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2239 lines
106 KiB
Plaintext
2239 lines
106 KiB
Plaintext
4 years ago
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Perhaps this book will be understood only by someone who has himself
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already had the thoughts that are expressed in it--or at least similar
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thoughts.--So it is not a textbook.--Its purpose would be achieved if it
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gave pleasure to one person who read and understood it.
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The book deals with the problems of philosophy, and shows, I believe, that
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the reason why these problems are posed is that the logic of our language
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is misunderstood. The whole sense of the book might be summed up the
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following words: what can be said at all can be said clearly, and what we
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cannot talk about we must pass over in silence.
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Thus the aim of the book is to draw a limit to thought, or rather--not to
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thought, but to the expression of thoughts: for in order to be able to draw
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a limit to thought, we should have to find both sides of the limit
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thinkable i.e. we should have to be able to think what cannot be thought.
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It will therefore only be in language that the limit can be drawn, and what
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lies on the other side of the limit will simply be nonsense.
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I do not wish to judge how far my efforts coincide with those of other
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philosophers. Indeed, what I have written here makes no claim to novelty in
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detail, and the reason why I give no sources is that it is a matter of
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indifference to me whether the thoughts that I have had have been
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anticipated by someone else.
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If this work has any value, it consists in two things: the first is that
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thoughts are expressed in it, and on this score the better the thoughts are
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expressed--the more the nail has been hit on the head--the greater will be
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its value.--Here I am conscious of having fallen a long way short of what
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is possible. Simply because my powers are too slight for the accomplishment
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of the task.--May others come and do it better.
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On the other hand the truth of the thoughts that are here communicated
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seems to me unassailable and definitive. I therefore believe myself to have
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found, on all essential points, the final solution of the problems. And if
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I am not mistaken in this belief, then the second thing in which the of
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this work consists is that it shows how little is achieved when these
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problem are solved.
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The world is all that is the case.
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The world is the totality of facts, not of things.
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The world is determined by the facts, and by their being all the
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facts.
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For the totality of facts determines what is the case, and also
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whatever is not the case.
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The facts in logical space are the world.
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The world divides into facts.
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Each item can be the case or not the case while everything else
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remains the same.
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What is the case--a fact--is the existence of states of affairs.
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A state of affairs a state of things is a combination of objects
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things.
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It is essential to things that they should be possible constituents
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of states of affairs.
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In logic nothing is accidental: if a thing can occur in a state of
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affairs, the possibility of the state of affairs must be written into the
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thing itself.
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It would seem to be a sort of accident, if it turned out that a
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situation would fit a thing that could already exist entirely on its own.
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If things can occur in states of affairs, this possibility must be in them
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from the beginning. Nothing in the province of logic can be merely
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possible. Logic deals with every possibility and all possibilities are its
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facts. Just as we are quite unable to imagine spatial objects outside
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space or temporal objects outside time, so too there is no object that we
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can imagine excluded from the possibility of combining with others. If I
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can imagine objects combined in states of affairs, I cannot imagine them
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excluded from the possibility of such combinations.
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Things are independent in so far as they can occur in all possible
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situations, but this form of independence is a form of connexion with
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states of affairs, a form of dependence. It is impossible for words to
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appear in two different roles: by themselves, and in propositions.
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If I know an object I also know all its possible occurrences in
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states of affairs. Every one of these possibilities must be part of the
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nature of the object. A new possibility cannot be discovered later.
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If I am to know an object, thought I need not know its external
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properties, I must know all its internal properties.
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If all objects are given, then at the same time all possible states
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of affairs are also given.
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Each thing is, as it were, in a space of possible states of affairs.
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This space I can imagine empty, but I cannot imagine the thing without the
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space.
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A spatial object must be situated in infinite space. A spatial
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point is an argument-place. A speck in the visual field, thought it need
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not be red, must have some colour: it is, so to speak, surrounded by colour-
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space. Notes must have some pitch, objects of the sense of touch some
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degree of hardness, and so on.
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Objects contain the possibility of all situations.
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The possibility of its occurring in states of affairs is the form of
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an object.
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Objects are simple.
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Every statement about complexes can be resolved into a statement
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about their constituents and into the propositions that describe the
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complexes completely.
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Objects make up the substance of the world. That is why they cannot
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be composite.
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If they world had no substance, then whether a proposition had sense
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would depend on whether another proposition was true.
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In that case we could not sketch any picture of the world true or
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false.
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It is obvious that an imagined world, however difference it may be
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from the real one, must have something-- a form--in common with it.
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Objects are just what constitute this unalterable form.
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The substance of the world can only determine a form, and not any
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material properties. For it is only by means of propositions that material
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properties are represented--only by the configuration of objects that they
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are produced.
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In a manner of speaking, objects are colourless.
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If two objects have the same logical form, the only distinction
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between them, apart from their external properties, is that they are
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different.
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Either a thing has properties that nothing else has, in which case
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we can immediately use a description to distinguish it from the others and
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refer to it; or, on the other hand, there are several things that have the
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whole set of their properties in common, in which case it is quite
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impossible to indicate one of them. For it there is nothing to distinguish
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a thing, I cannot distinguish it, since otherwise it would be distinguished
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after all.
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The substance is what subsists independently of what is the case.
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It is form and content.
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Space, time, colour being coloured are forms of objects.
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There must be objects, if the world is to have unalterable form.
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Objects, the unalterable, and the subsistent are one and the same.
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Objects are what is unalterable and subsistent; their configuration
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is what is changing and unstable.
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The configuration of objects produces states of affairs.
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In a state of affairs objects fit into one another like the links of a
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chain.
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In a state of affairs objects stand in a determinate relation to one
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another.
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The determinate way in which objects are connected in a state of
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affairs is the structure of the state of affairs.
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Form is the possibility of structure.
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The structure of a fact consists of the structures of states of
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affairs.
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The totality of existing states of affairs is the world.
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The totality of existing states of affairs also determines which
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states of affairs do not exist.
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The existence and non-existence of states of affairs is reality. We
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call the existence of states of affairs a positive fact, and their non-
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existence a negative fact.
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States of affairs are independent of one another.
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From the existence or non-existence of one state of affairs it is
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impossible to infer the existence or non-existence of another.
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The sum-total of reality is the world.
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We picture facts to ourselves.
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A picture presents a situation in logical space, the existence and non-
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existence of states of affairs.
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A picture is a model of reality.
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In a picture objects have the elements of the picture corresponding to
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them.
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In a picture the elements of the picture are the representatives of
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objects.
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What constitutes a picture is that its elements are related to one
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another in a determinate way.
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A picture is a fact.
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The fact that the elements of a picture are related to one another in
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a determinate way represents that things are related to one another in the
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same way. Let us call this connexion of its elements the structure of the
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picture, and let us call the possibility of this structure the pictorial
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form of the picture.
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Pictorial form is the possibility that things are related to one
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another in the same way as the elements of the picture.
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That is how a picture is attached to reality; it reaches right out
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to it.
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It is laid against reality like a measure.
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Only the end-points of the graduating lines actually touch the
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object that is to be measured.
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So a picture, conceived in this way, also includes the pictorial
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relationship, which makes it into a picture.
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These correlations are, as it were, the feelers of the picture's
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elements, with which the picture touches reality.
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If a fact is to be a picture, it must have something in common with
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what it depicts.
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There must be something identical in a picture and what it depicts,
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to enable the one to be a picture of the other at all.
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What a picture must have in common with reality, in order to be able
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to depict it--correctly or incorrectly--in the way that it does, is its
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pictorial form.
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A picture can depict any reality whose form it has. A spatial picture
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can depict anything spatial, a coloured one anything coloured, etc.
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A picture cannot, however, depict its pictorial form: it displays it.
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A picture represents its subject from a position outside it. Its
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standpoint is its representational form. That is why a picture represents
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its subject correctly or incorrectly.
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A picture cannot, however, place itself outside its representational
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form.
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What any picture, of whatever form, must have in common with reality,
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in order to be able to depict it--correctly or incorrectly--in any way at
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all, is logical form, i.e. the form of reality.
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A picture whose pictorial form is logical form is called a logical
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picture.
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Every picture is at the same time a logical one. On the other hand,
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not every picture is, for example, a spatial one.
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Logical pictures can depict the world.
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A picture has logico-pictorial form in common with what it depicts.
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A picture depicts reality by representing a possibility of existence
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and non-existence of states of affairs.
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A picture contains the possibility of the situation that it
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represents.
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A picture agrees with reality or fails to agree; it is correct or
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incorrect, true or false.
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What a picture represents it represents independently of its truth or
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falsity, by means of its pictorial form.
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What a picture represents is its sense.
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The agreement or disagreement or its sense with reality constitutes
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its truth or falsity.
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In order to tell whether a picture is true or false we must compare
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it with reality.
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It is impossible to tell from the picture alone whether it is true or
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false.
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There are no pictures that are true a priori.
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A logical picture of facts is a thought.
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'A state of affairs is thinkable': what this means is that we can
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picture it to ourselves.
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The totality of true thoughts is a picture of the world.
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A thought contains the possibility of the situation of which it is the
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thought. What is thinkable is possible too.
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Thought can never be of anything illogical, since, if it were, we
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should have to think illogically.
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It used to be said that God could create anything except what would
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be contrary to the laws of logic.The truth is that we could not say what an
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'illogical' world would look like.
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It is as impossible to represent in language anything that
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'contradicts logic' as it is in geometry to represent by its coordinates a
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figure that contradicts the laws of space, or to give the coordinates of a
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point that does not exist.
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Though a state of affairs that would contravene the laws of physics
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can be represented by us spatially, one that would contravene the laws of
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geometry cannot.
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It a thought were correct a priori, it would be a thought whose
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possibility ensured its truth.
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A priori knowledge that a thought was true would be possible only it
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its truth were recognizable from the thought itself without anything a to
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compare it with.
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In a proposition a thought finds an expression that can be perceived by
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the senses.
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We use the perceptible sign of a proposition spoken or written, etc.
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as a projection of a possible situation. The method of projection is to
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think of the sense of the proposition.
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I call the sign with which we express a thought a propositional
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sign.And a proposition is a propositional sign in its projective relation
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to the world.
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A proposition, therefore, does not actually contain its sense, but
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does contain the possibility of expressing it. 'The content of a
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proposition' means the content of a proposition that has sense. A
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proposition contains the form, but not the content, of its sense.
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What constitutes a propositional sign is that in its elements the
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words stand in a determinate relation to one another. A propositional sign
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is a fact.
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A proposition is not a blend of words.Just as a theme in music is
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not a blend of notes. A proposition is articulate.
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Only facts can express a sense, a set of names cannot.
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Although a propositional sign is a fact, this is obscured by the
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usual form of expression in writing or print. For in a printed proposition,
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for example, no essential difference is apparent between a propositional
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sign and a word. That is what made it possible for Frege to call a
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proposition a composite name.
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The essence of a propositional sign is very clearly seen if we
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imagine one composed of spatial objects such as tables, chairs, and books
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instead of written signs.
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Situations can be described but not given names.
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In a proposition a thought can be expressed in such a way that elements
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of the propositional sign correspond to the objects of the thought.
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I call such elements 'simple signs', and such a proposition 'complete
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analysed'.
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The simple signs employed in propositions are called names.
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A name means an object. The object is its meaning. 'A' is the same
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sign as 'A'.
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The configuration of objects in a situation corresponds to the
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configuration of simple signs in the propositional sign.
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Objects can only be named. Signs are their representatives. I can
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only speak about them: I cannot put them into words. Propositions can only
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say how things are, not what they are.
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The requirement that simple signs be possible is the requirement that
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sense be determinate.
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A proposition about a complex stands in an internal relation to a
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proposition about a constituent of the complex. A complex can be given only
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by its description, which will be right or wrong. A proposition that
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mentions a complex will not be nonsensical, if the complex does not exits,
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but simply false. When a propositional element signifies a complex, this
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can be seen from an indeterminateness in the propositions in which it
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occurs. In such cases we know that the proposition leaves something
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undetermined. In fact the notation for generality contains a prototype.
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The contraction of a symbol for a complex into a simple symbol can be
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expressed in a definition.
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A proposition cannot be dissected any further by means of a
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definition: it is a primitive sign.
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Every sign that has a definition signifies via the signs that serve
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to define it; and the definitions point the way. Two signs cannot signify
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in the same manner if one is primitive and the other is defined by means of
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primitive signs. Names cannot be anatomized by means of definitions. Nor
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can any sign that has a meaning independently and on its own.
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What signs fail to express, their application shows. What signs slur
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over, their application says clearly.
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The meanings of primitive signs can be explained by means of
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elucidations. Elucidations are propositions that stood if the meanings of
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those signs are already known.
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Only propositions have sense; only in the nexus of a proposition does a
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name have meaning.
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I call any part of a proposition that characterizes its sense an
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expression or a symbol. A proposition is itself an expression.
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Everything essential to their sense that propositions can have in common
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with one another is an expression. An expression is the mark of a form and
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a content.
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An expression presupposes the forms of all the propositions in which
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it can occur. It is the common characteristic mark of a class of
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propositions.
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It is therefore presented by means of the general form of the
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propositions that it characterizes. In fact, in this form the expression
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will be constant and everything else variable.
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Thus an expression is presented by means of a variable whose values
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are the propositions that contain the expression. In the limiting case the
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variable becomes a constant, the expression becomes a proposition. I call
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such a variable a 'propositional variable'.
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An expression has meaning only in a proposition. All variables can be
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construed as propositional variables. Even variable names.
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|
If we turn a constituent of a proposition into a variable, there is a
|
||
|
class of propositions all of which are values of the resulting variable
|
||
|
proposition. In general, this class too will be dependent on the meaning
|
||
|
that our arbitrary conventions have given to parts of the original
|
||
|
proposition. But if all the signs in it that have arbitrarily determined
|
||
|
meanings are turned into variables, we shall still get a class of this
|
||
|
kind. This one, however, is not dependent on any convention, but solely on
|
||
|
the nature of the pro position. It corresponds to a logical form--a logical
|
||
|
prototype.
|
||
|
|
||
|
What values a propositional variable may take is something that is
|
||
|
stipulated. The stipulation of values is the variable.
|
||
|
|
||
|
To stipulate values for a propositional variable is to give the
|
||
|
propositions whose common characteristic the variable is. The stipulation
|
||
|
is a description of those propositions. The stipulation will therefore be
|
||
|
concerned only with symbols, not with their meaning. And the only thing
|
||
|
essential to the stipulation is that it is merely a description of symbols
|
||
|
and states nothing about what is signified. How the description of the
|
||
|
propositions is produced is not essential.
|
||
|
|
||
|
Like Frege and Russell I construe a proposition as a function of the
|
||
|
expressions contained in it.
|
||
|
|
||
|
A sign is what can be perceived of a symbol.
|
||
|
|
||
|
So one and the same sign written or spoken, etc. can be common to
|
||
|
two different symbols--in which case they will signify in different ways.
|
||
|
|
||
|
Our use of the same sign to signify two different objects can never
|
||
|
indicate a common characteristic of the two, if we use it with two
|
||
|
different modes of signification. For the sign, of course, is arbitrary. So
|
||
|
we could choose two different signs instead, and then what would be left in
|
||
|
common on the signifying side?
|
||
|
|
||
|
In everyday language it very frequently happens that the same word
|
||
|
has different modes of signification--and so belongs to different symbols--
|
||
|
or that two words that have different modes of signification are employed
|
||
|
in propositions in what is superficially the same way. Thus the word 'is'
|
||
|
figures as the copula, as a sign for identity, and as an expression for
|
||
|
existence; 'exist' figures as an intransitive verb like 'go', and
|
||
|
'identical' as an adjective; we speak of something, but also of something's
|
||
|
happening. In the proposition, 'Green is green'--where the first word is
|
||
|
the proper name of a person and the last an adjective--these words do not
|
||
|
merely have different meanings: they are different symbols.
|
||
|
|
||
|
In this way the most fundamental confusions are easily produced the
|
||
|
whole of philosophy is full of them.
|
||
|
|
||
|
In order to avoid such errors we must make use of a sign-language
|
||
|
that excludes them by not using the same sign for different symbols and by
|
||
|
not using in a superficially similar way signs that have different modes of
|
||
|
signification: that is to say, a sign-language that is governed by logical
|
||
|
grammar--by logical syntax. The conceptual notation of Frege and Russell
|
||
|
is such a language, though, it is true, it fails to exclude all mistakes.
|
||
|
|
||
|
In order to recognize a symbol by its sign we must observe how it is
|
||
|
used with a sense.
|
||
|
|
||
|
A sign does not determine a logical form unless it is taken together
|
||
|
with its logico-syntactical employment.
|
||
|
|
||
|
If a sign is useless, it is meaningless. That is the point of Occam's
|
||
|
maxim. If everything behaves as if a sign had meaning, then it does have
|
||
|
meaning.
|
||
|
|
||
|
In logical syntax the meaning of a sign should never play a role. It
|
||
|
must be possible to establish logical syntax without mentioning the meaning
|
||
|
of a sign: only the description of expressions may be presupposed.
|
||
|
|
||
|
From this observation we turn to Russell's 'theory of types'. It can
|
||
|
be seen that Russell must be wrong, because he had to mention the meaning
|
||
|
of signs when establishing the rules for them.
|
||
|
|
||
|
No proposition can make a statement about itself, because a
|
||
|
propositional sign cannot be contained in itself that is the whole of the
|
||
|
'theory of types'.
|
||
|
|
||
|
The reason why a function cannot be its own argument is that the sign
|
||
|
for a function already contains the prototype of its argument, and it
|
||
|
cannot contain itself.
|
||
|
|
||
|
The rules of logical syntax must go without saying, once we know how
|
||
|
each individual sign signifies.
|
||
|
|
||
|
A proposition possesses essential and accidental features. Accidental
|
||
|
features are those that result from the particular way in which the
|
||
|
propositional sign is produced. Essential features are those without which
|
||
|
the proposition could not express its sense.
|
||
|
|
||
|
So what is essential in a proposition is what all propositions that
|
||
|
can express the same sense have in common. And similarly, in general, what
|
||
|
is essential in a symbol is what all symbols that can serve the same
|
||
|
purpose have in common.
|
||
|
|
||
|
So one could say that the real name of an object was what all
|
||
|
symbols that signified it had in common. Thus, one by one, all kinds of
|
||
|
composition would prove to be unessential to a name.
|
||
|
|
||
|
Although there is something arbitrary in our notations, this much is
|
||
|
not arbitrary--that when we have determined one thing arbitrarily,
|
||
|
something else is necessarily the case. This derives from the essence of
|
||
|
notation.
|
||
|
|
||
|
A particular mode of signifying may be unimportant but it is always
|
||
|
important that it is a possible mode of signifying. And that is generally
|
||
|
so in philosophy: again and again the individual case turns out to be
|
||
|
unimportant, but the possibility of each individual case discloses
|
||
|
something about the essence of the world.
|
||
|
|
||
|
Definitions are rules for translating from one language into another.
|
||
|
Any correct sign-language must be translatable into any other in accordance
|
||
|
with such rules: it is this that they all have in common.
|
||
|
|
||
|
What signifies in a symbol is what is common to all the symbols that
|
||
|
the rules of logical syntax allow us to substitute for it.
|
||
|
|
||
|
This serves to characterize the way in
|
||
|
which something general can be disclosed by the possibility of a specific
|
||
|
notation.
|
||
|
|
||
|
Nor does analysis resolve the sign for a complex in an arbitrary
|
||
|
way, so that it would have a different resolution every time that it was
|
||
|
incorporated in a different proposition.
|
||
|
|
||
|
A proposition determines a place in logical space. The existence of
|
||
|
this logical place is guaranteed by the mere existence of the constituents--
|
||
|
by the existence of the proposition with a sense.
|
||
|
|
||
|
The propositional sign with logical co-ordinates--that is the logical
|
||
|
place.
|
||
|
|
||
|
In geometry and logic alike a place is a possibility: something can
|
||
|
exist in it.
|
||
|
|
||
|
A proposition can determine only one place in logical space:
|
||
|
nevertheless the whole of logical space must already be given by it.
|
||
|
Otherwise negation, logical sum, logical product, etc., would introduce
|
||
|
more and more new elements in co-ordination. The logical scaffolding
|
||
|
surrounding a picture determines logical space. The force of a proposition
|
||
|
reaches through the whole of logical space.
|
||
|
|
||
|
A propositional sign, applied and thought out, is a thought.
|
||
|
|
||
|
4 A thought is a proposition with a sense.
|
||
|
|
||
|
The totality of propositions is language.
|
||
|
|
||
|
Man possesses the ability to construct languages capable of
|
||
|
expressing every sense, without having any idea how each word has meaning
|
||
|
or what its meaning is--just as people speak without knowing how the
|
||
|
individual sounds are produced. Everyday language is a part of the human
|
||
|
organism and is no less complicated than it. It is not humanly possible to
|
||
|
gather immediately from it what the logic of language is. Language
|
||
|
disguises thought. So much so, that from the outward form of the clothing
|
||
|
it is impossible to infer the form of the thought beneath it, because the
|
||
|
outward form of the clothing is not designed to reveal the form of the
|
||
|
body, but for entirely different purposes. The tacit conventions on which
|
||
|
the understanding of everyday language depends are enormously complicated.
|
||
|
|
||
|
Most of the propositions and questions to be found in philosophical
|
||
|
works are not false but nonsensical. Consequently we cannot give any answer
|
||
|
to questions of this kind, but can only point out that they are
|
||
|
nonsensical. Most of the propositions and questions of philosophers arise
|
||
|
from our failure to understand the logic of our language. They belong to
|
||
|
the same class as the question whether the good is more or less identical
|
||
|
than the beautiful. And it is not surprising that the deepest problems are
|
||
|
in fact not problems at all.
|
||
|
|
||
|
All philosophy is a 'critique of language' though not in Mauthner's
|
||
|
sense. It was Russell who performed the service of showing that the
|
||
|
apparent logical form of a proposition need not be its real one.
|
||
|
|
||
|
A proposition is a picture of reality. A proposition is a model of
|
||
|
reality as we imagine it.
|
||
|
|
||
|
At first sight a proposition--one set out on the printed page, for
|
||
|
example--does not seem to be a picture of the reality with which it is
|
||
|
concerned. But neither do written notes seem at first sight to be a picture
|
||
|
of a piece of music, nor our phonetic notation the alphabet to be a
|
||
|
picture of our speech. And yet these sign-languages prove to be pictures,
|
||
|
even in the ordinary sense, of what they represent.
|
||
|
|
||
|
It is obvious that a proposition of the form 'aRb' strikes us as a
|
||
|
picture. In this case the sign is obviously a likeness of what is
|
||
|
signified.
|
||
|
|
||
|
And if we penetrate to the essence of this pictorial character, we
|
||
|
see that it is not impaired by apparent irregularities such as the use
|
||
|
of musical notation. For even these irregularities
|
||
|
depict what they are intended to express; only they do it in a different
|
||
|
way.
|
||
|
|
||
|
A gramophone record, the musical idea, the written notes, and the
|
||
|
sound-waves, all stand to one another in the same internal relation of
|
||
|
depicting that holds between language and the world. They are all
|
||
|
constructed according to a common logical pattern. Like the two youths in
|
||
|
the fairy-tale, their two horses, and their lilies. They are all in a
|
||
|
certain sense one.
|
||
|
|
||
|
There is a general rule by means of which the musician can obtain
|
||
|
the symphony from the score, and which makes it possible to derive the
|
||
|
symphony from the groove on the gramophone record, and, using the first
|
||
|
rule, to derive the score again. That is what constitutes the inner
|
||
|
similarity between these things which seem to be constructed in such
|
||
|
entirely different ways. And that rule is the law of projection which
|
||
|
projects the symphony into the language of musical notation. It is the rule
|
||
|
for translating this language into the language of gramophone records.
|
||
|
|
||
|
The possibility of all imagery, of all our pictorial modes of
|
||
|
expression, is contained in the logic of depiction.
|
||
|
|
||
|
In order to understand the essential nature of a proposition, we
|
||
|
should consider hieroglyphic script, which depicts the facts that it
|
||
|
describes. And alphabetic script developed out of it without losing what
|
||
|
was essential to depiction.
|
||
|
|
||
|
We can see this from the fact that we understand the sense of a
|
||
|
propositional sign without its having been explained to us.
|
||
|
|
||
|
A proposition is a picture of reality: for if I understand a
|
||
|
proposition, I know the situation that it represents. And I understand the
|
||
|
proposition without having had its sense explained to me.
|
||
|
|
||
|
A proposition shows its sense. A proposition shows how things stand
|
||
|
if it is true. And it says that they do so stand.
|
||
|
|
||
|
A proposition must restrict reality to two alternatives: yes or no.
|
||
|
In order to do that, it must describe reality completely. A proposition is
|
||
|
a description of a state of affairs. Just as a description of an object
|
||
|
describes it by giving its external properties, so a proposition describes
|
||
|
reality by its internal properties. A proposition constructs a world with
|
||
|
the help of a logical scaffolding, so that one can actually see from the
|
||
|
proposition how everything stands logically if it is true. One can draw
|
||
|
inferences from a false proposition.
|
||
|
|
||
|
To understand a proposition means to know what is the case if it is
|
||
|
true. One can understand it, therefore, without knowing whether it is
|
||
|
true. It is understood by anyone who understands its constituents.
|
||
|
|
||
|
When translating one language into another, we do not proceed by
|
||
|
translating each proposition of the one into a proposition of the other,
|
||
|
but merely by translating the constituents of propositions. And the
|
||
|
dictionary translates not only substantives, but also verbs, adjectives,
|
||
|
and conjunctions, etc.; and it treats them all in the same way.
|
||
|
|
||
|
The meanings of simple signs words must be explained to us if we
|
||
|
are to understand them. With propositions, however, we make ourselves
|
||
|
understood.
|
||
|
|
||
|
It belongs to the essence of a proposition that it should be able to
|
||
|
communicate a new sense to us.
|
||
|
|
||
|
A proposition must use old expressions to communicate a new sense. A
|
||
|
proposition communicates a situation to us, and so it must be essentially
|
||
|
connected with the situation. And the connexion is precisely that it is its
|
||
|
logical picture. A proposition states something only in so far as it is a
|
||
|
picture.
|
||
|
|
||
|
In a proposition a situation is, as it were, constructed by way of
|
||
|
experiment. Instead of, 'This proposition has such and such a sense, we can
|
||
|
simply say, 'This proposition represents such and such a situation'.
|
||
|
|
||
|
One name stands for one thing, another for another thing, and they
|
||
|
are combined with one another. In this way the whole group--like a tableau
|
||
|
vivant--presents a state of affairs.
|
||
|
|
||
|
The possibility of propositions is based on the principle that
|
||
|
objects have signs as their representatives. My fundamental idea is that
|
||
|
the 'logical constants' are not representatives; that there can be no
|
||
|
representatives of the logic of facts.
|
||
|
|
||
|
It is only in so far as a proposition is logically articulated that
|
||
|
it is a picture of a situation. Even the proposition, 'Ambulo', is
|
||
|
composite: for its stem with a different ending yields a different sense,
|
||
|
and so does its ending with a different stem.
|
||
|
|
||
|
In a proposition there must be exactly as many distinguishable parts
|
||
|
as in the situation that it represents. The two must possess the same
|
||
|
logical mathematical multiplicity. Compare Hertz's Mechanics on
|
||
|
dynamical models.
|
||
|
|
||
|
This mathematical multiplicity, of course, cannot itself be the
|
||
|
subject of depiction. One cannot get away from it when depicting.
|
||
|
|
||
|
If, for example, we wanted to express what we now write as 'x.
|
||
|
fx' by putting an affix in front of 'fx'--for instance by writing 'Gen. fx'-
|
||
|
-it would not be adequate: we should not know what was being generalized.
|
||
|
If we wanted to signalize it with an affix 'g'--for instance by writing
|
||
|
'fxg'--that would not be adequate either: we should not know the scope of
|
||
|
the generality-sign. If we were to try to do it by introducing a mark into
|
||
|
the argument-places--for instance by writing 'G,G. FG,G' --it would
|
||
|
not be adequate: we should not be able to establish the identity of the
|
||
|
variables. And so on. All these modes of signifying are inadequate because
|
||
|
they lack the necessary mathematical multiplicity.
|
||
|
|
||
|
For the same reason the idealist's appeal to 'spatial spectacles' is
|
||
|
inadequate to explain the seeing of spatial relations, because it cannot
|
||
|
explain the multiplicity of these relations.
|
||
|
|
||
|
Reality is compared with propositions.
|
||
|
|
||
|
A proposition can be true or false only in virtue of being a picture
|
||
|
of reality.
|
||
|
|
||
|
It must not be overlooked that a proposition has a sense that is
|
||
|
independent of the facts: otherwise one can easily suppose that true and
|
||
|
false are relations of equal status between signs and what they signify. In
|
||
|
that case one could say, for example, that 'p' signified in the true way
|
||
|
what 'Pp' signified in the false way, etc.
|
||
|
|
||
|
Can we not make ourselves understood with false propositions just as
|
||
|
we have done up till now with true ones?--So long as it is known that they
|
||
|
are meant to be false.--No! For a proposition is true if we use it to say
|
||
|
that things stand in a certain way, and they do; and if by 'p' we mean Pp
|
||
|
and things stand as we mean that they do, then, construed in the new way,
|
||
|
'p' is true and not false.
|
||
|
|
||
|
But it is important that the signs 'p' and 'Pp' can say the same
|
||
|
thing. For it shows that nothing in reality corresponds to the sign 'P'.
|
||
|
The occurrence of negation in a proposition is not enough to characterize
|
||
|
its sense PPp = p. The propositions 'p' and 'Pp' have opposite sense, but
|
||
|
there corresponds to them one and the same reality.
|
||
|
|
||
|
An analogy to illustrate the concept of truth: imagine a black spot
|
||
|
on white paper: you can describe the shape of the spot by saying, for each
|
||
|
point on the sheet, whether it is black or white. To the fact that a point
|
||
|
is black there corresponds a positive fact, and to the fact that a point is
|
||
|
white not black, a negative fact. If I designate a point on the sheet a
|
||
|
truth-value according to Frege, then this corresponds to the supposition
|
||
|
that is put forward for judgement, etc. etc. But in order to be able to say
|
||
|
that a point is black or white, I must first know when a point is called
|
||
|
black, and when white: in order to be able to say,'"p" is true or false',
|
||
|
I must have determined in what circumstances I call 'p' true, and in so
|
||
|
doing I determine the sense of the proposition. Now the point where the
|
||
|
simile breaks down is this: we can indicate a point on the paper even if we
|
||
|
do not know what black and white are, but if a proposition has no sense,
|
||
|
nothing corresponds to it, since it does not designatea thing a truth-
|
||
|
value which might have properties called 'false' or 'true'. The verb of a
|
||
|
proposition is not 'is true' or 'is false', as Frege thought: rather, that
|
||
|
which 'is true' must already contain the verb.
|
||
|
|
||
|
Every proposition must already have a sense: it cannot be given a
|
||
|
sense by affirmation. Indeed its sense is just what is affirmed. And the
|
||
|
same applies to negation, etc.
|
||
|
|
||
|
One could say that negation must be related to the logical place
|
||
|
determined by the negated proposition. The negating proposition determines
|
||
|
a logical place different from that of the negated proposition. The
|
||
|
negating proposition determines a logical place with the help of the
|
||
|
logical place of the negated proposition. For it describes it as lying
|
||
|
outside the latter's logical place. The negated proposition can be negated
|
||
|
again, and this in itself shows that what is negated is already a
|
||
|
proposition, and not merely something that is prelimary to a proposition.
|
||
|
|
||
|
Propositions represent the existence and non-existence of states of
|
||
|
affairs.
|
||
|
|
||
|
The totality of true propositions is the whole of natural science or
|
||
|
the whole corpus of the natural sciences.
|
||
|
|
||
|
Philosophy is not one of the natural sciences. The word 'philosophy'
|
||
|
must mean something whose place is above or below the natural sciences, not
|
||
|
beside them.
|
||
|
|
||
|
Philosophy aims at the logical clarification of thoughts. Philosophy
|
||
|
is not a body of doctrine but an activity. A philosophical work consists
|
||
|
essentially of elucidations. Philosophy does not result in 'philosophical
|
||
|
propositions', but rather in the clarification of propositions. Without
|
||
|
philosophy thoughts are, as it were, cloudy and indistinct: its task is to
|
||
|
make them clear and to give them sharp boundaries.
|
||
|
|
||
|
Psychology is no more closely related to philosophy than any other
|
||
|
natural science. Theory of knowledge is the philosophy of psychology. Does
|
||
|
not my study of sign-language correspond to the study of thought-processes,
|
||
|
which philosophers used to consider so essential to the philosophy of
|
||
|
logic? Only in most cases they got entangled in unessential psychological
|
||
|
investigations, and with my method too there is an analogous risk.
|
||
|
|
||
|
Darwin's theory has no more to do with philosophy than any other
|
||
|
hypothesis in natural science.
|
||
|
|
||
|
Philosophy sets limits to the much disputed sphere of natural
|
||
|
science.
|
||
|
|
||
|
It must set limits to what can be thought; and, in doing so, to what
|
||
|
cannot be thought. It must set limits to what cannot be thought by working
|
||
|
outwards through what can be thought.
|
||
|
|
||
|
It will signify what cannot be said, by presenting clearly what can
|
||
|
be said.
|
||
|
|
||
|
Everything that can be thought at all can be thought clearly.
|
||
|
Everything that can be put into words can be put clearly. 4.12 Propositions
|
||
|
can represent the whole of reality, but they cannot represent what they
|
||
|
must have in common with reality in order to be able to represent it--
|
||
|
logical form. In order to be able to represent logical form, we should have
|
||
|
to be able to station ourselves with propositions somewhere outside logic,
|
||
|
that is to say outside the world.
|
||
|
|
||
|
Propositions cannot represent logical form: it is mirrored in them.
|
||
|
What finds its reflection in language, language cannot represent. What
|
||
|
expresses itself in language, we cannot express by means of language.
|
||
|
Propositions show the logical form of reality. They display it.
|
||
|
|
||
|
If two propositions contradict one another, then
|
||
|
their structure shows it; the same is true if one of them follows from the
|
||
|
other. And so on.
|
||
|
|
||
|
What can be shown, cannot be said.
|
||
|
|
||
|
Now, too, we understand our feeling that once we have a sign-
|
||
|
language in which everything is all right, we already have a correct
|
||
|
logical point of view.
|
||
|
|
||
|
In a certain sense we can talk about formal properties of objects and
|
||
|
states of affairs, or, in the case of facts, about structural properties:
|
||
|
and in the same sense about formal relations and structural relations.
|
||
|
Instead of 'structural property' I also say 'internal property'; instead
|
||
|
of 'structural relation', 'internal relation'. I introduce these
|
||
|
expressions in order to indicate the source of the confusion between
|
||
|
internal relations and relations proper external relations, which is very
|
||
|
widespread among philosophers. It is impossible, however, to assert by
|
||
|
means of propositions that such internal properties and relations obtain:
|
||
|
rather, this makes itself manifest in the propositions that represent the
|
||
|
relevant states of affairs and are concerned with the relevant objects.
|
||
|
|
||
|
An internal property of a fact can also be bed a feature of that
|
||
|
fact in the sense in which we speak of facial features, for example.
|
||
|
|
||
|
A property is internal if it is unthinkable that its object should
|
||
|
not possess it. This shade of blue and that one stand, eo ipso, in the
|
||
|
internal relation of lighter to darker. It is unthinkable that these two
|
||
|
objects should not stand in this relation. Here the shifting use of the
|
||
|
word 'object' corresponds to the shifting use of the words 'property' and
|
||
|
'relation'.
|
||
|
|
||
|
The existence of an internal property of a possible situation is not
|
||
|
expressed by means of a proposition: rather, it expresses itself in the
|
||
|
proposition representing the situation, by means of an internal property of
|
||
|
that proposition. It would be just as nonsensical to assert that a
|
||
|
proposition had a formal property as to deny it.
|
||
|
|
||
|
It is impossible to distinguish forms from one another by saying
|
||
|
that one has this property and another that property: for this presupposes
|
||
|
that it makes sense to ascribe either property to either form.
|
||
|
|
||
|
The existence of an internal relation between possible situations
|
||
|
expresses itself in language by means of an internal relation between the
|
||
|
propositions representing them.
|
||
|
|
||
|
Here we have the answer to the vexed question 'whether all relations
|
||
|
are internal or external'.
|
||
|
|
||
|
We can now talk about formal concepts, in the same sense that we
|
||
|
speak of formal properties. I introduce this expression in order to
|
||
|
exhibit the source of the confusion between formal concepts and concepts
|
||
|
proper, which pervades the whole of traditional logic. When something
|
||
|
falls under a formal concept as one of its objects, this cannot be
|
||
|
expressed by means of a proposition. Instead it is shown in the very sign
|
||
|
for this object. A name shows that it signifies an object, a sign for a
|
||
|
number that it signifies a number, etc. Formal concepts cannot, in fact,
|
||
|
be represented by means of a function, as concepts proper can. For their
|
||
|
characteristics, formal properties, are not expressed by means of
|
||
|
functions. The expression for a formal property is a feature of certain
|
||
|
symbols. So the sign for the characteristics of a formal concept is a
|
||
|
distinctive feature of all symbols whose meanings fall under the concept.
|
||
|
So the expression for a formal concept is a propositional variable in which
|
||
|
this distinctive f.
|
||
|
|
||
|
The propositional variable signifies the formal concept, and its
|
||
|
values signify the objects that fall under the concept.
|
||
|
|
||
|
Every variable is the sign for a formal concept. For every variable
|
||
|
represents a constant form that all its values possess, and this can be
|
||
|
regarded as a formal property of those values.
|
||
|
|
||
|
|
||
|
A formal concept is given immediately any object falling under it
|
||
|
is given. It is not possible, therefore, to introduce as primitive ideas
|
||
|
objects belonging to a formal concept and the formal concept itself. So it
|
||
|
is impossible, for example, to introduce as primitive ideas both the
|
||
|
concept of a function and specific functions, as Russell does; or the
|
||
|
concept of a number and particular numbers.
|
||
|
|
||
|
This is what Frege and Russell overlooked:
|
||
|
consequently the way in which they want to express general propositions
|
||
|
like the one above is incorrect; it contains a vicious circle. We can
|
||
|
determine the general term of a series of forms by giving its first term
|
||
|
and the general form of the operation that produces the next term out of
|
||
|
the proposition that precedes it.
|
||
|
|
||
|
To ask whether a formal concept exists is nonsensical. For no
|
||
|
proposition can be the answer to such a question. So, for example, the
|
||
|
question, 'Are there unanalysable subject-predicate propositions?' cannot
|
||
|
be asked.
|
||
|
|
||
|
Logical forms are without number. Hence there are no preeminent
|
||
|
numbers in logic, and hence there is no possibility of philosophical monism
|
||
|
or dualism, etc.
|
||
|
|
||
|
The sense of a proposition is its agreement and disagreement with
|
||
|
possibilities of existence and non-existence of states of affairs. 4.21 The
|
||
|
simplest kind of proposition, an elementary proposition, asserts the
|
||
|
existence of a state of affairs.
|
||
|
|
||
|
It is a sign of a proposition's being elementary that there can be no
|
||
|
elementary proposition contradicting it.
|
||
|
|
||
|
An elementary proposition consists of names. It is a nexus, a
|
||
|
concatenation, of names.
|
||
|
|
||
|
It is obvious that the analysis of propositions must bring us to
|
||
|
elementary propositions which consist of names in immediate combination.
|
||
|
This raises the question how such combination into propositions comes
|
||
|
about.
|
||
|
|
||
|
Even if the world is infinitely complex, so that every fact consists
|
||
|
of infinitely many states of affairs and every state of affairs is composed
|
||
|
of infinitely many objects, there would still have to be objects and states
|
||
|
of affairs.
|
||
|
|
||
|
It is only in the nexus of an elementary proposition that a name
|
||
|
occurs in a proposition.
|
||
|
|
||
|
When I use two signs with one and the same meaning, I express this by
|
||
|
putting the sign '=' between them. So 'a = b' means that the sign 'b' can
|
||
|
be substituted for the sign 'a'.A definition is a rule dealing with signs.
|
||
|
|
||
|
Expressions of the form 'a = b' are, therefore, mere representational
|
||
|
devices. They state nothing about the meaning of the signs 'a' and 'b'.
|
||
|
|
||
|
Can we understand two names without knowing whether they signify the
|
||
|
same thing or two different things?--Can we understand a proposition in
|
||
|
which two names occur without knowing whether their meaning is the same or
|
||
|
different? Suppose I know the meaning of an English word and of a German
|
||
|
word that means the same: then it is impossible for me to be unaware that
|
||
|
they do mean the same; I must be capable of translating each into the
|
||
|
other. Expressions like 'a = a', and those derived from them, are neither
|
||
|
elementary propositions nor is there any other way in which they have
|
||
|
sense. This will become evident later.
|
||
|
|
||
|
If an elementary proposition is true, the state of affairs exists: if
|
||
|
an elementary proposition is false, the state of affairs does not exist.
|
||
|
|
||
|
If all true elementary propositions are given, the result is a
|
||
|
complete description of the world. The world is completely described by
|
||
|
giving all elementary propositions, and adding which of them are true and
|
||
|
which false. For n states of affairs, there are possibilities of existence
|
||
|
and non-existence. Of these states of affairs any combination can exist and
|
||
|
the remainder not exist.
|
||
|
|
||
|
There correspond to these combinations the same number of
|
||
|
possibilities of truth--and falsity--for n elementary propositions.
|
||
|
|
||
|
Truth-possibilities of elementary propositions mean Possibilities of
|
||
|
existence and non-existence of states of affairs.
|
||
|
|
||
|
We can represent truth-possibilities by schemata of the following kind
|
||
|
'T' means 'true', 'F' means 'false'; the rows of 'T's' and 'F's' under the
|
||
|
row of elementary propositions symbolize their truth-possibilities in a way
|
||
|
that can easily be understood:
|
||
|
|
||
|
A proposition is an expression of agreement and disagreement with truth-
|
||
|
possibilities of elementary propositions.
|
||
|
|
||
|
Truth-possibilities of elementary propositions are the conditions of
|
||
|
the truth and falsity of propositions.
|
||
|
|
||
|
It immediately strikes one as probable that the introduction of
|
||
|
elementary propositions provides the basis for understanding all other
|
||
|
kinds of proposition. Indeed the understanding of general propositions
|
||
|
palpably depends on the understanding of elementary propositions.
|
||
|
|
||
|
For n elementary propositions there are ways in which a proposition
|
||
|
can agree and disagree with their truth possibilities.
|
||
|
|
||
|
We can express agreement with truth-possibilities by correlating the
|
||
|
mark 'T' true with them in the schema. The absence of this mark means
|
||
|
disagreement.
|
||
|
|
||
|
The expression of agreement and disagreement with the truth
|
||
|
possibilities of elementary propositions expresses the truth-conditions of
|
||
|
a proposition. A proposition is the expression of its truth-conditions.
|
||
|
Thus Frege was quite right to use them as a starting point when he
|
||
|
explained the signs of his conceptual notation. But the explanation of the
|
||
|
concept of truth that Frege gives is mistaken: if 'the true' and 'the
|
||
|
false' were really objects, and were the arguments in Pp etc., then Frege's
|
||
|
method of determining the sense of 'Pp' would leave it absolutely
|
||
|
undetermined.
|
||
|
|
||
|
The sign that results from correlating the mark 'I" with truth-
|
||
|
possibilities is a propositional sign.
|
||
|
|
||
|
It is clear that a complex of the signs 'F' and 'T' has no object or
|
||
|
complex of objects corresponding to it, just as there is none
|
||
|
corresponding to the horizontal and vertical lines or to the brackets.--
|
||
|
There are no 'logical objects'. Of course the same applies to all signs
|
||
|
that express what the schemata of 'T's' and 'F's' express.
|
||
|
|
||
|
For example, the following is a propositional sign: Frege's
|
||
|
'judgement stroke' '|-' is logically quite meaningless: in the works of
|
||
|
Frege and Russell it simply indicates that these authors hold the
|
||
|
propositions marked with this sign to be true. Thus '|-' is no more a
|
||
|
component part of a proposition than is, for instance, the proposition's
|
||
|
number. It is quite impossible for a proposition to state that it itself is
|
||
|
true. If the order or the truth-possibilities in a scheme is fixed once
|
||
|
and for all by a combinatory rule, then the last column by itself will be
|
||
|
an expression of the truth-conditions. If we now write this column as a
|
||
|
row, the propositional sign will become 'TT-T p,q' or more explicitly
|
||
|
'TTFT p,q' The number of places in the left-hand pair of brackets is
|
||
|
determined by the number of terms in the right-hand pair.
|
||
|
|
||
|
For n elementary propositions there are Ln possible groups of truth-
|
||
|
conditions. The groups of truth-conditions that are obtainable from the
|
||
|
truth-possibilities of a given number of elementary propositions can be
|
||
|
arranged in a series.
|
||
|
|
||
|
Among the possible groups of truth-conditions there are two extreme
|
||
|
cases. In one of these cases the proposition is true for all the truth-
|
||
|
possibilities of the elementary propositions. We say that the truth-
|
||
|
conditions are tautological. In the second case the proposition is false
|
||
|
for all the truth-possibilities: the truth-conditions are contradictory.
|
||
|
In the first case we call the proposition a tautology; in the second, a
|
||
|
contradiction.
|
||
|
|
||
|
Propositions show what they say; tautologies and contradictions show
|
||
|
that they say nothing. A tautology has no truth-conditions, since it is
|
||
|
unconditionally true: and a contradiction is true on no condition.
|
||
|
Tautologies and contradictions lack sense. Like a point from which two
|
||
|
arrows go out in opposite directions to one another. For example, I know
|
||
|
nothing about the weather when I know that it is either raining or not
|
||
|
raining.
|
||
|
|
||
|
Tautologies and contradictions are not, however, nonsensical. They
|
||
|
are part of the symbolism, much as '0' is part of the symbolism of
|
||
|
arithmetic.
|
||
|
|
||
|
Tautologies and contradictions are not pictures of reality. They do
|
||
|
not represent any possible situations. For the former admit all possible
|
||
|
situations, and latter none. In a tautology the conditions of agreement
|
||
|
with the world--the representational relations--cancel one another, so that
|
||
|
it does not stand in any representational relation to reality.
|
||
|
|
||
|
The truth-conditions of a proposition determine the range that it
|
||
|
leaves open to the facts. A proposition, a picture, or a model is, in the
|
||
|
negative sense, like a solid body that restricts the freedom of movement of
|
||
|
others, and in the positive sense, like a space bounded by solid substance
|
||
|
in which there is room for a body. A tautology leaves open to reality the
|
||
|
whole--the infinite whole--of logical space: a contradiction fills the
|
||
|
whole of logical space leaving no point of it for reality. Thus neither of
|
||
|
them can determine reality in any way.
|
||
|
|
||
|
A tautology's truth is certain, a proposition's possible, a
|
||
|
contradiction's impossible. Certain, possible, impossible: here we have
|
||
|
the first indication of the scale that we need in the theory of
|
||
|
probability.
|
||
|
|
||
|
The logical product of a tautology and a proposition says the same
|
||
|
thing as the proposition. This product, therefore, is identical with the
|
||
|
proposition. For it is impossible to alter what is essential to a symbol
|
||
|
without altering its sense.
|
||
|
|
||
|
What corresponds to a determinate logical combination of signs is a
|
||
|
determinate logical combination of their meanings. It is only to the
|
||
|
uncombined signs that absolutely any combination corresponds. In other
|
||
|
words, propositions that are true for every situation cannot be
|
||
|
combinations of signs at all, since, if they were, only determinate
|
||
|
combinations of objects could correspond to them. And what is not a
|
||
|
logical combination has no combination of objects corresponding to it.
|
||
|
Tautology and contradiction are the limiting cases--indeed the
|
||
|
disintegration--of the combination of signs.
|
||
|
|
||
|
Admittedly the signs are still combined with one another even in
|
||
|
tautologies and contradictions--i.e. they stand in certain relations to one
|
||
|
another: but these relations have no meaning, they are not essential to the
|
||
|
symbol.
|
||
|
|
||
|
It now seems possible to give the most general propositional form: that
|
||
|
is, to give a description of the propositions of any sign-language
|
||
|
whatsoever in such a way that every possible sense can be expressed by a
|
||
|
symbol satisfying the description, and every symbol satisfying the
|
||
|
description can express a sense, provided that the meanings of the names
|
||
|
are suitably chosen. It is clear that only what is essential to the most
|
||
|
general propositional form may be included in its description--for
|
||
|
otherwise it would not be the most general form. The existence of a general
|
||
|
propositional form is proved by the fact that there cannot be a proposition
|
||
|
whose form could not have been foreseen i.e. constructed. The general
|
||
|
form of a proposition is: This is how things stand.
|
||
|
|
||
|
Suppose that I am given all elementary propositions: then I can simply
|
||
|
ask what propositions I can construct out of them. And there I have all
|
||
|
propositions, and that fixes their limits.
|
||
|
|
||
|
Propositions comprise all that follows from the totality of all
|
||
|
elementary propositions and, of course, from its being the totality of
|
||
|
them all. Thus, in a certain sense, it could be said that all
|
||
|
propositions were generalizations of elementary propositions.
|
||
|
|
||
|
The general propositional form is a variable.
|
||
|
|
||
|
5 A proposition is a truth-function of elementary propositions. An
|
||
|
elementary proposition is a truth-function of itself.
|
||
|
|
||
|
Elementary propositions are the truth-arguments of propositions.
|
||
|
|
||
|
The arguments of functions are readily confused with the affixes of
|
||
|
names. For both arguments and affixes enable me to recognize the meaning of
|
||
|
the signs containing them. For example, when Russell writes '+c', the 'c'
|
||
|
is an affix which indicates that the sign as a whole is the addition-sign
|
||
|
for cardinal numbers. But the use of this sign is the result of arbitrary
|
||
|
convention and it would be quite possible to choose a simple sign instead
|
||
|
of '+c'; in 'Pp' however, 'p' is not an affix but an argument: the sense of
|
||
|
'Pp' cannot be understood unless the sense of 'p' has been understood
|
||
|
already. In the name Julius Caesar 'Julius' is an affix. An affix is
|
||
|
always part of a description of the object to whose name we attach it: e.g.
|
||
|
the Caesar of the Julian gens. If I am not mistaken, Frege's theory about
|
||
|
the meaning of propositions and functions is based on the confusion between
|
||
|
an argument and an affix. Frege regarded the propositions of logic as
|
||
|
names, and their arguments as the affixes of those names.
|
||
|
|
||
|
Truth-functions can be arranged in series. That is the foundation of
|
||
|
the theory of probability.
|
||
|
|
||
|
If all the truth-grounds that are common to a number of propositions
|
||
|
are at the same time truth-grounds of a certain proposition, then we say
|
||
|
that the truth of that proposition follows from the truth of the others.
|
||
|
|
||
|
In particular, the truth of a proposition 'p' follows from the truth
|
||
|
of another proposition 'q' is all the truth-grounds of the latter are truth-
|
||
|
grounds of the former.
|
||
|
|
||
|
The truth-grounds of the one are contained in those of the other: p
|
||
|
follows from q.
|
||
|
|
||
|
A proposition affirms every proposition that follows from it.
|
||
|
|
||
|
'p. q' is one of the propositions that affirm 'p' and at the same
|
||
|
time one of the propositions that affirm 'q'. Two propositions are opposed
|
||
|
to one another if there is no proposition with a sense, that affirms them
|
||
|
both. Every proposition that contradicts another negate it.
|
||
|
|
||
|
When the truth of one proposition follows from the truth of others, we
|
||
|
can see this from the structure of the proposition.
|
||
|
|
||
|
If the truth of one proposition follows from the truth of others,
|
||
|
this finds expression in relations in which the forms of the propositions
|
||
|
stand to one another: nor is it necessary for us to set up these relations
|
||
|
between them, by combining them with one another in a single proposition;
|
||
|
on the contrary, the relations are internal, and their existence is an
|
||
|
immediate result of the existence of the propositions.
|
||
|
|
||
|
If p follows from q, I can make an inference from q to p, deduce p
|
||
|
from q. The nature of the inference can be gathered only from the two
|
||
|
propositions. They themselves are the only possible justification of the
|
||
|
inference. 'Laws of inference', which are supposed to justify inferences,
|
||
|
as in the works of Frege and Russell, have no sense, and would be
|
||
|
superfluous.
|
||
|
|
||
|
All deductions are made a priori.
|
||
|
|
||
|
One elementary proposition cannot be deduced form another.
|
||
|
|
||
|
There is no possible way of making an inference form the existence of
|
||
|
one situation to the existence of another, entirely different situation.
|
||
|
|
||
|
There is no causal nexus to justify such an inference.
|
||
|
|
||
|
We cannot infer the events of the future from those of the present.
|
||
|
|
||
|
The freedom of the will consists in the impossibility of knowing
|
||
|
actions that still lie in the future. We could know them only if causality
|
||
|
were an inner necessity like that of logical inference.--The connexion
|
||
|
between knowledge and what is known is that of logical necessity. 'A knows
|
||
|
that p is the case', has no sense if p is a tautology.
|
||
|
|
||
|
If the truth of a proposition does not follow from the fact that it
|
||
|
is self-evident to us, then its self-evidence in no way justifies our
|
||
|
belief in its truth.
|
||
|
|
||
|
If one proposition follows from another, then the latter says more
|
||
|
than the former, and the former less than the latter.
|
||
|
|
||
|
If p follows from q and q from p, then they are one and same
|
||
|
proposition.
|
||
|
|
||
|
A tautology follows from all propositions: it says nothing.
|
||
|
|
||
|
Contradiction is that common factor of propositions which no
|
||
|
proposition has in common with another. Tautology is the common factor of
|
||
|
all propositions that have nothing in common with one another.
|
||
|
Contradiction, one might say, vanishes outside all propositions: tautology
|
||
|
vanishes inside them. Contradiction is the outer limit of propositions:
|
||
|
tautology is the unsubstantial point at their centre.
|
||
|
|
||
|
If Tr is the number of the truth-grounds of a proposition 'r', and if
|
||
|
Trs is the number of the truth-grounds of a proposition 's' that are at the
|
||
|
same time truth-grounds of 'r', then we call the ratio Trs : Tr the degree
|
||
|
of probability that the proposition 'r' gives to the proposition 's'. 5.151
|
||
|
In a schema like the one above in
|
||
|
|
||
|
|
||
|
There is no special object peculiar to probability propositions.
|
||
|
|
||
|
When propositions have no truth-arguments in common with one another,
|
||
|
we call them independent of one another. Two elementary propositions give
|
||
|
one another the probability 1/2. If p follows from q, then the proposition
|
||
|
'q' gives to the proposition 'p' the probability 1. The certainty of
|
||
|
logical inference is a limiting case of probability. Application of this
|
||
|
to tautology and contradiction.
|
||
|
|
||
|
In itself, a proposition is neither probable nor improbable. Either
|
||
|
an event occurs or it does not: there is no middle way.
|
||
|
|
||
|
Suppose that an urn contains black and white balls in equal numbers
|
||
|
and none of any other kind. I draw one ball after another, putting them
|
||
|
back into the urn. By this experiment I can establish that the number of
|
||
|
black balls drawn and the number of white balls drawn approximate to one
|
||
|
another as the draw continues. So this is not a mathematical truth. Now, if
|
||
|
I say, 'The probability of my drawing a white ball is equal to the
|
||
|
probability of my drawing a black one', this means that all the
|
||
|
circumstances that I know of including the laws of nature assumed as
|
||
|
hypotheses give no more probability to the occurrence of the one event
|
||
|
than to that of the other. That is to say, they give each the probability
|
||
|
1/2 as can easily be gathered from the above definitions. What I confirm by
|
||
|
the experiment is that the occurrence of the two events is independent of
|
||
|
the circumstances of which I have no more detailed knowledge.
|
||
|
|
||
|
The minimal unit for a probability proposition is this: The
|
||
|
circumstances--of which I have no further knowledge--give such and such a
|
||
|
degree of probability to the occurrence of a particular event.
|
||
|
|
||
|
It is in this way that probability is a generalization. It involves a
|
||
|
general description of a propositional form. We use probability only in
|
||
|
default of certainty--if our knowledge of a fact is not indeed complete,
|
||
|
but we do know something about its form. A proposition may well be an
|
||
|
incomplete picture of a certain situation, but it is always a complete
|
||
|
picture of something. A probability proposition is a sort of excerpt from
|
||
|
other propositions.
|
||
|
|
||
|
The structures of propositions stand in internal relations to one
|
||
|
another.
|
||
|
|
||
|
In order to give prominence to these internal relations we can adopt
|
||
|
the following mode of expression: we can represent a proposition as the
|
||
|
result of an operation that produces it out of other propositions which
|
||
|
are the bases of the operation.
|
||
|
|
||
|
An operation is the expression of a relation between the structures of
|
||
|
its result and of its bases.
|
||
|
|
||
|
The operation is what has to be done to the one proposition in order
|
||
|
to make the other out of it.
|
||
|
|
||
|
And that will, of course, depend on their formal properties, on the
|
||
|
internal similarity of their forms.
|
||
|
|
||
|
The internal relation by which a series is ordered is equivalent to
|
||
|
the operation that produces one term from another.
|
||
|
|
||
|
Operations cannot make their appearance before the point at which one
|
||
|
proposition is generated out of another in a logically meaningful way; i.e.
|
||
|
the point at which the logical construction of propositions begins.
|
||
|
|
||
|
Truth-functions of elementary propositions are results of operations
|
||
|
with elementary propositions as bases. These operations I call truth-
|
||
|
operations.
|
||
|
|
||
|
The sense of a truth-function of p is a function of the sense of p.
|
||
|
Negation, logical addition, logical multiplication, etc. etc. are
|
||
|
operations. Negation reverses the sense of a proposition.
|
||
|
|
||
|
An operation manifests itself in a variable; it shows how we can get
|
||
|
from one form of proposition to another. It gives expression to the
|
||
|
difference between the forms. And what the bases of an operation and its
|
||
|
result have in common is just the bases themselves.
|
||
|
|
||
|
An operation is not the mark of a form, but only of a difference
|
||
|
between forms.
|
||
|
|
||
|
The occurrence of an operation does not characterize the sense of a
|
||
|
proposition. Indeed, no statement is made by an operation, but only by its
|
||
|
result, and this depends on the bases of the operation. Operations and
|
||
|
functions must not be confused with each other.
|
||
|
|
||
|
A function cannot be its own argument, whereas an operation can take
|
||
|
one of its own results as its base.
|
||
|
|
||
|
It is only in this way that the step from one term of a series of
|
||
|
forms to another is possible from one type to another in the hierarchies
|
||
|
of Russell and Whitehead. Russell and Whitehead did not admit the
|
||
|
possibility of such steps, but repeatedly availed themselves of it.
|
||
|
|
||
|
|
||
|
The concept of successive applications of an operation is equivalent
|
||
|
to the concept 'and so on'.
|
||
|
|
||
|
One operation can counteract the effect of another. Operations can
|
||
|
cancel one another.
|
||
|
|
||
|
An operation can vanish e.g. negation in 'PPp' : PPp = p.
|
||
|
|
||
|
All propositions are results of truth-operations on elementary
|
||
|
propositions. A truth-operation is the way in which a truth-function is
|
||
|
produced out of elementary propositions. It is of the essence of truth-
|
||
|
operations that, just as elementary propositions yield a truth-function of
|
||
|
themselves, so too in the same way truth-functions yield a further truth-
|
||
|
function. When a truth-operation is applied to truth-functions of
|
||
|
elementary propositions, it always generates another truth-function of
|
||
|
elementary propositions, another proposition. When a truth-operation is
|
||
|
applied to the results of truth-operations on elementary propositions,
|
||
|
there is always a single operation on elementary propositions that has the
|
||
|
same result. Every proposition is the result of truth-operations on
|
||
|
elementary propositions.
|
||
|
|
||
|
All truth-functions are results of successive applications to
|
||
|
elementary propositions of a finite number of truth-operations.
|
||
|
|
||
|
At this point it becomes manifest that there are no 'logical objects'
|
||
|
or 'logical constants' in Frege's and Russell's sense.
|
||
|
|
||
|
The reason is that the results of truth-operations on truth-functions
|
||
|
are always identical whenever they are one and the same truth-function of
|
||
|
elementary propositions.
|
||
|
|
||
|
Even at first sight it seems scarcely credible that there should
|
||
|
follow from one fact p infinitely many others , namely PPp, PPPPp, etc. And
|
||
|
it is no less remarkable that the infinite number of propositions of logic
|
||
|
mathematics follow from half a dozen 'primitive propositions'. But in
|
||
|
fact all the propositions of logic say the same thing, to wit nothing.
|
||
|
|
||
|
Truth-functions are not material functions. For example, an
|
||
|
affirmation can be produced by double negation: in such a case does it
|
||
|
follow that in some sense negation is contained in affirmation? Does 'PPp'
|
||
|
negate Pp, or does it affirm p--or both? The proposition 'PPp' is not about
|
||
|
negation, as if negation were an object: on the other hand, the possibility
|
||
|
of negation is already written into affirmation. And if there were an
|
||
|
object called 'P', it would follow that 'PPp' said something different from
|
||
|
what 'p' said, just because the one proposition would then be about P and
|
||
|
the other would not.
|
||
|
|
||
|
If we are given a proposition, then with it we are also given the
|
||
|
results of all truth-operations that have it as their base.
|
||
|
|
||
|
If there are primitive logical signs, then any logic that fails to
|
||
|
show clearly how they are placed relatively to one another and to justify
|
||
|
their existence will be incorrect. The construction of logic out of its
|
||
|
primitive signs must be made clear.
|
||
|
|
||
|
If logic has primitive ideas, they must be independent of one
|
||
|
another. If a primitive idea has been introduced, it must have been
|
||
|
introduced in all the combinations in which it ever occurs. It cannot,
|
||
|
therefore, be introduced first for one combination and later reintroduced
|
||
|
for another. For example, once negation has been introduced, we must
|
||
|
understand it both in propositions of the form 'Pp' and in propositions
|
||
|
like 'Pp C q', 'dx. Pfx', etc. We must not introduce it first for the
|
||
|
one class of cases and then for the other, since it would then be left in
|
||
|
doubt whether its meaning were the same in both cases, and no reason would
|
||
|
have been given for combining the signs in the same way in both cases. In
|
||
|
short, Frege's remarks about introducing signs by means of definitions in
|
||
|
The Fundamental Laws of Arithmetic also apply, mutatis mutandis, to the
|
||
|
introduction of primitive signs.
|
||
|
|
||
|
The introduction of any new device into the symbolism of logic is
|
||
|
necessarily a momentous event. In logic a new device should not be
|
||
|
introduced in brackets or in a footnote with what one might call a
|
||
|
completely innocent air. Thus in Russell and Whitehead's Principia
|
||
|
Mathematica there occur definitions and primitive propositions expressed in
|
||
|
words. Why this sudden appearance of words? It would require a
|
||
|
justification, but none is given, or could be given, since the procedure is
|
||
|
in fact illicit. But if the introduction of a new device has proved
|
||
|
necessary at a certain point, we must immediately ask ourselves, 'At what
|
||
|
points is the employment of this device now unavoidable ?' and its place in
|
||
|
logic must be made clear.
|
||
|
|
||
|
All numbers in logic stand in need of justification. Or rather, it
|
||
|
must become evident that there are no numbers in logic. There are no pre-
|
||
|
eminent numbers.
|
||
|
|
||
|
In logic there is no co-ordinate status, and there can be no
|
||
|
classification. In logic there can be no distinction between the general
|
||
|
and the specific.
|
||
|
|
||
|
The solutions of the problems of logic must be simple, since they
|
||
|
set the standard of simplicity. Men have always had a presentiment that
|
||
|
there must be a realm in which the answers to questions are symmetrically
|
||
|
combined--a priori--to form a self-contained system. A realm subject to the
|
||
|
law: Simplex sigillum veri.
|
||
|
|
||
|
|
||
|
Though it seems unimportant, it is in fact significant that the
|
||
|
pseudo-relations of logic, such as C and z, need brackets--unlike real
|
||
|
relations. Indeed, the use of brackets with these apparently primitive
|
||
|
signs is itself an indication that they are not primitive signs. And surely
|
||
|
no one is going to believe brackets have an independent meaning. 5.4611
|
||
|
Signs for logical operations are punctuation-marks,
|
||
|
|
||
|
It is clear that whatever we can say in advance about the form of all
|
||
|
propositions, we must be able to say all at once. An elementary
|
||
|
proposition really contains all logical operations in itself. For 'fa' says
|
||
|
the same thing as 'dx. fx. x = a' Wherever there is compositeness,
|
||
|
argument and function are present, and where these are present, we already
|
||
|
have all the logical constants. One could say that the sole logical
|
||
|
constant was what all propositions, by their very nature, had in common
|
||
|
with one another. But that is the general propositional form.
|
||
|
|
||
|
The general propositional form is the essence of a proposition.
|
||
|
|
||
|
To give the essence of a proposition means to give the essence of
|
||
|
all description, and thus the essence of the world.
|
||
|
|
||
|
The description of the most general propositional form is the
|
||
|
description of the one and only general primitive sign in logic.
|
||
|
|
||
|
Logic must look after itself. If a sign is possible , then it is also
|
||
|
capable of signifying. Whatever is possible in logic is also permitted.
|
||
|
The reason why 'Socrates is identical' means nothing is that there is no
|
||
|
property called 'identical'. The proposition is nonsensical because we have
|
||
|
failed to make an arbitrary determination, and not because the symbol, in
|
||
|
itself, would be illegitimate. In a certain sense, we cannot make mistakes
|
||
|
in logic.
|
||
|
|
||
|
Self-evidence, which Russell talked about so much, can become
|
||
|
dispensable in logic, only because language itself prevents every logical
|
||
|
mistake.--What makes logic a priori is the impossibility of illogical
|
||
|
thought.
|
||
|
|
||
|
We cannot give a sign the wrong sense.
|
||
|
|
||
|
Occam's maxim is, of course, not an arbitrary rule, nor one that is
|
||
|
justified by its success in practice: its point is that unnecessary units
|
||
|
in a sign-language mean nothing. Signs that serve one purpose are logically
|
||
|
equivalent, and signs that serve none are logically meaningless.
|
||
|
|
||
|
Frege says that any legitimately constructed proposition must have a
|
||
|
sense. And I say that any possible proposition is legitimately constructed,
|
||
|
and, if it has no sense, that can only be because we have failed to give a
|
||
|
meaning to some of its constituents. Even if we think that we have done
|
||
|
so. Thus the reason why 'Socrates is identical' says nothing is that we
|
||
|
have not given any adjectival meaning to the word 'identical'. For when it
|
||
|
appears as a sign for identity, it symbolizes in an entirely different way--
|
||
|
the signifying relation is a different one--therefore the symbols also are
|
||
|
entirely different in the two cases: the two symbols have only the sign in
|
||
|
common, and that is an accident.
|
||
|
|
||
|
The number of fundamental operations that are necessary depends
|
||
|
solely on our notation.
|
||
|
|
||
|
All that is required is that we should construct a system of signs
|
||
|
with a particular number of dimensions--with a particular mathematical
|
||
|
multiplicity
|
||
|
|
||
|
It is clear that this is not a question of a number of primitive
|
||
|
ideas that have to be signified, but rather of the expression of a rule.
|
||
|
|
||
|
Every truth-function is a result of successive applications to
|
||
|
elementary propositions of the operation '-----TE,....'. This
|
||
|
operation negates all the propositions in the right-hand pair of brackets,
|
||
|
and I call it the negation of those propositions.
|
||
|
|
||
|
When a bracketed expression has propositions as its terms--and the
|
||
|
order of the terms inside the brackets is indifferent--then I indicate it
|
||
|
by a sign of the form 'E'. 'E' is a variable whose values are terms of
|
||
|
the bracketed expression and the bar over the variable indicates that it is
|
||
|
the representative of ali its values in the brackets. What the values of the
|
||
|
variable are is something that is stipulated. The stipulation is a
|
||
|
description of the propositions that have the variable as their
|
||
|
representative. How the description of the terms of the bracketed
|
||
|
expression is produced is not essential. We can distinguish three kinds of
|
||
|
description: 1.Direct enumeration, in which case we can simply substitute
|
||
|
for the variable the constants that are its values; 2. giving a function fx
|
||
|
whose values for all values of x are the propositions to be described; 3.
|
||
|
giving a formal law that governs the construction of the propositions, in
|
||
|
which case the bracketed expression has as its members all the terms of a
|
||
|
series of forms.
|
||
|
|
||
|
It is obvious that we can easily express how propositions may be
|
||
|
constructed with this operation, and how they may not be constructed with
|
||
|
it; so it must be possible to find an exact expression for this.
|
||
|
|
||
|
|
||
|
How can logic--all-embracing logic, which mirrors the world--use such
|
||
|
peculiar crotchets and contrivances? Only because they are all connected
|
||
|
with one another in an infinitely fine network, the great mirror.
|
||
|
|
||
|
Once a notation has been established, there will be in it a rule
|
||
|
governing the construction of all propositions that negate p, a rule
|
||
|
governing the construction of all propositions that affirm p, and a rule
|
||
|
governing the construction of all propositions that affirm p or q; and so
|
||
|
on. These rules are equivalent to the symbols; and in them their sense is
|
||
|
mirrored.
|
||
|
|
||
|
Must the sign of a negative proposition be constructed with that of
|
||
|
the positive proposition? Why should it not be possible to express a
|
||
|
negative proposition by means of a negative fact? E.g. suppose that "a'
|
||
|
does not stand in a certain relation to 'b'; then this might be used to say
|
||
|
that aRb was not the case. But really even in this case the negative
|
||
|
proposition is constructed by an indirect use of the positive. The positive
|
||
|
proposition necessarily presupposes the existence of the negative
|
||
|
proposition and vice versa.
|
||
|
|
||
|
If E has as its values all the values of a function fx for all values
|
||
|
of x, then NE = Pdx. fx.
|
||
|
|
||
|
I dissociate the concept all from truth-functions. Frege and Russell
|
||
|
introduced generality in association with logical productor logical sum.
|
||
|
This made it difficult to understand the propositions 'dx. fx' and 'x
|
||
|
. fx', in which both ideas are embedded.
|
||
|
|
||
|
What is peculiar to the generality-sign is first, that it indicates a
|
||
|
logical prototype, and secondly, that it gives prominence to constants.
|
||
|
|
||
|
The generality-sign occurs as an argument.
|
||
|
|
||
|
If objects are given, then at the same time we are given all objects.
|
||
|
If elementary propositions are given, then at the same time all elementary
|
||
|
propositions are given.
|
||
|
|
||
|
It is incorrect to render the proposition 'dx. fx' in the words,
|
||
|
'fx is possible ' as Russell does. The certainty, possibility, or
|
||
|
impossibility of a situation is not expressed by a proposition, but by an
|
||
|
expression's being a tautology, a proposition with a sense, or a
|
||
|
contradiction. The precedent to which we are constantly inclined to appeal
|
||
|
must reside in the symbol itself.
|
||
|
|
||
|
We can describe the world completely by means of fully generalized
|
||
|
propositions, i.e. without first correlating any name with a particular
|
||
|
object.
|
||
|
|
||
|
A fully generalized proposition, like every other proposition, is
|
||
|
composite. This is shown by the fact that in 'dx, O. Ox' we have to
|
||
|
mention 'O' and 's' separately. They both, independently, stand in
|
||
|
signifying relations to the world, just as is the case in ungeneralized
|
||
|
propositions. It is a mark of a composite symbol that it has something in
|
||
|
common with other symbols.
|
||
|
|
||
|
The truth or falsity of every proposition does make some alteration
|
||
|
in the general construction of the world. And the range that the totality
|
||
|
of elementary propositions leaves open for its construction is exactly the
|
||
|
same as that which is delimited by entirely general propositions. If an
|
||
|
elementary proposition is true, that means, at any rate, one more true
|
||
|
elementary proposition.
|
||
|
|
||
|
Identity of object I express by identity of sign, and not by using a
|
||
|
sign for identity. Difference of objects I express by difference of signs.
|
||
|
|
||
|
It is self-evident that identity is not a relation between objects.
|
||
|
This becomes very clear if one considers, for example, the proposition 'x
|
||
|
: fx. z. x = a'. What this proposition says is simply that only a
|
||
|
satisfies the function f, and not that only things that have a certain
|
||
|
relation to a satisfy the function, Of course, it might then be said that
|
||
|
only a did have this relation to a; but in order to express that, we should
|
||
|
need the identity-sign itself.
|
||
|
|
||
|
Russell's definition of '=' is inadequate, because according to it
|
||
|
we cannot say that two objects have all their properties in common.
|
||
|
|
||
|
Roughly speaking, to say of two things that they are identical is
|
||
|
nonsense, and to say of one thing that it is identical with itself is to
|
||
|
say nothing at all.
|
||
|
|
||
|
The identity-sign, therefore, is not an essential constituent of
|
||
|
conceptual notation.
|
||
|
|
||
|
This also disposes of all the problems that were connected with such
|
||
|
pseudo-propositions. All the problems that Russell's 'axiom of infinity'
|
||
|
brings with it can be solved at this point. What the axiom of infinity is
|
||
|
intended to say would express itself in language through the existence of
|
||
|
infinitely many names with different meanings.
|
||
|
|
||
|
In the general propositional form propositions occur in other
|
||
|
propositions only as bases of truth-operations.
|
||
|
|
||
|
This shows too that there is no such thing as the soul--the subject,
|
||
|
etc.--as it is conceived in the superficial psychology of the present day.
|
||
|
Indeed a composite soul would no longer be a soul.
|
||
|
|
||
|
The correct explanation of the form of the proposition, 'A makes the
|
||
|
judgement p', must show that it is impossible for a judgement to be a piece
|
||
|
of nonsense. Russell's theory does not satisfy this requirement.
|
||
|
|
||
|
To perceive a complex means to perceive that its constituents are
|
||
|
related to one another in such and such a way. This no doubt also explains
|
||
|
why there are two possible ways of seeing the figure as a cube; and all
|
||
|
similar phenomena. For we really see two different facts. If I look in the
|
||
|
first place at the corners marked a and only glance at the b's, then the
|
||
|
a's appear to be in front, and vice versa.
|
||
|
|
||
|
We now have to answer a priori the question about all the possible
|
||
|
forms of elementary propositions. Elementary propositions consist of names.
|
||
|
Since, however, we are unable to give the number of names with different
|
||
|
meanings, we are also unable to give the composition of elementary
|
||
|
propositions.
|
||
|
|
||
|
Our fundamental principle is that whenever a question can be decided
|
||
|
by logic at all it must be possible to decide it without more ado. And if
|
||
|
we get into a position where we have to look at the world for an answer to
|
||
|
such a problem, that shows that we are on a completely wrong track.
|
||
|
|
||
|
The 'experience' that we need in order to understand logic is not
|
||
|
that something or other is the state of things, but that something is :
|
||
|
that, however, is not an experience. Logic is prior to every experience--
|
||
|
that something is so.
|
||
|
|
||
|
And if this were not so, how could we apply logic? We might put it
|
||
|
in this way: if there would be a logic even if there were no world, how
|
||
|
then could there be a logic given that there is a world?
|
||
|
|
||
|
Russell said that there were simple relations between different
|
||
|
numbers of things individuals. But between what numbers? And how is this
|
||
|
supposed to be decided?--By experience? There is no pre-eminent number.
|
||
|
|
||
|
It would be completely arbitrary to give any specific form.
|
||
|
|
||
|
It is supposed to be possible to answer a priori the question
|
||
|
whether I can get into a position in which I need the sign for a 27-termed
|
||
|
relation in order to signify something.
|
||
|
|
||
|
But is it really legitimate even to ask such a question? Can we set
|
||
|
up a form of sign without knowing whether anything can correspond to it?
|
||
|
Does it make sense to ask what there must be in order that something can be
|
||
|
the case?
|
||
|
|
||
|
Clearly we have some concept of elementary propositions quite apart
|
||
|
from their particular logical forms. But when there is a system by which we
|
||
|
can create symbols, the system is what is important for logic and not the
|
||
|
individual symbols. And anyway, is it really possible that in logic I
|
||
|
should have to deal with forms that I can invent? What I have to deal with
|
||
|
must be that which makes it possible for me to invent them.
|
||
|
|
||
|
There cannot be a hierarchy of the forms of elementary propositions.
|
||
|
We can foresee only what we ourselves construct.
|
||
|
|
||
|
Empirical reality is limited by the totality of objects. The limit
|
||
|
also makes itself manifest in the totality of elementary propositions.
|
||
|
Hierarchies are and must be independent of reality.
|
||
|
|
||
|
If we know on purely logical grounds that there must be elementary
|
||
|
propositions, then everyone who understands propositions in their C form
|
||
|
must know It.
|
||
|
|
||
|
In fact, all the propositions of our everyday language, just as they
|
||
|
stand, are in perfect logical order.--That utterly simple thing, which we
|
||
|
have to formulate here, is not a likeness of the truth, but the truth
|
||
|
itself in its entirety. Our problems are not abstract, but perhaps the
|
||
|
most concrete that there are.
|
||
|
|
||
|
The application of logic decides what elementary propositions there
|
||
|
are. What belongs to its application, logic cannot anticipate. It is clear
|
||
|
that logic must not clash with its application. But logic has to be in
|
||
|
contact with its application. Therefore logic and its application must not
|
||
|
overlap.
|
||
|
|
||
|
If I cannot say a priori what elementary propositions there are,
|
||
|
then the attempt to do so must lead to obvious nonsense. 5.6 The limits of
|
||
|
my language mean the limits of my world.
|
||
|
|
||
|
Logic pervades the world: the limits of the world are also its limits.
|
||
|
So we cannot say in logic, 'The world has this in it, and this, but not
|
||
|
that.' For that would appear to presuppose that we were excluding certain
|
||
|
possibilities, and this cannot be the case, since it would require that
|
||
|
logic should go beyond the limits of the world; for only in that way could
|
||
|
it view those limits from the other side as well. We cannot think what we
|
||
|
cannot think; so what we cannot think we cannot say either.
|
||
|
|
||
|
This remark provides the key to the problem, how much truth there is
|
||
|
in solipsism. For what the solipsist means is quite correct; only it cannot
|
||
|
be said , but makes itself manifest. The world is my world: this is
|
||
|
manifest in the fact that the limits of language of that language which
|
||
|
alone I understand mean the limits of my world.
|
||
|
|
||
|
The world and life are one.
|
||
|
|
||
|
There is no such thing as the subject that thinks or entertains
|
||
|
ideas. If I wrote a book called The World as l found it , I should have to
|
||
|
include a report on my body, and should have to say which parts were
|
||
|
subordinate to my will, and which were not, etc., this being a method of
|
||
|
isolating the subject, or rather of showing that in an important sense
|
||
|
there is no subject; for it alone could not be mentioned in that book.--
|
||
|
|
||
|
The subject does not belong to the world: rather, it is a limit of
|
||
|
the world.
|
||
|
|
||
|
Where in the world is a metaphysical subject to be found? You will
|
||
|
say that this is exactly like the case of the eye and the visual field. But
|
||
|
really you do not see the eye. And nothing in the visual field allows you
|
||
|
to infer that it is seen by an eye.
|
||
|
|
||
|
For the form of the visual field is surely not like this
|
||
|
|
||
|
This is connected with the fact that no part of our experience is at
|
||
|
the same time a priori. Whatever we see could be other than it is. Whatever
|
||
|
we can describe at all could be other than it is. There is no a priori
|
||
|
order of things.
|
||
|
|
||
|
Here it can be seen that solipsism, when its implications are followed
|
||
|
out strictly, coincides with pure realism. The self of solipsism shrinks to
|
||
|
a point without extension, and there remains the reality co-ordinated with
|
||
|
it.
|
||
|
|
||
|
Thus there really is a sense in which philosophy can talk about the
|
||
|
self in a non-psychological way. What brings the self into philosophy is
|
||
|
the fact that 'the world is my world'. The philosophical self is not the
|
||
|
human being, not the human body, or the human soul, with which psychology
|
||
|
deals, but rather the metaphysical subject, the limit of the world--not a
|
||
|
part of it.
|
||
|
|
||
|
What this says is just that every proposition is a result of
|
||
|
successive applications to elementary propositions of the operation NE
|
||
|
|
||
|
If we are given the general form according to which propositions are
|
||
|
constructed, then with it we are also given the general form according to
|
||
|
which one proposition can be generated out of another by means of an
|
||
|
operation.
|
||
|
|
||
|
A number is the exponent of an operation.
|
||
|
|
||
|
The concept of number is simply what is common to all numbers, the
|
||
|
general form of a number. The concept of number is the variable number. And
|
||
|
the concept of numerical equality is the general form of all particular
|
||
|
cases of numerical equality.
|
||
|
|
||
|
The theory of classes is completely superfluous in mathematics. This
|
||
|
is connected with the fact that the generality required in mathematics is
|
||
|
not accidental generality.
|
||
|
|
||
|
The propositions of logic are tautologies.
|
||
|
|
||
|
Therefore the propositions of logic say nothing. They are the
|
||
|
analytic propositions.
|
||
|
|
||
|
All theories that make a proposition of logic appear to have content
|
||
|
are false. One might think, for example, that the words 'true' and 'false'
|
||
|
signified two properties among other properties, and then it would seem to
|
||
|
be a remarkable fact that every proposition possessed one of these
|
||
|
properties. On this theory it seems to be anything but obvious, just as,
|
||
|
for instance, the proposition, 'All roses are either yellow or red', would
|
||
|
not sound obvious even if it were true. Indeed, the logical proposition
|
||
|
acquires all the characteristics of a proposition of natural science and
|
||
|
this is the sure sign that it has been construed wrongly.
|
||
|
|
||
|
The correct explanation of the propositions of logic must assign to
|
||
|
them a unique status among all propositions.
|
||
|
|
||
|
It is the peculiar mark of logical propositions that one can
|
||
|
recognize that they are true from the symbol alone, and this fact contains
|
||
|
in itself the whole philosophy of logic. And so too it is a very important
|
||
|
fact that the truth or falsity of non-logical propositions cannot be
|
||
|
recognized from the propositions alone.
|
||
|
|
||
|
The fact that the propositions of logic are tautologies shows the
|
||
|
formal--logical--properties of language and the world. The fact that a
|
||
|
tautology is yielded by this particular way of connecting its constituents
|
||
|
characterizes the logic of its constituents. If propositions are to yield a
|
||
|
tautology when they are connected in a certain way, they must have certain
|
||
|
structural properties. So their yielding a tautology when combined in this
|
||
|
shows that they possess these structural properties.
|
||
|
|
||
|
It is clear that one could achieve the same purpose by using
|
||
|
contradictions instead of tautologies.
|
||
|
|
||
|
In order to recognize an expression as a tautology, in cases where
|
||
|
no generality-sign occurs in it, one can employ the following intuitive
|
||
|
method: instead of 'p', 'q', 'r', etc. I write 'TpF', 'TqF', 'TrF', etc.
|
||
|
Truth-combinations I express by means of brackets, e.g. and I use lines to
|
||
|
express the correlation of the truth or falsity of the whole proposition
|
||
|
with the truth-combinations of its truth-arguments, in the following way So
|
||
|
this sign, for instance, would represent the proposition p z q. Now, by way
|
||
|
of example, I wish to examine the proposition Pp.Pp the law of
|
||
|
contradiction in order to determine whether it is a tautology. In our
|
||
|
notation the form 'PE' is written as and the form 'E. n' as Hence the
|
||
|
proposition Pp. Pp. reads as follows If we here substitute 'p' for 'q'
|
||
|
and examine how the outermost T and F are connected with the innermost
|
||
|
ones, the result will be that the truth of the whole proposition is
|
||
|
correlated with all the truth-combinations of its argument, and its falsity
|
||
|
with none of the truth-combinations.
|
||
|
|
||
|
The propositions of logic demonstrate the logical properties of
|
||
|
propositions by combining them so as to form propositions that say nothing.
|
||
|
This method could also be called a zero-method. In a logical proposition,
|
||
|
propositions are brought into equilibrium with one another, and the state
|
||
|
of equilibrium then indicates what the logical constitution of these
|
||
|
propositions must be.
|
||
|
|
||
|
It follows from this that we can actually do without logical
|
||
|
propositions; for in a suitable notation we can in fact recognize the
|
||
|
formal properties of propositions by mere inspection of the propositions
|
||
|
themselves.
|
||
|
|
||
|
This throws some light on the question why logical propositions
|
||
|
cannot be confirmed by experience any more than they can be refuted by it.
|
||
|
Not only must a proposition of logic be irrefutable by any possible
|
||
|
experience, but it must also be unconfirmable by any possible experience.
|
||
|
|
||
|
Now it becomes clear why people have often felt as if it were for us
|
||
|
to 'postulate ' the 'truths of logic'. The reason is that we can postulate
|
||
|
them in so far as we can postulate an adequate notation.
|
||
|
|
||
|
It also becomes clear now why logic was called the theory of forms
|
||
|
and of inference.
|
||
|
|
||
|
Clearly the laws of logic cannot in their turn be subject to laws of
|
||
|
logic. There is not, as Russell thought, a special law of contradiction
|
||
|
for each 'type'; one law is enough, since it is not applied to itself.
|
||
|
|
||
|
The mark of a logical proposition is not general validity. To be
|
||
|
general means no more than to be accidentally valid for all things. An
|
||
|
ungeneralized proposition can be tautological just as well as a generalized
|
||
|
one.
|
||
|
|
||
|
The general validity of logic might be called essential, in contrast
|
||
|
with the accidental general validity of such propositions as 'All men are
|
||
|
mortal'. Propositions like Russell's 'axiom of reducibility' are not
|
||
|
logical propositions, and this explains our feeling that, even if they were
|
||
|
true, their truth could only be the result of a fortunate accident.
|
||
|
|
||
|
It is possible to imagine a world in which the axiom of reducibility
|
||
|
is not valid. It is clear, however, that logic has nothing to do with the
|
||
|
question whether our world really is like that or not.
|
||
|
|
||
|
The propositions of logic describe the scaffolding of the world, or
|
||
|
rather they represent it. They have no 'subject-matter'. They presuppose
|
||
|
that names have meaning and elementary propositions sense; and that is
|
||
|
their connexion with the world. It is clear that something about the world
|
||
|
must be indicated by the fact that certain combinations of symbols--whose
|
||
|
essence involves the possession of a determinate character--are
|
||
|
tautologies. This contains the decisive point. We have said that some
|
||
|
things are arbitrary in the symbols that we use and that some things are
|
||
|
not. In logic it is only the latter that express: but that means that logic
|
||
|
is not a field in which we express what we wish with the help of signs, but
|
||
|
rather one in which the nature of the absolutely necessary signs speaks for
|
||
|
itself. If we know the logical syntax of any sign-language, then we have
|
||
|
already been given all the propositions of logic.
|
||
|
|
||
|
It is possible--indeed possible even according to the old conception
|
||
|
of logic--to give in advance a description of all 'true' logical
|
||
|
propositions.
|
||
|
|
||
|
Hence there can never be surprises in logic.
|
||
|
|
||
|
One can calculate whether a proposition belongs to logic, by
|
||
|
calculating the logical properties of the symbol. And this is what we do
|
||
|
when we 'prove' a logical proposition. For, without bothering about sense
|
||
|
or meaning, we construct the logical proposition out of others using only
|
||
|
rules that deal with signs. The proof of logical propositions consists in
|
||
|
the following process: we produce them out of other logical propositions by
|
||
|
successively applying certain operations that always generate further
|
||
|
tautologies out of the initial ones. And in fact only tautologies follow
|
||
|
from a tautology. Of course this way of showing that the propositions of
|
||
|
logic are tautologies is not at all essential to logic, if only because the
|
||
|
propositions from which the proof starts must show without any proof that
|
||
|
they are tautologies.
|
||
|
|
||
|
In logic process and result are equivalent. Hence the absence of
|
||
|
surprise.
|
||
|
|
||
|
Proof in logic is merely a mechanical expedient to facilitate the
|
||
|
recognition of tautologies in complicated cases.
|
||
|
|
||
|
Indeed, it would be altogether too remarkable if a proposition that
|
||
|
had sense could be proved logically from others, and so too could a logical
|
||
|
proposition. It is clear from the start that a logical proof of a
|
||
|
proposition that has sense and a proof in logic must be two entirely
|
||
|
different things.
|
||
|
|
||
|
A proposition that has sense states something, which is shown by its
|
||
|
proof to be so. In logic every proposition is the form of a proof. Every
|
||
|
proposition of logic is a modus ponens represented in signs. And one
|
||
|
cannot express the modus ponens by means of a proposition.
|
||
|
|
||
|
It is always possible to construe logic in such a way that every
|
||
|
proposition is its own proof.
|
||
|
|
||
|
All the propositions of logic are of equal status: it is not the case
|
||
|
that some of them are essentially derived propositions. Every tautology
|
||
|
itself shows that it is a tautology.
|
||
|
|
||
|
It is clear that the number of the 'primitive propositions of logic'
|
||
|
is arbitrary, since one could derive logic from a single primitive
|
||
|
proposition, e.g. by simply constructing the logical product of Frege's
|
||
|
primitive propositions. Frege would perhaps say that we should then no
|
||
|
longer have an immediately self-evident primitive proposition. But it is
|
||
|
remarkable that a thinker as rigorous as Frege appealed to the degree of
|
||
|
self-evidence as the criterion of a logical proposition.
|
||
|
|
||
|
Logic is not a body of doctrine, but a mirror-image of the world.
|
||
|
Logic is transcendental.
|
||
|
|
||
|
Mathematics is a logical method. The propositions of mathematics are
|
||
|
equations, and therefore pseudo-propositions.
|
||
|
|
||
|
A proposition of mathematics does not express a thought.
|
||
|
|
||
|
Indeed in real life a mathematical proposition is never what we want.
|
||
|
Rather, we make use of mathematical propositions only in inferences from
|
||
|
propositions that do not belong to mathematics to others that likewise do
|
||
|
not belong to mathematics. In philosophy the question, 'What do we
|
||
|
actually use this word or this proposition for?' repeatedly leads to
|
||
|
valuable insights.
|
||
|
|
||
|
The logic of the world, which is shown in tautologies by the
|
||
|
propositions of logic, is shown in equations by mathematics.
|
||
|
|
||
|
If two expressions are combined by means of the sign of equality, that
|
||
|
means that they can be substituted for one another. But it must be manifest
|
||
|
in the two expressions themselves whether this is the case or not. When two
|
||
|
expressions can be substituted for one another, that characterizes their
|
||
|
logical form.
|
||
|
|
||
|
Frege says that the two expressions have the same meaning but
|
||
|
different senses. But the essential point about an equation is that it is
|
||
|
not necessary in order to show that the two expressions connected by the
|
||
|
sign of equality have the same meaning, since this can be seen from the two
|
||
|
expressions themselves.
|
||
|
|
||
|
And the possibility of proving the propositions of mathematics means
|
||
|
simply that their correctness can be perceived without its being necessary
|
||
|
that what they express should itself be compared with the facts in order to
|
||
|
determine its correctness.
|
||
|
|
||
|
It is impossible to assert the identity of meaning of two
|
||
|
expressions. For in order to be able to assert anything about their
|
||
|
meaning, I must know their meaning, and I cannot know their meaning without
|
||
|
knowing whether what they mean is the same or different.
|
||
|
|
||
|
An equation merely marks the point of view from which I consider the
|
||
|
two expressions: it marks their equivalence in meaning.
|
||
|
|
||
|
The question whether intuition is needed for the solution of
|
||
|
mathematical problems must be given the answer that in this case language
|
||
|
itself provides the necessary intuition.
|
||
|
|
||
|
The process of calculating serves to bring about that intuition.
|
||
|
Calculation is not an experiment.
|
||
|
|
||
|
Mathematics is a method of logic.
|
||
|
|
||
|
It is the essential characteristic of mathematical method that it
|
||
|
employs equations. For it is because of this method that every proposition
|
||
|
of mathematics must go without saying.
|
||
|
|
||
|
The method by which mathematics arrives at its equations is the method
|
||
|
of substitution. For equations express the substitutability of two
|
||
|
expressions and, starting from a number of equations, we advance to new
|
||
|
equations by substituting different expressions in accordance with the
|
||
|
equations.
|
||
|
|
||
|
|
||
|
The so-called law of induction cannot possibly be a law of logic,
|
||
|
since it is obviously a proposition with sense.---Nor, therefore, can it be
|
||
|
an a priori law.
|
||
|
|
||
|
The law of causality is not a law but the form of a law.
|
||
|
|
||
|
'Law of causality'--that is a general name. And just as in mechanics,
|
||
|
for example, there are 'minimum-principles', such as the law of least
|
||
|
action, so too in physics there are causal laws, laws of the causal form.
|
||
|
|
||
|
Indeed people even surmised that there must be a 'law of least
|
||
|
action' before they knew exactly how it went. Here, as always, what is
|
||
|
certain a priori proves to be something purely logical.
|
||
|
|
||
|
We do not have an a priori belief in a law of conservation, but rather
|
||
|
a priori knowledge of the possibility of a logical form.
|
||
|
|
||
|
All such propositions, including the principle of sufficient reason,
|
||
|
tile laws of continuity in nature and of least effort in nature, etc. etc.--
|
||
|
all these are a priori insights about the forms in which the propositions
|
||
|
of science can be cast.
|
||
|
|
||
|
Newtonian mechanics, for example, imposes a unified form on the
|
||
|
description of the world. Let us imagine a white surface with irregular
|
||
|
black spots on it. We then say that whatever kind of picture these make, I
|
||
|
can always approximate as closely as I wish to the description of it by
|
||
|
covering the surface with a sufficiently fine square mesh, and then saying
|
||
|
of every square whether it is black or white. In this way I shall have
|
||
|
imposed a unified form on the description of the surface. The form is
|
||
|
optional, since I could have achieved the same result by using a net with a
|
||
|
triangular or hexagonal mesh. Possibly the use of a triangular mesh would
|
||
|
have made the description simpler: that is to say, it might be that we
|
||
|
could describe the surface more accurately with a coarse triangular mesh
|
||
|
than with a fine square mesh or conversely, and so on. The different nets
|
||
|
correspond to different systems for describing the world. Mechanics
|
||
|
determines one form of description of the world by saying that all
|
||
|
propositions used in the description of the world must be obtained in a
|
||
|
given way from a given set of propositions--the axioms of mechanics. It
|
||
|
thus supplies the bricks for building the edifice of science, and it says,
|
||
|
'Any building that you want to erect, whatever it may be, must somehow be
|
||
|
constructed with these bricks, and with these alone.' Just as with the
|
||
|
number-system we must be able to write down any number we wish, so with the
|
||
|
system of mechanics we must be able to write down any proposition of
|
||
|
physics that we wish.
|
||
|
|
||
|
And now we can see the relative position of logic and mechanics. The
|
||
|
net might also consist of more than one kind of mesh: e.g. we could use
|
||
|
both triangles and hexagons. The possibility of describing a picture like
|
||
|
the one mentioned above with a net of a given form tells us nothing about
|
||
|
the picture. For that is true of all such pictures. But what does
|
||
|
characterize the picture is that it can be described completely by a
|
||
|
particular net with a particular size of mesh. Similarly the possibility of
|
||
|
describing the world by means of Newtonian mechanics tells us nothing about
|
||
|
the world: but what does tell us something about it is the precise way in
|
||
|
which it is possible to describe it by these means. We are also told
|
||
|
something about the world by the fact that it can be described more simply
|
||
|
with one system of mechanics than with another.
|
||
|
|
||
|
Mechanics is an attempt to construct according to a single plan all
|
||
|
the true propositions that we need for the description of the world.
|
||
|
|
||
|
The laws of physics, with all their logical apparatus, still speak,
|
||
|
however indirectly, about the objects of the world.
|
||
|
|
||
|
We ought not to forget that any description of the world by means of
|
||
|
mechanics will be of the completely general kind. For example, it will
|
||
|
never mention particular point-masses: it will only talk about any point-
|
||
|
masses whatsoever.
|
||
|
|
||
|
Although the spots in our picture are geometrical figures,
|
||
|
nevertheless geometry can obviously say nothing at all about their actual
|
||
|
form and position. The network, however, is purely geometrical; all its
|
||
|
properties can be given a priori. Laws like the principle of sufficient
|
||
|
reason, etc. are about the net and not about what the net describes.
|
||
|
|
||
|
If there were a law of causality, it might be put in the following
|
||
|
way: There are laws of nature. But of course that cannot be said: it makes
|
||
|
itself manifest.
|
||
|
|
||
|
One might say, using Hertt:'s terminology, that only connexions that
|
||
|
are subject to law are thinkable.
|
||
|
|
||
|
We cannot compare a process with 'the passage of time'--there is no
|
||
|
such thing--but only with another process such as the working of a
|
||
|
chronometer. Hence we can describe the lapse of time only by relying on
|
||
|
some other process. Something exactly analogous applies to space: e.g. when
|
||
|
people say that neither of two events which exclude one another can
|
||
|
occur, because there is nothing to cause the one to 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 to be found. And if such an
|
||
|
asymmetry is to be found, we can regard it as the cause of the occurrence
|
||
|
of the one and the non-occurrence of the other.
|
||
|
|
||
|
Kant's problem about the right hand and the left hand, which cannot
|
||
|
be made to coincide, exists even in two dimensions. Indeed, it exists in
|
||
|
one-dimensional space in which the two congruent figures, a and b, cannot
|
||
|
be made to coincide unless they are moved out of this space. The right hand
|
||
|
and the left hand are in fact completely congruent. It is quite irrelevant
|
||
|
that they cannot be made to coincide. A right-hand glove could be put on
|
||
|
the left hand, if it could be turned round in four-dimensional space.
|
||
|
|
||
|
What can be described can happen too: and what the law of causality
|
||
|
is meant to exclude cannot even be described.
|
||
|
|
||
|
The procedure of induction consists in accepting as true the simplest
|
||
|
law that can be reconciled with our experiences.
|
||
|
|
||
|
This procedure, however, has no logical justification but only a
|
||
|
psychological one. It is clear that there are no grounds for believing that
|
||
|
the simplest eventuality will in fact be realized.
|
||
|
|
||
|
It is an hypothesis that the sun will rise tomorrow: and this means
|
||
|
that we do not know whether it will rise.
|
||
|
|
||
|
There is no compulsion making one thing happen because another has
|
||
|
happened. The only necessity that exists is logical necessity.
|
||
|
|
||
|
The whole modern conception of the world is founded on the illusion
|
||
|
that the so-called laws of nature are the explanations of natural
|
||
|
phenomena.
|
||
|
|
||
|
Thus people today stop at the laws of nature, treating them as
|
||
|
something inviolable, just as God and Fate were treated in past ages. And
|
||
|
in fact both are right and both wrong: though the view of the ancients is
|
||
|
clearer in so far as they have a clear and acknowledged terminus, while the
|
||
|
modern system tries to make it look as if everything were explained.
|
||
|
|
||
|
The world is independent of my will.
|
||
|
|
||
|
Even if all that we wish for were to happen, still this would only be
|
||
|
a favour granted by fate, so to speak: for there is no logical connexion
|
||
|
between the will and the world, which would guarantee it, and the supposed
|
||
|
physical connexion itself is surely not something that we could will.
|
||
|
|
||
|
Just as the only necessity that exists is logical necessity, so too
|
||
|
the only impossibility that exists is logical impossibility.
|
||
|
|
||
|
For example, the simultaneous presence of two colours at the same
|
||
|
place in the visual field is impossible, in fact logically impossible,
|
||
|
since it is ruled out by the logical structure of colour. Let us think how
|
||
|
this contradiction appears in physics: more or less as follows--a particle
|
||
|
cannot have two velocities at the same time; that is to say, it cannot be
|
||
|
in two places at the same time; that is to say, particles that are in
|
||
|
different places at the same time cannot be identical. It is clear that
|
||
|
the logical product of two elementary propositions can neither be a
|
||
|
tautology nor a contradiction. The statement that a point in the visual
|
||
|
field has two different colours at the same time is a contradiction.
|
||
|
|
||
|
All propositions are of equal value.
|
||
|
|
||
|
The sense of the world must lie outside the world. In the world
|
||
|
everything is as it is, and everything happens as it does happen: in it no
|
||
|
value exists--and if it did exist, it would have no value. If there is any
|
||
|
value that does have value, it must lie outside the whole sphere of what
|
||
|
happens and is the case. For all that happens and is the case is
|
||
|
accidental. What makes it non-accidental cannot lie within the world, since
|
||
|
if it did it would itself be accidental. It must lie outside the world.
|
||
|
|
||
|
So too it is impossible for there to be propositions of ethics.
|
||
|
Propositions can express nothing that is higher.
|
||
|
|
||
|
It is clear that ethics cannot be put into words. Ethics is
|
||
|
transcendental. Ethics and aesthetics are one and the same.
|
||
|
|
||
|
When an ethical law of the form, 'Thou shalt...' is laid down, one's
|
||
|
first thought is, 'And what if I do, not do it?' It is clear, however, that
|
||
|
ethics has nothing to do with punishment and reward in the usual sense of
|
||
|
the terms. So our question about the consequences of an action must be
|
||
|
unimportant.--At least those consequences should not be events. For there
|
||
|
must be something right about the question we posed. There must indeed be
|
||
|
some kind of ethical reward and ethical punishment, but they must reside in
|
||
|
the action itself. And it is also clear that the reward must be something
|
||
|
pleasant and the punishment something unpleasant.
|
||
|
|
||
|
It is impossible to speak about the will in so far as it is the
|
||
|
subject of ethical attributes. And the will as a phenomenon is of interest
|
||
|
only to psychology.
|
||
|
|
||
|
If the good or bad exercise of the will does alter the world, it can
|
||
|
alter only the limits of the world, not the facts--not what can be
|
||
|
expressed by means of language. In short the effect must be that it becomes
|
||
|
an altogether different world. It must, so to speak, wax and wane as a
|
||
|
whole. The world of the happy man is a different one from that of the
|
||
|
unhappy man.
|
||
|
|
||
|
So too at death the world does not alter, but comes to an end.
|
||
|
|
||
|
Death is not an event in life: we do not live to experience death.
|
||
|
If we take eternity to mean not infinite temporal duration but
|
||
|
timelessness, then eternal life belongs to those who live in the present.
|
||
|
Our life has no end in just the way in which our visual field has no
|
||
|
limits.
|
||
|
|
||
|
Not only is there no guarantee of the temporal immortality of the
|
||
|
human soul, that is to say of its eternal survival after death; but, in any
|
||
|
case, this assumption completely fails to accomplish the purpose for which
|
||
|
it has always been intended. Or is some riddle solved by my surviving for
|
||
|
ever? Is not this eternal life itself as much of a riddle as our present
|
||
|
life? The solution of the riddle of life in space and time lies outside
|
||
|
space and time. It is certainly not the solution of any problems of
|
||
|
natural science that is required.
|
||
|
|
||
|
How things are in the world is a matter of complete indifference for
|
||
|
what is higher. God does not reveal himself in the world.
|
||
|
|
||
|
The facts all contribute only to setting the problem, not to its
|
||
|
solution.
|
||
|
|
||
|
It is not how things are in the world that is mystical, but that it
|
||
|
exists.
|
||
|
|
||
|
To view the world sub specie aeterni is to view it as a whole--a
|
||
|
limited whole. Feeling the world as a limited whole--it is this that is
|
||
|
mystical.
|
||
|
|
||
|
When the answer cannot be put into words, neither can the question be
|
||
|
put into words. The riddle does not exist. If a question can be framed at
|
||
|
all, it is also possible to answer it.
|
||
|
|
||
|
Scepticism is not irrefutable, but obviously nonsensical, when it
|
||
|
tries to raise doubts where no questions can be asked. For doubt can exist
|
||
|
only where a question exists, a question only where an answer exists, and
|
||
|
an answer only where something can be said.
|
||
|
|
||
|
We feel that even when all possible scientific questions have been
|
||
|
answered, the problems of life remain completely untouched. Of course there
|
||
|
are then no questions left, and this itself is the answer.
|
||
|
|
||
|
The solution of the problem of life is seen in the vanishing of the
|
||
|
problem. Is not this the reason why those who have found after a long
|
||
|
period of doubt that the sense of life became clear to them have then been
|
||
|
unable to say what constituted that sense?
|
||
|
|
||
|
There are, indeed, things that cannot be put into words. They make
|
||
|
themselves manifest. They are what is mystical.
|
||
|
|
||
|
The correct method in philosophy would really be the following: to say
|
||
|
nothing except what can be said, i.e. propositions of natural science--i.e.
|
||
|
something that has nothing to do with philosophy -- and then, whenever
|
||
|
someone else wanted to say something metaphysical, to demonstrate to him
|
||
|
that he had failed to give a meaning to certain signs in his propositions.
|
||
|
Although it would not be satisfying to the other person--he would not have
|
||
|
the feeling that we were teaching him philosophy--this method would be the
|
||
|
only strictly correct one.
|
||
|
|
||
|
My propositions are elucidatory in this way: he who understands me
|
||
|
finally recognizes them as senseless, when he has climbed out through them,
|
||
|
on them, over them. He must so to speak throw away the ladder, after he
|
||
|
has climbed up on it. He must transcend these propositions, and then he
|
||
|
will see the world aright.
|
||
|
|
||
|
What we cannot speak about we must pass over in silence.
|