infinitary logic the logic of expressions of infinite length. Quine has advanced the claim that firstorder logic (FOL) is the language of science, a position accepted by many of his followers. However, many important notions of mathematics and science are not expressible in FOL. The notion of finiteness, e.g., is central in mathematics but cannot be expressed within FOL. There is no way to express such a simple, precise claim as ‘There are only finitely many stars’ in FOL. This and related expressive limitations in FOL seriously hamper its applicability to the study of mathematics and have led to the study of stronger logics.
There have been various approaches to getting around the limitations by the study of so-called strong logics, including second-order logic (where one quantifies over sets or properties, not just individuals), generalized quantifiers (where one adds quantifiers in addition to the usual ‘for all’ and ‘there exists’), and branching quantifiers (where notions of independence of variables is introduced). One of the most fruitful methods has been the introduction of idealized ‘infinitely long’ statements. For example, the above statement about the stars would be formalized as an infinite disjunction: there is at most one star, or there are at most two stars, or there are at most three stars, etc. Each of these disjuncts is expressible in FOL.
The expressive limitations in FOL are closely linked with Gödel’s famous completeness and incompleteness theorems. These results show, among other things, that any attempt to systematize the laws of logic is going to be inadequate, one way or another. Either it will be confined to a language with expressive limitations, so that these notions cannot even be expressed, or else, if they can be expressed, then an attempt at giving an effective listing of axioms and rules of inference for the language will fall short. In infinitary logic, the rules of inference can have infinitely many premises, and so are not effectively presentable.
Early work in infinitary logic used cardinality as a guide: whether or not a disjunction, conjunction, or quantifier string was permitted had to do only with the cardinality of the set in question. It turned out that the most fruitful of these logics was the language with countable conjunctions and finite strings of first-order quantifiers. This language had further refinements to socalled admissible languages, where more refined set-theoretic considerations play a role in determining what counts as a formula.
Infinitary languages are also connected with strong axioms of infinity, statements that do not follow from the usual axioms of set theory but for which one has other evidence that they might well be true, or at least consistent. In particular, compact cardinals are infinite cardinal numbers where the analogue of the compactness theorem of FOL generalizes to the associated infinitary language. These cardinals have proven to be very important in modern set theory. During the 1990s, some infinitary logics played a surprising role in computer science. By allowing arbitrarily long conjunctions and disjunctions, but only finitely many variables (free or bound) in any formula, languages with attractive closure properties were found that allowed the kinds of inductive procedures of computer science, procedures not expressible in FOL. See also COMPACTNESS THEOREM, COM- PLETENESS , GÖDEL ‘S INCOMPLETENESS THEO – REMS , INFINITY , SECOND -ORDER LOGI. J.Ba.