Willard van Orman Quine
Willard van Orman Quine was the pre-eminent philosopher of logic at Harvard and perhaps in the whole country for several decades in the 20th century.

Analytic --Synthetic Distinction

In the early 1950's, Quine challenged the ancient analytic-synthetic distinction, arguing that in the end the "truth" of analytic statements, the proofs of mathematical theorems, and the use of logic, also depend on empirical verification.

The key idea of Quine's empiricism (and of David Hume's) is to deny the existence of any a priori knowledge of the world, whether analytic or synthetic.

As Charles Sanders Peirce had said, nothing is logically and necessarily true of the physical world. Logical truths like the Principles of Non-Contradiction and Bivalence (Excluded Middle) might be "true in all possible worlds," as Gottfried Leibniz had put it, but they tell us nothing about our physical world, until they are shown to be useful tools for reasoning by being empirically verified.

Epistemology Naturalized
Nearly twenty years later, Quine argued that epistemology, the justification of knowledge claims, should be "naturalized." All knowledge claims should be reduced to verification by the methods of natural science. "For suppose we hold," he says, "with the old empiricist Peirce, that the very meaning of a statement consists in the difference its truth would make to possible experience."
Every term and every sentence is a label attached to an idea, simple or complex, which is stored in the mind. When on the other hand we take a verification theory of meaning seriously, the indeterminacy would appear to be inescapable. The Vienna Circle espoused a verification theory of meaning but did not take it seriously enough. If we recognize with Peirce that the meaning of a sentence turns purely on what would count as evidence for its truth, and if we recognize with Duhem that theoretical sentences have their evidence not as single sentences but only as larger blocks of theory, then the indeterminacy of translation of theoretical sentences is the natural conclusion. And most sentences, apart from observation sentences, are theoretical. This conclusion, conversely, once it is embraced, seals the fate of any general notion of propositional meaning or, for that matter, state of affairs.

Philosophers have rightly despaired of translating everything into observational and logico-mathematical terms. They have despaired of this even when they have not recognized, as the reason for this irreducibility, that the statements largely do not have their private bundles of empirical consequences. And some philosophers have seen in this irreducibility the bankruptcy of epistemology. Carnap and the other logical positivists of the Vienna Circle had already pressed the term "metaphysics" into pejorative use, as connoting meaninglessness; and the term "epistemology" was next. Wittgenstein and his followers, mainly at Oxford, found a residual philosophical vocation in therapy: curing philosophers of the delusion that there were epistemological problems.

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.

Such a study could still include, even, something like the old rational reconstruction, to whatever degree such reconstruction is practicable; for imaginative constructions can afford hints of actual psychological processes, in much the way that mechanical simulations can. 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.

The old epistemology aspired to contain, in a sense, natural science; it would construct it somehow from sense data. Epistemology in its new setting, conversely, is contained in natural science, as a chapter of psychology. But the old containment remains valid too, in its way. We are studying how the human subject of our study posits bodies and projects his physics from his data, and we appreciate that our position in the world is just like his. Our very epistemological enterprise, therefore, and the psychology wherein it is a component chapter, and the whole of natural science wherein psychology is a component book — all this is our own construction or projection from stimulations like those we were meting out to our epistemological subject. There is thus reciprocal containment, though containment in different senses: epistemology in natural science and natural science in epistemology.

Although Quine's reciprocal containment suggested that epistemology might still play a foundational role in scientific understanding, his work appeared to many to be reductionist. It seems to deny the normative role of traditional epistemology, which hoped to justify all knowledge, including scientific knowledge. If epistemology ultimately depends on science, justifying or grounding science with epistemology would be circular.

Ontology
Quine began his famous essay "On What There Is" claiming it has a trivial answer,
A curious thing about the ontological problem is its simplicity. It can be put in three Anglo-Saxon monosyllables: 'What is there?' It can be answered, moreover, in a word—'Everything' —and everyone will accept this answer as true. However, this is merely to say that there is what there is. There remains room for disagreement over cases; and so the issue has stayed alive down the centuries.
Alexius Meinong disagreed, and in a most disagreeable way, insisting that "objects exist which do not exist," by which he meant things that do not have an ordinary material existence, such as abstract entities like numbers and Platonic Ideas. Meinong also meant impossible objects, like the "round square," which have meaning but do not have denotation, any reference to an example of such an object.
Quantified Modal Logic
Quine was perhaps best-known in the philosophy of logic for his views on quantification, which was an essential part of Aristotle's Prior Analytics and had been formalized first by Gottlob Frege in 1879 in his Begriffsschrift or "Concept Writing."

Frege replaced the familiar sentences (or statements or propositions) of the "first-order" predicate logic of Aristotle's syllogisms - "All men are mortal' - with the notion of quantification operators working on Propositional functions, formulas that include variables, some of which are "free" and others "bound" by the quantification operator.

The idea of "for all x" becomes ∀ x and is called the universal quantification operator. The notion of "for some x" is called the existential operator ∃ x. This is often read "there exists an x such that..."

These operators are followed by formulas describing the predicates, which may be properties or attributes of x. For example, ∀x Mx might describe "for all x"(the variable for a man) and "Mx" can be read "are mortal."

References
Quine, W. V. (1943). Notes on existence and necessity. The journal of philosophy, 40(5), 113-127.
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