Some Remarks on Quine's Logic (1)

In his Methods of Logic, Willard Van Orman Quine suggested to test the validity of formulae in Quantificational Logic by transcribing those formulae into Boolean Schemata and applying Truth-value Analysis and ANF. The way he suggested is quite different from the way suggested by I.M Copi in his Symbolic Logic, in which Copi suggested to provide a non-empty model with an normal interpretation to the quantificational schemata in question and test them by using truth-table method.

Moreover, In his Philosophy of Logic, in the chaper about logical truth, he tried to show that any schema-in-L that come out true under all substitution of sentences iff it is satisfied by all models, as long as the language L is reasonably rich, rich enough for elementary number theory, i.e. it can said about positive integers in terms of "+", "*", "=", "↓" and "(x)", but no sets.

The mereological attitude of Quine mentioned above can be explained by his "constructivism" that it is better to posit less on constructing theory.

However it is inevitable to posit sets if we want to simplify the ontology of numbers as set theories provide ways to express arithmetic with following three symbols: "↓", "(x)" and " ∈" and variables such as x, x', x'', x''',....etc., in this way it may be easier to say numbers are only theoretical constructs composed of logical connectives, quantifiers, membership and variables.