Logic, Law of

the general name given to the laws that form the basis of logical deduction.

The concept of logical law goes back to the ancient Greek concept of logos as the presupposition for the objective (“natural”) correctness of reasoning. The concept first received its true logical content in the works of Aristotle, who began the systematic description and the cataloging of such schemata of the logical connectives between arbitrary elementary propositions into complex propositions, whose convincingness (universal validity) resulted entirely from their form, more precisely, simply from the correct understanding of the meaning of the logical connectives, regardless of the truth value of the elementary propositions. Most of the laws of logic discovered by Aristotle were laws of the syllogism. Other laws were discovered later, and it was even established that the set of logical laws was infinite.

The different types of formal theories of logical inference have made it possible to survey, in some sense, the infinite set of logical laws. In such logical formalisms, or logical calculi, the laws of logic are expressed by specific types of formulas and are defined, each with respect to its own calculus, by deducible formulas of a given type (“universally valid formulas,” or the theorems of the calculi). The diversity of existing logical calculi has naturally given rise to the concept of the relativity of the laws of logic. However, at the same time, limits to this relativity are set by the specific type of logical calculus, since this type is not exclusively a matter of arbitrary choice but is dictated, or suggested, by a “logic of things” about which we wish to reason and also, in some sense, by a subjective certainty regarding one or another aspect of this logic.

All calculi based on the same hypothesis about the nature of the “logic of things” are equivalent in the sense that they describe or “generate” the same laws of logic. For example, calculi based on the principle of bivalence, namely, the calculi of classical logic, describe, in spite of their outward variety, the same “world” of the classical laws of logic, such as the law of truths by identity, which have long since received the generally accepted ontological and philosophical designation of “eternal truths” or “truths in all possible worlds.”

The laws of logic in intuitionist logic have not yet received a generally accepted ontological interpretation. The logic of mental mathematical constructions—that is, the logic of knowledge, and not the logic of existence—constitutes the “logic of things,” historically reflected by these laws of intuitionist logic.

The study of the laws of logic is the natural starting point for the logical analysis of the acceptable (“good”) methods of reasoning, that is, of inferences, since the very concept of an acceptable or logically correct argument is refined by the concept of a law of logic. The relation between logically correct arguments and the laws of logic is expressed in logic by the deduction theorem, which defines the special role, noted in classical philosophy by the Stoics, played by the laws of logic in the foundation and verification of our inferences. This is the case with regard to any assertion on the derivability of a conclusion B from the premises A₁, A₂, . . ., A_n whose truth is decided upon by finding, from among the laws of logic, the proposition A₁ ⊃ (A₂⊃ (…⊃ (A_n ⊃ B)…)), where ⊃ expresses the logical connective “if…then ... .” Such a relation between inferences and the laws of logic is of general theoretical significance and goes far beyond the limits of logic proper, providing a general method of formal proof by means of logic.

The term “law of logic” is used in traditional logic with regard to the “laws of thought,” such as the law of identity (“every entity coincides with itself), the law of noncontradiction (“no proposition can be at the same time true and false”), the principle of the excluded middle (“an arbitary proposition is either itself true or its negation is true”), and the law of sufficient reason (“every accepted proposition must be properly justified”). The first of these principles (the term “law” is inappropriate here) is an important presupposition of reasoning, which is related, however, not to logic but to ontology and the theory of knowledge; this presupposition, moreover, is always applied within precisely specified limits. The principle of sufficient reason is also not related to logic but rather is a clearly expressed methodological principle. The principle of the excluded middle does indeed belong to logic, although the corresponding formula (A ∨ ℸ A) is not universally valid in every logical system. Finally, the principle of noncontradiction—expressed in contemporary logic by the symbols ⌝ (A & ⌝ A)—is not only an assertion provable in any logical system but, to a certain extent, one that lies at the foundation of all contemporary formal logic.