Logic is the way we know something is true. Denoting rationality and reason, logic is a branch both of mathematics and of philosophy and lies at the very foundation of all intellectual pursuits. Aristotle is largely responsible for the development of the formal rules of logic which have become the basis for Western thought and science. Perhaps the best-known tenet of the so-called "laws of thought" or "Aristotelian logic" is the Law of the Excluded Middle.* This law holds that if a statement is true, then its negation cannot be true. If A is true, then "not-A" must be false. That is, there are only two choices -- yes and no. For example, if the statement "the sky is blue" is true, then its negation "the sky is not blue" must be false. All conditions of sky color are exhausted by the bimodal set of possibilities "blue" and "not blue."

This sort of reasoning seems intuitively obvious to humans. Aristotelian logic is somehow naturally suited to the way people think. After all, we ask, how could a sky both be blue and not blue at the same time? Well, it couldn’t, or could it?

The danger inherent in relying on any single logic system is that it tends to limit the number and kinds of problems amenable to analysis. The solution to any problem first requires that a question be posed. If the question does not appear tractable by our normal modes of logic, we try to reformulate it again and again until it is. When we do this, however, our thinking becomes limited by the capabilities of our logic system. One plus one does not always equal two.**

It is possible that human science today is beginning to feel the pinch of the limitations inherent in its ancient Aristotelian bimodal logic system. Perhaps the most striking examples occur in the field of quantum mechanics. Consider the following experiment. A solid plate with two small slits is placed in front of a beam of electrons. Behind the slits on the other side is a photographic screen able to record the arrival of electrons. During the experiment, electrons are sent toward the slits one by one, some bouncing off the blocking plate and others passing through the slits to be recorded when they hit the screen.

The Law of the Excluded Middle demands that any given electron must pass either "through the left slit" or "not through the left slit." These two choices define all of the logical possibilities for an electron-slit-passing event, and they are exclusive as well: If one is true, the other must necessarily be false.

Unfortunately,when nuclear physicists actually perform the "two slit experiment" they get seemingly impossible results. It turns out that the pattern recorded on the photographic screen could only have been generated if each electron passed through both slits simultaneously.

This is the classic "wavicle" problem in quantum physics. Contrary to traditional bimodal logic, the statements "through the left slit" and "not through the left slit" are both true at the same time. Aristotelian thinking cannot comprehend the problem because there is no "excluded middle" in the experiment.^{3015} The behavior of electrons must be impossible, and yet it occurs.

Perhaps the most important development for xenologicians in this century has been Gödel’s Incompleteness Theorem. In 1931 an American mathematician named Kurt Gödel devised a brilliant proof that any system of logic must necessarily either be internally inconsistent or incomplete.^{3027} In other words, Gödel’s proof demonstrated for the first time that there exist statements that are unprovable in any logic system, and that all arithmetic as we know it is at best incomplete, at worst inconsistent. it is logically impossible to construct a single grand "metalogic" capable of subsuming all other modes of logic while remaining consistent.^{3028} So hunian mathematics -- the language of the physical sciences -- is incomplete.

The implications in xenology are far-reaching indeed. We now know, for instance, that no single system of thinking (on Earth or anywhere else in the Galaxy) can hold, even in principle, all answers to all questions while remaining internally consistent. All logics must harbor unresolvable paradoxes. Therefore each new logic system we uncover in alien cultures most likely will teach us something new, some novel way of looking at the universe and understanding it in a consistent fashion which may be imperceptible -- even impossible -- from within our own system of logic. To this extent human thinking necessarily must be incomplete. Contact with alien minds will open new vistas of knowledge and beauty to mankind’s purview. Extraterrestrial logicians may find many of our most enduring paradoxes to be trivially solvable, and we may be able to resolve some of theirs equally effortlessly.

No non-Aristotelian logic system has yet been devised which resolves the "wavicle" paradox in the two slit experiment to the satisfaction of quantum physicists. However, mathematicians have imagined a wide variety of alternative logics which have been used successfully to resolve other paradoxical events recorded by human philosophers. The literature in this field is both difficult and extensive;913 no more than a brief smattering can be provided here.

Clearly a system with zero values is meaningless, and monovalue logics permit no choice. Such single-valued logic may turn out to be sufficient for genetic sentients, but if we wish to retain choice at least a two-valued (e.g., Aristotelian) system is required. We have seen, however, that two-choice logics. cannot explain many observable physical phenomena. As logician Clarence I. Lewis of Harvard University once noted: "The Law of the Excluded Middle is not writ in the heavens: It but reflects our rather stubborn adherence to the simplest of all possible modes of divi sion."^{902} Over the past century, human mathematicians have come up with "many-valued" logics which permit three or more states of truth instead of the Aristotelian two.^{908,909} An example of three-valued logic might involve the states "yes," "maybe," and "no." Alien computers could be programmed in trinary (rather than our binary) to handle this kind of computation; circuits might read "+," "0," and "-" rather than "on" and "off" as in normal binary digital machines. Another alternative system is the four-valued truth logic which is often used by Buddhist philosophers. (The four permissible truth states are "true," "false," "both," and "neither.")

Another kind of approach is to employ "modal" concepts rather than "truth" concepts. (See Bergmann,^{3031} Haack,^{913} Lewis and Langford,^{3029} Quine,^{3030} and von Wright.^{910}) These types of logic are customarily three-or four-valued, and are of four principle kinds.^{3032} The first of these are called alethic modes or modes of truth. Where Aristotelian logic permits only the truth values "true" and "false," alethic modal logic allows the following modes: "Necessarily true," "possibly true," "contingently true," and "impossible."^{3033} A second form of modal logic is called epistemic logic or modes of knowing, including the modes "verified" (that which is known to be true), "undecided" (that whose truth is unknown), and "falsified" (that which is known to be false). Third, there is deontic logic or modes of obligation, which work as follows: "the obligatory" (that which we ought to do), "the permitted" (that which we are allowed to do), "the indifferent" (that which makes no difference), and "the forbidden" (that which we must not do).^{3034} The fourth main group of modal logics is called existential logic or modes of existence, which include: "universality," "existence," and "emptiness."

Higher-valued logics have also been devised. The philosophy of the Jains of India uses a seven-valued truth logic. It is grounded in the religious beliefs of the sect and utilizes the following truth values:

1. True (a thing is); 2. False (a thing is not); 3. Indeterminate (impossible to say either is or is not); 4. Is and Is Not; 5. Is and Is Indeterminate; 6. Is Not and Is Indeterminate; and 7. Is and Is Not and Is Indeterminate.^{900}

Further permutations are possible, but these only change the way of saying and not the substance of what is said -- so are of no logical significance. (Note that Jam logic has an implicit Law of the Excluded Eighth.)

A few mathematicians have even formulated infinite-valued logics.^{901,899} Infinite logics range over a continuum of real numbers X such that 0 < X < 1. In this notation, "1" represents absolute truth and "0" represents complete falsity. Other peculiar systems thinking include plurality logic (using quantifiers such as "all," "some," and "none," or such as "all," "nearly all," "many," "not many," "few," and "none"),^{911} tense or temporal logic (systematizes reasoning with propositions that have a temporalized aspect and incorporate the axioms of time in general, such as "before" and "after" or "past," "present," and "future" relationships),^{912} probablistic logic, minimal logic, intuitionist logic, Chinese complementary logic, and so forth.^{913,894}

It is sobering to realize that all of the above described alternative logical systems have been devised by human minds. The human brain operates using neurons with a two-valued firing pattern. It may be that people -- indeed all Earthly forms of life - are hardwired or preadapted in some sense for Aristotelian modes of thinking. ETs on other worlds may have trivalue or higher-value neuronal firing patterns. To such minds the Aristotelian logic of humankind may seem horribly restrictive and primitive.

* Others include the Law of Identity (subject and predicate are identical) and the Law of Contradiction (nothing is both A and not-A).

** When 1.00 liter of water is added to 1.00 liter of ethyl alcohol, we get only 1.93 liters of solution -- not 2.00. There is a volume contraction of 3.5% due to intermolecular packing.^{3050}