Principia mathematica, vol. 1 (of 3)

The proofs of the earlier of the propositions of this number consist simply in noticing that they are instances of the general rules given in *1. In such cases, these rules are not premisses, since they assert any instance of themselves, not something other than their instances. Hence when a general rule is adduced in early proofs, it will be adduced in brackets[45], with indications, when required, as to the changes of letters from those given in the rule to those in the case considered. Thus "Taut " will mean what "Taut" becomes when is written in place of . If "Taut " is enclosed in square brackets before an asserted proposition, that means that, in accordance with "Taut," we are asserting what "Taut" becomes when is written in place of . The recognition that a certain proposition is an instance of some general proposition previously proved or assumed is essential to the process of deduction from general rules, but cannot itself be erected into a general rule, since the application required is particular, and no general rule can explicitly include a particular application.

Again, when two different sets of symbols express the same proposition in virtue of a definition, say *1·01, and one of these, which we will call (1), has been asserted, the assertion of the other is made by writing "[(1).(*1·01)]" before it, meaning that, in virtue of *1·01, the new set of symbols asserts the same proposition as was asserted in (1). A reference to a definition is distinguished from a reference to a previous proposition by being enclosed in round brackets.

The propositions in this number are all, or nearly all, actually needed in deducing mathematics from our primitive propositions. Although certain abbreviating processes will be gradually introduced, proofs will be given very fully, because the importance of the present subject lies, not in the propositions themselves, but (1) in the fact that they follow from the primitive propositions, (2) in the fact that the subject is the easiest, simplest, and most elementary example of the symbolic method of dealing with the principles of mathematics generally. Later portions—the theories of classes, relations, cardinal numbers, series, ordinal numbers, geometry, etc.—all employ the same method, but with an increasing complexity in the entities and functions considered.

I.e. implies that implies , i.e. a true proposition is implied by any proposition. This proposition is called the "principle of simplification" (referred to as "Simp"), because, as will appear later, it enables us to pass from the joint assertion of and to the assertion of simply. When the special meaning which we have given to implication is remembered, it will be seen that this proposition is obvious.

These four analogous propositions constitute the "principle of transposition," referred to as "Transp." They lead to the rule that in an implication the two sides may be interchanged by turning negative into positive and positive into negative. They are thus analogous to the algebraical rule that the two sides of an equation may be interchanged by changing the signs.

These two propositions are the source of the syllogism in Barbara (as will be shown later) and are therefore called the "principle of the syllogism" (referred to as "Syll"). The first states that, if follows from , then if follows from , follows from . The second states the same thing with the premisses interchanged.

In the last line of this proof, "(1).(2).*1·11" means that we are inferring in accordance with *1·11, having before us a proposition, namely , which, by (1), is implied by , which, by (2), is true. In general, in such cases, we shall omit the reference to *1·11.