Lejewski then goes on to offer a description of free logic, which he claims accommodates an answer to the problem.

Lejewski al

Lejewski also points out that free logic additionally can handle the problem of the empty set for statements like ${\displaystyle \forall x\,Fx\rightarrow \exists x\,Fx}$. Quine had considered the problem of the empty set unrealistic, which left Lejewski unsatisfied.^[33]

In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the "Quine–Putnam indispensability thesis," an argument for the reality of mathematical entities.^[34]

The form of the argument is as follows.

1. One must have ontological commitments to all entities that are indispensable to the best scientific theories, and to those entities only (commonly referred to as "all and only").
2. Mathematical entities are indispensable to the best scientific theories. Therefore,
3. One must have onto

   The form of the argument is as follows.

   The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry, but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.^[35]

   Epistemology

   Just as he challenged the dominant analytic–synthetic distinction, Quine also took aim at traditional normative epis

   Just as he challenged the dominant analytic–synthetic distinction, Quine also took aim at traditional normative epistemology. According to Quine, traditional epistemology tried to justify the sciences, but this effort (as exemplified by Rudolf Carnap) failed, and so we should replace traditional epistemology with an empirical study of what sensory inputs produce what theoretical outputs:^[36] "Epistemology, or something like it, simply falls into place as a chapter of psychology and hence of natural science. It studies a natural phenomenon, viz., a physical human subject. This human subject is accorded a certain experimentally controlled input—certain patterns of irradiation in assorted frequencies, for instance—and in the fullness of time the subject delivers as output a description of the three-dimensional external world and its history. The relation between the meager input and the torrential output is a relation that we are prompted to study for somewhat the same reasons that always prompted epistemology: namely, in order to see how evidence relates to theory, and in what ways one's theory of nature transcends any available evidence...But a conspicuous difference between old epistemology and the epistemological enterprise in this new psychological setting is that we can now make free use of empirical psychology." (Quine, 1969: 82–83)

   Quine's proposal is controversial among contemporary philosophers and has several critics, with Jaegwon Kim the most prominent among them.^[37]

   Jaegwon Kim the most prominent among them.^[37]