exempt from a deeper analysis of specific issues involved in its use. You might say that there can be no such consideration; either Y or not-Y may lead to valid conclusions, but certainly not both. Yet, a counter example is not at all difficult to construct!
Consider the following: through a point on a plane, one and only one straight line can be drawn parallel to a second straight line that does not contain the original point. This is called the parallel postulate and is a well known geometric axiom.
The early geometers always felt that it was more complex than the other Euclidean axioms and tried repeatedly to establish it as a provable theorem; they failed, but they never doubted its truth. Some mathematicians, among them Gauss, Lobachevsky, and Riemann, then thought to ask themselves, since we can't prove the postulate, maybe it is not even true. They looked to see what would happen if they accepted the contradictory of this postulate as an axiom.
They added its contradictory to the other axioms and analyzed the resulting system. To repeat, this system was identical to the Euclidean system with the exception that one axiom asserted as true in one was asserted as false in the other. Lo and behold, the new axiom system turned out to be fully as consistent as the old. Further, both were even `true' in reality, in different spheres of application.
Have we proven that the law of non-contradiction does not always hold? Not really; what we have shown is that a deep, meaningful identification of a concept, within the total context of its associated conceptual net is needed. In this specific instance, what is needed is a clearer understanding of the concept of a mathematical model and what it means for it to mirror reality.
The law of contradiction, in the form `P or not-P', is misapplied without a prior analysis of the full meaning of both `P' and `not-P' and a full analysis of the entire conceptual framework that contains `P' and `not-P'. Again, there are no shortcuts. Not even the law of non-contradiction is a shortcut to avoid an analysis of meaning.
Consistent identification, `A is A' or, if you will, `A' is your only standard, but it is a hard task master not a magic wand.
Ayn Rand was right when she urged ``check your premises.'' However, you must correctly interpret this injunction. It is a plea to be true to the deepest meaning and content of the entities and concepts being considered -- true to the law of identity. Accepting `obvious' axioms, yours or someone else's, is no substitute for hard thought.
The deepest meanings are not always translatable into tidy axiom systems. Bertrand Russell learned this lesson to his horror when his life's work, his hope to axiomatize all of mathematics and reduce it to formal logic alone, collapsed. Contradictions had arisen in the very heart of mathematics by an uncritical application of logic to the supposedly clear concept of `set', a concept pervading all of mathematics.
Suddenly mathematicians found that they could no longer safely reason axiomatically from premises that had served them well for hundreds of years. A truer identification had become necessary and meta-mathematics and meta-logic were created. These fields are nothing more than the correct identification of the nature of mathematics and logic. I think the point is clear; wherever you go, you cannot escape the law of identity.
One point that will be re-iterated later: contradictions cannot exist in metaphysics, that is, in reality. The very concept of a `contradiction' cannot even be understood when applied to reality itself; what else is there to contradict? What exists exists, period. Yet, epistemological contradictions can and do arise insidiously if we lose sight of identity.
Conceptual identity entails identification of the specific content of that particular concept. If the concept is at all complex, this identification will require a full integration of many auxiliary concepts. To neglect this logical net and violate the hierarchical structure of our knowledge opens us to contradiction or meaninglessness.
In other words, you must know not just what a concept asserts but on what previous concepts its meaning is based, that is, its full context. You may not `steal', or even borrow a concept, without also accepting its full context. As demonstrated in our discussion of morality and ethics, virtually every ethical system, except Objectivism, has this error at its root.
There is no means of acquiring and using knowledge that allows one to avoid integrating it logically into one's existing base of knowledge. Humans have no other method of understanding. All of our concepts are hierarchically constructed. This issue is fundamental to epistemology, the study of what knowledge is and how it is obtained and validated.
Let me return to an earlier issue, the claim that logic is sterile and inhuman. Logic is specifically human. If we understand that to use logic, meaning must be retained, no entity we know of, other than a human mind, is capable of it. All non-contradictory, meaningful knowledge relies on logic.
Logic does not prevent a person from following any course he might otherwise find appropriate. In fact, logic will allow him to judge what is appropriate and how best to effect it. Arguments such as ``love is not logical'' reflect a gross misunderstanding of what logic is, or for that matter, what love is.
A better argument would be that ``pleasure is not logical.'' True, while the experience of pleasure itself is not part of the field of logic, one can certainly use logic to obtain it.
Throw your Vulcan images aside. Pain, pleasure, happiness, and all perceptions and sensations are not part of logic itself. They are facts of reality and thus grist for logic's mill. A person who acts to benefit the person he loves acts logically. The person who ignores or hurts his love does not. Nothing is more profoundly human than meaning and logic.
Logic is the basis of understanding reality. It alone does not tell us what our goals should be, but it will help us to achieve them. There is not even a concept of should in logic alone. ``Shoulds'', ``oughts'' and the whole field of morality will have to wait until we discuss a particular fact of reality -- man's nature. This analysis itself, of course, will be embedded in logic.
To reiterate, ``A is A'' is not a vacuous tautology; it is a powerful demand that your concepts always retain genuine meaning and content. All other logical principles must bow to its primacy.
This means, in the most literal sense, that we must know what we are talking about -- to be something is to be something specific. Metaphysically, `what' means what existent entity (there being, of course, no other kind). Epistemologically, `what' means what concept, within what framework.
I have now established enough of a logical base to begin at the beginning, finally. Again, let me apologize for jumping out of sequence a little; I don't know how to avoid it.
In the next discussion, I will address the nature of reality from a logical base. You will see that this logical base itself rests on the nature of reality. We will then examine some logical fallacies often used to attack the validity of logic. I will then present some other logical fallacies for you to attempt to refute.
Entire contents Copyright (C) 1993-94 by Joel Katz, All rights reserved, except as below:
Permission is given to distribute this material electronically provided that it is unedited and presented in its entirety, including the copyright notice. Distribution in print or distribution of excerpts requires permission, address requests to djls@gate.net, no fee will be requested. I wish to be assured that I am not misrepresented and am made aware of where my work is being distributed. Quotation of brief excerpts is permitted so long as they are attributed.