Chapter 10 Chapter 10 Wittgenstein's Influence

Principia Mathematics was not very popular at first.Mathematical philosophy on the mainland is divided into two schools, the formalists and the intuitionists.Both schools completely deny that mathematics follows logic, and use contradictions to justify their denial. Formalists headed by Hilbert argued that the symbols in arithmetic are just marks on paper, meaningless, and arithmetic is formed by some arbitrary rules similar to the rules of playing chess. According to these rules, we can put Those notations are used operationally.This doctrine has the advantage of avoiding all philosophical controversy, but it also has the disadvantage of not being able to explain the use of numbers in calculations.If the o symbol is taken to mean a hundred or a thousand or any other finite number, all the rules of use proposed by the formalists are confirmed.This doctrine fails to explain what is meant by such simple propositions as "There were three men in this room" or "There were twelve apostles."This doctrine is quite sufficient for calculations, but not enough for the application of numbers.Since it is the application of numbers that counts, this doctrine of the formalists cannot but be seen as an unsatisfactory evasion.

The teachings of the intuitionists headed by Brauwe need to be discussed more seriously.The core of this doctrine is the denial of the law of the excluded middle.The doctrine holds that a proposition is only true or false if there is a way to determine whether that proposition is true or false.One of the common examples is the proposition: "There are three consecutive sevens in the decimal calculation of π".So far as the value of pi has been found, there are no three consecutive sevens, but there is no reason to suppose that this will not occur in a later place.If in the future it seems that there is indeed a place where three consecutive sevens appear, the problem will be solved, but if such a place is not reached, it does not prove that there will not be such a place later.So, while we may be perfectly able to prove that there are three consecutive sevens, we can never prove that there are no.This question is important for analysis.Infinite decimals sometimes follow a law that allows us to count as many terms as we like.

But sometimes (we must assume so) they do not follow any laws.According to generally accepted principles, the second case is infinitely more common than the first case.Moreover, if such decimals as "illegitimate" are not admitted, the whole theory of real mathematics collapses, and with it calculus and almost the whole of higher mathematics.Burrowe did not flinch in the face of this catastrophe, but most mathematicians considered it unbearable. The problem is much more general than the mathematical example above suggests.The question is: "Is there any point in saying a proposition is true or false if there is no way to decide whether it is true or false?" Or put another way: "Should 'true' and 'verifiable' be the same thing? ?” I don’t think we can say it’s the same thing, or we’d have to make some crude and unreasonable paradoxes.Take the following proposition as an example: "It snowed on the island of Manhetan on January 1, 1 AD."We cannot think of any way of seeing whether the proposition is true or false, but it seems absurd to maintain that it is neither true nor false.I shall not dwell on this subject now, as I have discussed it at length in Chapters 20 and 21 of "Inquiry into Meaning and Truth," which I discuss in this book. It will be covered in a later chapter of the book.At the same time, I think the doctrine of the intuitionists cannot but be rejected.

Both the intuitiveists and the formalists attacked the teachings of Principia Mathematica from the outside, and it does not seem very difficult to repel their attacks.The criticism of Wittgenstein and his school is another matter.These criticisms come from within and deserve respect. Wittgenstein had a profound influence on me.I gradually felt that my agreement with him on many points was too much.But I cannot fail to explain what the point of contention is. Wittgenstein's influence on me was divided into two parts: the first was before the First World War; the second was after the war he sent me the original manuscript of his Tractatus.His later teachings, presented in his Philosophical Investigations, did not affect me in the slightest.

At the beginning of 1914 Wittgenstein gave me a short typewritten essay containing notes on various logical problems.This article, and many conversations, influenced my thinking during the war years.He was in the Austrian army during the war, so I cut off all contact with him.What I know of his teachings at this time is entirely from unpublished sources.I do not know for certain that what I myself believed to be his opinion, then or later, was in fact his opinion.He always denied the interpretation of his teachings by others, even if these were his ardent disciples.The only exception I know of is E. p.Lemousse, this is the one I shall discuss shortly.

At the beginning of 1918 I gave a series of lectures in London.These lectures were later published in the journal The Monist (1918 and 1919).I prefaced these lectures with the following words of thanks to Wittgenstein: "The following essays, the first two of a series of eight lectures given in London during the first months of 1918, are mainly concerned with explaining the Some of the ideas learned by my friend and former student Ludwig Wittgenstein. Since August 1914 I have not had the opportunity to know his ideas. I do not even know whether he is alive or He is therefore not responsible for what was said in these lectures, except that he had originally supplied many of the theories in these lectures.

The other six lectures will appear in the next three issues of The Monist". It was in these lectures that I first adopted the name "logical atomism" to describe my philosophy.But it is not worth dwelling on this aspect, since Wittgenstein's 1914 theory is still at an immature stage. The important one was the Tractatus, of which he sent me a typescript shortly after the armistice, when he was a prisoner at Monte Cassino.I shall discuss the doctrines of the Tractatus, first as to their influence on me at that time, and secondly as to my thoughts on them afterwards. Perhaps the basic philosophical teaching of the Tractatus is that a proposition is a graph of the facts about which the proposition says.A map clearly conveys some knowledge, true or false; if the knowledge is true, it is because of structural similarities between the map and the places it relates to.

According to Wittgenstein, the same is true for asserting a fact with words.For example he said that if you use "aRb" This symbol represents the fact that a has a relation R to b. Your symbol works because it establishes a relation between "a" and "b" that represents the relationship between a and b. Relationship.This doctrine emphasizes the importance of structure.He says, for example: "The gramophone, the musical thought, the musical instrument, the sound wave, have that pictorial inner relation to each other. There is also this relation between language and the world. Logical structures have something in common with all of these."

"(Like the two youths in the story, their two horses and their lily. In a sense, they are all the same.)" (Tractatus, 4.014. ). Emphasizing the importance of structure, I still think he's right.However, as for the doctrine that a correct proposition must reproduce the structure of the facts concerned, I now find it very doubtful, although I recognized this doctrine at the time.In any case, even if the doctrine is true in some sense, I do not think it is of great importance.But Wittgenstein thought it was fundamental.He took it as the basis of a strange kind of logical mysticism.He maintains that the form common to a true proposition and its corresponding facts can only be shown, not spoken, since it is not another word in the language but words or something equivalent to them "Propositions can represent reality as a whole, but they cannot represent what they must have in common with reality in order to be able to represent it—the logical form.

"In order to represent logical forms, we should be able to place ourselves and propositions outside logic, that is, outside the world" (Tractatus 4.12.). This is the only point that is brought up. As close as I come to agreeing with Wittgenstein's claim, I am still not convinced. In my Introduction to the Tractatus I suggested that although in any language there are things that cannot be expressed by language, it is always possible to construct a higher language that can say those things.There are still things that cannot be said in this new language, but can be said in the next language, and so on ad infinitum.This suggestion, novel at the time, has now become an accepted logical banality.

This removes Wittgenstein's mysticism and, I think, also solves the new enigma posed by Gödel. Next, I will talk about what Wittgenstein said about identity.The importance of his statement may not be immediately apparent.To explain this theory, I have to first talk about the definition of identity in "Principles of Mathematics".Among the properties of a thing, Whitehead and I identify some that we call "narrative."These are some properties that have nothing to do with the general properties.For example, you could say, "Napoleon is a Corsican," or "Napoleon is fat."When you say this, you don't mean the properties of aggregates.But if you say "Napoleon had all the virtues of a general," or "Queen Elizabeth first had all the virtues of her father and grandfather without their faults," you mean the general nature.We distinguish between the nature of the totality and the role of narrative in order to avoid some contradictions. We say the definition "x and y are equivalent" to mean "y has all the stated properties of x", and it follows, in our system, that y has any of the properties that x has, Either narrative or not.Wittgenstein's objection to this point is as follows: "Russell's definition of "=" is not acceptable; because according to this definition, we cannot say that all the properties of two things are shared by them. (Even if This proposition is not correct at all, but it has meaning.) "Generally speaking: it is meaningless to say that two things are equal, and to say that one thing is equal to itself is to say nothing" ("Tractatus Logico-Philosophy", 5.5302 and 5.5303). At one point, I accepted this criticism, but I soon came to the conclusion that his criticism made mathematical logic impossible, and that Wittgenstein's criticism was in fact invalid. This is all the more obvious if we consider counting: if a and b had all the properties in common with them, you could never mention a without mentioning b, or count a without also counting b, no Count b as a separate item, but in the same act of counting. So there is no possibility that you will ever find that a and b are two. Wittgenstein's claim is to assume that difference is an indeterminate relationship, though I don't think he knew he was assuming it. But if he didn't, I see no reason for saying that two things have all their properties in common. meaningful. But if it is admitted that the difference exists, then, if a and b are two, a has a property that b does not have, that is, it is different from b. Therefore, I think Wiegand Stein's claim of identity is wrong. If so, it renders much of his system useless. Please take the definition of the number 2 as an example.We say that a class has terms, if the class has terms x and y, and x and y are not identical, and if z is a term of the class, then z is equal to either x or y.It is difficult to reconcile this definition with Wittgenstein's claim that we should never express the formula "x = y" or "xDy" in words, but by different letters stand for different things, and should never use two different letters to stand for the same thing.Apart from this technical difficulty, it is clear that, for the reasons given above, two things cannot be counted as two if all their properties are common to them, since counting as two One cannot fail to distinguish them, and thus give them different properties. Another consequence is that we cannot make an inner package that is both common and specific to a certain set of enumerated objects. Suppose, for example, that we have three objects, a, b, and c, then the property which is identical to a, identical to b, and identical to c is a property common and peculiar to these three objects.However, in Wittgenstein's system this approach is not applicable. Another point, which is very important, is that Wittgenstein does not endorse any statement about anything in the world.In Principia Mathematica, the totality of things is defined as all those classes of x's which are x's such that x=x, and to this class we may assign a number, (just as to any other class a number ), although of course we don't know what the correct number to specify is.Wittgenstein denies this.He says that a proposition like "There are more than three things in the world" is meaningless.When I was discussing the Tractatus with him in The Hague in 1919, I had a blank sheet of paper before me. I made three dots with ink on it.I asked him to admit: since there are these three points, there must be at least three things in the world; but he firmly refused.He does admit that there are three points on that paper, because that is a limited assertion, but he does not admit that any statement can be made about the world in general.This has something to do with his mysticism, but since he refuses to recognize equivalence, it is not surprising. There is another aspect related to the same kind of problem, which I call the "Axiom of Infinity".In a world containing only a finite number of things, that number is the largest possible number of things.In such a world, all advanced mathematics would collapse.How much stuff there is in the world seems to me to be a purely empirical question. I do not think that a mere logician should say anything on this question.Therefore, all those parts of mathematics that require an infinite number of things I take as hypothetical.All this seems to Wittgenstein extremely absurd.In his view, you could ask "How many people are there in London?" or "How many molecules are there in the sun?" But it doesn't make sense to infer that there are at least that many things in the world.As far as I can tell, this part of his theory must be wrong. Wittgenstein published two principles.It is very important that these two principles are true.That is, the principle of extensionality and the principle of atomicity. The principle of extensionality says that the truth or falsehood of any statement about a p proposition depends entirely on the truth or falsehood of p; the truth or falsehood of any statement involving a propositional function depends entirely on the extension of this function, then That is, depending on the range of values ​​that make this propositional function true.On the face of it, it is clear that this argument can be contested.Please take "A believes in p" as an example.Obviously, a person can believe some true propositions and disbelieve others, so the truth of "A believes p" does not depend entirely on the truth or falsehood of p.On this topic, Wittgenstein has a very mysterious passage.He said, "In the general propositional form some propositions occur only in a proposition on the basis of a truth-false operation." "At first sight there seems to be a different way in which one proposition appears in another". "Especially in some propositional forms in psychology such as 'A thinks that p is so', or 'A thinks p is true', etc." "Here, on the surface, it appears that the proposition p has a relation to the object A." "[Those propositions in modern epistemology (Russell, Moore, etc.) are held to be so]." "But it is clear that 'A believes p', 'A considers p to be true', 'A says p' are forms of 'p says p'; here we have no equivalence between facts and objects, but there is an equivalence relationship between them by virtue of the equivalence of their objects." "This shows that there is no such thing as the idea in contemporary superficial psychology, such as soul-subject, etc." (Tractatus 5.54ff.). Wittgenstein's thesis is that "A believes p" is not a function of p, but a function of the words A uses to denote the proposition p or the physical condition (whatever it is) that constitutes its belief.He was, as usual, dogmatic, and he poured out his opinion like a tsarist decree.But Xiaomin Kusano is not satisfied with this method.I have examined this question in detail in "Inquiry into Meaning and Truth" (pp. 267ff.), but am somewhat uncertain about the conclusions I have reached. Wittgenstein states the principle of atomicity in the following terms: "Every statement about compositions can be analyzed into a statement about their constituent parts, and into propositions which fully describe those compositions" (Philosophy of Logic On", 2.0201).This principle can be said to be the concrete manifestation of faith analysis.When Wittgenstein wrote the Tractatus, he believed (and, as far as I understand, he later disbelieved) that the world was made up of many simple things with various properties and relations.The simple properties and simple relations of simple things are "atomic facts", and assertions about them are "atomic propositions".The point of this principle is that if you know all atomic facts, and know that they are all atomic facts, and nothing else, you can deduce all other true propositions using logic alone, the important difficulty that this principle raises It is also related to some propositions such as "A believes in p", because here p is complex, it can be regarded as a compound.The peculiarity of such propositions is that they involve two verbs, one principal and one subordinate. Let's take a very simple example, say: "A believes that B is hot." Here "believes" is the main verb and "is" is the subordinate verb.The atomic principle requires us to try to express this fact without bringing up the subsidiary compound "B is hot".I have also discussed this principle in detail in "An Inquiry into Meaning and Truth" (from page 262). The conclusions I have come to with regard to these two principles are: "(1) The principle of extension cannot be falsified if interpreted strictly, and some sentences like 'A believes p' are analyzed; (2) the same analysis It cannot be proved that the atomic principle is false, but it is not enough to prove that it is true" "Inquiry into Meaning and Truth" (p. 273). A more general criticism of Wittgenstein's two principles is that there is no reason to believe in simple things and atomic facts.As far as I know, he thought so after all. But discussing this question takes us too far from the Tractatus.In a later chapter, I will also address this issue. Wittgenstein maintained that logic consists entirely of tautologies.I think he's right about this, although I didn't think so until I read what he had to say on the subject.There is also a very important point related to this, that is, all atomic propositions are independent of each other.It used to be thought that one fact could logically depend on another.This is only true if one of the facts is actually two facts put together. Logically, it follows from "A and B are persons" that A is a person.But that is because "A and B are persons" is actually two propositions put together.The consequence of the principle we have been discussing is that those selected atomic facts that are true in the actual world may be the entirety of the atomic facts that can be proved by logic, but it is clear that the principle of atomicity is necessary for this and, if it is not true, we cannot be sure that the simplest possible facts may sometimes be logically irrelevant. In the second edition of Principia Mathematica (1925) I considered some of Wittgenstein's teachings.I have adopted the principle of extension in a new Introduction, and have considered in the Appendix the apparently objections to this principle which, taken as a whole, I conclude to be invalid.In this new edition, my main aim is to reduce the use of the Axiom of Reducibility.This axiom (which I shall explain in a moment) seems to be necessary if we are to avoid contradictions on the one hand, and on the other hand to preserve all mathematics which is usually regarded as undisputed.But it is a debatable axiom because its truth can be doubted, and more importantly because, if it is true, its truth belongs to experience, not to logic. Whitehead and I recognize that this axiom is a weakness of our system, but I at least think it has something like the axiom of parallelism, which has always been considered a weakness of Euclidean geometry.I think sooner or later a way will be found to do away with this axiom while concentrating the difficulty on one point is a good thing.In the second edition of Principia Mathematica I succeeded in abolishing this axiom in many cases (which at first seemed indispensable), and especially in all uses of mathematical induction. I must now say what this axiom says, and why it seems indispensable.I have shown before the difference between properties that belong to some population of properties and those that do not.Properties that belong to populations of properties often cause trouble.Suppose, for example, that you come up with the definition: "A typical Englishman is one who has the qualities of most Englishmen".You will easily realize that most Englishmen do not have all the qualities that most Englishmen have.So, by your definition, a typical Brit is not typical.The trouble arises because the definition of the word "typical" means all qualities.Then it is taken as a property in itself.So it seems that, if you properly say "all properties", you don't really mean "all properties", but only "all properties that do not belong to the totality of properties".As I stated earlier, we speak of such properties as "assertive."The axiom of reducibility states that a property that is not an assertion is always formally equal to an asserted property. (Two properties are formally equal if they belong to the same group of things, or, more precisely, if their value of truth and falsity is the same for every subject.) In the first edition In his "Principles of Mathematics", we explain the reasons for accepting this axiom as follows: "The principle of reducibility is self-evident, which is a proposition that is difficult to support. But, in fact, self-evidence is just a reason for accepting an axiom part, is by no means essential. The reasons for accepting an axiom, as for any other, are always largely inductive, that is to say, many propositions that are almost beyond doubt can be deduced from the axiom, without The same plausible way to make these propositions true is that if the axiom is false, and nothing that could be false can be deduced from it. If the axiom appears to be self-evident, it actually means that it almost beyond doubt; for there are things which were supposed to be self-evident, but were later found to be false. If the axiom itself is almost beyond doubt, that only adds to the inductive evidence which follows from its results almost It comes from the fact that there is no doubt about it, and it cannot provide completely different new evidence. Absolute correctness can never be achieved, so every axiom and all its results always have some doubtful elements. In formal logic, more than most scientific There are few suspicious elements in it, but it is not without. This can be seen from this incident: the paradox comes from some premises that were previously unknown and needed to be limited. As far as the axiom of reducibility is concerned, the inductive evidence in its favor is strong, since both the inferences it allows and the results that follow are clearly valid.But while it is highly improbable that this axiom turns out to be an artifact, it is by no means impossible that it should be found to be deduced from some other, more fundamental and more obvious axiom.It is quite possible that the use of the principle of circularity (which is embodied in the stratumtypes described above) is overused, and the necessity of this axiom might be avoided if it were used less aggressively.Such variations, however, do not falsify anything asserted from the principles previously explained, but merely furnish more accessible proofs of the same theorems.There seems, therefore, no basis for fearing that the use of the axiom of reducibility will lead us into error" ("Introduction", Chapter II, Section VII). In the second edition we said: "One point which should obviously be improved is the axiom of reducibility. This axiom has only one purely pragmatic reason for its justification: it leads to the desired effect and nothing else. But it is not our Satisfactory axioms. However, regarding this problem, it cannot be said that a satisfactory solution can be obtained. Leon Tristek resolutely abolished this axiom and did not adopt any replacement. Judging from his research , it is clear that his approach entails sacrificing a great deal of ordinary mathematics. Another approach (recommended by Wittgenstein for philosophical reasons) is to assume that the functions of propositions are always true-false functions, And a function can appear in a proposition only by its value. There are difficulties with such a view, but these difficulties may not be insurmountable. This view has the consequence that all functions of functions are extensional.It requires us to assert that "A believes p" is not a function of p.In the Tractatus (cited above, and pp. 19-21) it is shown how this is possible.We are not going to conclude that this doctrine is indeed correct, but it seems worthwhile to set out its results in the following pages.It appears that everything in Book I is still correct (although new proofs are often required); the doctrine of inductive cardinal numbers and ordinal numbers continues; and the real numbers in general can no longer be properly resolved.And Kantor's proof of 2n>n also collapses, unless n is finite.Perhaps there is some other axiom which is less satisfactory than the axiom of reducibility to produce these results, but we have not yet found such an axiom (Introduction, p. XIV). Not long after the second edition of "Principles of Mathematics" was published, F. p.Lemousse took up the problem of the axiom of reduction in two very important articles, The Foundations of Mathematics, published in 1925, and Mathematical Logic, published in 1926 year.Unfortunately, Lemousse's early death prevented his opinion from fully developing.But what he has achieved is important and deserves serious consideration.His main argument was that mathematics had to be made purely extensional, and that the trouble with Principia Mathematics arose from the intrusion into the introductory point of view.Whitehead and I maintain that a class can only be specified by a propositional function, and this applies even to classes that seem to be specified for enumerations.For example, the class of three individuals a, b, and c is determined by the propositional function "x=a or x=b or x=c".Wittgenstein's rejection of identity (which Lemousse acknowledged) made this impossible, but, on the other hand, Lemouse argued that there is no way to use enumerations to define an infinite class. logical objection.We cannot define an infinite class in this way, because we are always mortal, but it is an empirical fact that we are not exempt from death, and this empirical fact should be disregarded by logicians.According to this, he argues, the axiom of multiplication is a tautology.For example, go back to the millionaire with infinite pairs of socks.Lemousse argues that there is no need for a rule for choosing one of each pair of socks.He argues that, logically, an infinite number of arbitrary choices is as admissible as a finite number of choices. He applies a similar idea to the notion of changing propositional functions.Whitehead and I considered a propositional function to be an expression with an undetermined variable that becomes an ordinary sentence as soon as a value is assigned to the variable.For example, "x is human", once we replace "x" with a proper name, it becomes an ordinary sentence.Looking at propositional functions in this way, they are inclusive (with the exception of or about variables). The words "is human" form part of many ordinary sentences, and propositional functions are one way of making several of these sentences.The value of a function is determined by the different values ​​of the variable, and the variable has different values ​​because of the intrinsic properties of the sentence. Lemousse's idea of ​​the propositional function is quite different. He sees propositional functions simply as a means of correlating propositions and values ​​of variables.In addition to the notion of a predicate-function defined previously (which we will still need for some purposes), we use extension to define a new notion of a propositional function (rather an illustration, since In our system, it must be considered undefinable).Such a function of an individual is caused by any one-many relation in extension between the proposition and the individual; it may also be said to be a mutual relation (whether practical or not) which puts a unique proposition Associated to each individual, the individual is the subject of the function, and the proposition is its value. For example, A (Socrates) may be Queen Ann is dead, A (Plato) may be Einstein is a great man; AxE is just an arbitrary combination of Ax propositions and xF individuals ("Fundamentals of Mathematics", p. 52 Page). Applying this new interpretation to the concept of the "propositional function" he was able to abolish the axiom of reducibility and to denote "x = y" by something that does not differ in sign from the definition in Principia Mathematica. Define, although that definition now has a new interpretation.In this way he managed to keep the symbolic part of Principia Mathematica almost unchanged. Regarding this part of the symbol, he says, "In form, it has hardly changed; but its meaning has changed greatly. In this way of preserving form and changing interpretation, I follow that school of mathematical logicians who rescued mathematics from the hands of skeptics and provided a rigorous argument for propositions by means of an astonishing series of definitions.Only in this way can we save mathematics from the Bolshevik threat of Brauer and Weiler" Fundamentals of Mathematics, p. 56). I have a hard time making up my mind about the validity of Lemousse's new interpretation of the concept of a "propositional function." It seems to me that a completely arbitrary correlation of entities to propositions is unsatisfactory.Take the reasoning from "fx is true for all values ​​of x" to "fa" as an example.According to Lemousse's interpretation of the concept "fx", we do not know what "ea" could be.Instead, before we can know what "fx" means, we must know "fa" and "fb" and "fc" and so on, throughout the universe.General propositions lose their raison d'être, since they can be asserted only by enumerating all the individual instances.Whatever your opinion of this objection, Lemousse's suggestion is indeed a coincidence, and, if not quite solving all the difficulties, is probably on the right track.Lemousse himself had his doubts.He said, "Although I try to modify Whitehead and Russell's claims, I think I have overcome many difficulties, but I can't think that this modification is completely satisfactory" ("Mathematical Logic", p. 81). On another matter, I think Lemousse's research must be admitted to be correct.I have enumerated various contradictions, one kind of example is the person who said "I lied", and another kind of example is the question of whether there is a maximum cardinality.莱穆塞证明，前一类是和一个字或语句之于其意义的关系有关，是把二者弄混的结果。如果避免了这种混乱，这类的矛盾就没有了。莱穆塞主张，另一类矛盾只能用类型学说来解决。在《数学原理》里，有两种不同的层型。有外延阶层：个体，个体的类，个体的类的类，等等。莱穆塞保留这个阶层。但是还有另外一个阶层，正是这另外的那个阶层使可化归性公理成为必需的。这就是某一对象的某一主目或性质的函项阶层。先是断言阶层，这个阶层不指任何函项总体；其次是指断言函项总体的函项，如，“拿破仑具有大将的一切特长”。我们可以称这些为“第一级函项”。然后是指第一级函项总体的函项，这样下去，以至于无穷。莱穆塞用他对于“命题函项”这个概念的新解释，取消了这个阶层，这样就只留有外包阶层。我希望他的学说是有效的。 虽然他是以维根斯坦的一个门人来写书，并且除了维根斯坦的神秘主义之外，一切都跟着他走，他探索这个问题的途径却是非常不同的。维根斯坦发表一些格言，让读者测量其高深。他的一些格言从字面上看是和符号逻辑的存在很难相合的。正相反，即使莱穆塞追随维根斯坦追随得很紧，他却极其小心地说明，（不管所讲的是什么学说，）如何能把这个学说配合到数理逻辑的主体里去。 有大量的、深奥的文献论述数理逻辑的基础。除了在《对意义与真理的探讨》中讨论外延性和原子性原理和排中律以外，自一九二五年出版第二版《数学原理》以后，我没有做纯是逻辑的研究。所以，后来关于这个科目的研究没有影响我在哲学上的发展，因而也就不属于本书的范围。