Notes on Logic: Difference between revisions

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In philosophy there are no deductions; it is purely descriptive. The word "philosophy" ought always to designate something over or under, but not beside, the natural sciences. [''See'' 4.111.] Philosophy gives no pictures of reality, and can neither confirm nor confute scientific investigations. It consists of logic and metaphysics, the former its basis. Epistemology is the philosophy of psychology. [''See'' 4.1121.] Distrust of grammar is the first requisite for philosophizing. Philosophy is the doctrine of the logical form of scientific propositions (not primitive propositions only). A correct explanation of the logical propositions must give them a unique position as against all other propositions. [6.112.]

Frege said "propositions are names"; Russell said "propositions correspond to complexes". Both are false; and especially false is the statement "propositions are names of complexes". Facts cannot be named. [''Cf.'' 3.144.] The false assumption that propositions are names leads us to believe there must be "logical objects": for the meaning of logical propositions would have to be such things.

A proposition is a standard with reference to which facts behave, but with names it is otherwise. Just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition; it is thus bi-polarity and sense come in. In this theory p has the same meaning as not-p but opposite sense. The meaning is the fact. A proper theory of judgment must make it impossible to judge nonsense. [''Cf.'' 5.5422.] The "sense of" an ab­function of a proposition is a function of its sense. [''Cf.'' 5.2341.] In not-p, p is exactly the same as if it stands alone (this point is absolutely fundamental). Among the facts which make "p or q" true there are also facts which make "p and q" true; hence, if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical. But in fact their sense is different, and we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where their b­ function stands under a generality sign or enters into another function such as "I believe that", etc., because then the sense enters.

It may be doubted whether, if we formed all possible atomic propositions, "the world would be completely described if we declared the truth or falsehood of each" (Russell). [''Cf.'' 4.26.]

Every statement about complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. [''Cf.'' 2.0201.] How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic. To repeat: every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e. that proposition which is equivalent to saying the complex exists.

Whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions. Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms. All that is essential about molecular functions is their T-F (true-false) schema (i.e. the statement of the cases where they are true and cases where they are false). It is ''a priori'' likely that the introduction of atomic propositions is fundamental for the under­ standing of all other kinds of propositions. In fact, the understanding of general propositions obviously depends on that of atomic propositions. [''Cf.'' 4.411.]

Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which, in general, the ''symbolizing'' facts will be the same in several members. If now we find an ab-function of such a kind that by repeated applications of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by the application of this function. Such a function is ~p ∨ ~q. It is easy to suppose a contradiction in the fact that, on the one hand, every possible complex proposition is a simple ab-function of simple propositions, and that, on the other hand, the repeated application of one ab-function suffices to generate all these propositions. If, e.g., an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does "p" deny "not-p" or assert "p", or both? [''See'' 5.44.] And how do matters stand with the definition of "⊃" by "∨" and "~", or of "∨" by "~" and "⊃"? And how, e.g., shall we introduce p|q (i.e. ~p ∨ ~q), if not by saying that this expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p|q is merely a mechanical instrument for constructing all possible ''symbols'' of ab-functions. The symbols arising by repeated application of the symbol "|" do ''not'' contain the symbol "p|q". We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them, e.g., as those symbols of functions which can be generated by repeated application of the operation "|". And we say now: For all p's and q's, "p|q" says something indefinable about the sense of those simple propositions which are contained in p and q.

Just as people used to struggle to bring all propositions into the subject-predicate form, so now it is natural to conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully satisfied by Russell's theory of manufactured relations.

A very natural objection to the way in which I have introduced, e.g., propositions of the form xRy is that by it propositions such as (∃x,y)xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. ''But'' when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y)φ(x,y) for all φ's in any way which makes the sense of these propositions dependent on the sense of all propositions of the form φ(a,b), and thereby the justification of our procedure is established.

It is easy to suppose only such symbols are complex as contain names of objects, and that accordingly "(x,φ)φx" or "(∃x,y)xRy" must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what ''is'' the meaning, e.g., of "~(∃x,y).xRy"? Can we put "not" before a name? Alternate indefinability shows the indefinables have not yet been reached. [''Cf.'' 5.42.] The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot, therefore, introduce it first for one combination, then for another; e.g. if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as (∃x,y)xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases and there could be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, ''mutatis mutandis'', that Frege has said for the introduction of symbols by definitions. [''Cf.'' 5.451.]

It can never express the common characteristic of two objects that we designate them by the same name but otherwise by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where, then, would be the common element in the designations? [''Cf.'' 3.322.] Nevertheless, one is always tempted, in a difficulty, to take refuge in different ways of designation.

No proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types. [''Cf.'' 3.332.]