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an inference or reasoning that substantiates some known incongruity, absurdity, or paradoxical statement contradicting generally accepted notions.
Aristotle called sophisms false proofs in which the grounds for the conclusion are only apparent and derive from a purely subjective impression caused by insufficient logical or semantic analysis. The initial persuasiveness of many sophisms—their “logical” character—is usually related to a well-concealed error on the semiotic or logical level. Errors on the semiotic level may involve metaphorical speech, homonymy and polysemy, and amphibology, which violate the unambiguous character of the thought and lead to a confusion of the meanings of terms. Errors on the logical level include deceptive substitution of the basic thought (thesis) of the proof, the taking of false premises for true ones, violation of the permissible methods of reasoning (the rules of logical inference), and the use of “unauthorized” or even “forbidden” rules or actions, such as division by zero in a mathematical sophism. The last-mentioned mistake may also be considered semiotic because it is related to agreement concerning “correctly constructed formulas.”
The ancient “horn sophism,” which is ascribed to Eubulides, runs as follows; “That which you have not lost, you have. You have not lost horns. Therefore, you have horns.” In this case the ambiguity of the major premise is concealed. If the major premise is understood to be universal and to mean “everything that you have not lost,” the inference is logically flawless but uninteresting because the major premise is obviously false; if the major premise is understood to be particular, the conclusion does not follow logically. This became known, however, only after Aristotle had created logic.
A modern sophism has it that as we grow older the “years of life” not only seem shorter but are in fact shorter: “Every year of your life is 1/n of your life, where n is the number of years you have lived. But n + 1 > n. Therefore, 1/(n + 1) < 1/n.”
Historically, the concept of sophism has always been associated with the idea of deliberate falsification, this in line with Protagoras’ admission that the task of the Sophist is to represent the worst argument as the best by devising clever tricks in speech and reasoning, ignoring the truth, and concentrating on practical advantage or success in debate. (Of course, Protagoras himself fell victim to the “Euatlos’ sophism.”) Usually associated with this same idea is the “criterion of reason” formulated by Protagoras: man’s opinion is the measure of truth. Plato remarked that reason must not involve a person’s subjective will, lest it become necessary to recognize the validity of contradictions—which, incidentally, the Sophists argued—and therefore to consider all statements sound.
This thought of Plato’s was developed in the Aristotelian principle of noncontradiction and, in modern logic, in interpretations demanding that absolute consistency be proved. Brought from the field of pure logic into the field of “factual truths,” it gave rise to a special “style of thinking” that ignored the dialectics of “interval situations.” In interval situations, Protagoras’ criterion, more broadly understood as the relativity of truth to the conditions and means of knowing truth, proves very significant. This is why many lines of reasoning that lead to paradoxes but are otherwise flawless are defined as sophisms even though they essentially demonstrate the interval nature of the corresponding epistemo-logical situations. An example is seen in the “pile sophism”: “One grain is not a pile. If n grains are not a pile, then n + 1 grains are also not a pile. Therefore, any number of grains is not a pile.” This is just one of the “paradoxes of transitivity” that arise in situations of “indistinguishability.” A situation of indis-tinguishability is a typical example of an interval situation in which the property of the transitivity of equality is not preserved during the transition from one “interval of indistinguishability” to another, thus making the principle of mathematical induction inapplicable in such situations.
The attempt to see in this an “intolerable contradiction” that is characteristic of experience and that mathematical thought “overcomes” in the abstract concept of the number continuum (J. H. Poincaré) is not supported, however, by the general proof of the removability of such situations in the sphere of mathematical thinking and experience. Suffice it to say that the description and practice of using the “laws of identity” (equality), which are so important in this sphere, generally depend, as in the empirical sciences, on the meaning given to the expression “one and the same object” and on the means or criteria of identity that are used. In other words, whether one speaks of mathematical objects or, for example, objects of quantum mechanics, the answers to questions of identity are necessarily linked to interval situations. Moreover, it is by no means always possible to oppose a solution “above this interval” to a particular solution to the question “within” the interval of indistinguishability, that is, to substitute the abstraction of identity for the abstraction of indistinguishability. But only in the latter case can one say the contradiction has been “overcome.”
Apparently the Sophists themselves were the first to understand the importance of the semiotic analysis of sophisms. Prodicus considered the doctrine of speech and the correct use of names to be paramount. Sophisms are frequently analyzed and used as examples in Plato’s dialogues. Aristotle wrote a specialized work entitled Sophistical Refutations, and the mathematician Euclid wrote the Pseudaria, a distinctive catalog of sophisms in geometric proofs.
REFERENCESAkhmanov, A. S. Logicheskoe uchenie Aristotelia. Moscow, 1960.
Bradis, V. M., V. L. Minkovskii, and A. K. Kharcheva. Oshibki v matematicheskikh rassuzhdeniiakh, 3rd ed. Moscow, 1967.
M. M. NOVOSELOV