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Text of the page (random words):
for example would therefore present the proposition p q now i will proceed to inquire whether such a proposition as p p the law of contradiction is a tautology the form ξ is written in our notation the form ξ η thus hence the proposition p q runs thus if here we put p instead of q and examine the combination of the outermost t and f with the innermost it is seen that the truth of the whole proposition is co ordinated with all the truth combinations of its argument its falsity with none of the truth combinations 6 121 the propositions of logic demonstrate the logical properties of propositions by combining them into propositions which say nothing this method could be called a zero method in a logical proposition propositions are brought into equilibrium with one another and the state of equilibrium then shows how these propositions must be logically constructed 6 122 whence it follows that we can get on without logical propositions for we can recognize in an adequate notation the formal properties of the propositions by mere inspection 6 1221 if for example two propositions p and q give a tautology in the connexion p a then it is clear that q follows from p e g that q follows from p q p we see from these two propositions themselves but we can also show it by combining them to p q p q and then showing that this is a tautology 6 1222 this throws light on the question why logical propositions can no more be empirically confirmed than they can be empirically refuted not only must a proposition of logic be incapable of being contradicted by any possible experience but it must also be incapable of being confirmed by any such 6 1223 it now becomes clear why we often feel as though logical truths must be postulated by us we can in fact postulate them in so far as we can postulate an adequate notation 6 1224 it also becomes clear why logic has been called the theory of forms and of inference 6 123 it is clear that the laws of logic cannot themselves obey further logical laws there is not as russell supposed for every type a special law of contradiction but one is sufficient since it is not applied to itself 6 1231 the mark of logical propositions is not their general validity to be general is only to be accidentally valid for all things an ungeneralized proposition can be tautologous just as well as a generalized one 6 1232 logical general validity we could call essential as opposed to accidental general validity e g of the proposition all men are mortal propositions like russell s axiom of reducibility are not logical propositions and this explains our feeling that if true they can only be true by a happy chance 6 1233 we can imagine a world in which the axiom of reducibility is not valid but it is clear that logic has nothing to do with the question whether our world is really of this kind or not 6 124 the logical propositions describe the scaffolding of the world or rather they present it they treat of nothing they presuppose that names have meaning and that elementary propositions have sense and this is their connexion with the world it is clear that it must show something about the world that certain combinations of symbols which essentially have a definite character are tautologies herein lies the decisive point we said that in the symbols which we use something is arbitrary something not in logic only this expresses but this means that in logic it is not we who express by means of signs what we want but in logic the nature of the essentially necessary signs itself asserts that is to say if we know the logical syntax of any sign language then all the propositions of logic are already given 6 125 it is possible also with the old conception of logic to give at the outset a description of all true logical propositions 6 1251 hence there can never be surprises in logic 6 126 whether a proposition belongs to logic can be calculated by calculating the logical properties of the symbol and this we do when we prove a logical proposition for without troubling ourselves about a sense and a meaning we form the logical propositions out of others by mere symbolic rules we prove a logical proposition by creating it out of other logical propositions by applying in succession certain operations which again generate tautologies out of the first and from a tautology only tautologies follow naturally this way of showing that its propositions are tautologies is quite unessential to logic because the propositions from which the proof starts must show without proof that they are tautologies 6 1261 in logic process and result are equivalent therefore no surprises 6 1262 proof in logic is only a mechanical expedient to facilitate the recognition of tautology where it is complicated 6 1263 it would be too remarkable if one could prove a significant proposition logically from another and a logical proposition also it is clear from the beginning that the logical proof of a significant proposition and the proof in logic must be two quite different things 6 1264 the significant proposition asserts something and its proof shows that it is so in logic every proposition is the form of a proof every proposition of logic is a modus ponens presented in signs and the modus ponens can not be expressed by a proposition 6 1265 logic can always be conceived to be such that every proposition is its own proof 6 127 all propositions of logic are of equal rank there are not some which are essentially primitive and others deduced from these every tautology itself shows that it is a tautology 6 1271 it is clear that the number of primitive propositions of logic is arbitrary for we could deduce logic from one primitive proposition by simply forming for example the logical product of frege s primitive propositions frege would perhaps say that this would no longer be immediately self evident but it is remarkable that so exact a thinker as frege should have appealed to the degree of self evidence as the criterion of a logical proposition 6 13 logic is not a theory but a reflexion of the world logic is transcendental 6 2 mathematics is a logical method the propositions of mathematics are equations and therefore pseudo propositions 6 21 mathematical propositions express no thoughts 6 211 in life it is never a mathematical proposition which we need but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics in philosophy the question why do we really use that word that proposition constantly leads to valuable results 6 22 the logic of the world which the propositions of logic show in tautologies mathematics shows in equations 6 23 if two expressions are connected by the sign of equality this means that they can be substituted for one another but whether this is the case must show itself in the two expressions themselves it characterizes the logical form of two expressions that they can be substituted for one another 6 231 it is a property of affirmation that it can be conceived as double denial it is a property of 1 1 1 1 that it can be conceived as 1 1 1 1 6 232 frege says that these expressions have the same meaning but different senses but what is essential about equation is that it is not necessary in order to show that both expressions which are connected by the sign of equality have the same meaning for this can be perceived from the two expressions themselves 6 2321 and that the propositions of mathematics can be proved means nothing else than that their correctness can be seen without our having to compare what they express with the facts as regards correctness 6 2322 the identity of the meaning of two expressions cannot be asserted for in order to be able to assert anything about their meaning i must know their meaning and if i know their meaning i know whether they mean the same or something different 6 2323 the equation characterizes only the standpoint from which i consider the two expressions that is to say the standpoint of their equality of meaning 6 233 to the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition 6 2331 the process of calculation brings about just this intuition calculation is not an experiment 6 234 mathematics is a method of logic 6 2341 the essential of mathematical method is working with equations on this method depends the fact that every proposition of mathematics must be self evident 66 24 the method by which mathematics arrives at its equations is the method of substitution for equations express the substitutability of two expressions and we proceed from a number of equations to new equations replacing expressions by others in accordance with the equations 6 241 thus the proof of the proposition 2x2 4 runs ω ν μ x ω ν μ x def ω2 2 x ω2 2 x ω2 1 1 x ω2 ω2 x ω1 1 ω1 1 x ω ω ω ω x ω ω ω ω x ω1 1 1 1 6 3 logical research means the investigation of all regularity and outside logic all is accident 6 31 the so called law of induction cannot in any case be a logical law for it is obviously a significant proposition and therefore it cannot be a law a priori either 6 32 the law of causality is not a law but the form of a law 1 6 321 law of causality is a class name and as in mechanics there are for instance minimum laws such as that of least action so in physics there are causal laws laws of the causality form 6 3211 men had indeed an idea that there must be a law of least action before they knew exactly how it ran here as always the a priori certain proves to be something purely logical 6 33 we do not believe a priori in a law of conservation but we know a priori the possibility of a logical form 6 34 all propositions such as the law of causation the law of continuity in nature the law of least expenditure in nature etc etc all these are a priori intuitions of possible forms of the propositions of science 6 341 newtonian mechanics for example brings the description of the universe to a unified form let us imagine a white surface with irregular black spots we now say whatever kind of picture these make i can always get as near as i like to its description if i cover the surface with a sufficiently fine square network and now say of every square that it is white or black in this way i shall have brought the description of the surface to a unified form this form is arbitrary because i could have applied with equal success a net with a triangular or hexagonal mesh it can happen that the description would have been simpler with the aid of a triangular mesh that is to say we might have described the surface more accurately with a triangular and coarser than with the finer square mesh or vice versa and so on to the different networks correspond different systems of describing the world mechanics determine a form of description by saying all propositions in the description of the world must be obtained in a given way from a number of given propositions the mechanical axioms it thus provides the bricks for building the edifice of science and says whatever building thou wouldst erect thou shalt construct it in some manner with these bricks and these alone as with the system of numbers one must be able to write down any arbitrary number so with the system of mechanics one must be able to write down any arbitrary physical proposition 6 342 and now we see the relative position of logic and mechanics we could construct the network out of figures of different kinds as out of triangles and hexagons together that a picture like that instanced above can be described by a network of a given form asserts nothing about the picture for this holds of every picture of this kind but this does characterize the picture the fact namely that it can be completely described by a definite net of definite fineness so too the fact that it can be described by newtonian mechanics asserts nothing about the world but this asserts something namely that it can be described in that particular way in which as a matter of fact it is described the fact too that it can be described more simply by one system of mechanics than by another says something about the world 6 343 mechanics is an attempt to construct according to a single plan all true propositions which we need for the description of the world 6 3431 through their whole logical apparatus the physical laws still speak of the objects of the world 6 3432 we must not forget that the description of the world by mechanics is always quite general there is for example never any mention of particular material points in it but always only of some points or other 6 35 although the spots in our picture are geometrical figures geometry can obviously say nothing about their actual form and position but the network is purely geometrical and all its properties can be given a priori laws like the law of causation etc treat of the network and not of what the network describes 6 36 if there were a law of causality it might run there are natural laws but that can clearly not be said it shows itself 6 361 in the terminology of hertz we might say only uniform connexions are thinkable 6 3611 we cannot compare any process with the passage of time there is no such thing but only with another process say with the movement of the chronometer hence the description of the temporal sequence of events is only possible if we support ourselves on another process it is exactly analogous for space when for example we say that neither of two events which mutually exclude one another can occur because there is no cause why the one should occur rather than the other it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry and if there is such an asymmetry we can regard this as the cause of the occurrence of the one and of the non occurrence of the other 6 36111 the kantian problem of the right and left hand which cannot be made to cover one another already exists in the plane and even in one dimensional space where the two congruent figures a and b cannot be made to cover one another without moving them out of this space the right and left hand are in fact completely congruent and the fact that they cannot be made to cover one another has nothing to do with it a right hand glove could be put on a left hand if it could be turned round in four dimensional space 6 362 what can be described can happen too and what is excluded by the law of causality cannot be described 6 363 the process of induction is the process of assuming the simplest law that can be made to harmonize with our experience 6 3631 this process however has no logical foundation but only a psychological one it is clear that there are no grounds for believing that the simplest course of events will really happen 6 36311 that the sun will rise to morrow is an hypothesis and that means that we do not know whether it will rise 6 37 a necess...