Nelson Goodman

In his book Fact, Fiction, and Forecast, Goodman introduced the "new riddle of induction", so-called by analogy with Hume's classical problem of induction. He accepted Hume's observation that inductive reasoning (i.e. inferring from past experience about events in the future) was based solely on human habit and regularities to which our day-to-day existence has accustomed us. Goodman argued, however, that Hume overlooked the fact that some regularities establish habits (a given piece of copper conducting electricity increases the credibility of statements asserting that other pieces of copper conduct electricity) while some do not (the fact that a given man in a room is a third son does not increase the credibility of statements asserting that other men in this room are third sons).

Hempel's confirmation theory argued that the solution is to differentiate between hypotheses, which apply to all things of a certain class, and evidence statements, which apply to only one thing. Goodman's famous counterargument was to introduce the predicate grue, which applies to all things examined before a certain time t just in case they are green, but also to other things just in case they are blue and not examined before time t. If we examine emeralds before time t and find that emerald a is green, emerald b is green, and so forth, each will confirm the hypothesis that all emeralds are green. However, emeralds a, b, c,..etc. also confirm the hypothesis that all emeralds are grue. Thus, before time t, the apparently law-like statements "All emeralds are green" and "All emeralds are grue" are equally well confirmed by observation, but obviously "All emeralds are grue" is not a law-like statement.

Goodman's example showed that the difficulty in determining what constitutes law-like statements is far greater than previously thought, and that once again we find ourselves facing the initial dilemma that "anything can confirm anything".

Goodman, along with Stanislaw Lesniewski, is the founder of the contemporary variant of nominalism, which argues that philosophy, logic, and mathematics should dispense with set theory. Goodman's nominalism was driven purely by ontological considerations. After a long and difficult 1947 paper coauthored with W. V. O. Quine, Goodman ceased to trouble himself with finding a way to reconstruct mathematics while dispensing with set theory – discredited as sole foundations of mathematics as of 1913 (Russell and Whitehead, in Principia Mathematica).

The program of David Hilbert to reconstruct it from logical axioms was proven futile in 1931 by Gödel. Because of this and other failures of seemingly fruitful lines of research, Quine soon came to believe that such a reconstruction was impossible, but Goodman's Penn colleague Richard Milton Martin argued otherwise, writing a number of papers suggesting ways forward.

According to Thomas Tymoczko's afterword in New directions in the philosophy of mathematics, Quine had "urged that we abandon ad hoc devices distinguishing mathematics from science and just accept the resulting assimilation", putting the "key burden on the theories (networks of sentences) that we accept, not on the individual sentences whose significance can change dramatically depending on their theoretical context." In so doing, Tymoczko claimed, philosophy of mathematics and philosophy of science were merged into quasi-empiricism: the emphasis of mathematical practice as effectively part of the scientific method, an emphasis on method over result.

The Goodman–Leonard (1940) calculus of individuals is the starting point for the American variant of mereology. While the exposition in Goodman and Leonard invoked a bit of naive set theory, the variant of the calculus of individuals that grounds Goodman's 1951 The Structure of Appearance, a revision and extension of his PhD thesis, makes no mention of the notion of set (while his PhD thesis still did).[8] Simons (1987) and Casati and Varzi (1999) show that the calculus of individuals can be grounded in either a bit of set theory, or monadic predicates, schematically employed. Mereology is accordingly "ontologically neutral" and retains some of Quine's pragmatism (which Tymoczko in 1998 carefully qualified as American Pragmatism).