A type of deductive theory, such as those used in mathematics, of which Euclid's Elements is one of the early forms. Long a model for scientific theorizing, the axiomatic system has been studied intensively only since the end of the 19th century, and this in conjunction with the development of mathematical, or symbolic, logic in research on the foundations of logic and of mathematics (see logic, symbolic). In its earlier sense the axiomatic system was considered as having a meaning content, whereas more recently it has been understood in a purely formal sense—as practically synonymous, in fact, with the formal system (see formalism). According to H. B. Curry, the notion of formal system is more restricted than that of axiomatic system, whereas for other authors the formal system is more general since it lacks the conditions of effectiveness that should characterize the axiomatic system.
General Characterization. A scientific theory is made up of propositions that are in turn composed of terms. One can establish the validity of a proposition by deducing it from other propositions, but it is impossible to proceed in this way to infinity. In the same manner, one can explain the sense of a term by defining it through the use of other terms, but again it is impossible to proceed to infinity. To build a theory it is therefore necessary to start from terms that are accepted without definition and from propositions that are considered as valid without demonstration. To these primitive elements are then added rules of definition, with whose help it is possible to define new terms from the undefined primitive terms or from terms already defined, and rules of deduction, with whose help it is possible to obtain new propositions from the primitive propositions or from propositions already deduced. The primitive propositions are called the axioms of the theory. The propositions that can be deduced by means of the rules of deduction are said to be proved or demonstrated. The axioms and the proved propositions are the theorems of the theory.
According to the earlier view, an axiomatic system expresses a certain order of truth. A deductive science is based on postulates or first principles that express self-evident truths or on propositions demonstrated by a superior science. The whole order of demonstrative knowledge is thus founded on self-evident propositions. The primitive terms are taken with their natural meaning, while the rules of deduction are those of ordinary logic and are not considered to be part of the theory.
According to the modern view, however, an axiomatic system is considered simply as expressing an order of possible deductions. The axioms are not considered to be true, but only as propositions provisionally accepted as valid. An axiomatic system is thus nothing but a hypotheticodeductive system. The undefined terms are not understood with regard to their intuitive meaning, but are understood in terms of what is asserted about them in the axioms. On the other hand, the logic that is used is explicitly incorporated into the system. This complete formalization of the deductive process is what is referred to as formal axiomatic method.
Interpretation. A formal axiomatic system must be considered as a pure system of deduction. Such a system, however, has interest only to the extent that it can be used to investigate the properties of theories having a factual content, i.e., to the extent that there are relationships between this system and "contensive statements," to use Curry's terminology. A contensive statement is a statement that pertains to some factual domain and is such that its truth or falsity can be established by appropriate methods that take account of the facts such statements concern; these may be mathematical or empirical in nature. An interpretation of an axiomatic system is a many-toone correspondence between some propositions of the system and some contensive statements belonging to a given field. An interpretation is valid if every contensive statement corresponding to a theorem is true.
The advantage of the formal axiomatic method is that it offers the possibility of studying in a synthetic manner the properties of all theories having the same form, such as those of the different empirical domains of the physical sciences. The construction of such theories raises the problem of the choice of the appropriate axioms, which itself is related to the problem of induction. Such theories must also be located with reference to empirical facts by way of interpretation, and this is connected with the problem of verification.
Abstract Characterization. Curry refers to axiomatic systems as deductive theories, themselves a particular type of theory. A theory is defined as a class of statements belonging to a certain definite class of statements that are previously postulated. (A statement must be distinguished from a sentence, which is a linguistic expression designating a statement.) A class is said to be definite if some effective process exists that makes it possible to determine whether a given object is a member of the class or not. The statements that belong to a theory are called its theorems. A theory is said to be deductive when it constitutes an inductive class of statements. An inductive class is a class whose elements are generated from certain initial elements (forming a definite class) by means of certain specified modes of combination that have an "effective character"—in the sense that some effective method exists that makes it possible to determine whether a given element has been actually produced from given elements of the class. (It should be noted that an inductive class is not necessarily definite.) The initial statements of a deductive theory form a definite class of statements; they are called the axioms of the theory. The modes of combination of a deductive theory are its rules of deduction; when applied to an appropriate number of theorems, such a rule produces a new theorem. Every construction of a theorem from given theorems by means of the rules of deduction is called a demonstration. The theorems of the deductive theory are then the statements for which demonstrations exist.
Formal System. When it is specified that the statements assert that certain formal objects have particular properties or stand in particular relation to each other, a theory is called a system or, more explicitly, a formal system. To describe such a system, one identifies a particular class of objects, called formal objects, and a particular class of predicates, called basic predicates. The statements of the system are formed by applying a basic predicate to an appropriate number of formal objects.
Types. There are two types of systems: syntactical systems and ob systems. In a syntactical system, the formal objects are taken as the expressions of a particular language. In an ob system, the formal objects form a monotectonic inductive class—monotectonic in the sense that, for every element, there exists only one construction that produces it. The elements of this class are called obs; its initial elements, atoms; and the modes of combination, operations.
Epitheory. When a system is constituted, it can be studied as a given object; such a study is then referred to as an epitheory. The main epitheoretical questions pertaining to an axiomatic system concern its consistency (the impossibility of deducing within the system both a proposition and its negation), its independence (the impossibility of deducing any axiom from the others), and its completeness (every proposition formulable in the terms of the system is provable or refutable, in which case its negation is provable, within the system). Also of interest is the decision problem, which is concerned with formulating an effective method that makes it possible to determine, for every proposition formulable in the terms of the system, whether the proposition is provable in the system. One of the most famous epitheoretical theorems is the incompleteness theorem of K. Gödel: in every consistent system that is sufficient to formalize ordinary arithmetic there are undecidable propositions, i.e., propositions that are neither provable nor refutable.
Bibliography: a. tarski, Introduction to Logic and to the Methodology of Deductive Sciences (New York 1941), first pub. in Polish 1936. h. b. curry, Foundations of Mathematical Logic (New York 1963). j. a. ladriÈre, Les Limitations internes des formalismes (Louvain 1957).
[j. a. ladriÈre]