Our primitive idea, now, is a certain truth-function called "incompatibility," which we will denote by . Negation can be at once defined as the incompatibility of a proposition with itself, i.e. "not-" is defined as "." Disjunction is the incompatibility of not- and not-, i.e. it is . Implication is the incompatibility of and not-, i.e. . Conjunction is the negation of incompatibility, i.e. it is . Thus all our four other functions are defined in terms of incompatibility. It is obvious that there is no limit to the manufacture of truth-functions, either by introducing more arguments or by repeating arguments. What we are concerned with is the connection of this subject with inference. [Pg 148] If we know that is true and that implies , we can proceed to assert . There is always unavoidably something psychological about inference: inference is a method by which we arrive at new knowledge, and what is not psychological about it is the relation which allows us to infer correctly; but the actual passage from the assertion of to the assertion of is a psychological process, and we must not seek to represent it in purely logical terms. In mathematical practice, when we infer, we have always some expression containing variable propositions, say and , which is known, in virtue of its form, to be true for all values of and ; we have also some other expression, part of the former, which is also known to be true for all values of and ; and in virtue of the principles of inference, we are able to drop this part of our original expression, and assert what is left. This somewhat abstract account may be made clearer by a few examples. Let us assume that we know the five formal principles of deduction enumerated in Principia Mathematica. (M. Nicod has reduced these to one, but as it is a complicated proposition, we will begin with the five.) These five propositions are as follows:— (1) " or " implies —i.e. if either is true or is true, then is true. (2) implies " or "—i.e. the disjunction " or " is true when one of its alternatives is true. (3) " or " implies " or ." This would not be required if we had a theoretically more perfect notation, since in the conception of disjunction there is no order involved, so that " or " and " or " should be identical. But since our symbols, in any convenient form, inevitably introduce an order, we need suitable assumptions for showing that the order is irrelevant. (4) If either is true or " or " is true, then either is true or " or " is true. (The twist in this proposition serves to increase its deductive power.) [Pg 149] (5) If implies , then " or " implies " or ."