## Tractatus Logico-Philosophicus

### Posted by Ali Reda | Posted in | Posted on 10/23/2013

### Bertrand Russell’s Preface

###
As one with a long experience of the difficulties of logic and of the deceptiveness of theories which seem irrefutable, I find myself unable to be sure of the rightness of a theory, merely on the ground that I cannot see any point on which it is wrong. But to have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance. This merit, in my opinion, belongs to Mr Wittgenstein’s book, and makes it one which no serious philosopher can afford to neglect.

### Preface

###
This book will perhaps only be understood by those who have themselves already thought the thoughts which are expressed in it or similar thoughts. It is therefore not a text-book. Its object would be attained if there were one person who read it with understanding and to whom it afforded pleasure.

The book deals with the problems of philosophy and shows, as I believe, that the method of formulating these problems rests on the misunderstanding of the logic of our language. Its whole meaning could be summed up somewhat as follows: What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent.

The book will, therefore, draw a limit to thinking, or rather—not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought). The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense.

How far my efforts agree with those of other philosophers I will not decide.Indeed what I have here written makes no claim to novelty in points of detail; and therefore I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another. On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved.

### Overview

###

There are seven main propositions in the text. These are:
- The world is everything that is the case.
- What is the case (a fact) is the existence of states of affairs.
- A logical picture of facts is a thought.
- A thought is a proposition with a sense.
- A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)
- The general form of a proposition is the general form of a truth function, which is: [\bar p,\bar\xi, N(\bar\xi)]. This is the general form of a proposition.
- Whereof one cannot speak, thereof one must be silent.

- The world is everything that is the case.
- What is the case (a fact) is the existence of states of affairs.
- A logical picture of facts is a thought.
- A thought is a proposition with a sense.
- A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)
- The general form of a proposition is the general form of a truth function, which is: [\bar p,\bar\xi, N(\bar\xi)]. This is the general form of a proposition.
- Whereof one cannot speak, thereof one must be silent.

### Questions of Ontology and Logical Atomism

###

World is the totality of all positive facts. Facts are made of states of affairs (atomic facts), Positive facts are the existence of states of affairs (what is the case) and from knowing what is the case, we can know what is not the case (negative facts); states of affairs are state of a combination of objects (objects and relations between them). Objects are simple, unanalyzable, and mutually independent. Facts exist in what Wittgenstein calls "logical space". Logical space is the domain of everything that is logically possible. For instance, though it is not true that Toronto is the capital of Canada, there is nothing illogical about supposing that it might be, so its possibility exists in logical space. Some items in logical space (for instance, "Ottawa is the capital of Canada") are true, while some items in logical space are false. True or false, everything in logical space is possible. "Love is purple" is not an item in logical space, because it is not logically possible (love is not the kind of thing to which we can ascribe a color).

Through Kenny's chess analogy, we can see Wittgenstein's logical atomism. For the sake of this analogy, the chess pieces (objects) and their positions (relations between objects) constitute game status (states of affairs) and therefore facts and the totality of facts is the entire particular game of chess. We can communicate such a game of chess in the exact way that Wittgenstein says a proposition represents the world. We might make a report for every piece's position using chess shortcuts like say "WR/KR1" to communicate a white rook's being on the square commonly labeled as king's rook 1. The logical form of our reports must be the same logical form of the chess pieces and their arrangement on the board in order to be meaningful. Our communication about the chess game must have as many possibilities for constituents and their arrangement as the game itself.

An object has internal and external properties. The internal properties are its logical form: what kind of object it is and how it can combine with\relate to other objects in states of affairs, it shows us which states of affairs the object can occur in. To return to the earlier example, we don't know what the number two is if we say it is purple. Thus, objects and their internal properties are what make up the substance of the world. But the material properties of the world are determined by the objects' external properties. For instance, the internal properties of yellow and red are indistinguishable (i.e they need light to be colored, so they are colorless internally without anything external) and they can both occur in the same sorts of states of affairs. The only way we can distinguish red from yellow is by their external properties, by saying that certain things are true of yellow that are not true of red, and vice versa. Substance is what subsists independently of what is the case.

### Picture Theory of Representation

###

The logical picture of the facts is the thought. We can only think logically because thoughts must share the logical form of what they are about which are facts. Thoughts contain the possibility of the state of affairs which it thinks. We can imagine worlds other than this one, but we cannot imagine worlds that do not have the same logical form as this one. That is, we can imagine a world where horses speak and grass is pink, but we cannot imagine a world without space, time, or color. We can think contradictory things (e.g. "It is raining and it is not raining"), but we cannot think illogical thoughts that have no sense. I cannot think, "The number two is purple". A thought is a proposition with a sense\meaning.
The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.

A propositional sign is any form of communication by a perceptible sign (written, spoken or signs) used in transmitting thoughts. A proposition is a propositional sign’s projective relation to the world. The combination of objects in the state of affairs corresponds to the combination of objects of the thoughts corresponds to the combination of simple signs in the propositional sign. All share the same form and logical structure. The simple sign is the name of the object. The name means the object. The object is its meaning.
To understand a proposition means to know what is the case if it is true. It is understood by anyone who understands its constituents.

To reach the level of the state of affairs and its objects, we analyze the complex objects of everyday speech into simpler parts by means of definitions. In the analysis of "Plato talks with his pupils", "Plato" needs to be replaced with something like "the man who was the teacher of Aristotle". Wittgenstein went further and argued that it must be possible to continue this kind of analysis to a point at which no more subdivision would be possible. Ultimately, simple symbols—the symbols for names—cannot be further defined: they are fully analyzed so we cannot say what they mean. The meaning of a name is external to the name. The meaning of a name is the object it denotes, and there is nothing in the name itself (as a written or spoken sign) that can tell us what object it denotes. Rather, we learn the meaning of a name by observing how and in reference to what it is used. The meaning of a name lies outside the name. We must show what they mean by using what Wittgenstein calls "elucidations”. They are propositions which contain the names.

Propositions make "pictures" of facts in the world. The picture is a model of reality; it represents a fact by having its logical structure. The logical structure of the picture is the relation between the elements of the picture- the internal properties of these objects-. The logical structure of the fact is its states of affairs, the relation between its objects. I.e. The elements of a picture correspond to the objects of a fact. The situation represented by a picture is the sense of the picture. Comparing the picture of a fact and the reality – the sense of the picture agrees with reality or not- shows the truthfulness of a fact in the world.
It is laid against reality like a measure.

This comparison implies that the truthfulness of a fact requires empirical evidence. But this empirical experience is subjective, i.e it depends on the subject's beliefs, sensory organs. like one person who sees illusions and another one who doesn't.

Wittgenstein calls the possibility that things are related to one another -the possibility of this logical structure- "pictorial form".
A picture cannot, depict (describe) its pictorial form: it displays it.

He is making the important distinction between saying and showing.
What can be shown cannot be said.

In this way, linguistic expression can be seen as a form of geometric projection, where the many languages are the different forms of projection but the logical structure of the expression is the unchanging geometric relationships. We cannot say with language what is common in the structures, rather it must be shown, because any language we use will also rely on this relationship, and so we cannot step out of our language with language and in general stepping out of language is stepping out of our thoughts or out of the logical space which is impossible. And it is only because of this commonality that speech can be understood.
What expresses itself in language, we cannot express by means of language.

The sense of a proposition is internal to the proposition. While propositions can depict all of reality, they cannot depict its logical form.
A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.

Wittgenstein was inspired for this theory by the way that traffic courts in Paris reenact automobile accidents. A toy car is a representation of a real car, a toy truck is a representation of a real truck, and dolls are representations of people. In order to convey to a judge what happened in an automobile accident, someone in the courtroom might place the toy cars in a position like the position the real cars were in, and move them in the ways that the real cars moved. In this way, the elements of the picture (the toy cars) are in spatial relation to one another, and this relation itself pictures the spatial relation between the real cars in the automobile accident.

The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.

A propositional sign is any form of communication by a perceptible sign (written, spoken or signs) used in transmitting thoughts. A proposition is a propositional sign’s projective relation to the world. The combination of objects in the state of affairs corresponds to the combination of objects of the thoughts corresponds to the combination of simple signs in the propositional sign. All share the same form and logical structure. The simple sign is the name of the object. The name means the object. The object is its meaning.

To understand a proposition means to know what is the case if it is true. It is understood by anyone who understands its constituents.

Propositions make "pictures" of facts in the world. The picture is a model of reality; it represents a fact by having its logical structure. The logical structure of the picture is the relation between the elements of the picture- the internal properties of these objects-. The logical structure of the fact is its states of affairs, the relation between its objects. I.e. The elements of a picture correspond to the objects of a fact. The situation represented by a picture is the sense of the picture. Comparing the picture of a fact and the reality – the sense of the picture agrees with reality or not- shows the truthfulness of a fact in the world.

It is laid against reality like a measure.

This comparison implies that the truthfulness of a fact requires empirical evidence. But this empirical experience is subjective, i.e it depends on the subject's beliefs, sensory organs. like one person who sees illusions and another one who doesn't.

Wittgenstein calls the possibility that things are related to one another -the possibility of this logical structure- "pictorial form".

A picture cannot, depict (describe) its pictorial form: it displays it.

He is making the important distinction between saying and showing.

What can be shown cannot be said.

In this way, linguistic expression can be seen as a form of geometric projection, where the many languages are the different forms of projection but the logical structure of the expression is the unchanging geometric relationships. We cannot say with language what is common in the structures, rather it must be shown, because any language we use will also rely on this relationship, and so we cannot step out of our language with language and in general stepping out of language is stepping out of our thoughts or out of the logical space which is impossible. And it is only because of this commonality that speech can be understood.

What expresses itself in language, we cannot express by means of language.

The sense of a proposition is internal to the proposition. While propositions can depict all of reality, they cannot depict its logical form.

A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.

### Propositions

###

Every part of a proposition which characterizes its sense is an expression (a symbol). Wittgenstein accepts Frege's and Russell's view of a proposition as a function of the expressions. For instance, the variable function "the x is on the y." only gives sense when propositional variables are filled with expressions such as "hat" or "table" thus the variable function becomes a proposition.

The expressions themselves are meaningless outside the context of a proposition. This is an important remark to show that Language is the Source of Philosophical Confusion, when philosophers get confused over questions like whether the Good is more or less identical than the Beautiful or of what knowledge is; they are not confused because the essence of knowledge is difficult to identify. Rather, they are confused because they have abstracted a word from the contexts in which it has a function and find that, outside these contexts, the word loses its meaning. Another source of confusion is in the language of everyday life it very often happens that the same word signifies in two different ways—and therefore belongs to two different symbols. For example the word “is” appears as the copula, as the sign of equality, and as the expression of existence. Another source of confusion is that two words, which signify in different ways, are apparently applied in the same way in the proposition. For example, in the proposition “Green is green”—where the first word is a proper name as the last an adjective.

It is not surprising that the deepest problems are in fact not problems at all.

In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying a sign in the same way which signify different meanings. Two functions being used in different ways cannot possibly have the same meaning, i.e. be the same function. So no proposition can make a statement about itself, because a propositional sign cannot be contained in itself or else it would signify two different meanings.

Wittgenstein criticizes Russell's Theory of Types saying it can be disposed of simply by recognizing that a proposition that makes a statement about itself is being used in two different ways, and so it cannot be the same proposition. The "F" in "F(fx)" and the first "F" in "F(F(fx))" range over different kinds of variables, the letter by itself signifies nothing. He is saying that the theory of types is implicit in logic and a priori, Russell’s' paradox shouldn't happen in the first place.

The rules of logical syntax must go without saying, once we know how each individual sign signifies.

Wittgenstein introduces the notion of formal, or internal, properties, those properties that show themselves (as opposed to being spoken about) in a proposition. These properties define the logical structure of propositions, facts, and objects. We can’t think of the object as not having them. It would be as nonsensical (illogical) to ascribe a formal property to a proposition as to deny it the formal property. Propositions have the same internal properties as the facts they depict.

Ferge said that any proposition of the form "x is a y," "x" will represent an object and "y" will represent a concept. According to Wittgenstein, there is a fundamental difference between "x is a horse" and "x is a concept." Only grammar leads us to think the two are equivalent. A formal concept defines the formal properties of an object, state of affairs, or fact. Like "x is a number" We cannot say that x is a number: it being a number shows itself. Any attempt to use a formal concept in a proposition (e.g. "two is a number," "purple is a color") will result in a nonsensical pseudo-proposition. A concept proper is a function and can feature in propositions, like "x is a horse”.

It is not surprising that the deepest problems are in fact not problems at all.In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying a sign in the same way which signify different meanings. Two functions being used in different ways cannot possibly have the same meaning, i.e. be the same function. So no proposition can make a statement about itself, because a propositional sign cannot be contained in itself or else it would signify two different meanings.

The rules of logical syntax must go without saying, once we know how each individual sign signifies.

Wittgenstein introduces the notion of formal, or internal, properties, those properties that show themselves (as opposed to being spoken about) in a proposition. These properties define the logical structure of propositions, facts, and objects. We can’t think of the object as not having them. It would be as nonsensical (illogical) to ascribe a formal property to a proposition as to deny it the formal property. Propositions have the same internal properties as the facts they depict.

Ferge said that any proposition of the form "x is a y," "x" will represent an object and "y" will represent a concept. According to Wittgenstein, there is a fundamental difference between "x is a horse" and "x is a concept." Only grammar leads us to think the two are equivalent. A formal concept defines the formal properties of an object, state of affairs, or fact. Like "x is a number" We cannot say that x is a number: it being a number shows itself. Any attempt to use a formal concept in a proposition (e.g. "two is a number," "purple is a color") will result in a nonsensical pseudo-proposition. A concept proper is a function and can feature in propositions, like "x is a horse”.

### Logic and Propostions as Truth Functions

###

Propositions are built up as truth-functions of elementary propositions. An elementary proposition is a truth-function of itself. A proposition that shares all the truth- grounds of one or several other propositions is said to follow from those propositions. If one proposition follows from another, we can say the sense of the former is contained in the sense of the latter. For instance, the truth-grounds for "p" are contained in the truth-grounds for "p.q" ("p" is true in all those cases where "p.q" is true), so we can say that "p" follows from "p.q" and that the sense of "p" is contained in the sense of "p.q."

Wittgenstein emphasizes that the truth-value of a proposition has no bearing on its sense. True or false, it still makes a picture of the world, and we can still draw logical inferences from that picture. The propositions p and ~p depict the same possible situation, only they have opposite sense one says that the picture presented is the case, and the other says that it is not the case.

Wittgenstein's work here is that the sense of a proposition is given if its truth conditions are given. If we know under what circumstances a proposition is true and under what circumstances it is false, then we know all there is to know about that proposition. This is exactly what truth tables do. Any proposition, according to Wittgenstein, consists of one or more elementary proposition, each of which can be true or false independently of any other. If we put all the elementary propositions that constitute a given proposition into a truth table that lists all the possible combinations of true or false that can hold between them, we will have an exhaustive list of the truth-conditions of the given proposition. Thus, a truth table can show us the sense of the proposition. In logic process and result are equivalent. Wittgenstein did not invent truth tables, but their use in modern logic is usually traced to his introduction of them in the Tractatus.

The great advantage of this notation is that it expresses the sense of a proposition without any of the connectives (Logical constants) we normally find in logical notation, such as "and," "or," and "if… then." Frege builds his entire system from the "primitive" connective "not" and "if…then." Russell builds his from "not" and "or." These "primitive" connectives are in fact interchangeable (Frege's "if p then q" can be expressed in Russell's system as "q or not p," and Russell's "p or q" can be expressed by Frege's "if not p then q"). If the same proposition can be expressed in a handful of different ways then clearly, none of these connectives are essential to the sense of the proposition, thus giving credence to Wittgenstein's "fundamental idea" that "the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts". Since all these propositions and axioms are equivalent so there are no differences between them, so as a tautology says nothing, all the propositions of logic say the same thing: nothing. Wittgenstein is trying to dissociate the importance of notation from logic itself. In a truth table, the connections between elementary propositions "show" themselves, and so need not be said.
Even the negation that occurs in a proposition, is no characteristic of its sense because if (ssp = p) then denial is already contained in affirmation. And if there was an object called “s”, then “ssp” would have to say something other than “p”.
Logic must look after itself. What makes logic a priori is the impossibility of illogical thought.

In this statement, he is alluding to a further difference between his conception of logic and the Universalist conception espoused by Frege and Russell. According to the Universalist conception, certain logical axioms must be laid out as fundamental "laws" of logic. Wittgenstein explains that his method can "show" the workings of logical inference, thus rendering unnecessary the "laws of inference" that both Frege and Russell had built into their axiomatic systems. One proposition follows from a second proposition if the first is true whenever the second is true. If we express "p or q" as "(TTTF)(p,q)" and "p and q" as "(TFFF)(p,q)" we can see that the former follows from the latter by comparing their truth-grounds: where there is a "T" in the latter proposition, there is a corresponding "T" in the former proposition. We don't need a law of inference to tell us this: it shows itself plainly in the truth-grounds of the two propositions. We should not need external laws to tell us how proceed with logic since there is nothing external to logic. Logic should not stand in need of justification.

Hence, there can never be surprises in logic.

His conception of logic is explained:

"The propositions of logic describe the scaffolding of the world, or rather they represent it."

The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.

Logic is the common structure that links our minds to the universe but what is its relation to mathematics?

The metaphor of scaffolding brings to light four principal aspects of Wittgenstein's conception of logic. First, scaffolding is a framework structure: it is a skeleton of joints rather than a building with walls and rooms. Similarly, logic does not consist of propositions with a sense, but only provides a framework within which propositions with a sense may fit. Second, the framework of scaffolding is used to construct a more substantial building, just as logic provides a framework within which the substantial facts about the world may fit. Third, scaffolding has points of contact with the building it is placed against, but it does not overlap with the building, nor is it a part of the building. Logic has points of contact with the world in that both logic and the world share a logical form, but the content (as opposed to the form) of facts themselves has no analogue in logic. Fourth, scaffolding is only a tool used in construction: a sturdy and complete building has no need of scaffolding. Similarly, we do not need logic or philosophy when language is functioning normally. These tools are only needed to provide clarity when language misfires and attempts to speak nonsense.

The “experience” which we need to understand logic is not that such and such is the case, but that something is; but that is no experience. Logic precedes every experience—that something is so. It is before the how, not before the what.

A proposition that is true no matter what (e.g. "(TTTT)(p,q)") is called a "tautology" and a proposition that is false no matter what (e.g. "(FFFF)(p,q)") is called a "contradiction". Tautologies and contradictions lack sense in that they do not represent any possible situations (state of affairs) so they are senseless but they are not nonsense either, because they consist of elementary propositions and are held together in a logical way. It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colours at the same time is a contradiction.

The operation is that which must happen to a proposition in order to make another out of it. Like combining the two propositions, "p" and "q," to form a new proposition "p.q". An operation combining elementary propositions in a truth- function is a truth-operation. An operation is not a form or object in its own right; it simply expresses the difference between the forms of two propositions or in other words the relation that stands between the structure of the base proposition and the structure of the resulting proposition.

Wittgenstein attempts to rid logical notation of the signs for generality and identity. Whenever a variable is given, that variable expresses all objects that can take that variable place, so generality is already given when a variable is given. We don't need an additional sign to denote generality. As for identity, to say of two things that they are identical is senseless because they can’t have two different signs referring to one meaning and to say of one thing that it is identical with itself is to say nothing at all.

A proposition of the form "A believes that p" does not actually involve a relationship between A and the proposition "p". For A to think, believe, or say that p is the case. They relate p to the verbal expression of p, so that what we are really saying is "'p' says that p". And the internal similarity between the words and the proposition is obvious. Wittgenstein further infers that there is no such thing as a "soul" where thoughts and beliefs reside.

Wittgenstein gives a general form for expressing a term in a particular series as "[a, x, O'x]," where "a" stands for the first term in the series, "x" stands for an arbitrarily selected term, and "O'x" stands for the term that immediately follows "x." The "O'" is the operation by which a term in the series is generated out of another. So, for instance, we could express the series of square numbers as [1, x,(sqr(x) + one)^2].
Wittgenstein takes the successive application of an operation as the model of a proposition. His definition of the general propositional form as "[?p, ??, N(??)]" is a variation of the general form for expressing a term in a series: "[a, x, O'x]." The "?p" is the collection of elementary propositions that a given proposition is composed of, and thus is the first term in the series of operations that generates a complex operation. The "??" is a complex proposition in this series of successive negations, and "N(??)" shows us how the next term in the series will be generated, namely by negating all the terms in "??."

Wittgenstein observes that all propositions can be derived by means of successive applications of the operation (——T)(?,….), that is by means of negating all the terms in the right hand pair of brackets. When Wittgenstein claims that all propositions can be derived by successive applications of a negating operation, he is alluding to the "Sheffer stroke," a logical constant discovered in the early 20th century. While Frege develops a system that relies only on the logical constants "not" and "if…then," and Russell develops a system that relies only on the logical constants "not" and "or," it was discovered that the Sheffer stroke—usually symbolized as a vertical bar, "|"—was a logical constant that could stand on its own. The proposition "p|q" is equivalent to "~p.~q." Thus, "~p" can be expressed "p|p," "p v q" can be expressed "(p|q)|(p|q)," and so on. Wittgenstein draws on the Sheffer stroke to show that a single operation can be used to derive any proposition from any other proposition. Given the elementary propositions, we can generate all other propositions using the Sheffer stroke.

Frege's search for something more certain than pure intuition to ground the concepts of was largely against Kant, who argued that our knowledge of mathematics is based on pure intuition. Any given number could be generated, according to Kant, by adding a certain number of ones: 4 = 1 + 1 + 1 + 1, while 98 = 1 + 1 + 1 + …. Pure intuition is necessary for the concept of "and so on" that makes it possible to add infinitely many ones together. Frege claimed that he could make pure intuition unnecessary to mathematics by giving a definition of number based in logic that would provide a general rule more rigorous than "and so on" for adding successive ones. Frege and Russell both developed ingenious systems to prove that the laws of mathematics could be inferred from basic logical axioms, they were largely successful.

In defining mathematics as a "method of logic", Wittgenstein suggests that numbers are not objects that can be constructed out of logical forms. Numbers are exponents of operations: they constitute an index for expressing how many times an operation has been applied. Thus, the propositions of mathematics do not say anything about the world, but only reflect the method in which propositions are constructed. So the logic of the world which the propositions of logic show in tautologies, mathematics shows in equations. Mathematical propositions express no thoughts. The propositions of mathematics are equations, and therefore pseudo-propositions. Because they contain properties of affirmation; two expressions may have same meaning but both are used unlike logic. It is a property of “1+1+1+1” that it can be conceived as “(1+1)+(1+1)”. The equation characterizes only the standpoint from which I consider the two expressions, that is to say the standpoint of their equality of meaning. The curious thing about Wittgenstein's philosophy of mathematics in the Tractatus is that it relies on the concept of "and so on" that Frege had gone to such lengths to eliminate.

Wittgenstein emphasizes that the truth-value of a proposition has no bearing on its sense. True or false, it still makes a picture of the world, and we can still draw logical inferences from that picture. The propositions p and ~p depict the same possible situation, only they have opposite sense one says that the picture presented is the case, and the other says that it is not the case.

Even the negation that occurs in a proposition, is no characteristic of its sense because if (ssp = p) then denial is already contained in affirmation. And if there was an object called “s”, then “ssp” would have to say something other than “p”.

Logic must look after itself. What makes logic a priori is the impossibility of illogical thought.

Hence, there can never be surprises in logic.His conception of logic is explained:

"The propositions of logic describe the scaffolding of the world, or rather they represent it."

The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.Logic is the common structure that links our minds to the universe but what is its relation to mathematics?

The metaphor of scaffolding brings to light four principal aspects of Wittgenstein's conception of logic. First, scaffolding is a framework structure: it is a skeleton of joints rather than a building with walls and rooms. Similarly, logic does not consist of propositions with a sense, but only provides a framework within which propositions with a sense may fit. Second, the framework of scaffolding is used to construct a more substantial building, just as logic provides a framework within which the substantial facts about the world may fit. Third, scaffolding has points of contact with the building it is placed against, but it does not overlap with the building, nor is it a part of the building. Logic has points of contact with the world in that both logic and the world share a logical form, but the content (as opposed to the form) of facts themselves has no analogue in logic. Fourth, scaffolding is only a tool used in construction: a sturdy and complete building has no need of scaffolding. Similarly, we do not need logic or philosophy when language is functioning normally. These tools are only needed to provide clarity when language misfires and attempts to speak nonsense.

Wittgenstein gives a general form for expressing a term in a particular series as "[a, x, O'x]," where "a" stands for the first term in the series, "x" stands for an arbitrarily selected term, and "O'x" stands for the term that immediately follows "x." The "O'" is the operation by which a term in the series is generated out of another. So, for instance, we could express the series of square numbers as [1, x,(sqr(x) + one)^2].

Wittgenstein takes the successive application of an operation as the model of a proposition. His definition of the general propositional form as "[?p, ??, N(??)]" is a variation of the general form for expressing a term in a series: "[a, x, O'x]." The "?p" is the collection of elementary propositions that a given proposition is composed of, and thus is the first term in the series of operations that generates a complex operation. The "??" is a complex proposition in this series of successive negations, and "N(??)" shows us how the next term in the series will be generated, namely by negating all the terms in "??."

Wittgenstein observes that all propositions can be derived by means of successive applications of the operation (——T)(?,….), that is by means of negating all the terms in the right hand pair of brackets. When Wittgenstein claims that all propositions can be derived by successive applications of a negating operation, he is alluding to the "Sheffer stroke," a logical constant discovered in the early 20th century. While Frege develops a system that relies only on the logical constants "not" and "if…then," and Russell develops a system that relies only on the logical constants "not" and "or," it was discovered that the Sheffer stroke—usually symbolized as a vertical bar, "|"—was a logical constant that could stand on its own. The proposition "p|q" is equivalent to "~p.~q." Thus, "~p" can be expressed "p|p," "p v q" can be expressed "(p|q)|(p|q)," and so on. Wittgenstein draws on the Sheffer stroke to show that a single operation can be used to derive any proposition from any other proposition. Given the elementary propositions, we can generate all other propositions using the Sheffer stroke.

Frege's search for something more certain than pure intuition to ground the concepts of was largely against Kant, who argued that our knowledge of mathematics is based on pure intuition. Any given number could be generated, according to Kant, by adding a certain number of ones: 4 = 1 + 1 + 1 + 1, while 98 = 1 + 1 + 1 + …. Pure intuition is necessary for the concept of "and so on" that makes it possible to add infinitely many ones together. Frege claimed that he could make pure intuition unnecessary to mathematics by giving a definition of number based in logic that would provide a general rule more rigorous than "and so on" for adding successive ones. Frege and Russell both developed ingenious systems to prove that the laws of mathematics could be inferred from basic logical axioms, they were largely successful.

Frege's search for something more certain than pure intuition to ground the concepts of was largely against Kant, who argued that our knowledge of mathematics is based on pure intuition. Any given number could be generated, according to Kant, by adding a certain number of ones: 4 = 1 + 1 + 1 + 1, while 98 = 1 + 1 + 1 + …. Pure intuition is necessary for the concept of "and so on" that makes it possible to add infinitely many ones together. Frege claimed that he could make pure intuition unnecessary to mathematics by giving a definition of number based in logic that would provide a general rule more rigorous than "and so on" for adding successive ones. Frege and Russell both developed ingenious systems to prove that the laws of mathematics could be inferred from basic logical axioms, they were largely successful.

In defining mathematics as a "method of logic", Wittgenstein suggests that numbers are not objects that can be constructed out of logical forms. Numbers are exponents of operations: they constitute an index for expressing how many times an operation has been applied. Thus, the propositions of mathematics do not say anything about the world, but only reflect the method in which propositions are constructed. So the logic of the world which the propositions of logic show in tautologies, mathematics shows in equations. Mathematical propositions express no thoughts. The propositions of mathematics are equations, and therefore pseudo-propositions. Because they contain properties of affirmation; two expressions may have same meaning but both are used unlike logic. It is a property of “1+1+1+1” that it can be conceived as “(1+1)+(1+1)”. The equation characterizes only the standpoint from which I consider the two expressions, that is to say the standpoint of their equality of meaning. The curious thing about Wittgenstein's philosophy of mathematics in the Tractatus is that it relies on the concept of "and so on" that Frege had gone to such lengths to eliminate.

### Language and Solipsism

###
The limits of my language mean the limits of my world. Logic fills the world: the limits of the world are also its limits.

The language which I understand, is the representation of my world, we cannot therefore say in logic: This and this there is in the world, that there is not. For that would apparently presuppose that we exclude certain possibilities and this cannot be the case since otherwise logic must get outside the limits of the world.

What we cannot think, that we cannot think: we cannot therefore say what we cannot think.

This observation leads Wittgenstein to reflect on the limited truth of solipsism. The term "solipsism" defines a number of related philosophical positions, all of which claim that the objects and people in the world only exist as objects of my awareness, that only I, as a thinking consciousness, truly exist. Wittgenstein draws the analogy between the relationship between the metaphysical subject and the world on one hand, and the relationship between the eye and the visual field on the other. I cannot see my eye anywhere in my visual field, but the existence of a visual field presupposes the existence of the eye. Similarly, myself is not something I encounter in the world, but my experience of the world presupposes that there is a self to experience it.

However, I cannot talk about this self because it is outside the limits of the world, and hence outside the limits of language. The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world much as the eye is the limit of the visual field. So there are no objects or elementary propositions that correspond to this "I": there are no propositions with sense, true or false, relating to it.

So both language and the world share the same limits leads to the reflection that solipsism is correct in the claim that "the world is my world. However, the thesis of solipsism cannot be put into language, but can only show itself. At this point, however, there is no real distinction between solipsism and pure realism, The I in solipsism shrinks to an extensionless point and there remains the reality co-ordinated with it.

The limits of my language mean the limits of my world. Logic fills the world: the limits of the world are also its limits.The language which I understand, is the representation of my world, we cannot therefore say in logic: This and this there is in the world, that there is not. For that would apparently presuppose that we exclude certain possibilities and this cannot be the case since otherwise logic must get outside the limits of the world.

What we cannot think, that we cannot think: we cannot therefore say what we cannot think.

### Logic and Laws of Nature

###

Logic determines the form that laws of nature can take, but it does not it make any claims regarding nature. Scientific laws themselves do not belong to logic, because they make claims about experience and do not hold a priori. The law of induction, the law of causality, and other such scientific principles are not exactly empirical facts, either.
All such propositions, including the principle of sufficient reason, the 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.

They define the framework within which we can talk about natural phenomena. It is clear that there are no grounds for believing that the simplest course of events will really happen.
That the sun will rise tomorrow is a hypothesis; and that means that we do not know whether it will rise. A necessity for one thing to happen because another has happened does not exist. There is only logical necessity.

Wittgenstein when discussing the concept of causality, that to comprehend it we need to understand the two actions in sequence as relative to each other and not a law of causality itslef. He gives an example on measuring time, we cannot compare any process with the “passage of time”—there is no such thing—but only with another process (say, with the movement of the chronometer). Hence we can describe the lapse of time only by relying on some other process. Time is proved as a dimension after the special relativity which renders his example a fault, none the less he is correct.

The Kantian problem of the right and left hand which cannot be made to cover one another already exists in the plane, and even in one-dimensional space; where the two congruent figures a and b cannot be made to cover one another without moving them out of this space. The right and left hand are in fact completely congruent. And the fact that they cannot be made to cover one another has nothing to do with it. A right-hand glove could be put on a left hand if it could be turned round in four-dimensional space.

Wittgenstein compares the laws of nature to a square mesh laid out over a surface of black and white spots. This mesh allows us describe the surface by saying of each square in the mesh whether it is black or white. Of course, a triangular mesh or a hexagonal mesh could be used just as well as a square mesh, though certain kinds of mesh will likely provide a simpler and more accurate description of the surface than others. And while the mesh itself can tell us nothing about the distribution of black and white on the surface, we can learn about the surface by observing what kinds of mesh describe it most accurately. The laws of nature do not tell us anything about the world, nor are they necessarily true of the world. Rather, they are tools we can use to make sense of the world in order to understand its regularities with greater clarity in order to be able to describe reality. Science is ultimately descriptive, not explanatory.

Logic determines the form that laws of nature can take, but it does not it make any claims regarding nature. Scientific laws themselves do not belong to logic, because they make claims about experience and do not hold a priori. The law of induction, the law of causality, and other such scientific principles are not exactly empirical facts, either.

All such propositions, including the principle of sufficient reason, the 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.

They define the framework within which we can talk about natural phenomena. It is clear that there are no grounds for believing that the simplest course of events will really happen.

That the sun will rise tomorrow is a hypothesis; and that means that we do not know whether it will rise. A necessity for one thing to happen because another has happened does not exist. There is only logical necessity.

### Answerable Questions and the Mystical

###

Questions can only be answered when the questions themselves can be framed in words. Thus, we can only ask questions and get answers regarding facts about the world, and not about anything transcendent.
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.

For an answer which cannot be expressed, the question too cannot be expressed. The riddle does not exist. If a question can be put at all, then it can also be answered.

Skepticism is not irrefutable, but palpably senseless, if it would doubt where a question cannot be asked. For doubt can only exist where there is a question; a question only where there is an answer, and this only where something can be said.

Specifically, Wittgenstein limits propositions to making claims, true or false, about how things stand in the world, which is the business of natural science. To think of philosophy as made up of propositions is a common error. The role of philosophy is not to state truths about the universe, but to clarify the truths stated by the other sciences. The business of philosophy is not of saying, but of showing: philosophy clarifies the logical structure of our propositions that is clouded by everyday language. Philosophy is an "activity" and not one of the natural sciences. Since philosophy cannot step outside the boundaries of language, it should act as a watchdog at those boundaries, clarifying vague propositions. He concludes that the only correct method in philosophy is to confine oneself to what can be spoken, and, whenever others try to say the unsayable (ethics, aesthetics, metaphysics, etc.), to point out to them that they are speaking nonsense. Wittgenstein asserts that most philosophical confusion arises from trying to speak about things that can only be shown.
Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries.

Philosophy limits the disputable sphere of natural science. It should limit the thinkable and thereby the unthinkable. It should limit the unthinkable from within through the thinkable. It will mean the unspeakable by clearly displaying the speakable. Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.

Anything transcendental can’t be said because there is no perspective external to the world from which we can talk about the world or its contents generally. For Wittgenstein, ethical `propositions' are absolute judgments of value of the form, ethics pervades all of life; Ethics are transcendental because no aspect of life is untouched by ethics. Our attitude toward the world shapes the world we live in. Thus, we cannot talk about ethics since logical language can only reflect the world, any discussion of the mystical that which lies outside of the metaphysical subject's world, is meaningless. Actions are not good or bad because of their consequences, but because of the overall attitude toward life that they embody. While the exercise of the will has no direct effect on the world itself, this exercise of the will defines the kind of world a person inhabits:
"The world of the happy man is a different one from that of the unhappy man".

Next he talks about the value of the world and its meaning. Because all propositions are of equal value in the world so they cannot express anything higher. Accidental is what isn't necessary. He concludes:
The sense of the world must lie outside the world. In it there is no value—and if there were, it would be of no value.If there is a value which is of value, it must lie outside all happening and being-so. For all happening and being-so is accidental. What makes it non-accidental cannot lie in the world, for otherwise this would again be accidental. It must lie outside the world.

The Sensible can be said because it is logical, It can be thought of and it is possible in our world. They can be pictured as the elements in a picture. It's truthfulness can be confirmed by comparing with the real world so it can be true or false.

What can be shown and not said it what is outside language and thought, what can't be defined, what can't be thought about, like the logical form of the world or elucidations, they are tautologies so they are senseless. It is the relation between the pictures' elements so It can't be described by any picture but can only be shown in a picture.

‘A state of affairs is thinkable’: what this means is that we can picture it to ourselves.

What can't be be put into words, (inexpressible so can neither be said nor shown), is what lies out of the world, is illogical or nonsensical, it also what can't be defined, what can't be thought about and what can't be pictured nor be shown in a picture, because there is no mapping between elements of the picture and the objects. For example God has no propositions and its components can't be defined so they can't be put into words or be thought about so they are nonsensical.

Then he elaborates on the mystical:
There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical. It is not how things are in the world that is mystical, but that it exists. Feeling the world as a limited whole—it is this that is mystical.

Questions can only be answered when the questions themselves can be framed in words. Thus, we can only ask questions and get answers regarding facts about the world, and not about anything transcendent.

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.

For an answer which cannot be expressed, the question too cannot be expressed. The riddle does not exist. If a question can be put at all, then it can also be answered.

Skepticism is not irrefutable, but palpably senseless, if it would doubt where a question cannot be asked. For doubt can only exist where there is a question; a question only where there is an answer, and this only where something can be said.

Specifically, Wittgenstein limits propositions to making claims, true or false, about how things stand in the world, which is the business of natural science. To think of philosophy as made up of propositions is a common error. The role of philosophy is not to state truths about the universe, but to clarify the truths stated by the other sciences. The business of philosophy is not of saying, but of showing: philosophy clarifies the logical structure of our propositions that is clouded by everyday language. Philosophy is an "activity" and not one of the natural sciences. Since philosophy cannot step outside the boundaries of language, it should act as a watchdog at those boundaries, clarifying vague propositions. He concludes that the only correct method in philosophy is to confine oneself to what can be spoken, and, whenever others try to say the unsayable (ethics, aesthetics, metaphysics, etc.), to point out to them that they are speaking nonsense. Wittgenstein asserts that most philosophical confusion arises from trying to speak about things that can only be shown.

Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries.

Philosophy limits the disputable sphere of natural science. It should limit the thinkable and thereby the unthinkable. It should limit the unthinkable from within through the thinkable. It will mean the unspeakable by clearly displaying the speakable. Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.

Anything transcendental can’t be said because there is no perspective external to the world from which we can talk about the world or its contents generally. For Wittgenstein, ethical `propositions' are absolute judgments of value of the form, ethics pervades all of life; Ethics are transcendental because no aspect of life is untouched by ethics. Our attitude toward the world shapes the world we live in. Thus, we cannot talk about ethics since logical language can only reflect the world, any discussion of the mystical that which lies outside of the metaphysical subject's world, is meaningless. Actions are not good or bad because of their consequences, but because of the overall attitude toward life that they embody. While the exercise of the will has no direct effect on the world itself, this exercise of the will defines the kind of world a person inhabits:

"The world of the happy man is a different one from that of the unhappy man".

Next he talks about the value of the world and its meaning. Because all propositions are of equal value in the world so they cannot express anything higher. Accidental is what isn't necessary. He concludes:

The sense of the world must lie outside the world. In it there is no value—and if there were, it would be of no value.If there is a value which is of value, it must lie outside all happening and being-so. For all happening and being-so is accidental. What makes it non-accidental cannot lie in the world, for otherwise this would again be accidental. It must lie outside the world.The Sensible can be said because it is logical, It can be thought of and it is possible in our world. They can be pictured as the elements in a picture. It's truthfulness can be confirmed by comparing with the real world so it can be true or false.

What can be shown and not said it what is outside language and thought, what can't be defined, what can't be thought about, like the logical form of the world or elucidations, they are tautologies so they are senseless. It is the relation between the pictures' elements so It can't be described by any picture but can only be shown in a picture.

‘A state of affairs is thinkable’: what this means is that we can picture it to ourselves.What can't be be put into words, (inexpressible so can neither be said nor shown), is what lies out of the world, is illogical or nonsensical, it also what can't be defined, what can't be thought about and what can't be pictured nor be shown in a picture, because there is no mapping between elements of the picture and the objects. For example God has no propositions and its components can't be defined so they can't be put into words or be thought about so they are nonsensical.

Then he elaborates on the mystical:

There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical. It is not how things are in the world that is mystical, but that it exists. Feeling the world as a limited whole—it is this that is mystical.

### Conclusion

###

The controversial ending proposition of the book:

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 surmount these propositions; then he sees the world rightly.

There are, at present, two dominant ways to read Wittgenstein's Tractatus Logico-Philosophicus (TLP).

- The irresolute reading takes what is called a view of nonsense: it takes Wittgenstein's propositions in the TLP to be nonsensical in that they are trying to express what, according to Wittgenstein, can only be shown, e.g. propositions about the logical form of propositions. The traditional interpretation, perhaps best represented by P. M. S. Hacker, takes Wittgenstein to be pointing out that the kinds of subject matter he treats of lies outside the realm of sensible discourse. The Tractatus treats of things that cannot be said, but can only be shown.
- The resolute reading on the other hand takes what is called an austere view of nonsense: this takes the propositions in the TLP to be actual, irredeemable nonsense. In this sense the whole of TLP becomes a quasi-ironic argument against transcendental idealism. The locus classicus of this reading is Cora Diamond's The Realistic Spirit.

As a result, several distinctions can be made between the two readings. For example, if we are irresolute then we take Wittgenstein to be a realist, whereas if we are resolute the question of realism/anti-realism does not arise.

However, we can try to understand the frame of mind that would be necessary to think that these propositions make sense. They are evident as logic itself. Wittgenstein is not trying to tell us a number of things that we did not already know; he is trying to instruct us in a way of thinking that will help us out of philosophical muddles. While the propositions of the Tractatus may themselves be nonsense, Wittgenstein hopes that they have served their instructive purpose. We are expected to put down this book not with the knowledge that the world is made up of objects and states of affairs and those propositions depict facts, but with an understanding of why it is impossible to say these sorts of things. The goal of the Tractatus, as Wittgenstein claims in his preface, is "to draw a limit … to the expression of thoughts."

Wittgenstein concludes:

"What we cannot speak about we must pass over in silence".

The controversial ending proposition of the book:

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 surmount these propositions; then he sees the world rightly.There are, at present, two dominant ways to read Wittgenstein's Tractatus Logico-Philosophicus (TLP).

- The irresolute reading takes what is called a view of nonsense: it takes Wittgenstein's propositions in the TLP to be nonsensical in that they are trying to express what, according to Wittgenstein, can only be shown, e.g. propositions about the logical form of propositions. The traditional interpretation, perhaps best represented by P. M. S. Hacker, takes Wittgenstein to be pointing out that the kinds of subject matter he treats of lies outside the realm of sensible discourse. The Tractatus treats of things that cannot be said, but can only be shown.
- The resolute reading on the other hand takes what is called an austere view of nonsense: this takes the propositions in the TLP to be actual, irredeemable nonsense. In this sense the whole of TLP becomes a quasi-ironic argument against transcendental idealism. The locus classicus of this reading is Cora Diamond's The Realistic Spirit.

As a result, several distinctions can be made between the two readings. For example, if we are irresolute then we take Wittgenstein to be a realist, whereas if we are resolute the question of realism/anti-realism does not arise.

However, we can try to understand the frame of mind that would be necessary to think that these propositions make sense. They are evident as logic itself. Wittgenstein is not trying to tell us a number of things that we did not already know; he is trying to instruct us in a way of thinking that will help us out of philosophical muddles. While the propositions of the Tractatus may themselves be nonsense, Wittgenstein hopes that they have served their instructive purpose. We are expected to put down this book not with the knowledge that the world is made up of objects and states of affairs and those propositions depict facts, but with an understanding of why it is impossible to say these sorts of things. The goal of the Tractatus, as Wittgenstein claims in his preface, is "to draw a limit … to the expression of thoughts."

Wittgenstein concludes:

"What we cannot speak about we must pass over in silence".

Are you the author of this analysis??

I wrote a part of it. The rest is compiled from many sources merged with my comments and analysis. May I ask why? :)