Willard Van Orman Quine

Quine and Logic 277
of Methods of Logic (1950, 1959, 1972, and 1982), each after the first
a considerable enlargement of the one before it. I have used all four
editions of the text in what seem myriad offerings of my class in
basic symbolic logic. I first learned the subject from Quine, studying
its mimeographed predecessor.
Methods of Logic is like no other text in the subject. First of all,
it is very appropriately titled. It serves up a veritable feast of methods,
thoughtfully prepared for application and well garnished with
their rationales. The book encompasses all that Quine sees as falling
within logic’s realm, and indeed somewhat more. Methods builds on
no abstract system with formal rules. In it Quine gives us machinery
ready to use and not bogged down in technicalities that serve no
practical purpose. Still, the book is crisp, precise, and painstaking,
imparting regard for rigorous thinking at every turn.
Early we find Quine shunning the distinction between a sentence
letter and its double negation. Since this is classical logic, the two
will cash out in the same way (although in chap. 13’s axiom system,
a brief departure, they need to be distinguished). Similarly, it
is efficient to take both conjunction and alternation (Quine favors
‘alternation’ over ‘disjunction’) as capable of combining any finite
plurality of components, not just two. Quine is eschewing fussiness
that would slow application of the mechanism.
Since the symbolic expressions that are the heart of the subject
are not part of a formal system, Quine does not call them formulas.
He emphasizes that what they do is schematically represent the sentences
that realize them; they serve as “logical diagrams,” and he
calls them schemata. Thus ‘pvqvr’ is a schema, representing any
threefold alternation. A schema is fleshed out by taking actual
sentences, and later predicates, to supplant its schematic letters. We
think of the letters as proxies. Quine’s schemata are ready to use.
They are symbolizations that might be given to the sentences, however
complex, that they represent.
That view of logic’s symbolic expressions lays bare what is always
implicit in the subject’s development: The logical truths are
just those that realize valid schemata, or, turned around more usefully,
a sentence is a logical truth just in case it has some correct
symbolization that is valid.
For testing validity or consistency of truth-functional schemata,
Quine favors his method of truth-value analysis. In addition to giving
all the information of truth tables, this technique allows us to
Cambridge Companions Online © Cambridge University Press, 2006