Notes on Logic: Difference between revisions

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.

What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false. What we know when we understand a proposition is this: we know what is the case if it is true and what is the case if it is false. But we do not necessarily know whether it is actually true or false. [''Cf.'' 4.024.]

Every proposition is essentially true-false. Thus a proposition has two poles (corresponding to case of its truth and case of its falsity). We call this the ''sense'' of a proposition. The ''meaning'' of a proposition is the fact which actually corresponds to it. The chief characteristic of my theory is: ''p has the same meaning as not-p'' (constituent = particular, component = particular or relation, etc.). [''Cf.'' 4.0621.]

Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols. It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition. To under­stand a proposition p it is not enough to know that p implies "p is true", but we must also know that ~p implies "p is false". This shows the bi-polarity of the proposition. We understand a proposition when we understand its constituents and forms. [''Cf.'' 4.024.] If we know the meaning of "a" and "b" and if we know what "<span class="nowrap">x R y</span>" means for all x's and y's, then we also understand "<span class="nowrap">a R b</span>". I understand the proposition "<span class="nowrap">a R b</span>" when I know that either the fact that <span class="nowrap">a R b</span> or the fact that not <span class="nowrap">a R b</span> corresponds to it; but this is not to be confused with the false opinion that I understand "<span class="nowrap">a R b</span>" when I know that "<span class="nowrap">a R b</span> or not <span class="nowrap">a R b</span>" is the case.

If we overlook the fact that propositions have a ''sense'' which is independent of their truth or falsehood, it easily seems as if true and false were two equally justified relations between the sign and what is signified. (We might then say, e.g., that "q" ''signifies'' in the true way what "not-q" ''signifies'' in the false way.) But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? No, for a proposition is true when it is as we assert in the proposition; and accordingly if by "q" we mean "not-q", and it is as we mean to assert, then in the new interpretation "q" is actually true and ''not'' false. [4.061-2.] But it is important that we ''can'' mean the same by "q" as by "not-q", for it shows that neither to the symbol "not" nor to the manner of its combination with "q" does a characteristic of the denotation of "q" correspond. [''Cf.'' 4.0621.]

An analogy for the theory of truth: Consider a black patch on white paper. Then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact; to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege's "truth-values"), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that "p" is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the ''sense'' of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper which is white and black, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value) whose properties might be called "false" or "true". The verb of a proposition is not "is true" or "is false", as Frege believes, but what is true must already contain the verb. [''Cf.'' 4.063.]

One is tempted to interpret "not-p" as "everything else, only not p". That from a single fact p an infinity of others, not-not-p, etc., follow is hardly credible. [''See'' 5.43.] Man possesses an innate capacity for constructing symbols with which ''some'' sense can be expressed without having the slightest idea what each word signifies. [''Cf.'' 4.002.] The best example of this is mathematics, for man has until recently used the symbols for numbers without knowing what they signify or that they signify nothing.

The assertion-sign is logically quite without significance. It only shows, in Frege and in Whitehead and Russell, that these authors hold the 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 pro­positions. 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 "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 divides 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.

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 ab­ function 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 ab-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.]

If there were a world created in which the principles of logic were true, in that world the whole of mathematics holds. No world can be created in which a proposition is true, unless the constituents of the proposition are created also. [''Cf.'' 5.123.]

Indefinables are of two sorts: names and forms. Propositions cannot consist of names alone, they cannot be classes of names. [''Cf '' 3.142.] A name cannot only occur in two different propositions, but can occur in the same way in both. Propositions, which are symbols having reference to facts, are themselves facts (that this inkpot is on this table may express that I sit in this chair). We must be able to understand propositions we have never heard before. But every proposition is a new symbol. Hence we must have ''general'' indefinable symbols; these are unavoidable if propositions are not all indefinable. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads us to an impenetrable thicket.

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 understanding of all other kinds of propositions. In fact, the understanding of general propositions obviously depends on that of atomic propositions. [''Cf.'' 4.411.]

One reason for thinking the old notation wrong is that it is very unlikely that from every proposition p, an infinite number of other propositions not-not-p, not-not-not-not-p, etc., should follow. [''Cf.'' 5.43.] The very possibility of Frege's explanations of "not-p" and "if p then q", from which it follows that "not-not-p" denotes the same as p, makes it probable that there is some method of designation in which "not-not-p" corresponds to the same symbol as "p". But if this method of designation suffices for logic, it must be the right one.

Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the "manner of deduction" is indifferent. [''Cf.'' 5.132.]

The reason why "~Socrates" means nothing is that "~x" does not express a property of x. Signs of the forms "p ∨ ~p" are senseless, but not the proposition "(p) p ∨ ~p". If I know that this rose is either red or not red, I know nothing. [''Cf.'' 4.461.] The same holds of all ab-functions. The assumption of the existence of logical objects makes it appear remarkable that in the sciences propositions of the form "p ∨ q", "p ⊃ q", etc., are only then not provisional when "∨" and "⊃" stand within the scope of a generality-sign (apparent variable). That "or" and "not", etc., are not relations in the same sense as "right" and "left", etc., is obvious to the plain man. The possibility of cross-definition in the old logical indefinables shows, of itself, that these are not the right indefinables, and even more conclusively, that they do not denote relations. [''Cf.'' 5.42.] Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are "not" and "or", like judgment, ''analogous'' to predicates and relations, because they do not introduce anything new.

The sense of an ab-function of p is a function of the sense of p. [''Cf.'' 5.2341.] The ab-functions use the discrimination of facts which their arguments bring forth in order to generate new discriminations. The ab-notation shows the dependence of ''or'' and ''not,'' and thereby that they are not to be employed as simultaneous indefinables.

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.

It is easy to suppose only such symbols are complex as contain names of objects, and that accordingly "(x,ϕ)ϕx" or "<span class="nowrap">(∃ x, y)x R y</span>" 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 "<span class="nowrap">~(∃ x, y).x R y</span>"? 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 <span class="nowrap">x R y</span> has been introduced, it must henceforth be understood in propositions of the form <span class="nowrap">a R b</span> just in the same way as in propositions such as (∃x,y)<span class="nowrap">x R y</span> 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.

It is to be remembered that names are not things but classes: "A" is the same letter as "A". [''Cf.'' 3.203.] This has the most important consequences for every symbolic language.

One must not say "The complex sign '<span class="nowrap">a R b</span>'" says that a stands in the relation R to b; but that "a" stands in a certain relation to "b" says ''that'' <span class="nowrap">a R b</span>. [''Cf.'' 3.1432.]

Only facts can express sense, a class of names cannot. [''Cf.'' 3.142.] This is easily shown. In <span class="nowrap">a R b</span> it is not the complex that symbolizes but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolized by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.

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.]

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.]

We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other. Types can never be distinguished from each other by saying (as is currently done) that one has these ''but'' the other has those properties, for this presupposes that there is a ''meaning'' in asserting all these properties of both types. [''Cf.'' 4.1241.] And, from this it follows that, at least, these properties may be types, but certainly not the objects of which they are asserted.