Natural deduction and arbitrary objects

This paper is an abridged and simplified version of my monograph Reasoning with Arbitrary Objects [4]. It may be read by the diligent as a preparation for the longer work or by the indolent as a substitute for it. But the reader, in either case, may find it helpful to consult the paper, A Defence of Arbitrary Objects [3], for general philosophical orientation. This paper deals with certain problems in understanding natural deduction and ordinary reasoning. As is well known, there exist in ordinary reasoning certain procedures for arguing to a universal conclusion and from an existential premiss. We may establish that all objects have a given property by showing that an arbitrary object has the property; and having shown that there exists an object with a given property, we feel entitled to give it a name and infer that it has the property. For example: we may establish that all triangles have interior angles summing to 180 " by showing of an arbitrary triangle that its interior angles sum to 180'; and having established that there exists a bisector to an angle, we feel entitled to give it a name and infer that it is a bisector to the angle. These informal procedures correspond to certain of the quantificational rules in systems of natural deduction. Corresponding to the first is the rule of universal generalisation, which allows us to infer VXI&) from p(a) under suitable restrictions. Corresponding to the second is the rule of existential instantiation, which allows us to infer cp(a) from 3x&), again under suitable restrictions. In these inferences, certain terms play a crucial role; and it is natural to ask how it should be understood. What role should be attributed to the term a in inferences from natural deduction? What is to be made of our talk of arbitrary triangles or indefinite bisectors in ordinary reasoning? We here take seriously the idea that the critical terms in these inferences refer to arbitrary or representative objects. The term a in the inferences from natural deduction functions as a name of a suitable arbitrary object. And our talk of arbitrary triangles or indefinite triangles is to be taken at its face value as evincing reference to arbitrary objects. Our principle concern will be to apply this hypothesis to two main systems of natural deduction: the one of Quine's Methods of Logic [ 141; the