Willard Van Orman Quine

Source: Wikipedia. Pages: 26. Chapters: Cognitive synonymy, Confirmation holism, Duhem-Quine thesis, Formative epistemology, Hold come what may, Indeterminacy of translation, Inscrutability of reference, Naturalized epistemology, Neurathian bootstrap, New Foundations, Plato's beard, Quine's paradox, Quine (computing), Quine-McCluskey algorithm, Radical translation, Two Dogmas of Empiricism. Excerpt: In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998). The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality () and membership (). TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number n, type n+1 objects are sets of type n objects; sets of type n have members of type n-1. Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules: and (notation to be improved). The axioms of TST are: If is a formula, then the set exists.In other words, given any formula , the formula is an axiom where represents the set .This type theory is much less complicated than the one first set out in the Principia Mathematica, which included types for relations whose arguments were not necessarily all of the same type. In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchy of sets described here. The well-formed formulas of New Foundations (NF) are the same as the well-formed formula of TST, but with the type annotations are erased. The axioms of NF are: By convention, NF's Comprehension schema is stated using the concept of stratified formula and making no direct reference to types. A formula is said to be stratified if there exists a function f from pieces of syntax to the natural numbers, such that for any atomic subformula of we have f(y) = f(x) + 1, while for any atomic subformula of , we have f(x) = f(y). Comprehension then becomes: ex