LUDWIG WITTGENSTEIN EXPLAINED: TRACTATUS LOGICO-PHILOSOPHICUS (1921)

Ludwig Wittgenstein's Tractatus Logico-Philosophicus was published in 1921. It immediately become one of the most important works of 20th century analytic philosophy. It is also one of the most difficult and mysterious. It features prominently in my favorite visual novel, Subarashiki Hibi 〜 Furenzoku Sonzai. I was largely motivated to study it by that VN. I want to share my knowledge of it with anyone else who loves Subarashiki Hibi like I do, and hopefully everyone else can pick up something along the way!

2. Background to the Tractatus Logico-Philosophicus

2.a. Ludwig Wittgenstein's Life Up to 1921

For a full biography of Ludwig Wittgenstein, I highly recommend reading Ludwig Wittgenstein: The Duty of Genius by Ray Monk. One thing is clear when you read his biography: Wittgenstein was a very odd person. He probably had a mental disorder of some kind. I've heard potential diagnoses ranging from autism spectrum to schizoid personality disorder. He was a very private person and often difficult to deal with even for those who knew him closest. He had serious problems sustaining intimate relationships, even though he clearly deeply cared for certain individuals in his life. We know that his family seemed to have a pattern of depression. Wittgenstein was the youngest of nine children, and no less than three of his older brothers committed suicide. The odds also seem to have been high that he was a closeted homosexual. In any case, Wittgenstein's life is fascinating, amusing, and sad all at once. But for now we can shift to those details which are most important in understanding his entry to philosophy and thus to the writing of the Tractatus Logico-Philosophicus.

Wittgenstein's father was a steel magnate who amassed a fabulous fortune. He was a new breed of cultural elite in Vienna, one that was no longer "aristocratic" in the sense of blood, but a "self-made" businessman-aristocrat of sorts. In that sense, the Wittgensteins were more comparable to American gilded age elites like the Carnegies and Rockefellers than European royals. Ludwig was the youngest of nine children and grew up in a world of extreme sophistication. He was actually considered to be sort of dull and slow compared to his older siblings. He eventually moved to England to study aeronautics. This means that his background was primarily formed by reading literature and poetry and by studying scientific and mathematic works. His experience with philosophy was quite meager. Some say he was thus one of the last major philosophers to enter as a true outsider to the field. His work bears some influence from this: a mix of both clinical scientific taciturnity and strangely poetic expressions.

Wittgenstein became interested in the philosophical foundations of mathematics while in England. He soon read and began communicating with two major figures of the time who he would end up critiquing throughout the Tractatus: Gottlob Frege and Bertrand Russell. I will thoroughly introduce the philosophical insights of these two figures when they are introduced in the Tractatus proper, but some word about their place in the philosophical tradition is required.

2.b. Frege, Russell, and a Review of Logical Calculus

Gottlob Frege was a German mathematics professor whose role in the history of philosophy is of tremendous importance but was to some degree forgotten about in the English-speaking world until interest in Wittgenstein caused a renewal of interest in his influence Frege along with him. Frege's chief accomplishment was introducing the first system of propositional logical calculus to the Western philosophical tradition. Anyone who has taken a logic class has a long tradition going back to Frege to thank for it, even if Frege's system was never widely used since it is cumbersome and awkward. For that reason, we will only use the more modern notation of Wittgenstein during this lecture. And it is necessary to be clear about how to understand this logical calculus. [1]

To say that "Either John or Mary (or both) are going to the store today" we write:

The v symbol represents that one or the other is the case, without excluding both being true.

Now we'll try to write "John is going to the store today, but Mary isn't." It looks like:

Wittgenstein uses a period to signify that this is true and this is also true. Modern logic often uses an & symbol instead, but Wittgenstein uses the . so extensively that we should stick with it to make the text easier to understand. The ~ symbol signifies a negation, that it is NOT the case.

Lastly, we want to represent "The store is open today and either John or Mary is going." We will write:

Just like in a mathematical equation, we use parentheses to make it clear where our symbols are distributed.

Frege is most remembered for introducing a propositional calculus similar to the above (but a lot messier) to philosophy. But he also had positions about the metaphysics of mathematics and logic. But these would first be investigated and critiqued by the most important philosophical influence on Wittgenstein, Bertrand Russel.

Bertrand Russell was a very important philosophical influence on Wittgenstein. He first studied under Russell as a student, but soon proved himself to be capable enough that the two became colleagues and mutual mentors of each other, as well as close friends. By the mid-1920s or so, the two began to grow distant. Russell was not just a logical theorist. He was just as engaged in social criticism, political activism, and other humanitarian pursuits. He was known for his skeptical arguments against theism and liberal social attitudes. Wittgenstein had no interest in these aspects of Russell's thought and in fact thought very negatively of them in many ways.

I will address the relevant thoughts of Russell when they come up in the Tractatus itself. They are primarily related to the ontology of facts in logic, the representational role of language, and the role of axioms and logic and mathematics.

2.c. The Tractatus's Place in Wittgenstein's Philosophical Career

Wittgenstein began studying under Bertrand Russell in 1911 and dedicated himself with strenuous ardor to logical and mathematical conundrums in philosophy. He enlisted in the army upon the outbreak of World War I in 1914 and served in various positions all the way to the war's end in 1918. Not only did he return from the line of duty with honors for bravery, but also with one of the most unique, beguiling, and challenging philosophical works ever written. This work, honed to perfection over four grueling years of war, would become known as the Tractatus Logico-Philosophicus. With it, Wittgenstein was quite confident that he had solved all problems of philosophy.

Wittgenstein was overwhelmed with a drive toward finding truth and perfecting his thought. He was a highly systematic thinker, but his perfectionism meant that he completed almost no works in full. In fact, the Tractatus Logico-Philosophicus is the only work of his which can be said to be "complete" in the exact state Wittgenstein wanted it to be. Having believed he has solved all problems of philosophy, he spent most of the 1920s teaching mathematics to elementary schoolers in rural Austria. But eventually he did return to philosophy as he felt his work in the field was not in fact done. He has other manuscripts and collections of notes which are substantial enough to rival the Tractatus in terms of importance, most notably the Philosophical Investigations which were published shortly after his death in 1951.

These two works seem to present very different philosophies, or at least treat very different philosophical problems in fundamentally different ways. For that reason it is customary to split Wittgenstein's work into the "early" Wittgenstein, represented by the Tractatus and his early notebooks, and the "later" Wittgenstein, represented by just about everything published after. There is a question in modern Wittgenstein scholarship about how much of a difference there really is between the two "eras." To me the biggest difference seems to be that his later work is more concerned with questions that are, in his words, situated in "the stream of life" and thus more attuned towards things like psychology and epistemology. With that in mind, this lecture will entirely be about the "early" Wittgenstein, but it is undecided how much this division is really legitimate in the first place.

3. Structure of the Tractatus Logico-Philosophicus

The format of the Tractatus is very different from an average philosphical text. It is very short, running only about 70 pages. But it is dense. Very dense. It contains almost no arguments, examples, or direct proofs but rather consists of a number of propositions that Wittgenstein assumes will be taken as self-evident, with the reasoning behind them as being implicit. One of the most common words used by writers to describe the Tractatus is "oracular" in virtue of its declarative tone and enigmatic content. It does not have the long, rambling sentences of some more difficult philosophers. In fact, its compact writing style and rigorously ordered structure make it appear misleadingly direct and easy to grasp. But it is because of how much is NOT said in the Tractatus that it is so hard to interpret.

1. The world is all that is the case.

2. What is the case--a fact--is the existence of states of affairs.

3. A logical picture of facts is a thought.

4. A thought is a proposition with a sense.

5. A proposition is a truth-function of elementary propositions.
(An elementary proposition is a truth-function of itself.)

6. The general form of a truth-function is: [‾p, ‾ξ, N(‾ξ)].
This is the general form of a proposition.

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

These seven propositions are, in its shortest form, the Tractatus. But what do they mean? Each one, with the exception of Proposition 7, is joined by several sub-hierarchies of sub-propositions and clauses for each one. For an example of what this looks like, here's the structure of Proposition 1:

The main proposition, 1, has two sub-propositions: 1.1 and 1.2. 1.1 has three sub-propositions (1.11, 1.12, and 1.13) and 1.2 has one sub-proposition (1.21). Proposition 1 is very short and so it doesn't branch out very far. The other Propositions (minus 7) commonly branch out to sub-sub-sub-sub-propositions like 2.0121, 2.0122, 2.0123, etc. They start with a basic proposition and then give further elaboration and specification for each point, and they can get pretty thorough.

Note that my quotes from the text may also skip over certain sub-propositions for the sake of making the point clearer. For example, if a section reads 2.02 > 2.0201 > 2.021 > 2.0211 > 2.0212 > 2.022, I may simply quote it as 2.02 > 2.021 > 2.022 to follow the "higher-level" line of argument. The Tractatus is not like a standard philosophical text where it can only be read in the exact linear order. In fact, its tree structure makes it arguably more well-suited to a hypertext format than to written pages. For this reason, I recommend using THIS SITE to look at it even if you have a hard copy.

The Tractatus was completed in 1918 but was not published until 1921. The reason was simple. No one understood it! The two people who probably had the best chance of understanding it in the world at the time were Frege and Russell. Frege found it incomprehensible. Russell felt that he understood it after a careful reading side by side with Wittgenstein. Wittgenstein said he did not understand it at all.

Publishers required some form of explanation of how to interpret the work in order for it to be published. Wittgenstein had written a preface, but people thought it was only more confusing. He refused to budge at first, thinking that an introduction from anyone else would destroy the beauty and integrity of the work which had such a pure simplicity.

Eventually, however, he relented and allowed Russell to write an introduction for the general reader. This remains in most editions of the Tractatus. Wittgenstein claimed that Russell's introduction fundamentally misunderstood the Tractatus, and for that reason I don't rely on it too much. But it is very useful in explaining the philosophical background of positions that Wittgenstein was responding to in most of the work. Wittgenstein wrote his own introduction to the work which was much shorter and much less technical. I will return to this introduction and its value at the end of this lecture.

I actually think one of the best introductory guides for the Tractatus is its entry on SparkNotes, of all places. It uses a thematic division which doesn't exactly map on to Wittgenstein's division into the 7 propositions, but I think it is very helpful for mapping the topics and positions of the text. I thus structure this explanation by the same format as the SparkNotes page and will otherwise make reference to it when necessary. Now we can finally begin to look at the Tractatus itself. All quotations are from the translation by David Pears and Brian McGuiness printed in the 2001 Routledge Classics edition unless otherwise noted.

4. Summary of the Tractatus Logico-Philosophicus

4.a. Propositions 1-2.0141: Ontology of Facts and States of Affairs

Wittgenstein begins by mapping out a kind of ontology or idea of all that exists. Three terms repeatedly occur: facts, states of affairs, and objects. There is no direct definition or example of these three, which makes this section somewhat frustrating and difficult to parse. Regardless, there is a hierarchy of sorts:

While we don't have Wittgenstein's definitions of these terms and thus should be hesitant to provide our own, this section will be easier to understand if we give an example of what a fact, state of affairs, or object could be. A fact is something that could be true or false and it depends on states of affairs. The Ogden translation translates the term in question ("Sachverhalten") as "atomic fact," which helps to get at its meaning. A state of affairs is a fact so simple that its truth or falsehood does not depend on any other fact. To use an example from Bertrand Russell's introduction:

The fact that Socrates is a wise Athenian depends on the more basic states of affairs that Socrates is wise and that Socrates is an Athenian. To use an example from SparkNotes, the fact that my house is safe from burglars could depend on the state of affairs that the only pair of keys to my house is in my pocket. [2] Now what about "objects?" In these examples, they would have to be the most fundamental things that could combine in any proposition. This proves hard to think of. What can exist in a form so simple that it does not depend on anything else? I will come back to this.

Facts and states of affairs exist in what Wittgenstein calls "logical space." This refers to the realm of everything that is logically possible, including both what is true and what is false but possible. For another SparkNotes example: [3]

The fact "Ottawa is the capital of Canada" is true. The fact "Toronto is the capital of Canada" is false, but there is nothing illogical about it. "Fact" might be a somewhat misleading translation of the German term ("Tatsache") as in English a fact is by definition true, but Wittgenstein uses it here to refer to both to things that are true and those that are false but logically possible. That is, even though Toronto is not the capital of Canada, we can imagine a world where this was the case. "Love is purple" however, is simply not something that can be ascribed truth or falsehood. The objects "love" and "purple" simply do not combine in a way that makes sense because it makes no sense to ascribe color to a concept like "love."

SparkNotes uses the helpful metaphor of logical space as a grid of light bulbs to illustrate the idea: [4]

Each light bulb represents a possible state of affairs. And each one is fed by a wires. The different kinds of wires represent different kinds of objects. The wires connect with sockets that are fit for that particular kind of object. Each light bulb has a number of wires plugging into it just as each state of affairs is made up of a number of objects. Depending on how the wires combine they will either conflict with each other and prevent the light bulb from receiving power or combine successfully and turn it on. So, for example, the wires for "Ottawa," "capital," and "Canada" would successfully combine because those represent a true state of affairs, while the wires for "Toronto," "capital," and "Canada" would conflict with each other as they represent a false state of affairs.

Now, what exactly are these "objects?" Well, I used the examples "Ottawa," "capital," and "Canada" for simplicity's sake, but in truth these could not be proper examples of objects. Objects would have to be simple enough that they did not depend on anything else to exist. "Ottawa," "capital," and "Canada" cannot be examples of objects, because these are concepts that exist based on human constructs and in dependence on other things. That is, "Ottawa" is a concept that exists because there are buildings, streets, a population, and so on. So it seems difficult to give an example for what an object could be. The next section will address them more thoroughly. But there are a few more important things we can say about them here.

A state of affairs is a combination of objects. Every object has a logical form that determines how and in what states of affairs it can occur. For example, if "love" was an object:

The logical form of "love" would dictate that it can occur in certain ways in some propositions (like "love is the opposite of hate") but not in others (like "love is purple"). Wittgenstein makes it clear that objects don't just happen to have the possibility of contituting states of affairs. That is, they aren't just substances that happen to exist and can potentially be combined in states of affairs. He says that constituting states of affairs is actually the "essence" of objects. That is, it is what makes them what they are. The essence of any object is not that certain properties hold of it it, but the POSSIBILITY of certain properties holding of it.

Wittgenstein uses an analogy to make this clear (2.0131): A visual object need not be any specific color, but it must be SOME color or another. It exists in a so-called "color space." A tone must have SOME pitch or other. It exists in a so-called "sound space." In this way, an object need not exist in any specific state of affairs, but it must exist in SOME states of affairs. So objects exist in "logical space."

With this in mind, we should return to the metaphor of the grid of light bulbs as it helps make several points clearer:

1. States of affairs are mutually independent: Any light bulb can be lit or unlit without affecting any other light bulb.

2. The essence of an object is its possibility of existing in certain states of affairs: The significance of the wire lies in which light bulbs it plugs into and how it does so.

3. The world is the totality of FACTS, not of things. The world is the totality of lit light bulbs, not the totality of the objects that exist as wires powering them.

The consequence of these three positions is a massive limit to the meaningful statements we can make about the ontology of the world. We cannot say that certain things exist or don't exist. We can only state facts that relate these things to others. For example, if we have an ontological question such as whether or not unicorns exist, the only way to answer it is to list all the facts that there are about unicorns. We can say that there has never been a sighting of a unicorn or that unicorns have only been encountered in fiction. But it doesn't even make sense to exclude unicorns from our ontology, because it is an ontology of FACTS and not of things. The only way we can say that unicorns don't exist is by saying that there are no true facts asserting anything about them as flesh-and-blood creatures. [5]

This ontology sets the stage for the rest of the Tractatus. It seems to propose a form of what has been called "logical atomism." This term originates from Bertrand Russell, who seems to have developed similar ideas after the early influence of Wittgenstein. However, the two held very different views about what logical atomism ultimately meant.

Logical atomism, like physical atomism, holds that there are fundamental, indivisible particulars that our experience can be analyzed down into. Russell analyzes the world into a threefold division just like Wittgenstein does:

The major difference is that Russell breaks down "objects" into "particulars" and "universals." A particular is something that applies to one atomic fact like a name or specified object. A universal is a particular quality like color, shape, disposition, etc. Wittgenstein doesn't break objects into particulars and universals because he sees all states of affairs as composed of objects that are true or false independently of one another. Russell's idea of particulars and universals denies this independence because it requires objects to be seen as shared or excluded between different facts.

4.b. Propositions 2.02-2.063: Objects Combine to Form States of Affairs

One again, we never get an exact definition of what an "object" or "state of affairs" is. But we know that objects are the simplest kind of thing that there are and that states of affairs are the simplest kind of fact that there are. Neither can be further analyzed or reduced because they form the fundamental basis of the ontology. I used examples for objects in the above section, but in truth none of those can really be examples of objects. Things like keys or chairs or tables or capitals cannot be objects, as those things can be divided into smaller parts. But does anything really exist that cannot be further analyzed and reduced into component parts?

This point remains unsettled, but there are three basic interpretations we can entertain at this point for what "objects" are: [6]

1. Objects are the basic elements of sense-data. This follows Russell, who argued that we experience particulars and universals as elements of sense-data that we are directly acquainted with. But there is a problem to this interpretation. Wittgenstein says that objects are things that can combine in states of affairsas being either true or false, but a speck in the visual field can be many different colors. So it is hard to see how we could attribute "truth" or "falsity" to it. And if Wittgenstein truly did mean sense-data when he talked about objects, he probably would have just said so.

2. Objects are not anything as specific as sense-data, but are a basic underlying structure to the universe. They are the basic building blocks of reality even if we cannot say exactly what they are.

3. Objects have no independent existence at all. Wittenstein repeatedly says that objects only exist within states of affairs, that the essence of objects is to combine in states of affairs, and that the world consists of states of affairs and not objects.

We might ask where the idea of simple objects existing comes from in the first place. In his words, objects form the substance of the world, so they cannot be compound (2.021). "Substance" refers to what is the underlying, unchanging, and indestructible nature of the world. It is what stays the same even if everything else changes. Wittgenstein's conception of substance must be somewhat in line with this.

He claims that objects are "simple" (2.02) and that "the fixed, the existent, and the object are one" (2.027). But despite behing unchanging and indestructible, these "objects" are nothing but empty logical forms. They don't reveal anything about the world until they are combined in states of affairs. There is an unchanging, indestructible, and necessary logical form to the universe for Wittgenstein, but everything that is the case depends on states of affairs which are enitrely contingent.

So if everything that is the case is completely contingent, what does this logical form of the world consist of? Remember the light bulb analogy. The logical form determines how these states of affairs are created (which wires can plug into which lightbulbs). "Purple," for example, is a color and so can only be used in contexts where a color word is called for. States of affairs like what kind of a color purple is and what sorts of things in the world are purple are contingent facts that could be otherwise. But the fact that purple is a color is a necessary aspect of the logical form of "purple."

Objects have internal and external properties. The internal properties are the logical form of the object, meaning what kind of object it is and how it can combine with other objects in states of affairs). The external properties are what IS true of it and what states of affairs it DOES in fact occur in. The internal properties are a part of logical form and thus remain true no matter what. There will always be these objects with these logical forms. They are the substance of the world. We might be able to imagine worlds other than this one, but not worlds with a different logical form from this one. To use the SparkNotes example, we could imagine a world where horses speak or where grass was pink, but not a world without space, time, or color. [7]

If the world had no substance or logical form, then "purple is a color" would be a contingent fact, just like "that lampshade is purple." But purple being a color cannot be contingent, because that would mean it could have been otherwise and could have been something like a number. And that would be nonsense. If the world had no substance, then whether a proposition was true would depend on whether another proposition was true. That would mean that a proposition like "that lampshade is purple" couldn't be true or false on its own. But Wittgenstein claims that all facts and states of affairs are true or false on their own with no influence from any other.

Although there are objects with internal properties that are always the same, it is important to remember that this says nothing about what is actually the case. The internal properties of "yellow" and "red" are indistinguishable since both are colors. They can only occur in the same kind of states of affairs because they have the same logical form (that is, they can only occcur where a color word is appropriate). The only way "yellow" and "red" are distinguishable is by their external properties. That is, they can only be distinguished by what happens to be true of them (what things do happen to be yellow or red).

This is why Wittgenstein never gives an example of what an object is. There is nothing to be said about objects. Asking "What is an object?" is like asking what everything has in common, and the only answer to this seems to be that everything shares a logical form that allows it to occur in states of affairs. Objects are the simplest, most basic things that there are, and the only thing that all things have in common is that we can say something about them.

Remember that it's not just that objects HAVE certain internal properties. It's actually the internal properties that determine the ESSENCE of objects. These properties are constitutive of the objects themselves. That means there is no need for an external, metaphysical "glue" of sorts to put objects together in a state of affairs. In his words, they fit together "like the links of a chain" (2.03). The objects combine on their own with no need for external laws or rules to hold them together, because their properties of being able to combine (their logical form) are the very constituents of the objects themselves.

With the status of objects made clear, we can return to what was already established. Some states of affairs are the case and will be called "positive" facts. Some are not the case and will be called "negative" facts. Every state of affairs is the case or not the case independently of any other. The world is the sum total of thoes states of affairs that are the case.

4.c. Propositions 2.1-3.144: The Picture Theory of Facts

This section shifts the Tractatus from its ontology to a discussion of how we make sense and communicate what there is via language, thought, and representation. We should note that the discussion of thought in this section is not concerned with anything like psychology or epistemology. Wittgenstein is interested in how things ARE, not how we perceive them to be. In fact, he expresses a concern over Russell and Frege for introducing too much psychology and epistemology into logic. Wittgenstein is not concerned with how thoughts work or where they come from, but what their FORM Is. And that form is the same form that everything else has: logical form.

Wittgenstein claims that we cannot think anything that is illogical (3.03). We cannot things that have no sense. That is, we cannot think in a way that puts together objects that cannot combine. For example, a thought such as "the number 2 is purple" would not have the correct logical form. It's not even clear what that thought would be, so it can't even really count as a thought.

So what about things that do have a correct logical form? How do the true and false states of affairs that make up the world relate to our thoughts of them? Wittgenstein's answer to this has become known as the "picture theory." According to him, propositions are logical "pictures" of facts. A proposition is a representation of a fact, but note that it is not as concrete as something like a sentence. We might say that a proposition is the meaning behind a sentence. If I translate the same sentence into a number of different languages or think it in my head or express it through sign language, the same meaning that is conveyed behind them all will be the proposition. A sentence is what is UTTERED, but a proposition is what is SAID. Wittgenstein would call a sentence a propositional "sign." In other words, it is a written or spoken manifestation of a proposition.

The word "picture" in the Tractatus is somewhat technical, with both a literal and metaphorical use. He doesn't so much give the word "picture" a new usage as much as expand its current one. The German word he uses is "Bild," which means "picture" or "image" in the visual sense that we normally tihnk of but also broadly is used for any kind of appearance or form and is often used that way in compound words. Wittgenstein says that we "picture" facts to ourselves (2.1). Conceiving of something is a matter of picturing it to ourselves.

If something can be the case due to having a correct logical form (i.e. existing in logical space and being true or false facts), we can conceive of it and thus make what Wittgenstein calls a "logical picture" of it. That is, we make a representation of it that uses logical form. We represent facts to ourselves by means of pictures. The elements of a picture correspond to the elements of a fact. That is, it reflects the objects that constitute it in some manner.

The genesis of this idea came to Wittgenstein during his service in World War I. He happened upon a newspaper article about a court case involving a traffic accident. The way the accident actually happened was difficult to explain without a visual aid, so the court used small models of the building, cars, and so on to display what happened. The idea struck Wittgenstein because people in the court could understand what was the case in virtue of a shared form.

The models of the cars and buildings only have their FORM in common with the real cars and buildings, but because of that bit of overlap they have the same limits of representation and can show what is possible. For example, we can use the models to represent one car on its side, one car on top of the other, one car broken in half, one car on the roof of a building, and so on. But we cannot use them to represent something logically impossible, such as two cars existing in the same exact place at the same exact time.

In Wittgenstein's words, we can display something that is contrary to the laws of physics but not to the laws of geometry (3.0321). What this means would be something like the following: We could imagine one car floating in midair with those models, even if this is not in accordance with our physical laws. But we could not imagine a car existing outside of space or being in two different places at the same time. This metaphor does not appear in the Tractatus proper, but it is a very helpful illustration of the concept of the picture theory. A picture helps us apprehend a fact by having a similar logical form, just as the models do in reference to the real cars and buildings.

He isn't talking just about visual pictures, but any kind of shared form that allows us to picture something or imagine it, comprehend it, or apprehend it to ourselves. He would call these "logical pictures." There is a direct correspondence between logical pictures and facts. Every fact that exists, whether it is positive or negative, there is only one logical picture that corresponds to it. We can tell which fact a logical picture depicts because it shares the same form as the fact, just as a visual picture illustrates something visual by having the same form with the thing it depicts, like a face having the same objects (eye, nose, mouth, etc.) as the picture of it in roughly the same order.

Wittgenstein uses the example of a ruler laid against its object to mesaure its length (2.1512). The ruler and the object have nothing in common except that they both have length, but because of that commonality, we can relate one to the other. There only needs to be one point of contact to relate different objects to one another. Because both a ruler and a measured object have length, it is possible to relate different aspects of the objects to graduated lines and numbers on the ruler. And in the same way, because a logical picture and a fact have logical form in common, it is possible to relate elements of the fact to those of the logical picture.

Note, however, that since logical pictures represent logical form, they determine was what is POSSIBLE. To return to our earlier metaphor, they only determine which light bulbs could exist and not which ones happen to be turned on or not. A picture is either true or false, but there is no picture which is a priori true. We can only know if it is true or not by comparing it with reality and deciding if the sense of the picture agrees with what happens to be the case or not (2.22).

A picture cannot depict its own pictorial form. That is, a picture can have the same logical form as a fact, but it cannot depict that logical form proper. The logical form SHOWS itself in the picture. If we remember the opening sections, we said that the only things we can say about the world are individual facts and that facts exist in virtue of their logical form. So facts are the things which can be SAID while form is that which can be SHOWN. And just as there is only one matching logical picture for each and every fact, there will be one and only one thought matching each logical picture.

A thought is a logical picture of a possible situation because thoughts share the same logical form as logical pictures and facts. It is possible to have a false thought but not an illogical one. To use an analogy of geometry, we could display a circule of any circumfrence regardless of whether or not that circle truly existed or not. But we could never decpit one that contradicted the laws of space (3.032). So in our thoughts we cannot represent a thought that contradicts the laws of logic.

To complete the chain from fact to logical picture to thought, we represent thoughts by means of propositions. A proposition is the meaning that we express in sentences, thoughts, etc. in the sense that it is conveyed through propositional signs such as speech, writing, and body language. Like a picture, a proposition represents a possible state of affairs by sharing the form in common with it. This means that its elements are arranged in a similar way. This is why a sentence has a sense but a random string of words does not. The latter has no structure or internal coherence in its arrangement.

Sentences can picture propositions because they share the same logical form as them. What exactly this means is not entirely clear. But one idea might be as simple as this: If we have a proposition like "Othello loves Desdemona," it might display that state of affairs by virtue of "Othello" being at the beginning, "Desdemona" being at the end, and "loves" being in the middle. [8]. Whatever form this mirroring takes, Wittgenstein claims that that sentences and other propositional signs share the same logical picture as propositions, which share the same logical picture as thoughts, which share the same logical picture as facts.

Here, aRb stands as a logical representation of a statment that says that "a" stands in relation "R" to "b." The proposition does not say what relation holds between its elements. Rather, that relation is what makes the proposition say-able in the first place. Facts are something we can SAY, while their logical form is something that is SHOWN. As facts can exist in virtue of a logical form that is SHOWN, so the proposition can only SHOW the logical form of relation between its elements. It can only DISPLAY them. This distinction between saying and showing will have profound implications in the next section.

4.d. Propositions 3.2-3.5: Language and Logical Form

When I first read the Tractatus, this is the first area that I got lost in. It's one of the first where the engagement with Frege and Russell becomes much more direct and technical. The concern is now with the representation of propositions in logical calculus. For that reason, this section will be nearly incomprehensible without some background on Frege and Russell's work in this field.

Wittgenstein assumes a familiarity with two related areas of Frege and Russell's thought: denoting phrases in semiotics and set theory in logic and mathematics. Though not quite chronological, the best place to find an opening might be "On Denoting," a short and paradigmatic essay published by Russell in 1905. This work has clear connections to the Tractatus in that it too is interested in the logical form of propositions, how we apprehend them, and especially how this happens in language. This is the same concern of Wittgenstein in this section.

Like Frege, Russell realized that the predicate form of grammar often masks the true logical form of a proposition. We read sentences as being composed of subjects and predicates, when they are actually composed of functions and variable place-holders. For an illustration of what this means, take the following sentence:

At first glance, the structure of this sentence looks like a simple statement of a subject and predicate: "X is Y." (X = "all horses," is = "are," Y = "mammals.") But what ts actually represents logically is something like this:

This sentence does not talk about "all horses" as one object being some predicate or another. This might not seem particularly interesting, but the philosophical value of this discovery is that a lot of our philosophical confusions can be cleared up by becoming clearer about the "inner" logical structure of propositions as opposed to the "outer" impression of our grammar. [9]

One example of a "problem" that is solved via this method is presented. In a classical logical frame, all propositions can be judged as either true or false. Something must either be the case or not be the case. But how can we establish this for propositions that refer to things that do not exist? For example:

France has not had a king since the 17th century. There is thus no such thing as a "present king of France." The term simply refers to something that does not exist. So we cannot look at that individual and find an answer about whether he is bald or not. In fact, how do we even understand what this phrase means in the first place? Certailny we can understand the phrase "present king of France" enough to know that there isn't one, at the very least! But how do we do this without a real object to compare it to?

Frege's opinion was that we must admit that propositions like this are simply neither true nor false. But Russell was uncomfortable with this idea as it denied a basic logical axiom: That a proposition must only be true or false. Russell thus came up with a way to show how we can still comprehend the meaning of this sentence and others like it and thus establish it as true or false.

The sentence "The present king of France is bald" appears as a simple "X is Y" sentence, just as "All horses is mammals" does. But what it asserts is even more complex than "For all X, if X is a horse, then X is a mammal." What it asserts is actually three claims:

1. "There is an x such that x is currently the king of France." (i.e. there is a present king of France)

2. "For any x and y, if x is currenly the king of France and y is currently the king of France, then x=y." (i.e. there is only one present king of France)

There is no object called "present king of France" in these sentences. So the truth and falsity of these sentences can be easily assessed. As we can deduce that the first of these three claims is false, the entire proposition can thus be rejected. That said, how would we express this? It would not be accurate to say "the present king of France is not bald," this still presupposes the non-existent entity "present king of France." But this is also corrected with proper logical notation:

The proposition "The present king of France is bald" with its three claims looks like this in logical notation:

And these are the wrong and correct way to deny the truth of this proposition, respectively:

The first one affirms the existence of a present king of France but denies that he is bald. The second denies the entire proposition and thus denies the existence of a present king of France. And so what seemed like an insurmountable problem (establishing the truth or falsity of a sentence with a non-existent entity in it) is really a matter of being misled by the grammatical form of a sentence and not understanding its real logical structure.

This ability to analyze the form of a proposition also makes quantifier logic possible and allows us to represent open formulas. That means that we can make statements in logic like "there exists an x such that..." or "for all x..." that specify the number of values that can apply for any function. For example, take the function "X is a horse:"

There are many values of X that would satisfy this function, such as Mr. Ed, Pegasus, that animal in the petting zoo, and so on. For Frege, all values of X that satifsy the function "X is a horse" is an "extension" of the concpet "is a horse" (i.e. all horses).

Russell follows Frege in defining collections of things that satisy a cerain function as "sets" or "calsses." For example, the "set" of all horses is the set of all objects that satisfy the function "X is a horse." For the sake of completeness in logical analysis, we should be able to make sets of anything at all: [10]

That seems fine so far, but once we accept this, a problem emerges. If there are sets of everything, there must be sets of sets. For example, there is a set of all sets, a set of all sets with at least one member that begins with the letter "a," and a set of all sets with two members (this is Russell's definition of the number 2). And if that is the case, there must also be sets that contain themselves. For example, the set of all sets that begin with the letter "s" must contain itself. If there can be sets that contain themselves, there must be set of all sets that contain themselves as well as a set of all sets that do not contain themselves. But now a paradox emerges:

Does the set of all sets that do not contain themselves contain itself? If we answer yes, then it will in fact contain itself and this leads to a contradiction. If we answer no, then it will also by definition contain itself and this leads to a contradiction. This is known as Russell's Paradox. Since it is derived from the most basic laws of Frege's logic, it was devastating to Frege's entire career. That said, Russell did offer his own potential solution: the Theory of Types.

First-order sets can only contain objects, second-order sets can only contain objects and first-order sets, third-order sets can only contain objects, first-order sets, and second-order sets, and so on. This would mean that we wouldn't have to worry about the possibility of a set trying to contain itself and thus wouldn't run into the paradox. We would however need to invent a symbolism to distinguish the symbols for objects from those for first-order sets and so on. At the end of the day, this seems to be more of a workaround than a real attempt to solve the paradoxical nature of the problem itself.

This sets the stage for this section of the Tractatus, where Wittgenstein shifts to a discussion of the logical properties of language and representation.

Our everyday speech features complex objects that can be analyzed into simpler parts via definitions, but also simple names that cannot.

opprobrious ─────────────────────► "expressing scorn, disgrace, or contempt"

If I don't know what a word like "opprobrious" means, someone can define it in simpler words. But there are also simple symbols for names that cannot be further defined. They are fully analyzed. In a fully analyzed proposition, they will directly mean the object they refer to with nothing standing in the way (3.203).

Wittgenstein does not define exactly what objects are, but we can have some idea of what they are if we think of them as are something that cannot be further analyzed and relates directly to the thing they name. This might seem impossible. A quick glance at the dictionary shows that there don't seem to be names that we can't define. But for Wittgenstein a name would have to be too simple to be defined.

However, the "definitions" in a dictionary are not really "definitions" in a pure way. True names of objects are too simple to be defined. So instead of saying what they mean, we have to SHOW what they mean. And that is generally how a dictionary definition works. It shows a word's meaning pointing us to what kind of sentences it is used in rather than appealing to some kind of eternal logical form. This kind of definition is an "elucidation" in Wittgenstein's terms. And it rests on one of his primary assertions: that a sign's meaning IS its use in propositions.

We know the nature of objects based on what states of affairs they appear in. The same is true of signs. That is, things that have a sense that we use to represent ideas, facts, words, sentences, ideas, etc. Names and words are all meaningless outside of the propositions they occur in. If our language of signs was perfect, the signs we use to express propositions and their variables would always be completely clear so that we are never confused about their meaning. But natural language is never perfectly logical. We often use the same sign in different ways or different signs in the same way, which makes the logic of our language hard to se. But natural language is never perfectly logical. We often use the same sign in different ways or different signs in the same way, which makes the logic of our language hard to see (3.321-3.322).

In 3.323, Wittgenstein makes the point that the word "is" can be used in three completely logically distinct ways:

1. "is" as a copula: "John is tall." (Tj)
2. "is" as an expression of identity: "John is John." (j=j)
3. "is" as an existential quantification: "There is a person named John." ((∃x)Jx)

A sign like "is" has no meaning apart from its use. To get an idea from what he means by this look at a proposition like:

This is a function of "The X is on the Y." "Hat" and "table" are two values that can be used in the function "The X is on the Y." That itself does not tell us much. But the fact that "hat" and "table" can be used as values for the function while "love" and "purple" cannot DOES tell us something. That is, the fact that "The hat is on the table" conveys a meaning while "The love is on the purple" does not is what is gives meaning to signs like "is." The meaning comes from when and with what objects they can be used.

This also solves Russell's Paradox. Wittgenstein says that Russell makes the mistake of giving his signs a "meaning" of their own. There is no need to posit a Theory of Types when we recognize that a proposition that makes a statement about itself is not being used in the same way as the original proposition. A function cannot be used to talk about itself (a set cannot contain itself), because this would be giving it two different uses. This would mean it wouldn't really be talking about "itself" to begin with, as its meaning is defined by its use. This is hard to understand until we look at it in logical notation:

A proposition making a statement about itself (a set containing itself):
F(F(fx))

The logical form of F(fx) and F(F(fx)) are different. This means that their use is different and thus that their meaning is different. They are as different as when we use "is" as a copula and when we use "is" as a sign of identity. This is why Russell's Paradox does not occur.

And this brings home the important point of this section: the sense of a proposition is entirely internal to the proposition itself. The elements of a proposition are only related to one another and not to anything external to them. These two propositions are identical in their essential features:

They are both forms of "The X is on the Y." They only vary in the contingent fact of what values are given to their variables. That the hat is on the table happens to be true or that the book is on the shelf happens to be false tells us nothing about the proposition, because these are contingent facts about the world and are not a priori truths. However, the fact that these are kind of values that CAN be used in "The X is on the Y" does help us elucidate the meaning of the proposition.

"The X is on the Y" does not have a meaning that can be said. We cannot say its logical form. But we can SHOW its position in logical space by understanding what values can meaningfully use for X and Y (3.3421). This is in contrast to Frege, who claimed that every proposition is a complex name for one of two objects, the true and the false, which exist in a kind of higher Platonic realm. For Wittgenstein, no proposition true or false by any metaphysical or a priori truth. It only has an internal meaning, which it gains in virtue of what kinds of values can fill out its variables. [11]

4.e. Propositions 4-4.116: Philosophy as the Logical Clarificatoin of Thoughts

In the above section, we established that the sense of a proposition is internal to the proposition itself while the meaning of a name is external to the name. The meaning of the name is the thing that it denotes and there is nothing in the name itself that can tell us what object it denotes. We learn the meaning of the name only by observing how and in reference to what it is used. This is why the meaning of names is their use. The meaning of the name lies outside of the name itself and so it must be made clear by means of elucidations.

The totality of all propositions is the sum total of what we are capable of expressing in language (4.001). Our everyday language does possess a logical structure of its own, but it rarely makes its logical form obvious, which leads to all kinds of philosophical confusion.

A proposition does not require elucidation like a name does because of the above Picture Theory. A proposition and the reality that it depicts share a logical form, and that is enough for the one to depict the other. There is nothing external to the proposition that can make the connection between it and what it depicts any clearer than it already is.

Wittgenstein uses the analogy of sheet music and a symphony. If you can read music, you don't need anything else to help you translate written notes into sounds besides the shared form of the notes and the music. In fact, it isn't even POSSBLE for anything else to make it clearer. It is the same with a proposition. Its shared form with the picture of reality is how we come to understand the proposition. Nothing can make that connection clearer than it already is.

Do we have to follow certain rules in order to understand the sense of a proposition? Are there rules on how to interpret those rules? Wittgenstein takes up the question of rule-following much more thoroughly in his later writings such as the Philosophical Investigations, but in the Tractatus he simply says that elucidations like rules and interpretations for propositions are not necessary when the common logical form already shows itslef. This avoids the infinite regress of needing rules to interpret a proposition, then rules into interpret those rules, and so on. There is no need for rules when the logical form can SHOW itself without being said and thus without needing to be interpreted.

The point here about showing and not saying is crucial. Wittgenstein thinks that not only do we not NEED to say what the sense of a proposition that shows itself is, we are in fact UNABLE to. The proposition shares a logical form with the reality it depicts. The commonality of that logical form is what enables us to understand speech in the first place.

And it should be emphasized that the truth-value of a proposition has nothing to do with its sense. A proposition makes a picture of the world irrelevant of whether that picture is true or false. And we can draw logical inferences from that picture simply in virtue of the fact that it can be expressed.

This means that "logical constants" like "is," "and" (.), "not" (~), and so on cannot be represented or talked about in language. Take these two propositions for example:

These propositions have the same sense (p=~~p). This is because the sign ~ ("not") doesn't mean anything. He equates it to a punctuation sign or to the parentheses or line numbers that a proposition occurs in. The propositions "p" and "~p" might have opposite senses but they display the same logical picture. One of them simply says that the picture presented is the case and the other that it is not. The picture itself remains the same.

Wittgenstein goes so far as to call the claim that logical constants cannot be represented his "fundamental idea" (4.0312). Indeed, the distinction between saying and showing is most clearly formulated here. Note that this claim is not about the world or about language but about LOGIC.

Logical connectives like "is" and "not" (~) cannot be represented or talked about in language. This makes them meaningless. The shared form between propositions and logical form is like that between notes on a sheet and music or names and objects. It can only be shown and not expressed in language, described, or talked about. But the upshot of this is extreme and perhaps even troubling. Much of Frege's and Russell's philosophy was centered around a discussion of things like the nature of logical inference and logical relations. But for Wittgenstein, these are all fruitless endeavors that are born from a fundamentally impossible desire: to talk about what can only be shown.

Logic is shown in the way the world is held together. It is simply impossible for us to say what these workings are or to make them clearer in language than they already are. When this point becomes clear, we realize just how radically Wittgenstein limits that which can be meaningfully said. Propositions can only make claims about how things stand in the world, whether true or false. That is the business of natural science. He says that philosophy is not at all what people have thought it was: a series of propositions about the true nature of reality. Thinking that this is the aim of philosophy has been the greatest cause of philosophical confusion (4.003). He clarifies what the true aim of philosophy is in 4.112:

Philosophy is an "activity." Its business is showing rather than saying. Philosophy does not make statements about what is the case in some way that stands outside of reality and the world. It is a method of clarifying the logical structure of our propositions which are so clouded by our everyday language. Everything that can be said is the whole of natural science. And when everything that can be said is made clear, everything that cannot be said will also be made clear. And this means that no one part of natural science (meaning no part of what can be said about how things stand in the world) is more closely related to philosophy than any other, be it ecology, psychology, evolutionary theory, quantum physics, or whatever else (4.1122).

4.f. Propositions 4.12-4.128: What Can Be Shown Cannot Be Said

We can't meaningfully talk about things like logical relations. Propositions can only make statements about how things stand in the world. Propositions can depict all of reality but they cannot depict the logical form of reality. A proposition can only depict something external to it, so in order to depict logical form it would have to do so from a standpoint that is outside of logical space. That is impossible. We can only SHOW this logical form in virtue of the shared form between the proposition and the reality it depicts.

In this section, Wittgenstein introduces a distinction between what he calls "formal concepts" and "concepts proper." This follows from the claim that only propositions that picture states of affairs in the world have meaning. But to fully understand the idea, we have look at the ideas of Frege that Wittgenstein is responding to.

Frege analyzed the logical properties of language and made a distinction between objects and concepts. For example:

In this proposition, we have an object and a concept. The object is "the president of America," as this denotes a specific thing in the world to which we can ascribe properties. The concept is "Texan," as this is a category into which one or more object may fall. We could say that "the president of America" is a value of the function "is a Texan." This can be generalized. "Socrates" is a function of the value "is an Athenian." Most broadly, in any proposition of the form "X is a Y," the X will represent an object and the Y will represent a concept.

But this distinction falls apart immediately when we try to apply it to logical properties themselves. What do we make of a proposition like this?

This follows the "X is a Y" format, but the function of X is now "the concept of a horse." If we can say things about "the concept of a horse" and ascribe properties to it, it must be an object. But that is contradictory. It thus seems that no true proposition can have the form "X is a concept." This is because the idea of a "concept" is something that can only SHOW itself.

To be clear, we have to make a distinction between a formal concept and a concept proper (4.1272):

A concept proper can be spoken about meaningfully. We concept of "horse" makes sense in everyday propositions that speak about how things stand in the world. But a formal concept cannot be spoken about in any meaningful way at all. It can only SHOW its meaning. We cannot say that "X is a number." We can only come to understand that X is a number by observing what kinds of propositions it occurs in. Any attempt to use a formal concept in a proposition, such as "Two is a number" or "Purple is a color," will result in nonsense. "X is a horse" and "X is a concept" are identical grammatically. Thus they appear to be logically equivalent on the surface. But they are not. Only "X is a horse" has sense. Any attempt to say something like "X is a concept" results in nonsense.

Unlike Frege and Russell, Wittgenstein asserts that formal concepts are not represented in logical notation as sets or functions. This is why Russell's Paradox does not occur. They cannot be introduced in the same way that objects are (4.12721). That is to say, we can say "There is an X such that..." but not "There is an object such that..." Frege and Russell might try to express X being an object as a function like O(x). But for Wittgenstein, formal concepts can only show themselves as variables. The variable X in "X is a horse" SHOWS that X is an object because it holds the place of an object in the proposition. That is the most we can do. [12]

Because we cannot talk about formal concepts, we cannot talk about the number of them that exist or what kinds of formal objects there are. The logical forms are also "without number" (4.128). These logical forms make no claim to anything concrete in the world at all. This means that we cannot meaningfully speka of ideas like philosophical monism or dualism. We can only talk about objects and states of affairs in the world, and formal concepts have no objects or states of affairs corresponding to them.

This is an extremely harsh conclusion. It means that not only can we not make abstract philosophical statements, but that even statements like "Two is a number" or "Purple is a color" have no sense because they describe formal properties of concepts or objects. We might think that surely we understand what someone means when they say that two is a number! But Wittgenstein says no. We only think we understand because we are being misled by a familiar grammatical structure. To test whether a proposition has sense or not, we need to ask what possible situation in the world it represents. Since there is no situation corresponding to "Two is a number," this is nonsense. We can describe things that are colors or concepts or objects, but we cannot speak meaningfully of colors or concepts or objects themselves.

That said, there is immediately a problem here: We are not supposed to be able to make meaningful statements about formal concepts. So what do we make of all the propositions of the Tractatus? Almost all of them speak about formal concepts like objects,states of affairs, and facts. They do not speak about specific ways that things stand in the world like the propositions of natural science do. In fact, it was only by using these propositions that we arrived at the distinction between formal concepts and concepts proper in the first place. This tension is only truly dealt with at the end of the Tractatus, so I will ask you to wait until then to see how this is dealt with. But keep it in the back of your mind that Wittgenstein already seems to pull the rug out from under his own feet.

4.g. Propositions 4.2-5.156: Elementary Propositions and Truth-Grounds

4.2. The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.
4.3. Truth-possibilities of elementary propositions mean possibilities mean possibilities of existence and non-existence of states of affairs.
4.4. A proposition is an expression of agreement and disagreement with truth-possibilities of elementary propositions.
4.5. 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.
5. A proposition is a truth-function of elementary propositions.
(An elementary proposition is a truth-function of itself.)
5.1. Truth-functions can be arranged in series.
That is the foundation of the theory of probability.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus 4.2-5.156

All there is to say about reality is propositions that happen to be the case or not. These stand independently of one another. They can only depict possible situations that are in the world, not formal concepts that are abstract. The latter can only SHOW their form. There are no logical constants like "and" (.), "not" (~), etc. that hold them together as a kind of logical glue, but only objects that combine like links in a chain. They combine in certain ways. In this section, Wittgenstein explains how these combinations themselves convey logical form on their own without the need of logical constants.

No picture or fact is a priori true. We only know that something is true or false by comparing it with reality. And that itself is the entire meaning of the proposition. If we know when a proposition is true and when it is false, we know what a proposition means in its entirety. We can say that "Ottawa is the capital of Canada" is true or false only in virtue of knowing when that would be true and when it would be false.

Wittgenstein now introduces the term "elementary proposition." Any proposition consists of one or more elementary propositions. This section can get confusing, but all that Wittgenstein means by elementars propositions are the fundamental basics of a proposition. For example:

A proposition like "Socrates is a wise Athenian" depends on the elementary propositions "Socrates is wise" and "Socrates is an Athenian." At least this gives us an idea of what it would look like. What makes these elementary propositions so fundamental is that they are independent of each other and that their truth or falsity does not depend on the truth or falsity of any other proposition. Once again, Wittgenstein gives us no examples of what these might look like. We are interested in the necessary logical components of the system, no matter what a particular example might look like. So we will refer to them by simple names like p, q, r, etc. A proposition like p.q will depend on the elementary propositions p and q.

If we take all the elementary propositions that constitute a proposition, list all the possible combinations of truth and falsity they could have, and whether these would make the proposition itself true or false, we would have a complete understanding of the proposition. This sense would be entirely internal to the proposition itself. It would SHOW itself without needing to be said. And there is a tool that does exactly this: truth-tables.

Any given elementary proposition is either true or false. If we have two elementary propositions, p and q, there are four truth-possibilities:

These tables will double in size every time we add another elementary proposition:

For the sake of simplicity, we will continue to work with only two elmentary propositions, p and q. We can express the truth-conditions of a proposition involving p and q, such as "if p then q," by writing all the possibilities of p and q on the left and whether "if p then q" could be logically possible under those conditions on the right.

Assuming a fixed order in our notation for these truth-possibilities, we can express this proposition as the values of the rightmost column in a parenthetical string:

(TTTT) (p, q): Tautology (if p then p; and if q then q) [p⊃p.q⊃q]
(FTTT) (p, q): Not both p and q [~(p.q)]
(TFTT) (p, q): If q, then p [q⊃p]
(TTFT) (p, q): If p, then q [p⊃q]
(TTTF) (p, q): p or q [p v q]
(FFTT) (p, q): Not q [~q]
(FTFT) (p, q): Not p [~p]
(FTTF) (p, q): p or q, but not both [(p.~q) v (q.~p)]
(TFFT) (p, q): If p, then q and if q, then p [p≡q]
(TFTF) (p, q): p
(TTFF) (p, q): q
(FFFT) (p, q): Neither p nor q [p|q]
(FFTF) (p, q): p and not q [p.~q]
(FTFF) (p, q): q and not p [q.~p]
(TFFF) (p, q): q and p [q.p]
(FFFF) (p, q): Contradiction (p and not p; and q and not q) [(p.~p).(q.~q)]

Pay spectial attention to the first and last: a tautology is something that is always necessarily true and so its string only has "T" values (TTTT). A contradiction is something that is always necessarily false and so its string only has "F" values (FFFF). The rest show the varied circumstances under which the proposition is true and false. Wittgenstein calls these parenthetical strings "truth-grounds." If we know the truth-grounds of a proposition, we know the sense of the proposition in full.

We can understand a proposition via its truth-grounds alone. We do not need any connectives such as "and" (.), "or" (v), "if then" (⊃), and so on. None of these are essential to the sense of a proposition, because the sense can be fully expressed in a truth table without these symbols. This is right in line with the proposal that logical constants do not represent anything, because they don't show anything that can't already be seen in a truth table. This system of truth tables and truth-grounds thus accounts for all the workings of logical inference.

Frege and Russell thought that we needed "laws of inference" to follow in their axiomatic systems. But Wittgenstein shows that there is no need to appeal to any kind of "law" to infer things. A proposition follows from another if the first is true whenever the second is true. And this directly shows itself when we write out their truth-grounds:

We don't need a "law" of inference to tell us that the one proposition follows from the other. It shows itself plainly when the notation is clear.

The two limiting cases are tautologies (such as p.p), which are always true, and contradictions (such as p.~p), which are always false. Wittgenstein describes these as "sinloss" or "senseless." Note that this is not the same as "unsinnig" or "nonsensical." "Nonsensical" is the word that he uses to describe sentences like "the number two is purple" because they are completely outside of the realm of logic. However, tautologies and contradictions are called "senseless" because they do not convey any sense or state of affairs in the world, but they are held together in a logical manner as they consist of truth-possibilities for elementary propositions.

A contradiction like "It is raining and it is not raining" is never true. But there is nothing "illogical" about it since it can be expressed in a truth table and in logical notation (p.~p). The only propositions that should be excluded from logic are those that cannot be written in logical notation or established as true or false whatsoever. Wittgenstein claims that tautologies and contradictions belong in the realm of logic in the same way that the value 0 belongs in arithmetic. This is a stance against Frege and Russell, who held that contradictions are "illogical" and thus not a part of logic. Wittgenstein says that this is an arbitrary limitation. Something being logical does not mean it needs to be actual. It only needs to be possible to express it in a correct logical manner. So there should be no reason to exclude propositions which are always false, such as contradictions.

The last area to note in this section is the status of probability. The logic of inference is the basis for probability. Take as example these two propositions once more:

If "p and q" is true, then there is a 1/3 chance that "p or q" is true, based on nothing but the possibilities of their truth-grounds. But this is only a theoretical procedure based on psychological grounds. In reality, there are no degrees of probability. Propositions are either true or false (5.153).

4.h. Propositions 5.2-5.4611: The Irrelevance of Notation in Logic

5.2. The structures of propositions stand in internal relations to one another.
5.3. 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.
5.4. At this point is becomes manifest that there are no 'logical objects' or 'logical constants' (in Frege's and Russell's sense).
5.41. 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.
5.42. It is self-evident that v, ⊃, etc. are not relations in the sense in which right and left etc. are relations.
The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations.
And it is obvious that the '⊃' defined by means of '~' and 'v' is identical with the one that figures with '~' in the definition of 'v'; and that the second 'v' is identical with the first one; and so on.
5.43. Even at first sight it seems scarcely credible that there should follow from one fact p infinitely many others, namely ~~p, ~~~~p, 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.
5.44. Truth-fucntions 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 '~~p' negate ~p, or does it affirm p--or both?
The proposition '~~p' 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 '~', it would follow that '~~p' said something different from what p said, just because the one proposition would then be about ~ and the other would not.
5.45. 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.
5.46. If we introduced logical signs properly, then we should also have introduced at the same time the sense of all combinations of them; i.e. not only 'p v q' but '~(p v ~q)' as well, etc. etc. We should also have introduced at the same time the effect of all possible combinations of brackets. And thus it would have been made clear that the real general primitive signs are not 'p v q', '(∃x).fx', etc. but the most general form of their combinations.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus 5.2-5.4611

In the previous section, Wittgenstein criticized the idea that we need laws of inference in our logic because all propositions show their sense by nothing more than the truth-grounds of their elementary propositions. Here he attacks something else even more fundamental to Frege's and Russell's work in logic: the idea of logical axioms and logical objects.

Frege and Russell defined logic as a supremely general set of laws in the form of propositions. Just as the lwas of chemistry pertain to all chemical interactions and the laws of physics pertain to all natural phenomena, the laws of logic would pertain to everything, including other laws and to themselves. The laws of logic dictate the form that other sets of laws can take. For example, we might imagine that there were physical laws other than the ones we have. To use the SparkNotes example, we could imagine that massive bodies would repel each other instead of attracting one another. But we cannot imagine physical laws that are illogical. [13] This is why logic is prior to psychology, to mathematics, to metaphysics, and to anything else in philosophy or science. And by this understanding, anything is rational that follows the laws of logic.

For Frege and Russell, logic is an axiomatic system. There are certain axioms, logical objects or connectives, and laws of inference. If p⊃q is true and p is true, then q is true. This is true of any and all propositions that could ever exist. This is a fundamental axiom wihch is made up of fundamental objets like "if-then" (⊃) and "and" (.) that are self-evidently true. Laws of inference tell us how to deduce all propositions from fundamental axioms. (p v ~p) for example is an axiom. p has to be the case or ~p has to be the case. The argument goes that (p v ~~~p) is not an axiom in the same way that p.~p is, but it follows from it (because ~~~p=~p). What allows us to understand that is the laws of inference.

Wittgenstein critiques this view. For him, this all tries to give too much of a metaphysical background to logic. In his words, logic must "look after itself" (5.473). Frege and Russell try to say what can only be shown. The truth is that (p v ~p) and (p v ~~~p) have the exact same truth-grounds when drawn up in a truth table:

And because a proposition is nothing more than its truth-grounds, these are just the same proposition in the first place. There is no need for a law of inference at all since these don't say anything different in the first place. They have the same sense. Russell tries to ground logic with a kind of metaphysics when the very definition of logic is something that has to come before all metaphysics and stand on its own with no need for a ground.

In fact, it's not even correct to say that (p v ~p) and (p v ~~~p) have the same "sense," because these are both tautologies. This means that they don't express any "sense" at all. In fact, all propositions of logic are tautologies by definition. This means that all axioms and propositions of logic are equivalent and that there can be no preeminent ones. He says that "all the propositions of logic say the same thing, to wit nothing" (5.43).

Wittgenstein dissociates the importance of notation from logic completely. All that is essential to a proposition is its sense. If (p.~q) has the same sense as ~(q v ~p), then they are the same proposition. There is no need to explain how one proposition like p can "imply" an infinite number of others like ~~p and ~~~~p, because these are all really just the same proposition.

We might think that there is some process where we combine elementary propositions like p and q to make a proposition like p.q and that we need to find some way to explain how these can come together. An operation can be applied successively to create a series of new propositions. Sometimes it cancels itself out:

But in any case, all propositions are nothing more than a number of operations performed on elementary propositions. These operations combine or modify the truth-functions of elementary propositions. So we call them "truth-operations."

Where Free and Russell make a mistake is that they think that operations are a kind of object or form. Operations are not an action of some kind that we "take," since propositions are ultimately nothing more than their truth-grounds. Operations are nothing more than a shorthand explanation of the difference between the truth-grounds of different propositions. p and p.q have different truth-grounds. And the "adding" of q is simply a way to express that. There is no real "operation" going on. It is just a way to explain the difference of the truth-grounds between p and p.q. Any operation that expresses the same relation is the same operation in the same way that any notation that expresses the same sense is the same proposition.

This is why Wittgenstein concludes that there is something fundamentally flawed about the ideas of logical objects and logical constants. Frege built his entire system on the "primitive" connectives "not" (~) and "if-then" (⊃), meaning that all other logical operations can be defined in terms of them. Russell did the same but built his from "not" (~) and "or" (v). But these are in fact interchangeable. Frege's (p⊃q) is the same as Russell's (q v ~p). Russell's (p v q) is the same as Frege's (~p⊃q). And that means that there is nothing "primitive" or "fundamental" about these connectives. All propositions of logic say the same thing, which is nothing.

4.i. Propositions 5.47-5.54: Superfluous Signs in Logic

5.47. 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 '(∃x).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.
5.5. Every truth-function is a result of successive applications to elementary propositions of the operation '(-----T)(ξ,....)'.
This operation negates all the propositions in the right-hand pair of brackets, and I call it the negation of those propositions.
5.51. If ξ has only one value, then N(‾ξ) = ~p (not p); if it has two values, then N(‾ξ) = ~p.~q (neither p nor q).
5.52. If ξ has as its values all the values of a function fx for all values of x, then N(‾ξ) = ~(∃x).fx.
5.53. 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.
5.54. In the general propositional form propositions occur in other propositions only as bases of truth-operations.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus 5.47-5.54

Logic must "look after itself." We don't need rules or laws to tell us how logic should function. Whatever is ruled out by logic will be impossible or inconceivable and we do not need rules or laws to tell us what that is. If something lacks sense, it is only because we haven't given a meaning to the signs in the proposition. For example, the proposition "Socrates is identical" says nothing because we have not given a meaning to the word "identical" when used as an adjective (5.4733). The "laws of logic" are given all at once and not in a hierarchy like Frege or Russell thought. This means that "(p.q)" means the same thing as "~(~p v q)" and that "fa" means the same thing as "(∃x).fx.x=a." It is only the sense that matters.

If all logical constants are given at once, it must be because they share a basic propositional form. Wittgenstein calls this the "essence" of a proposition (5.471). It should serve as the sole logical constant and render all other constants superfluous. What Wittgenstein has in mind is an application of a negating operation. He alludes to the Sheffer stroke, a logical constant discovered in the early 20th century. Frege developed a logical system defining everything in terms of "not" (~) and "if-then" (⊃). Russell developed one relying only on "not" (~) and "or" (v). But the Sheffer stroke, symbolized by a vertical bar (|), is functionally complete on its own.

The Sheffer stroke is a negation of disjunction. It means that one or other can be the case, both can neither be the case, but both of them cannot be the case. It is called NAND for short (not-and). I will prove its complete functionality before moving on.

The truth-grounds of p|q are "(FTTTT) (p, q)." Note that these are the same truth-grounds as ~(p.q) and (~p v ~q), meaning that these are logically equivalent to p|q.

To be functionally complete, the Sheffer stroke needs to be able to represent the following four operations by means of nothing but its own NAND operations:

1. negation (~p)
2. conjunction (p.q)
3. disjunction (p v q)
4. implication (p⊃q)

The truth-grounds of p|q are "(FTTT) (p, q)." Compare this, however, with the truth-grounds for (p.q):

We can see that they are the exact opposite. Whenever p|q is true, p.q is false, and vice versa. For that reason, we can say that the Sheffer stroke has the same value as negation. Observe the following:

The truth-grounds of p|p are (FT) (p). These are the exact same truth-grounds as ~p. Therefore, for any well-formed formula, we can say that ~φ=(φ|φ).

Since the Sheffer stroke is the equivalent of negation, we can replace the inner negation with a NAND operation:

Since negation is logically equivalent to NAND operations, we can replace these negations with their NAND equivalents:

And since a p|q and ~(p.q) are logically equivalent, this can also be written as a NAND operation:

1. negation: ~p=(p|p)
2. conjunction: p.q=(p|q)|(p|q)
3. disjunction: p v q=(p|p)|(q|q)
4. implication: p⊃q=p|(q|q)

On the basis of the above, Wittgenstein concludes that all non-Sheffer stroke signs in logic are ultimately redundant. Joint negation functions as the sole logical constant. With that in mind, he introduces the operatipn that all propositions can be derived by the successive application of the operation "(-----T)(ξ,....)." This operation negates all the terms in the righthand pair of brackets. For instance, when there is one elementary proposition, p, it becomes "~p". When there are two elementary propositions, p and q, it becomes "~p.~q". And so on. Wittgenstein abbreviates this terminology to "N(‾ξ)." N stands for negation and ‾ξ stands for all the propositions in the righthand pair of brackets (5.502). ξ is simply a placeholder value and the bar on top signifies that this applies for ALL values that could take its place.

There is no need for laws of inference, logical axioms, or logical objects. We then end this section with a very difficult section. In traditional logic there is a distinction between truth-functional logic and quantified logic. Truth-functional logic talks about single propositions joined to form more complex propositions. For example, we form "p.q" from "p" and "q." Quantifier logic, on the other hand, makes generalizations about entire classes of propositions. It uses symbols like the universal quantifier (∀x = "For each x...") and the existential quantifier (∃x = There exists an x such that...").

Wittgenstein's truth-tables works very well for establishing the meaning of individual propositions, but Frege's and Russell's logic contains quantifiers like these. It seems that we can only say that some individual proposition is true or false, but not that a proposition is true or false for all values of x or that there exists an x such that the proposition is true or false. But we don't need the universal or existential quantifiers at all. Observe these logically equivalent alternatives:

(∀x).~fx = N(fx) ◄─── Universal negation (fx is false for all values of x)

(∃x).fx = N(N(fx)) ◄─── Existential generation (there is at least one value of x that makes fx true)

(∃x).~fx = f(N(x)) ◄─── Existential negation (there is at least one value of x that makes fx false)

(∀x).fx = N(f(N(x))) ◄─── Universal generalization (fx is true for all values of x)

Wittgenstein further wants to get rid of the sign of identity (=), seeing it as another superfluous sign in logic. And all that is superfluous in logic is meaningless. To say that two different things are identical (p=q) is nonsensical in logic, as in an ideal logic the same thing should never have two different names. And to say that the same thing is identical (p=p) is senseless as it gives us no new information. Any attempt to give meaning to the sign of identity is to try to say something that can only be shown.

4.j. Propositions 5.541-5.641: The "Truth" in Solipsism

5.541. At first sight it looks as if it were also posisble for one proposition to occur in another in a different way.
Particularly with certain forms of proposition in psychology, such as 'A believes that p is the case' and 'A has the thought p', etc.
For if there are considered superficially, it looks as if the proposition p stood in some kind of relation to an object A.
(And in modern theory of knowledge (Russell, Moore, etc.) these propositions have actually been construed in this way.)
5.542. It is clear, however, that 'A believes that p', 'A has the thought p', and 'A says p' are of the form '"p" says p': and this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects.
5.55. 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.
5.6. The limits of my language mean the limits of my world.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus 5.541-5.641

This section at first seems like a random deviation as it begins talking about the nature of the self or subject. The reason this pivot occurs is one kind of proposition which at first glance seems like a problem for Wittgenstein's idea of logic. These are propositions of the form "A believes that p," "A says that p," "A hopes that p," and so on. According to Wittgenstein, elementary propositions can only occur in other propositions as a basis of their truth functions. That means that they occur as parts of the truth and falsity values in their truth-tables. That means that the truth or falsity of any elementary proposition must have an influence on the truth or falsity of the whole.

"A believes that p" seems to present us with a snag. After all, someone can believe that it wil rain tomorrow whether or not it will in fact rain tomorrow. If "A believes that p" is a composite of the proposition p, among other things, then we would expect p to have some bearing on the truth or falsity of the whole. But in this case, it does not seem to.

The answer to this conundrum is that a proposition of the form "A believes that p" does not actually involve a relationship between "A" and the proposition "p." He says that it is actually of the form "'p' says p" and that "this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects" (5.542). What he says here is that the proposition "p" somehow occurs in whatever form (thought, hope, utterance, etc.) and that the correlation that emerges is between the proposition as it manifests and that which it pictures. "A" simply drops out of consideration as a necessary part of this.

"A" cannot coherently refer to a "soul" or "self" in which thoughts and beliefs reside. He compares it to a a two-dimensional image of a cube that gives the illusion of three dimensions (5.5423). The cube does not really exist as a cube, but is only an impression given from a number of lines in a particular order. Similarly, the "soul" is a composite made up of numerous thoughts, beliefs, and attitudes. And in Wittgenstein's words, "a composite soul would no longer be a soul" (5.5421). As the soul cannot stand for a unified "A" in a proposition, we should not read this proposition as existing on the basis of a belief and a unified consciousness, but existing on the basis of a proposition that is expressed and the fact that it pictures.

Because there is no metaphysical subject, we cannot learn a priori what kinds of objects or elementary propositoins there are. Logic is prior to any particular experience, but it is not prior to experience itself. As SparkNotes puts it, it is the shape that experience takes. [14] Logic can teach us that there are objects and elementary propositions, but it is by applying logic that we can learn about what objects and elementary propositions there are. There is no such thing as "logical experience" that we can consult regarding the various forms of elementary propositions (5.552). Experience has no place in logical analysis. Wittgenstein says that if we have to look at the world in order to apply logic, that shows us that we have made a fundamental mistake (5.551).

Lastly, Wittgenstein says that "the limits of my language mean the limits of my world" (5.6). The limits of our language are determined by the totality of elementary propositions and the limits of the world are determined by the totality of facts. There is a 1:1 correspondence between facts and elementary propositions, so we cannot say what lies outside the limits of the world, just as we cannot say anything that lies outside the limits of logic (5.61). Everything that is in the world is logical and therefore capable of being expressed in language. Just as we cannot speak of anything outside the world, we cannot so to say put ourself in a place outside the world to look at it from the exterior.

If we view everything from the "inside" so to speak and cannot take a step "outside" of it, we have to ask: What is the "self?" What is the "soul?" What is the "human being?" Wittgenstein here addresses the idea of solipsism. Solipsism is the idea that "I" am the only thing that exists, that the whole world is my own mind or consciousness. Wittgenstein claims that there is a truth in solipsism. We come to believe that solipsism is true because we only know of the existence of objects and other people because we are conscious of them. Wittgenstein says that what the solipsist MEANS is correct, but because he tries to put something that can only be SHOWN into language, he misfires and begins to speak nonsense.

The limits of my language mean the limits of my world. This means that I am the limit of my own world. But the problem emerges when we then conclude that I am the only thing that exists. If that is the case, what is this "I?" What is this thing that is the only thing that exists? What am I even referring to? The truth is that I cannot find "myself" anywhere in my own experience. I can only be conscious of some thing, some object, and not exist as a pure consciousness. Since I cannot find my own "consciousness" anywhere in my conscious experience, I'm suddenly at a loss to explain what this "I" is that is the only thing that exists. In the language of the Tractatus, this can be expressed as there being no object or elementary proposition that corresponds to this "I." There are no propositions with sense that are true or false relating to it.

We might think of the relation between the metaphysical subject and the world as that of the eye and the visual field (5.633). I cannot see my eye anywhere in the visual field, but the existence of a visual field presupposes the existence of an eye to see it. I never encounter "myself" anywhere in my own experience. As Wittgenstein puts it:

The idea of a "self" is not something that can be talked about as an object. It is a "limit" to the world. As discussed above, the "self" is nothing more than the thoughts, ideas, and beliefs that constitute it. There is nothing that is distinct from and more essential to the self than this. To use our earlier analogy, we could say that the self is nothing more than the number of lit light bulbs that manage to make up our "world" at any time. This does not mean that there's some "self" in a higher dimension that we can access. It is a "limit," but for that reason is insubstantial.

Of course, at this point there is no actual distinction between solipsism and pure realism. The idea that there are in fact objects and people in the world in just the way we think there are normally. The solipsist's idea of the "self" is not something that can be expressed in language. Therefore it does not rule out any factual statements that can be made about the world, i.e. any statements with actual sense. Since these statements about how things stand in the world is the only real way to make sense of the world, the realist and the solipsist only argue over pseudo-propositions that don't mean anything.

The self is the same thing as the world, logic, and language: It is the limit to all that there is. The solipsist wants to express this truth, but cannot do so meaningfully because we cannot make general statements about the nature of the world, logic, or language in a metaphysical way. These are not things that can be pictured and have logical form. So now we are back where we started: Only propositions expressing truth-functions of elementary propositions have sense. And no proposition that involves the "self" or "subject" can be one of these.

4.k. Propositions 6-6.241: The General Form of a Proposition

Wittgenstein now wants to sketch out the general form of all propositions. All logical operations are in truth nothing more than repeated applications of the Scheffer stroke, which is a form of joint negation. Therefore, we can say that all logical propositions are simply a series of negations on certain elementary propositions. In Wittgenstein's notation, the general form of a proposition reads similarly to a mathematical series. In that sense it is like any other series, such as the seriews of square numbers which goes 1, 4, 9, 16, etc. In 5.2522, Wittgenstein gives his form for expressing a term in a particular series:

"a" stands for the first term in the series, "x" stands for an arbitrarily selected term in the series, and "O'x" stands for the term that immediately follows "x." The O symbolizes the operation by which the next term in the series is generated. For example, we would represent the series of square numbers as:

The bar over the symbol symbolizes that it applies for ALL values that it represents. So ‾p refers to all the elementary propositions of a proposition. ‾ξ is a complex proposition in the series of successive negations. And N(‾ξ) refers to the negation of the prior term, ‾ξ, which is how the next term in the series is generated. This is the general form of any proposition.

This section now moves on to a consideration of mathematics based on the conclusions already established. Wittgenstein believes that mathematics is a method of logic, but denies that numbers are logical objects constructed out of logical forms. He claims that a number is the exponent of an operation (6.021). In other words, numbers are a shorthand for expressing how many times an operation has been applied. A mathematical expression like "+5" is merely a shorthand to express the difference in truth-grounds between 7 and 12, for example.

The full-scale attack on Frege's and Russell's conception of logic ends with a metaphor which claims that logic is a kind of "scaffolding" of the world (6.124). There's a lot about this metaphor which is informative. Scaffolding is nothing but a framework and it tells us nothing about the actual interior of the building with its rooms and furniture. In the same way, logic provides a logical form to the world but decrees nothing about the actual way things stand in the world. And most importantly, a sturdy and complete building has no need of scaffolding. [15]. And as Wittgenstein says:

We have no need for logic or philosophy when our language is functioning normally. They only become useful when we misfire and begin speaking nonsense. The propositions of logic are all tautologies. They say nothing. Any attempt to give content to them is misguided. The only time we need to "prove" them is the same time we need mathematical proofs: When they are complex enough that their being tautologies is not immediately apparent.

4.l. Propositions 6.3-6.3751: Natural Laws Are Not Logical

For Wittgenstien, logical and mathematical propositions do not express any thoughts. Metaphysical claims about abstract entities are, strictly speaking, nonsense. The only thing he seems to have left us able to say are individual facts about what holds or not in the world. They have to be able to represent a state of affairs that we can picture through logical form. But this leaves an unresolved area of language and thought.

We know that we can speak of the propositions of natural science (how things stand in the world). But what about the LAWS of natural science? The law of causality states that all processes occur through causes and effects. The law of conservation states that matter or energy can never be created or destroyed. The law of least expenditure states that the true trajectory of a particle is that which takes the least action. These are not logical facts since they make specific claims about things that happen in the world. They could be other than they are. But they certainly are not simple statements about things that are such and such the case in the world. To paraphrase SparkNotes, it's one thing to say that I kicked the ball and the ball moved. It's another thing to say that my kicking of it CAUSED the ball to move. This is more than a simple "fact" about how things stand in the world.

Wittgenstein here introduces a metaphor: The laws of nature are like a mesh grid divided into small squares that we lay over a surface of black and white spots. We can only describe the surface by saying that each individual square is either black or white. The method here is optional. We could use a mesh that was made of triangles or hexagons instead of squares. But one kind of mesh is more likely to provide a description that is more accurate and easier to understand. The mesh itself does not tell us anything about the surface. But we can infer things about the surface by seeing what kinds of mesh describe it more accurately.

The laws of nature are just like this mesh. They do not tell us anything about the world. They are only tools that we use to make sense of the world. They are not even necessarily TRUE of the world. The only thing that is necessarily true is what is logically certain. And this is not the case for our "laws" of nature. To use the example from SparkNotes, let's say I put a coin into a candy machine. The first ten times I do so, a piece of candy comes out. The 11th time, I put a coin in exactly the way I did before, but nothing comes out. I assume that something must have gone wrong in the machine and that it was not mere chance that nothing happened. [16] But there is nothing based on "logic" at the core of this belief. The following quotes make this point explicitly:

What is clear, however, is that we need the law of causality in order to make any sense of the world whatsoever. He makes the following point:

What he means here is that the so-called "law of causality" is nothing more than the belief that things happen for a reason and that it is not just chance that the sun rises every morning and that the candy didn't come out the 11th time I put the coin in. For that reason, causality cannot even be said to be a law but the "form of a law." That is, we need to accept causality if we can even have "laws" of nature in the first place.

Wittgenstein claims we cannot compare any process with the "passage of time," but only with another process, like the movement of a chronometer (6.3611). We can only describe a process by appealing to another process. This illustrates the point that we can only make sense of natural phenomena if we see them as conforming to some kind of regularity. We never discover causes and effects in nature. Causality does not exist in the world as any sort of "thing." It is only a tool that we use in trying to understand the regularity of nature.

In the modern world, we have become deluded into thinking we have found the answer to everything and have discovered the fundamental truths about the workings of the world when we discuss things like laws of nature. But all we are getting clear about is the framework through which we understand the world, not the world itself. This framework of natural laws is more sophisticated and reliable than superstition, but it works by exactly the same principle: Observing things that happen in the world and proposing explanations for why they do. These explanations are never discovered in nature. Wittgenstein summarizes this observation in a pithy fashion:

We should note that Wittgenstein is not anti-science. He does not think that scientific truths are useless and, more importantly, he does not think that they are simply matters of convention. What he wants to suggest is that the laws we use to explain natural phenomena are not themselves things that have been empirically discovered. In fact there is nothing in the world that has any claim on what is possible or impossible. There is only logical necessity, and there is only logical possibility.

4.m. Propositions 6.4-7: Ethics, the Mystical, and the Tractatus Itself

The final sections of the Tractatus are, far and away, the most important parts of the entire text. Most interpretations of its ultimate meaning somehow or other rely on the propositions in this section. They are extraordinarily difficult and enigmatic, and I can only recommend that you first read them for yourself. They very quickly address ethics, aesthetics, religion and the "mystical," death, and the ultimate status of philosophy and the Tractatus itself. I can only offer my own interpretation.

Proposition 6.4 claims that "all propositions are of equal value." What he gets at here is that statements of "value" are like statements about the self or soul. They cannot be expressed in language. Nothing "higher" can be expressed, because it would require us to stand outside of logic, language, and the world. There is simply not a higher standpoint to express this from. We can only state what is the case and is not. This immediately rules out "ethical" and "aesthetic" statements from being put into words. Wittgenstein says that they are "transcendental." And somewhat mysteriously, he claims that "Ethics and aesthetics are one and the same" (6.421).

This seems to be connected to the idea of the self as a limit to the world. For Wittgenstein claims that a good or bad will cannot alter the world itself. That is, it cannot change the things that can be expressed in language. It can only change the world insofar as it causes it to "wax and wane as a whole" (6.43). To me, this seems to be implying that our "self" is nothing more than the number of allegorical light bulbs that make up our "world" and that a will can do nothing more than change those boundaries.

Our death also is not something we experience. The "world" does not alter in death, but merely comes to an end. Nor is there any "value" or "meaning to be found in it. As Wittgenstein says in 6.4312, not only is the immortality of the human soul not guaranteed, but it in no way provides the answer to what we want. No riddle is solved if we live forever. Our eternal life will be just as enigmatic as our present one. The answer we seek cannot be answered in language.

The "mystical" for Wittgenstein seems to be the human drive for creating meanings in religion or philosophy. He claims that they are born from viewing the world "sub specie aeterni" or "under the sign of infinity" (a phrase borrowed from Spinoza). In a sense, it seems to be that we inherently understand that there is something more to the world than what we can express in language. We know that whatever we can express in language is limited.

The upshot is even more radical. When the answer cannot be put into words, the question cannot be put into words either. These questions about the meaning of life and value of existence are thus also things that cannot have sense. And what this ultimately means is that the solution to the problems of our life come not when these questions are answered, but when they simply vanish. As he puts it: Is this not the reason why those who have come to understand the sense of life been unable to put it into words? (6.521)

At that point, all we have left are the propositions of natural science. The philosophical "problems" we worried about are now unasked, but also unneccessary. This is why he says that the "correct" method of philosophy would ultimately be to say nothing but what can be said, that is the propositions of natural science. We could only practice philosophy by demonstrating when people have gone wrong and begun talking nonsense by saying things that did not correctly picture a state of affairs in the world. This would not feel like we were "doing philosophy" at all, but it is all that is left at this point. (6.53)

That said, the Tractatus has certainly not been following this method at all! If we just look at the overarching propositions of the Tractatus, they say things like "The world is all that is the case" and "What is the case--a fact--is the existence of states of affairs." This is why Wittgenstein says that anyone who understands him will ultimately have to recognize his very propositions as nonsensical and to cast them aside as well. And he famously ends the Tractatus with the following:

5. Interpreting the Tractatus Logico-Philosophicus

5.a. The "Standard" and "Resolute" Readings

Any interpretation of the Tractatus has to start by dealing with these last couple propositions. What are we to do with a work of philosophy that says that to properly understand it we have to realize that its propositions are nonsense? Bertrand Russell was the first one who felt this tension. In his words, "Mr. Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the skeptical reader that possibly there may be some loophole" (p. xxi). Wittgenstein was adamant that Russell did not understand the Tractatus, however, which seems to rule out the idea of a loophole.

What role can the propositions of the Tractatus play if they cannot convey sense in the way the propositions of natural science can? If we remember Proposition 4, remember that Wittgenstein claimed that things like numbers, objects, and colors can only be shown and not said. That said, he does not totally get rid of the concepts of numbers, objects, and numbers. He simply says that they can be shown but not said.

A standard interpretation of the Tractatus represented by figures such as P.M.S. Hacker is than that all the "metaphysical" assertions of the Tractatus are similar. There are indeed fundamental truths about the nature of the world, but they can only make themselves manifest and not be captured in words. It is a matter of dispute if things like Wittgenstein's objects are real things in the world or parts of our experience or even just ideas, but this standard conception claims that there are truths that the Tractatus shows which can be more or less understood in the terms Wittgenstein spells out.

A rival interpretation has gained traction since the early 1990s which is called the "new" or "resolute" reading. It was first expressed by Cora Diamond and is also taken by other notable figures in the field of Wittgenstein scholarship like James Conant. It is called the "resolute" reading because it claims that we should read the end of the Tractatus with firm resolution. That is, when Wittgenstein says that his propositions are "nonsense," he means it. He means it utterly and fully. They are as nonsensical as "The number two is purple." They cannot convey any meaning or truth at all.

This sounds suprising. It seems like we can make sense of them. But we cannot use any of them to construct a senseical proposition of the form "Wittgenstein says that 'p'" where "p" is one of the propositions in the Tractatus. None of them can be put in a correct logical form because they all refer to formal concepts and not concepts proper. We might think we can, but this is an illusion.

We should note the language in 6.54. When referring to his propositions, Wittgenstein does not say "anyone who understands my propositions eventually recognizes them as nonsensical." He says "anyone who understands ME eventually recognizes them as nonsensical." We cannot understand the propositions because there is nothing in them to understand. But we can understand the state of mind we would have to be in to THINK that the propositions make sense.

Under this reading, the entire Tractatus minus its epilogue comes to be read as a sort of semi-ironic prologue. We are meant to "fall for it" and play along. But then it ultimately fails, like all "metaphysical" systems. And we realize exactly what Wittgenstein proposes: When there are no questions left, just this is the answer. The standard reading reads Wittgenstein as believing that things like objects and states of affairs do really exist, but can only be shown and not said. The resolute reading denies that this kind of metaphysical thinking makes sense at all in the first place.

Looking at Wittgenstein's statements about the Tractatus don't make it easy to distinguish which of these two readings should be considered "accurate" insofar as being authentic to what Wittgenstein himself thought. Wittgenstein was very quiet on the meaning of his work. He thought that he couldn't express his thoughts any more clearly than he already had in the Tractatus itself. Many of his statements about the work only make it harder to interpret, in fact. But some of his statements do lend credence to the resolute reading.

One of the most valuable places to discover this is the introduction that Wittgenstein himself wrote to the Tractatus. It is much shorter than Russell's and doesn't give much insight to what people really wanted the introduction to do: actually explain the book's contents. However, it does say something revealing. In it, he claims that: "The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence" (p. 3). He continues by saying that the work will "draw a limit to the expression of thoughts," that this limit will be drawn in language, and that what lies on the other side of the limit will simply be nonsense. Lastly, he says that the value of the work lies both in the fact that it has solved all philosophical problems, and that in doing it shows how little is truly accomplished when they are solved.

Proponents of the resolute reading claim that readers have relied on Russell's introduction too much and ignored Wittgenstein's own words in his and have distorted it by trying to link it too much with Russell's and Frege's work. They have ignored a lot of Wittgenstein's own words about the book. However, I should note that the resolute reading ignores a substantial amount of Wittgenstein's own words about his work as well. Wittgenstein later spoke of his earlier belief in logical atomism as something he held but had to give up. It would be strange for him to speak like this if he only presented it as an example of nonsense. Wittgenstein was also quite dejected when figures like Frank Ramsey made more technical critiques to the "metaphysical" parts of the Tractatus and expressed frustration at having to "start from scratch" after having these parts proved wrong. And that also would make no sense if these propositions were only supposed to be nonsense.

5.b. The "Ethical" Nature of the Tractatus Logico-Philosophicus

Both the standard and resolute readings have advantages and weaknesses and it doesn't seem that we can prove either definitively based on Wittgenstein's own words about the work. One thing that is even more confusing is found in a letter he wrote where he claimed that the Tractatus was primarily an "ethical" work. This is very strange on both the standard and resolute readings. What does anything in the Tractatus have to do with ethics?

Michael Kremer offers an interesting perspective on this. He does not fall completely in line with the resolute reading, but he sees the standard reading as having a problem at its core related to the idea of saying and showing. Wittgenstein repeatedly asserts that logic must "look after itself." His primary critique of Russell and Frege is that they attempt to give a grounding for logic that is metaphysical such as making laws of logical inference or logical objects. None of this is acceptable, because logic has to be utterly simple and self-justifying, no matter how badly we want to give it some kind of justification and grounding.

However, Kremer sees the standard interpretation as reintroducing the same problem in a covert way. The standard reading is that Wittgenstein says that there are ineffable proposition-like insights in philosophy that show themselves without being said. This is tempting because it serves as the ground that we yearn for so desperately without having to need further justification or grounding. But this all rests on a misunderstanding of the word "show." To use Kremer's example, if you are going to take care of my cat while I am away, I can show you that there is enough food for the cat until I return, then I can show you how to make the cat purr by stroking its chin. These are two different uses of the word "show." The first points to something that is a fact that could be true or false. But the second does not. It shows HOW.

The traditional reading is that logic SHOWS its truths without saying them. But this confuses these two uses of "show." It reads a "showing how" as a "showing that," but there is no "that" there. And this is is what he means by nonsense. This is nothing but the same error as Russell and Frege: trying to provie a justification for logic instead of letting logic look after itself. Logic needs no ground for its truth, even if that truth is ineffable and self-revealing.

And how does this relate to ethics? Wittgenstein mentions ethics very briefly in the Tractatus in 6.421, where he claims that ethics tries to make statements about the whole world or life from a perspective "outside" of the world and life. It tries to say something about the whole of the world when all we can meaningfully talk about is particular facts within the world being true or not. But Wittgenstein does not say that ethics and aesthetics are "nonsensical" or "meaningless." He says they are "transcendental." There is only one other thing that he describes as "transcendental" in the Tractatus, which is logic. So we can say that ethics must have a similar position as logic.

Kremer then proposes that ethics must also "look after itself." It is groundless and can only be so. It is not that it provides its own justification, it is that it cannot be justified at all because it stands before all justification. With this in mind, the clarity it not the clarity of a proposition or ineffable truth, but a clearness about how we should live. And that is essentially to free us from the need to "justify" ethics and "define" ethics and just to let us be ethical. [17]

6. Closing Words

Wittgenstein said that the purpose of the Tractatus would be fulfilled if it gave pleasure to one who read it. I hope I was able to help towards this purpose. Go play Subarashiki Hibi 〜 Furenzoku Sonzai.

FOOTNOTES

1. Greg Baker, 2013, School of Computing Science - Simon Fraser University, "Discrete Math," "Propositional Logic."

17. Michael Kremer [Michael Beaney ed.], Oxford Handbook of the History of Analytic Philosophy, "The Whole Meaning of a Book of Nonsense: Introducing Wittgenstein's Tractatus"