Notes on Logic: Difference between revisions

The assertion-sign is logically quite without significance. It only shows, in Frege and in Whitehead and Russell, that these authors hold t e propositions so indicated to be true. "<math>\vdash</math>", therefore, belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true. [''Cf.'' 4.442.] Assertion is merely psychological. There are only unasserted pro­ positions. Judgment, command and question all stand on the same level; but all have in common the propositional form, and that alone interests us. What interests logic are only the unasserted propositions. When we say A judges that, etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form but not in the proper order. This shows that a proposition itself must occur in the statement to the effect that it is judged. For instance, however "not-p" may be explained, the question <nowiki>''</nowiki>What is negated?"  must have a meaning.  In "A judges (that) p", p cannot be replaced by a proper name. This is apparent if we substitute "A judges that p is true and not-p is false". The proposition "A judges (that) p" consists of the proper name A, the proposition p with its two poles, and A's being related to both these poles in a certain way. This is obviously not a relation in the ordinary sense. Every right theory of judgment must make it impossible for me to judge that "this table penholders the book" (Russell's theory does not satisfy this requirement). [''Cf.'' 5.5422.] The structure of the proposition must be recognized and then the rest is easy. But ordinary language conceals the structure of the proposition: in it relations look like predicates, and predicates like names, etc.

Names are points, propositions arrows-they have ''sense''. [''Cf.'' 3.144.] The sense of a proposition is determined by the two poles ''true'' and ''false.'' The form of a proposition is like a straight line, which vides all points of a plane into right and left. The line does this automatically, the form of the proposition only by convention. It is wrong to conceive every proposition as expressing a relation. A natural attempt at such a solution consists in regarding "not-p" as the opposite of "p", where, then, "opposite" would be the indefinable relation. But it is easy to see that every such attempt to replace functions with sense (ab-functions) by descriptions, must fail.

The form of a proposition may be symbolized in the following way: Let us consider symbols of the form "xRy", to which correspond primarily pairs of objects of which one has the name "x", the other the name "y". The x's and y's stand in various relations to each other, and among other relations the relation R holds between some but not between others. I now determine the sense of "xRy" by laying down the rule: when the facts behave in regard to "xRy" so that the meaning of "x" stands in the relation R to the meaning of "y", then I say that these facts are "of like sense" (''gleichsinnig'') with the proposition "xRy"; otherwise, "of opposite sense" (''entgegengesetzt''). I correlate the facts to the symbol "xRy" by thus dividing them into  those of  like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form "xRy" when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as by a name, I extract its meaning from among all possible things.

If  p = not-not-p,  etc.,  this shows  that  the traditional  method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that in analysing such propositions, we must not be guided by Russell's method of symbolizing.

In place of every proposition "p" let us write "<math>^{a}_{b}p</math>". Let every correlation of propositions to  each other or of names to  propositions be effected by a correlation of their poles "a" and "b". Let this correlation be transitive. Then accordingly "<math>^{a-a}_{b-b}p</math>" is  the same symbol  as "<math>^{a}_{b}p</math>". Let  n propositions  be given.  I  then  call a "class  of poles" of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it.

There are no propositions containing real variables. Those symbols which are called propositions in which "variables occur are in reality not propositions at all, but only schemes of  propositions, which  do not become propositions unless we replace the variables by constants. There is no proposition  which is expressed  by "x = x",  for "x"  has no signification. But there is a proposition "(x).x = x", and propositions such as "Socrates = Socrates",  etc.  In  books on  logic no variables ought to occur, but  only general  propositions_ which justify the use of variables. It follows  that the so-called  definitions in  logic  are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions. And similarly, the so-called primitive ideas (''Urzeichen'') of logic are not primitive ideas but schemes of them. The mistaken idea that there are ''things'' called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which, e.g., expresses that q  follows from  p and p ⊃ q is this: p. p ⊃ q. ⊃_p,q.q.

A proposition cannot occur in itself. This is the fundamental  truth  of the theory of types. [''Cf.'' 3.332.] In a proposition convert all indefinables into variables, there then remains a class of propositions which does not include all propositions, but does include  an entire type. If we change a constituent a of a proposition φ(a) into a variable, then there is a class <math>\hat{p}[( \exists x ) . \phi x = p]</math>. This class, in general, still depends upon what, by an ''arbitrary convention'', we mean by "φx". But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is not now dependent upon any convention, but only upon the nature of the symbol "φx". It corresponds to a logical type. [''Cf.'' 3.315.]

There are two ways in which signs are similar. The names "Socrates" and "Plato" are similar: they are both names. But whatever they have in common must not be introduced  before  "Socrates"  and "Plato" are introduced. The same applies to a subject-predicate form, etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e. types are not indefinables.