Parity of reasoning is another process Of simulating induction. This process of inference is applied in formal science like mathematics. Particularly it is applied in geometry.
Though parity of reasoning gives the appearance of making a generalization, in reality it is not at all a form of induction. On the contrary it is substantially a deductive form of reasoning and has nothing to do with induction. Hence it is considered as simulating induction.
Induction by parity of reasoning is the generalization of a truth on the basis that the same reasoning which has proved a particular case will apply in every other similar case.
Suppose we prove in geometry that the two adjacent angles make two right angles; or that the three interior angles of a triangle make two right angles.
The proof that is given in one case is applicable in every other similar case. Whatever proof justifies the case that two adjacent angles are 180°, will be applicable in every case of adjacent angles. Or the proof that justifies that the interior angles of a triangle are 180°, will hold good in case of every triangle.
In algebra we also apply the same procedure of reasoning. Suppose we prove that a2 - b2 = (a + b) (a - b). On this basis we conclude that the difference of the square of any two numbers is equal to the product of their sum and difference. The symbolic expression of this form of reasoning is: ' •
Si is necessarily P.
All S's are necessarily P.
It is to be noted that conclusion in parity of reasoning is a mathematical proposition. And a mathematical proposition is necessarily true as it is drawn from some axioms, definitions and theorems.
The conclusion appears as if there is a generalization but it is not on the basis of observation of facts. The conclusion is a deductive one and the basis for this deduction is similarity in reasoning.
The reasoning or proof that justifies a mathematical proposition will also apply in every other similar case. In fact any single proof makes a theorem. In the examples, the concept of an adjacent triangle stands for any adjacent angle or a triangle represents every triangle.
Further in parity of reasoning there is no observation of facts. By the help of some axioms, definitions and corollaries different theorems are proved. A triangle, for instance, is not a particular instance of triangles rather stands for the whole class of triangles. It is an abstract idea. So here there is no observation of facts of experience.
As there is no observation of facts, there is no inductive leap in parity of reasoning. When we draw a diagram of triangle and prove that its interior angles make two right angles it is not that we are passing from known case to unknown cases.
On the other hand what is proved in case of a particular diagram is a statement about all similar diagrams. Further the idea of uniformity of nature or law of causation has nothing to do in this form of reasoning.
Hence parity of reasoning is not at all inductive, rather it is deductive in nature. As it is deductive its conclusion is logically certain and its opposite is self-contradictory. While the conclusion of an induction is probable, the conclusion in parity of reasoning is certain. *
Thus parity of reasoning is not based on observation of facts. It has no inductive leap nor is its conclusion probable. It is not an inductive form of reasoning but it only simulates induction.