Volume 1 / Part 1 — *4. EQUIVALENCE AND FORMAL RULES.
Reading edition
*4. EQUIVALENCE AND FORMAL RULES.
Summary of *4.
In this number, we shall be concerned with rules analogous, more or less, to those of ordinary algebra. It is from these rules that the usual "calculus of formal logic" starts. Treated as a "calculus," the rules of deduction are capable of many other interpretations. But all other interpretations depend upon the one here considered, since in all of them we deduce consequences from our rules, and thus presuppose the theory of deduction. One very simple interpretation of the "calculus" is as follows: The entities considered are to be numbers which are all either or ; "" is to have the value if is and is ; otherwise it is to have the value ; is to be if is , and if is ; is to be if and are both , and is to be in any other case; is to be if and are both , and is to be in any other case; and the assertion-sign is to mean that what follows has the value . Symbolic logic considered as a calculus has undoubtedly much interest on its own account; but in our opinion this aspect has hitherto been too much emphasized, at the expense of the aspect in which symbolic logic is merely the most elementary part of mathematics, and the logical prerequisite of all the rest. For this reason, we shall only deal briefly with what is required for the algebra of symbolic logic.
When each of two propositions implies the other, we say that the two are equivalent, which we write "." We put
*4·01.
It is obvious that two propositions are equivalent when, and only when, both are true or both are false. Following Frege, we shall call the truth-value of a proposition truth if it is true, and falsehood if it is false. Thus two propositions are equivalent when they have the same truth-value.
It should be observed that, if , may be substituted for without altering the truth-value of any function of which involves no primitive ideas except those enumerated in *1. This can be proved in each separate case, but not generally, because we have no means of specifying (with our apparatus of primitive ideas) that a function is one which can be built up out of these ideas alone. We shall give the name of a truth-function to a function whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas of *1. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, " believes " may be true for one true value of and false for another.
The principal propositions of this number are the following:
*4·1.
*4·11.
These are both forms of the "principle of transposition."
*4·13.
This is the principle of double negation, i.e. a proposition is equivalent to the falsehood of its negation.
*4·2.
*4·21.
*4·22.
These propositions assert that equivalence is reflexive, symmetrical and transitive.
*4·24.
*4·25.
I.e. is equivalent to " and " and to " or ," which are two forms of the law of tautology, and are the source of the principal differences between the algebra of symbolic logic and ordinary algebra.
*4·3.
This is the commutative law for the product of propositions.
*4·31.
This is the commutative law for the sum of propositions.
The associative laws for multiplication and addition of propositions, namely
*4·32.
*4·33.
The distributive law in the two forms
*4·4.
*4·41.
The second of these forms has no analogue in ordinary algebra.
*4·71.
I.e. implies when, and only when, is equivalent to . This proposition is used constantly; it enables us to replace any implication by an equivalence.
*4·73.
I.e. a true factor may be dropped from or added to a proposition without altering the truth-value of the proposition.
*4·01.
*4·02.
This definition serves merely to provide a convenient abbreviation.
*4·1.
*4·11.
*4·12.
*4·13.
*4·14.
*4·15.
*4·2.
*4·21.
*4·22.
Dem.
Note. The above three propositions show that the relation of equivalence is reflexive (*4·2), symmetrical (*4·21), and transitive (*4·22). Implication is reflexive and transitive, but not symmetrical. The properties of being symmetrical, transitive, and (at least within a certain field) reflexive are essential to any relation which is to have the formal characters of equality.
*4·24.
Dem.
*4·25.
Note. *4·24·25 are two forms of the law of tautology, which is what chiefly distinguishes the algebra of symbolic logic from ordinary algebra.
*4·3.
Note. Whenever we have, whatever values and may have, we have also For
*4·31.
*4·32.
Dem.
Note. Here "(1)" stands for "," which is obtained from the above steps by *4·22. The use of *4·22 will often be tacit, as above. The principle is the same as that explained in respect of implication in *2·31.
*4·33.
The above are the associative laws for multiplication and addition. To avoid brackets, we introduce the following definition:
*4·34.
*4·36.
*4·37.
*4·38.
*4·39.
*4·4.
This is the first form of the distributive law.
Dem.
*4·41.
This is the second form of the distributive law—a form to which there is nothing analogous in ordinary algebra. By the conventions as to dots, "" means "."
Dem.
*4·42.
Dem.
*4·43.
Dem.
*4·44.
Dem.
*4·45.
The following formulae are due to De Morgan, or rather, are the propositional analogues of formulae given by De Morgan for classes. The first of them, it will be observed, merely embodies our definition of the logical product.
*4·5.
*4·51.
*4·52.
*4·53.
*4·54.
*4·55.
*4·56.
*4·57.
The following formulae are obtained immediately from the above. They are important as showing how to transform implications into sums or into denials of products, and vice versa. It will be observed that the first of them merely embodies the definition *1·01.
*4·6.
*4·61.
*4·62.
*4·63.
*4·64.
*4·65.
*4·66.
*4·67.
*4·7.
Dem.
*4·71.
Dem.
The above proposition is constantly used. It enables us to transform every implication into an equivalence, which is an advantage if we wish to assimilate symbolic logic as far as possible to ordinary algebra. But when symbolic logic is regarded as an instrument of proof, we need implications, and it is usually inconvenient to substitute equivalences. Similar remarks apply to the following proposition.
*4·72.
Dem.
*4·73.
This proposition is very useful, since it shows that a true factor may be omitted from a product without altering its truth or falsehood, just as a true hypothesis may be omitted from an implication.
*4·74.
*4·76.
*4·77.
*4·78.
Dem.
*4·79.
Dem.
Note. The analogues, for classes, of *4·78·79 are false. Take, e.g. *4·78, and put = English people, = men, = women. Then is contained in or , but is not contained in and is not contained in .
*4·8.
*4·81.
*4·82.
*4·83.
Note. *4·82·83 may also be obtained from *4·43, of which they are virtually other forms.
*4·84.
*4·85.
*4·86.
*4·87.
*4·87 embodies in one proposition the principles of exportation and importation and the commutative principle.