Footnotes to Plato

Tuesday, April 20, 2021

A review of Tractatus by Wittgenstein

When you see X ⊃ Y, you are to read "X implies Y" or "If X, then Y." Sometimes you may see the modern form, X —> Y, in my comments.
When you see X ≡ Y, you should read "X equivalent to Y" or "X strictly implies Y" or "If and only if X, then Y" or "If X, then Y and if Y, then X" (these forms are sometimes shortened to "iff"). "X ≡ Y" can also be read: "Y is necessary for X and X is sufficient for Y." Occasionally one also sees "X ≡ Y" interpreted as "Y holds just in case X holds" or "Y holds only in case X holds" and the converse, "X holds just in case Y holds." Sometimes my comments use the modern X <—> Y.
For the quantifier all, you will see ( ), as in (x)Fx. In my comments I sometimes use the modern symbol ∀, as in ∀xFx. The existential quantifer ∃, as in ∃xFx, is used in W's text and in my comments. P●Q should be read "P and Q." Often the dot is dropped, as in PQ. Occasionally dots are used instead of parentheses.

In a tart response to a critic, Bertrand Russell once pointed out that he no longer held the philosophical views in question.

Yet, the Library of Living Philosophers printed the critic's analysis anyway. The editor of that series, Paul Arthur Schilpp, was following the time-honored rule that even disavowed philosophical positions could be worthy of discussion.

Well-known is the fact that Ludwig Wittgenstein later changed his mind about Tractatus, having been convinced by Frank Ramsay that it did not, after all, solve all major questions. Wittgenstein then went on to a philosophy that is neatly summed by Marshall Macluhan's aphorism, "the medium is the message."

Tractatus, with its quasi-engineering format and style, made a powerful impact on the "hard-headed realists," who saw it as an antidote to wooly-minded metaphysics and just what the doctor ordered as a backup for 20th Century scientific materalism or materialist science – though three years after its publication the quantum mechanics revolution would throw a monkey wrench into the whole materialist agenda. Nevertheless, the "logical positivists" – who counted Wittgenstein as much as Comte among their forebears – carried the banner of "scientific" philosophy for much of the century.

Yes, it is true that logical positivism no longer is in vogue. Yet, it will be conceded that the "system" of logical positivism underlies the unspoken assumptions of a great many people among the educated and semi-educated classes.

To me, that alone provides adequate reason to revisit Tractatus with a view to questioning its assertions and assumptions.

Clearly the brilliance of Wittgenstein's contribution, flawed though it is, makes it worthy of discussion even today. Yet we must concede that the century since publication has seen an enormous proliferation of work in mathematical logic (sometimes called logistic) and set theory, thus making some of these comments passe. Hopefully, not all.

A few remarks on relations

[This section to be polished up later.]

Though, as the Bradley-Russell debate shows, relations constitute an imperfect vehicle for representing human reality, they are nevertheless integral to human communication. I would even go so far as to define mathematics as "the science of exact binary relations." (Rxyz can be put into binary terms, as in the ordered pairs < x, < y,z > >.)

I regard a relation as a verb-like construct that pairs two noun-like constructs. The noun can represent an abstract picture of some ponderable object or of an "action." An action is abstracted from a number of specific events. For example, there is no such thing as running devoid of runners. We may say that runs relates some object, usually some animal, to something else.

But on that ground, one might argue that Joline runs doesn't seem to be a relation.

In such a case, we have two options: We may say that for aRb (or R〈a,b〉) , b = ∅. Or we may note that Joline runs abbreviates the picture Joline moves (by running) her body.

And we must concede that there is nothing to stop an infinite regress type of relation, as in

R〈a, 〈 b,〈 c, 〈 ... 〉 〉 〉 〉 etc. Or, we can write aRb, b=cSd, d=eTf... and so on. In fact, such relations are typical in mathematics. The philosophical problem arises in trying to define a relation in terms of another relation.

That problem, in fact, has much of Russell's paradox about it, at least insofar as the self-referencing aspect is concerned.

Then there is aR(aRb), which also brings up the self-referencing issue.

Despite these cautions, the way we ordinarily communicate with each other is well expressed as a system of relations, both empirical and logical/formal.

We may take every word in the Oxford dictionary as the set A and form the pairings A X A. We then take every word in that dictionary that has even a vague possibility of representing some action-type notion and call that a relation.

Among subsets of relations are R〈 a,k〉 and R〈k,b〉 where a and b represent variables and k is held constant.

All such relations represent propositions or assertions.

Empirical assertions are judged true or false according to whether we believe they accord with a purported objective reality.

Logical propositions follow rules laid down by logicians that reflect commonly accepted means of human reasoning. Such formal propositions are true or false based on their internal consistency. AB true requires that both A and B be adjudged true, and that this judgment must stem from axioms (though even that proviso is not altogether so, as Goedel showed). A~A is false by almost universal agreement.

I don't wish to rehearse all the arcana of the theory of logic here. My point is that the notion of sets of relations covers the field of logic very well.

Well, yes, it is true that many relation pairs are gibberish – at present – though someone may pair two words/images/ideas in some technical or poetic way that would be meaningful to a subset of hearers/viewers. Thus, we may say that most if not all pairs and their relation potentially are meaningful. By meaningful, we are saying that a binary truth value is applicable to the specific relation/proposition.^†

But it follows that only some small subset C of A X B would represent meaningful propositions. That is, if we regard p as the specific relation R〈 a,b〉, where here a and b are each instances, we find that p is a member of A X A, where A is the set of all words and concept symbols used in English. A proposition however is regarded as carrying a truth value. Most members of A² are not regarded by anyone as meaningful relation pairs and so are not propositions, though in future any non-proposition may take on a sense that can be affirmed or denied, thus becoming a proposition.

Denial can only be applied to members of the set of propositional relations P.

That is, ~p is a member of P^c, or ~p ∈ P^c <—> p ∈ P. P^c is not a subset of A². P^c is a subset of (A²)^c.

In a sense, can we not say that the formalism of relations and the generally accepted rules of logic show without further ado that the human brain/mind is wired to a primary in-built grammar. What human language cannot be cast in the form of ordered pairs of relation sets? This is not to say that the work of Chomsky et al is pointless, but to say that it seems that that endeavor was long prefigured by the work of the logicians and philosophers.

Russell did not like Bradley's objection that while relations have a practical value, they cannot exist intrinsically. He favored the idea that any infinite regress was harmless –> though he certainly did not feel that way about his own epynonymous paradox.

My view is that, as the relations system fully encompasses all of ordinary logic (including mathematical logic), Wittgenstein's attempt to define away Russell's bugbear cannot be held to have succeeded. That is, the edifice of logic contains the very sort of self-referencing difficulty elucidated by Russell for set theory and by Goedel for formal systems in general.

This further suggests that humans can not "know" the truths of their existence based on logic alone.

As the set of propositions B is a subset of A X A, it follows that B is finite. This means that each proposition can be numbered, reducing it to being represented without quantifier or variable. But, it is more convenient to write ∃xFx, which represents a subset F< a, x >, where F and a are held constant and x varies, or F< x, a>. Multiple relations can be expressed as pairs, of course. a < b < c is the less-than relation L, written L(a, (b,c)) [Here I use the rounded parenthetical mark in order to avoid confusion between "<" for "less than" and "< , >" for "ordered pair set."]

On the matter of the validity of relations as a means of discussing verbalization, I would point out that we really have no alternative. That's how most of us speak/think: via relations. That is not to say that idealists don't have a point when they deny that validity. In other words, it may well be that relations do not yield a proper conception of reality. Yet, in our day-to-day affairs, they work well enough.

Hence the dispute over "internal" versus "external" relations might be seen in that light. W, though not an idealist, seems to be aware of the concern about these distinctions.

A.C. Ewing, in Idealism, A Critical Survey (Methuen 1933), writes that the controversy arose from the idealist belief that "all reality is an interconnected system such that a difference in one part would always involve a difference in others."^‡

Now, concerning the relation "equals," Tractatus, as others, take that to mean "same as." That can be a definition.

But consider the relation set =<2+2,4>, or that is, "2+2 = 4". Crudely, we teach children that "2 and 2 is the same as 4." But that is an abbreviated way of saying something more complex:

One way of expressing this would be

2 = 2.
This says that the set named 2 is bijective with itself (which is not quite saying: "the same as itself.")

The "+" operation requires that either the set named m or the set named n must be used to form a set with which it is bijective. So m X ∅ will do, producing elements of
form < x, ∅ >. Calling that set m', we now make m' ∪ n, which (not proved here) is bijective with some x ∈ N. That x is bijective with the sum "m + n," which is the set m' ∪ n.

That is, 2 + 2 --> 2 ∪ 2', which is bijective with 4.

So the relation "=" here indicates more than "same as." It indicates a procedure which must be followed (at least inferentially) in order to match the left side of the relation with the right.

Thus I suggest that "=" be taken as a special case of the equivalence relation "≡". I.e. "2 ∪ 2' ≡ 4" in this case means "2 + 2 = 4".

As Tarski pointed out "truth" is an indefinable term in mathematics. But some form of the concept is still necessary, whether it be called "satisfiability" or some such. In our view of relations, by "truth" we imply that relation R poses a yes or no question for the pair x and y: "Does x 'go with' y in the order < x,y >?" Or is there a reason to say "x certainly does not belong with y in the specified order"? E.g., "Does 2+2 go with 4 in the equals relation?" Yes. "Does 2+1 go with 4 in the equals relation"? No. [Here R is symmetric.] That is, < 2+1,4 > ∈ R^c where R ⊂ N X N (and any sum n+m ∈ N).

Moreover, one can see how our example draws together the correspondence and coherence theories of truth. That is, we say xRy corresponds to a truth value of T "because" x goes with y under R. Yet, we cannot blindly say that that is so. First, we have to justify that claim with a set of other relation/assertions. Of course, we cannot insist that all truths derive from a few interdependent axioms, as perhaps the idealists might say. In a pluralist cosmos, the axioms would not necessarily be interdependent. Also, as a practical matter, we can only assess parts of "reality" and cannot really find a perfect set of propositional relations that reflects a purported Theory of Everything. In other words, perhaps the coherence theory is valid, but we for the most part must make do with the correspondence notion.

A word on the axiom of infinity

W never brings up this axiom in Tractatus but it seems to me that the whole thrust of his critique demands that he say something about it. He does bring up infinity, without acknowledging the philosophical problem posed by the concept, though his aim is to solve or eject all such problems through correct reasoning and language.

In any case, a modern form of the axiom is, as Wikipedia reports:

∃N(∅ ∈ N ● ∀n ∈ N((n ∪ {n}) ∈ N)

Wikipedia says that the set N is presumed infinite. Zermelo's original axiom, as I understand it, actually says that N, which designates the natural numbers, "is a set." Zermelo wished to avoid the paradoxes by tightening the definition of set, which could not now be any aggregate of objects, as defined intensionally. Otherwise one has, say, X = {x|x ∉ x}, which leads to is X ∈ X? If yes, no. If no, yes.

But rather than taking a positive approach to saying what a set can be, it seems to me better to say what it cannot be, So we require an axiom schema that says

~(x ∈ y)(y ∈ z)...(t ∈ x).

The dots mean any string of (u_i ∈ v_i) of finite length. In words, "if x is in y, then no set that y is in may be in x." A picture is a sequence of nested containers. The largest box cannot be in any smaller one.

By formalizing what cannot be a set in this way, we don't need to axiomatize the concept of N The successor algorithm given above tells us that any natural number may be so defined. We give the null set, ∅, the name 0. 1 is defined as ∅ ∪ {∅}, which reduces to {∅}. 2 is the name given to {∅} ∪ {{∅}} = {{∅},{{∅}}}. You can see that any natural number will follow upon iteration.

We do not say axiomatically that there is an infinity. We say that there is a set N that contains all the naturals defined as shown. The set N is the name given for the entire collection. N is barred from being a member of N (that claim rests on a different axiom though if we define N that way, we can in this instance bypass that axiom also). We then define finite as a quality of any member of N. Any set that is bijective with an n is finite. Hence, N is infinite. (The quality is that n —> 1+n, but N -/-> 1+N.)

This formulation is found in NBG set theory, for example. In any case, we may argue that infinity no longer is axiomatic. Infinity is defined as a non-finite set. Finite is held to be descriptive of the cardinal for any member of N.

Another standard way to define infinity set theoretically:

If A is a proper subset of B and A and B are bijective, then B (and A) is infinite.

Is an axiom needed? Isn't the subset axiom⁺⁺ sufficient?

Of course, the inductive definition above dovetails with the subset definition.

N = {n|n --> n+1}, where n is defined recursively.

X = {n²|n² --> (n+1)²}

Since X ⊂ N and since for every n there is exactly one n², N must be infinite.

I recall Barkeley Rosser making sure to tack onto his book on logic an appendix with a new proof that the axiom of infinity could be derived from other axioms, though I haven't had time to decode the proof. His axiom differs from others in that his book incorporates W.V. Quine's New Foundations approach to logic.

Rosser's axiom scheme 13, along with each statement obtained from it by prefixing some set of universal quantifiers, is given as:

∀(m,n) m,n ∈ N ● m+1 = n+1 ● --> ● m = n

In general, he refers to this statement as the "axiom of infinity."

In fact, I suggest that, despite the proof (by Spector) that the axiom could be derived from other NF axioms, philosophically it was unnecessary in Rosser's approach. That is because Rosser in that book makes a point of distinguishing between "all" and "any." Of course, usually the truth value of a proposition that reads (all x)Px or (any x)Px is the same in either case, which is why we use only one universal quantifier ∀ . Yet, it is possible to define a set using an "any" specifier: α, as in αn(n --> n+1). We then give this aggregate the name N and observe that this α form defines the aggregate by intensionally specifying its elements, and does not violate the "hierarchy axiom" cited above. Hence, it is a set. (Before we know that a particular aggregate is a set, we say that x is not necessarily a member of the aggregate but only an affiliate of it.) Of course, one may suppose that that axiom appears to imply infinity. But we may apply the α modifier to that axiom, without requiring an understanding of completion or closure. That is, the α modifier may be read to imply closure, but it need not be read that way.

In any case, these observations, along with Rosser's unique approach to logic in his Logic for Mathematicians (1953), reminds one that there are many roads in formal logic, not one – though of course the basics are, for the most part, unchanging.

In addition, I suppose it might be mentioned that the status of Tractatus has suffered considerable decline among mathematical logicians. Kurt Gödel noted that he had given it only "superficial" attention, though he was involved in the Vienna Circle as a young man. He minimized W's influence on that philosophical group.

By 1983, an important anthology of mathematical philosophy had dropped Tractatus from its second edition in favor of the work of others.^* It seems, then, that despite Russell's sponsorship, Tractatus was more influential among those logical positivists who were not primarily mathematicians or mathematical logicians. An important exception was Rudolf Carnap,^†† whose notion that logical syntax could replace outworn philosophy echoes the thrust of Tractatus. In fact, Carnap focused the Vienna Circle on Tractatus in 1926 and 1927, according to Hao Wang.^‡‡ Gödel, in fact, suppressed a detailed review he wrote of Carnap's philosophy for the Library of Living Philosophers. Carnap's attempt to carry out what he took to be W's vision is not generally accepted as having succeeded.

In any case, I don't wish to undervalue Tractatus, which was thought out by a young man in a prisoner of war camp. As I mentioned, the aim here is to throw a spotlight on one of the supposed philosophical pillars of the modern educated elite, most of whom of course have only the vaguest inkling of its import.

So then, how firm is the foundation on which they stand?

Further

There is no claim here that I have fairly addressed all the abstruse points raised by W in Tractatus or that I am competent to examine the numerous contributions made by mathematical logicians since its publication. Rather, I have tried to mine this work for what nuggets there are that may be of value a century later. That is, this critique is a highly ego-centric one, I must concede.

A positive summary
of Tractatus and W's other work is found at The School of Life, a YouTube channel:
https://www.youtube.com/watch?v=pQ33gAyhg2c

Freeman Dyson,
the physicist, relates a less-than-heartwarming encounter with Wittgenstein.
https://www.youtube.com/watch?v=byi3vOnVodQ
Dyson calls W a "charlatan" who liked to torment people.

Argues that W slipped up on Goedel theorem
http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate

Kurt Gödel thought
W's later philosophy had "taken a step backward" from Tractatus, according to Hao Wang in Reflections on Kurt Gödel (MIT 1987). Wang adds that in 1972 Gödel noted that he had read Tractatus in 1927, though "never thoroughly."

†. In probability theory, the truth value system ought remain binary. Middle values pertain to degree of uncertainty. There is a case for a third truth value of 0, for provably undecidable. But, a problem with barring the logical rule of excluding the middle is that it eviscerates whole sections of mathematics that mathematicians are loath to give up.

Footnotes

++. The subset axiom is the axiom of Zermelo-Fraenkel set theory which asserts the existence for any set a and a formula A(y) of a set x consisting of all elements of a satisfying A(y).

∃x∀y (y ∈ x --> y ∈ a ● A(y))

Wolfram MathWorld notes that that axiom is called the subset axiom by Enderton (1977), while Kunen (1980) calls it the comprehension axiom. Itô (1986) terms it the axiom of separation, but this name appears to not be used widely in the literature and to have the additional drawback that it is potentially confusing with the separation axioms of Hausdorff arising in topology.

This axiom was introduced by Zermelo.

Points of interest:

(i) The formula A(y) would in most cases be intensional, that is, would specify properties common to the y's.
(ii) The axiom of choice is implied by the subset axiom.

That is, "A(y)" is not itself a formula, but stands for the set of applicable formulas. Thence y_o ∈ x is arbitrary. To say that an arbitrary y can be specified is to effectively express the axiom of choice. Yet, I can imagine that many would disagree with that assessment.

‡ . Current physics denies that this can always hold because of the claim that some parts of the cosmos are too far away and moving too fast relative to us to ever be able to interact with our sector in any manner.

* . Philosophy of Mathematics: Selected Readings edited by Paul Benacerraf and Hilary Putnam (2d ed. Cambridge 1983; 1st ed. Prentice-Hall 1964).

† † . Reflections on Kurt Gödel by Hao Wang (MIT 1987).

‡‡ . According to André Carus's article on Carnap at the Stanford Encyclopedia of Philosophy (2020):

The Wittgensteinian program favored by the Vienna Circle ... had collapsed in 1930. But Carnap soon recovered, and during a sleepless night on 21 January 1931, conceived of an entirely new basis for the Vienna Circle’s characteristic doctrines (Awodey & Carus 2009). Instead of trying to fuse Hilbert and Wittgenstein, Carnap now dropped Wittgenstein altogether and pursued a Hilbertian approach. “Meaning” was no longer rooted in the correspondence between configurations of elementary facts and their linguistic representations. In fact, meaning was banished altogether, at least in our statements about the language of science (our metalinguistic “elucidations” such as those in the Tractatus itself or the Aufbau).

Tractatus 3

3
The logical picture of the facts is the thought.

3.001
“An atomic fact is thinkable”⁠—means: we can imagine it.

3.01
The totality of true thoughts is a picture of the world.

3.02
The thought contains the possibility of the state of affairs which it thinks. What is thinkable is also possible.

3.03
We cannot think anything unlogical, for otherwise we should have to think unlogically.

3.031
It used to be said that God could create everything, except what was contrary to the laws of logic. The truth is, we could not say of an “unlogical” world how it would look.

3.032
To present in language anything which “contradicts logic” is as impossible as in geometry to present by its coordinates a figure which contradicts the laws of space; or to give the coordinates of a point which does not exist.

This claim appears to have been refuted by Goedel's undecidability theorem, as well as being denied by other paradoxes. Even in ordinary language we have the liar's paradox and Richard's paradox.

Here are three paradoxes, all based on the same idea, taken from a Cambridge web page
https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html

1. Let A be the set of all positive integers that can be defined in under 100 words. Since there are only finitely many of these, there must be a smallest positive integer n that does not belong to A. But haven't I just defined n in under 100 words?

2. Let B be the set of all reasonably interesting positive integers. Let n be the smallest integer not belonging to B. But surely the defining property of n makes it reasonably interesting.

3. Let X be the set of all definable real numbers. Since there are only countably many definitions, X is countable. Indeed, we can explicitly count X - just list the elements in alphabetical order of their definitions. Now apply to this list some explicit diagonal process, obtaining a number y that does not belong to X. But haven't I just defined y?

We should observe that when he wrote Tractatus, Wittgenstein did not have access to Principia Mathematica, in which the various paradoxes are laid out in detail, but only a detailed summary of that massive work. Yet the young war veteran was certainly highly focused on Russell's paradox.

3.0321
We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.

3.04
An a priori true thought would be one whose possibility guaranteed its truth.

3.05
Only if we could know a priori that a thought is true if its truth was to be recognized from the thought itself (without an object of comparison).

3.1
In the proposition the thought is expressed perceptibly through the senses.

3.11
We use the sensibly perceptible sign (sound or written sign, etc.) of the proposition as a projection of the possible state of affairs.

The method of projection is the thinking of the sense of the proposition.

3.12
The sign through which we express the thought I call the propositional sign. And the proposition is the propositional sign in its projective relation to the world.

3.13
To the proposition belongs everything which belongs to the projection; but not what is projected.

Therefore the possibility of what is projected but not this itself.

In the proposition, therefore, its sense is not yet contained, but the possibility of expressing it.

(“The content of the proposition” means the content of the significant proposition.)

In the proposition the form of its sense is contained, but not its content.

3.14
The propositional sign consists in the fact that its elements, the words, are combined in it in a definite way.

The propositional sign is a fact.

3.141
The proposition is not a mixture of words (just as the musical theme is not a mixture of tones).

The proposition is articulate.

3.142
Only facts can express a sense, a class of names cannot.

3.143
That the propositional sign is a fact is concealed by the ordinary form of expression, written or printed.

For in the printed proposition, for example, the sign of a proposition does not appear essentially different from a word.

(Thus it was possible for Frege to call the proposition a compounded name.)

3.1431
The essential nature of the propositional sign becomes very clear when we imagine it made up of spatial objects (such as tables, chairs, books) instead of written signs.

The mutual spatial position of these things then expresses the sense of the proposition.

3.1432
We must not say, “The complex sign ‘aRb’ says ‘a stands in relation R to b’ ”; but we must say, “That ‘a’ stands in a certain relation to ‘b’ says that aRb”.

3.144
States of affairs can be described but not named.

(Names resemble points; propositions resemble arrows, they have sense.)

3.2
In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign.

3.201
These elements I call “simple signs” and the proposition “completely analysed.”

3.202
The simple signs employed in propositions are called names.

3.203
The name means the object. The object is its meaning. (“a” is the same sign as “a”.)

3.21
To the configuration of the simple signs in the propositional sign corresponds the configuration of the objects in the state of affairs.

3.22
In the proposition the name represents the object.

3.221
Objects I can only name. Signs represent them. I can only speak of them. I cannot assert them. A proposition can only say how a thing is, not what it is.

3.23
The postulate of the possibility of the simple signs is the postulate of the determinateness of the sense.

3.24
A proposition about a complex stands in internal relation to the proposition about its constituent part.

A complex can only be given by its description, and this will either be right or wrong. The proposition in which there is mention of a complex, if this does not exist, becomes not nonsense but simply false.

That a propositional element signifies a complex can be seen from an indeterminateness in the propositions in which it occurs. We know that everything is not yet determined by this proposition. (The notation for generality contains a prototype.)

The combination of the symbols of a complex in a simple symbol can be expressed by a definition.
3.25
There is one and only one complete analysis of the proposition.

3.251
The proposition expresses what it expresses in a definite and clearly specifiable way: the proposition is articulate.

3.26
The name cannot be analysed further by any definition. It is a primitive sign.

3.261
Every defined sign signifies via those signs by which it is defined, and the definitions show the way.

Two signs, one a primitive sign, and one defined by primitive signs, cannot signify in the same way. Names cannot be taken to pieces by definition (nor any sign which alone and independently has a meaning).

3.262
What does not get expressed in the sign is shown by its application. What the signs conceal, their application declares.

3.263
The meanings of primitive signs can be explained by elucidations. Elucidations are propositions which contain the primitive signs. They can, therefore, only be understood when the meanings of these signs are already known.

3.3
Only the proposition has sense; only in the context of a proposition has a name meaning.

3.31
Every part of a proposition which characterizes its sense I call an expression (a symbol).

(The proposition itself is an expression.)

Expressions are everything⁠—essential for the sense of the proposition⁠—that propositions can have in common with one another.

An expression characterizes a form and a content.

3.311
An expression supposes the forms of all propositions in which it can occur. It is the common characteristic mark of a class of propositions.

3.312
It is therefore represented by the general form of the propositions which it characterizes.

And in this form the expression is constant and everything else variable.

3.313
An expression is thus presented by a variable, whose values are the propositions which contain the expression.

(In the limiting case the variable becomes constant, the expression a proposition.)

I call such a variable a “propositional variable.”

3.314
An expression has meaning only in a proposition. Every variable can be conceived as a propositional variable.

(Including the variable name.)

3.315
If we change a constituent part of a proposition into a variable, there is a class of propositions which are all the values of the resulting variable proposition. This class in general still depends on what, by arbitrary agreement, we mean by parts of that proposition. But if we change all those signs, whose meaning was arbitrarily determined, into variables, there always remains such a class. But this is now no longer dependent on any agreement; it depends only on the nature of the proposition. It corresponds to a logical form, to a logical prototype.

3.316
What values the propositional variable can assume is determined.

The determination of the values is the variable.

3.317
The determination of the values of the propositional variable is done by indicating the propositions whose common mark the variable is.

The determination is a description of these propositions.

The determination will therefore deal only with symbols not with their meaning.

And only this is essential to the determination, that it is only a description of symbols and asserts nothing about what is symbolized.

The way in which we describe the propositions is not essential.

3.318
I conceive the proposition⁠—like Frege and Russell⁠—as a function of the expressions contained in it.

3.32
The sign is the part of the symbol perceptible by the senses.

3.321
Two different symbols can therefore have the sign (the written sign or the sound sign) in common⁠—they then signify in different ways.

3.322
It can never indicate the common characteristic of two objects that we symbolize them with the same signs but by different methods of symbolizing. For the sign is arbitrary. We could therefore equally well choose two different signs and where then would be what was common in the symbolization?

3.323
In the language of everyday life it very often happens that the same word signifies in two different ways⁠—and therefore belongs to two different symbols⁠—or that two words, which signify in different ways, are apparently applied in the same way in the proposition.

Thus the word “is” appears as the copula, as the sign of equality, and as the expression of existence; “to exist” as an intransitive verb like “to go”; “identical” as an adjective; we speak of something but also of the fact of something happening.

3.324
Thus there easily arise the most fundamental confusions (of which the whole of philosophy is full).

3.325
In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying signs in the same way which signify in different ways. A symbolism, that is to say, which obeys the rules of logical grammar⁠—of logical syntax.

(The logical symbolism of Frege and Russell is such a language, which, however, does still not exclude all errors.)

3.326
In order to recognize the symbol in the sign we must consider the significant use.

3.327
The sign determines a logical form only together with its logical syntactic application.

3.328
If a sign is not necessary then it is meaningless. That is the meaning of Occam’s razor.

(If everything in the symbolism works as though a sign had meaning, then it has meaning.)

3.33
In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.

3.331
From this observation we get a further view⁠—into Russell’s “Theory of Types.” Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.

3.332
No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).

3.333
A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.

If, for example, we suppose that the function F⁡(f⁡x) could be its own argument, then there would be a proposition “F⁡(F⁡(f⁡x))”, and in this the outer function F and the inner function F must have different meanings; for the inner has the form φ⁡(f⁡x), the outer the form ψ⁡(φ⁡(f⁡x)). Common to both functions is only the letter “F”, which by itself signifies nothing.

This is at once clear, if instead of “F⁡(F⁡u)” we write “(∃φ):F⁡(φ⁡u).φ⁡u=F⁡u”.

Herewith Russell’s paradox vanishes.

3.333 makes sense for the liar's paradox. But Russell's paradox arises from the naive definition of set. Perhaps W. is right about rules of definition, but we know that set theorists have preferred axioms that prevent the paradox from arising, such as the prohibition of any set being an element of itself.

3.334
The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.

3.34
A proposition possesses essential and accidental features.

Accidental are the features which are due to a particular way of producing the propositional sign. Essential are those which alone enable the proposition to express its sense.

3.341
The essential in a proposition is therefore that which is common to all propositions which can express the same sense.

And in the same way in general the essential in a symbol is that which all symbols which can fulfill the same purpose have in common.

3.3411
One could therefore say the real name is that which all symbols, which signify an object, have in common. It would then follow, step by step, that no sort of composition was essential for a name.

3.342
In our notations there is indeed something arbitrary, but this is not arbitrary, namely that if we have determined anything arbitrarily, then something else must be the case. (This results from the essence of the notation.)

3.3421
A particular method of symbolizing may be unimportant, but it is always important that this is a possible method of symbolizing. And this happens as a rule in philosophy: The single thing proves over and over again to be unimportant, but the possibility of every single thing reveals something about the nature of the world.

3.343
Definitions are rules for the translation of one language into another. Every correct symbolism must be translatable into every other according to such rules. It is this which all have in common.

3.344
What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.

3.3441
We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, can be replaced by the notations of “~p” (“not p”) and “p∨q” (“p or q”).

(Herewith is indicated the way in which a special possible notation can give us general information.)

3.3442
The sign of the complex is not arbitrarily resolved in the analysis, in such a way that its resolution would be different in every propositional structure.

3.4
The proposition determines a place in logical space: the existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the significant proposition.

3.41
The propositional sign and the logical coordinates: that is the logical place.

3.411
The geometrical and the logical place agree in that each is the possibility of an existence.

3.42
Although a proposition may only determine one place in logical space, the whole logical space must already be given by it.

(Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements⁠—in coordination.)

(The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)

3.5
The applied, thought, propositional sign, is the thought.

Tractatus 2

2
What is the case, the fact, is the existence of atomic facts.

2.01
An atomic fact is a combination of objects (entities, things).

2.011
It is essential to a thing that it can be a constituent part of an atomic fact.

2.012
In logic nothing is accidental: if a thing can occur in an atomic fact the possibility of that atomic fact must already be prejudged in the thing.

2.0121
It would, so to speak, appear as an accident, when to a thing that could exist alone on its own account, subsequently a state of affairs could be made to fit.

If things can occur in atomic facts, this possibility must already lie in them.

(A logical entity cannot be merely possible. Logic treats of every possibility, and all possibilities are its facts.)

Just as we cannot think of spatial objects at all apart from space, or temporal objects apart from time, so we cannot think of any object apart from the possibility of its connection with other things.

If I can think of an object in the context of an atomic fact, I cannot think of it apart from the possibility of this context.

2.0122
The thing is independent, in so far as it can occur in all possible circumstances, but this form of independence is a form of connection with the atomic fact, a form of dependence. (It is impossible for words to occur in two different ways, alone and in the proposition.)

2.0123
If I know an object, then I also know all the possibilities of its occurrence in atomic facts.

(Every such possibility must lie in the nature of the object.)

A new possibility cannot subsequently be found.

I interpret this thus: Any object -- or its representation -- is a member of some relation. Hence, the object implies all the relations of which it is a member.

By relation I mean the subset R_i< a,b >. If a is the object, then there is may be a subset in which b varies. We also have other relations, as in R_j< a,b >. So he is saying, I suppose, that all the relevant b's and R's are implied.

I note that W. is not always clear as to the distinction between an object and its representation. Of course, in the idealist and semi-idealist view (as for example in my paper Toward a Signal Model of Perception), the lines blur between exterior and mental object, between object and concept).

2.01231
In order to know an object, I must know not its external but all its internal qualities.

2.0124
If all objects are given, then thereby are all possible atomic facts also given.

2.013
Every thing is, as it were, in a space of possible atomic facts. I can think of this space as empty, but not of the thing without the space.

2.0131
A spatial object must lie in infinite space. (A point in space is an argument place.)

A speck in a visual field need not be red, but it must have a colour; it has, so to speak, a colour space round it. A tone must have a pitch, the object of the sense of touch a hardness, etc.

2.014
Objects contain the possibility of all states of affairs.

2.0141
The possibility of its occurrence in atomic facts is the form of the object.

2.02
The object is simple.

These objects are primitives, rather like Leibniz's monads, except of course W. has not souls in mind.

2.0201
Every statement about complexes can be analysed into a statement about their constituent parts, and into those propositions which completely describe the complexes.

2.021
Objects form the substance of the world. Therefore they cannot be compound.

2.0211
If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true.

2.0212
It would then be impossible to form a picture of the world (true or false).

2.022
It is clear that however different from the real one an imagined world may be, it must have something⁠—a form⁠—in common with the real world.

2.023
This fixed form consists of the objects.

So does W. means mental objects or physical things "out there in the real world" in a form of naive realism?

I suppose he is in effect saying that what imaginary and real worlds have in common are what might be called thought forms, or just thoughts, or concepts.

2.0231
The substance of the world can only determine a form and not any material properties. For these are first presented by the propositions⁠—first formed by the configuration of the objects.

2.0232
Roughly speaking: objects are colourless.

2.0233
Two objects of the same logical form are⁠—apart from their external properties⁠—only distinguished from one another in that they are different.

An object is a simple but it has a logical form? The only logical forms I know of are essentially relations. It would be the relations that are composed of simples.

I suppose he might be saying that aRb and aRc represent simples in that a and b are specific instances (held constant).

May we say that objects are to atomic facts what ur elements are to some set theories?

2.02331
Either a thing has properties which no other has, and then one can distinguish it straight away from the others by a description and refer to it; or, on the other hand, there are several things which have the totality of their properties in common, and then it is quite impossible to point to any one of them.

For if a thing is not distinguished by anything, I cannot distinguish it⁠—for otherwise it would be distinguished.

2.024
Substance is what exists independently of what is the case.

2.025
It is form and content.

2.0251
Space, time and colour (colouredness) are forms of objects.

2.026
Only if there are objects can there be a fixed form of the world.

2.027
The fixed, the existent and the object are one.

2.0271
The object is the fixed, the existent; the configuration is the changing, the variable.

2.0272
The configuration of the objects forms the atomic fact.

2.03
In the atomic fact objects hang one in another, like the links of a chain.

2.031
In the atomic fact the objects are combined in a definite way.

2.032
The way in which objects hang together in the atomic fact is the structure of the atomic fact.

2.033
The form is the possibility of the structure.

2.034
The structure of the fact consists of the structures of the atomic facts.

2.04
The totality of existent atomic facts is the world.

2.05
The totality of existent atomic facts also determines which atomic facts do not exist.

2.06
The existence and nonexistence of atomic facts is the reality.

2.061
Atomic facts are independent of one another.

2.062
From the existence or nonexistence of an atomic fact we cannot infer the existence or nonexistence of another.

2.063
The total reality is the world.

2.1
We make to ourselves pictures of facts.

2.11
The picture presents the facts in logical space, the existence and nonexistence of atomic facts.

2.12
The picture is a model of reality.

2.13
To the objects correspond in the picture the elements of the picture.

2.131
The elements of the picture stand, in the picture, for the objects.

2.14
The picture consists in the fact that its elements are combined with one another in a definite way.

2.141
The picture is a fact.

2.15
That the elements of the picture are combined with one another in a definite way, represents that the things are so combined with one another.

This connection of the elements of the picture is called its structure, and the possibility of this structure is called the form of representation of the picture.

2.151
The form of representation is the possibility that the things are combined with one another as are the elements of the picture.

2.1511
Thus the picture is linked with reality; it reaches up to it.

2.1512
It is like a scale applied to reality.

2.15121
Only the outermost points of the dividing lines touch the object to be measured.

2.1513
According to this view the representing relation which makes it a picture, also belongs to the picture.

2.1514
The representing relation consists of the coordinations of the elements of the picture and the things.

2.1515
These coordinations are as it were the feelers of its elements with which the picture touches reality.

2.16
In order to be a picture a fact must have something in common with what it pictures.

2.161
In the picture and the pictured there must be something identical in order that the one can be a picture of the other at all.

2.17
What the picture must have in common with reality in order to be able to represent it after its manner⁠—rightly or falsely⁠—is its form of representation.

2.171
The picture can represent every reality whose form it has.

The spatial picture, everything spatial, the coloured, everything coloured, etc.

2.172
The picture, however, cannot represent its form of representation; it shows it forth.

2.173
The picture represents its object from without (its standpoint is its form of representation), therefore the picture represents its object rightly or falsely.

2.174
But the picture cannot place itself outside of its form of representation.

2.18
What every picture, of whatever form, must have in common with reality in order to be able to represent it at all⁠—rightly or falsely⁠—is the logical form, that is, the form of reality.

2.181
If the form of representation is the logical form, then the picture is called a logical picture.

2.182
Every picture is also a logical picture. (On the other hand, for example, not every picture is spatial.)

2.19
The logical picture can depict the world.

2.2
The picture has the logical form of representation in common with what it pictures.

2.201
The picture depicts reality by representing a possibility of the existence and nonexistence of atomic facts.

2.202
The picture represents a possible state of affairs in logical space.

2.203
The picture contains the possibility of the state of affairs which it represents.

2.21
The picture agrees with reality or not; it is right or wrong, true or false.

2.22
The picture represents what it represents, independently of its truth or falsehood, through the form of representation.

2.221
What the picture represents is its sense.

2.222
In the agreement or disagreement of its sense with reality, its truth or falsity consists.

2.223
In order to discover whether the picture is true or false we must compare it with reality.

2.224
It cannot be discovered from the picture alone whether it is true or false.

2.225
There is no picture which is a priori true.

Tractatus 1

1¹ The world is everything that is the case.

1.1
The world is the totality of facts, not of things.

Do we have here an indication of naive realism? Well, at least W's world is defined as composed of facts, which must be mental in nature. That is, a fact is considered to carry the the truth value true. It tells the hearer something about his world that holds or doesn't accord with what we call reality. Non-facts I presume are propositions or relations that are regarded as not true or as presenting a silly association.

1.11
The world is determined by the facts, and by these being all the facts.

1.12
For the totality of facts determines both what is the case, and also all that is not the case.

1.13
The facts in logical space are the world.

Logical space? Is this a Platonic space of pure forms? Yes, I would say it is. I daresay Plato would argue that his pure forms are equivalent to W's pure logical forms.

1.2
The world divides into facts.

I agree that the human mind carves up Creation into little mental compartments for utilitarian reasons (homeostasis).

1.21
Any one can either be the case or not be the case, and everything else remain the same.

Yes, the mind holds input from the environment constant while it considers a particular "fact" as either true or false.

1. The decimal figures as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3, etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on.

Tractatus Logico-Philosophicus

Dedicated to the Memory of My Friend David H. Pinsent
Motto: … und alles, was man weiss, nicht bloss rauschen und
brausen gehört hat, lässt sich in drei Worten sagen. –KÜRNBERGER.

Preface

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

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

The book will, therefore, draw a limit to thinking, or rather⁠—not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought).

The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense.

How far my efforts agree with those of other philosophers I will not decide. Indeed what I have here written makes no claim to novelty in points of detail; and therefore I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another.

I will only mention that to the great works of Frege and the writings of my friend Bertrand Russell I owe in large measure the stimulation of my thoughts.

If this work has a value it consists in two things. First that in it thoughts are expressed, and this value will be the greater the better the thoughts are expressed. The more the nail has been hit on the head.⁠—Here I am conscious that I have fallen far short of the possible. Simply because my powers are insufficient to cope with the task.⁠—May others come and do it better.

On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved. And if I am not mistaken in this, then the value of this work secondly consists in the fact that it shows how little has been done when these problems have been solved.

L. W.
Vienna, 1918

Introduction to Tractatus

Translated by C.K. Ogden

By Bertrand Russell,
F. R. S.

Mr. Wittgenstein’s Tractatus Logico-Philosophicus, whether or not it prove to give the ultimate truth on the matters with which it deals, certainly deserves, by its breadth and scope and profundity, to be considered an important event in the philosophical world. Starting from the principles of Symbolism and the relations which are necessary between words and things in any language, it applies the result of this inquiry to various departments of traditional philosophy, showing in each case how traditional philosophy and traditional solutions arise out of ignorance of the principles of Symbolism and out of misuse of language.

The logical structure of propositions and the nature of logical inference are first dealt with. Thence we pass successively to Theory of Knowledge, Principles of Physics, Ethics, and finally the Mystical (das Mystische).

In order to understand Mr. Wittgenstein’s book, it is necessary to realize what is the problem with which he is concerned. In the part of his theory which deals with Symbolism he is concerned with the conditions which would have to be fulfilled by a logically perfect language. There are various problems as regards language. First, there is the problem what actually occurs in our minds when we use language with the intention of meaning something by it; this problem belongs to psychology. Secondly, there is the problem as to what is the relation subsisting between thoughts, words, or sentences, and that which they refer to or mean; this problem belongs to epistemology. Thirdly, there is the problem of using sentences so as to convey truth rather that falsehood; this belongs to the special sciences dealing with the subject-matter of the sentences in question. Fourthly, there is the question: what relation must one fact (such as a sentence) have to another in order to be capable of being a symbol for that other? This last is a logical question, and is the one with which Mr. Wittgenstein is concerned. He is concerned with the conditions for accurate Symbolism, i.e. for Symbolism in which a sentence “means” something quite definite. In practice, language is always more or less vague, so that what we assert is never quite precise. Thus, logic has two problems to deal with in regard to Symbolism: (1) the conditions for sense rather than nonsense in combinations of symbols; (2) the conditions for uniqueness of meaning or reference in symbols or combinations of symbols. A logically perfect language has rules of syntax which prevent nonsense, and has single symbols which always have a definite and unique meaning. Mr. Wittgenstein is concerned with the conditions for a logically perfect language—not that any language is logically perfect, or that we believe ourselves capable, here and now, of constructing a logically perfect language, but that the whole function of language is to have meaning, and it only fulfills this function in proportion as it approaches to the ideal language which we postulate.

The essential business of language is to assert or deny facts. Given the syntax of language, the meaning of a sentence is determined as soon as the meaning of the component words is known. In order that a certain sentence should assert a certain fact there must, however the language may be constructed, be something in common between the structure of the sentence and the structure of the fact. This is perhaps the most fundamental thesis of Mr. Wittgenstein’s theory. That which has to be in common between the sentence and the fact cannot, he contends, be itself in turn said in language. It can, in his phraseology, only be shown, not said, for whatever we may say will still need to have the same structure.

The first requisite of an ideal language would be that there should be one name for every simple, and never the same name for two different simples. A name is a simple symbol in the sense that it has no parts which are themselves symbols. In a logically perfect language nothing that is not simple will have a simple symbol. The symbol for the whole will be a “complex”, containing the symbols for the parts. (In speaking of a “complex” we are, as will appear later, sinning against the rules of philosophical grammar, but this is unavoidable at the outset. “Most propositions and questions that have been written about philosophical matters are not false but senseless. We cannot, therefore, answer questions of this kind at all, but only state their senselessness. Most questions and propositions of the philosophers result from the fact that we do not understand the logic of our language. They are of the same kind as the question whether the Good is more or less identical than the Beautiful” (4.003).) What is complex in the world is a fact. Facts which are not compounded of other facts are what Mr. Wittgenstein calls Sachverhalte, whereas a fact which may consist of two or more facts is a Tatsache: thus, for example “Socrates is wise” is a Sachverhalt, as well as a Tatsache, whereas “Socrates is wise and Plato is his pupil” is a Tatsache but not a Sachverhalt.

He compares linguistic expression to projection in geometry. A geometrical figure may be projected in many ways: each of these ways corresponds to a different language, but the projective properties of the original figure remain unchanged whichever of these ways may be adopted. These projective properties correspond to that which in his theory the proposition and the fact must have in common, if the proposition is to assert the fact.

In certain elementary ways this is, of course, obvious. It is impossible, for example, to make a statement about two men (assuming for the moment that the men may be treated as simples), without employing two names, and if you are going to assert a relation between the two men it will be necessary that the sentence in which you make the assertion shall establish a relation between the two names. If we say “Plato loves Socrates”, the word “loves” which occurs between the word “Plato” and the word “Socrates” establishes a certain relation between these two words, and it is owing to this fact that our sentence is able to assert a relation between the persons named by the words “Plato” and “Socrates”. “We must not say, the complex sign ‘aRb’ says that ‘a stands in a certain relation R to b’; but we must say, that ‘a’ stands in a certain relation to ‘b’ says that aRb” (3.1432).

Mr. Wittgenstein begins his theory of Symbolism with the statement (2.1): “We make to ourselves pictures of facts.” A picture, he says, is a model of the reality, and to the objects in the reality correspond the elements of the picture: the picture itself is a fact. The fact that things have a certain relation to each other is represented by the fact that in the picture its elements have a certain relation to each other. “In the picture and the pictured there must be something identical in order that the one can be a picture of the other at all. What the picture must have in common with reality in order to be able to represent it after its manner—rightly or falsely—is its form of representation” (2.161, 2.17).

We speak of a logical picture of a reality when we wish to imply only so much resemblance as is essential to its being a picture in any sense, that is, when we wish to imply no more than identity of logical form. The logical picture of a fact, he says, is a Gedanke. A picture can correspond or not correspond with the fact and be accordingly true or false, but in both cases it shares the logical form with the fact. The sense in which he speaks of pictures is illustrated by his statement: “The gramophone record, the musical thought, the score, the waves of sound, all stand to one another in that pictorial internal relation which holds between language and the world. To all of them the logical structure is common. (Like the two youths, their two horses and their lilies in the story. They are all in a certain sense one)” (4.014). The possibility of a proposition representing a fact rests upon the fact that in it objects are represented by signs. The so-called logical “constants” are not represented by signs, but are themselves present in the proposition as in the fact. The proposition and the fact must exhibit the same logical “manifold”, and this cannot be itself represented since it has to be in common between the fact and the picture. Mr. Wittgenstein maintains that everything properly philosophical belongs to what can only be shown, or to what is in common between a fact and its logical picture. It results from this view that nothing correct can be said in philosophy. Every philosophical proposition is bad grammar, and the best that we can hope to achieve by philosophical discussion is to lead people to see that philosophical discussion is a mistake. “Philosophy is not one of the natural sciences. (The word ‘philosophy’ must mean something which stands above or below, but not beside the natural sciences.) The object of philosophy is the logical clarification of thoughts. Philosophy is not a theory but an activity. A philosophical work consists essentially of elucidations. The result of philosophy is not a number of ‘philosophical propositions’, but to make propositions clear. Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred” (4.111 and 4.112). In accordance with this principle the things that have to be said in leading the reader to understand Mr. Wittgenstein’s theory are all of them things which that theory itself condemns as meaningless. With this proviso we will endeavour to convey the picture of the world which seems to underlie his system.

The world consists of facts: facts cannot strictly speaking be defined, but we can explain what we mean by saying that facts are what makes propositions true, or false. Facts may contain parts which are facts or may contain no such parts; for example: “Socrates was a wise Athenian”, consists of the two facts, “Socrates was wise”, and “Socrates was an Athenian.” A fact which has no parts that are facts is called by Mr. Wittgenstein a Sachverhalt. This is the same thing that he calls an atomic fact. An atomic fact, although it contains no parts that are facts, nevertheless does contain parts. If we may regard “Socrates is wise” as an atomic fact we perceive that it contains the constituents “Socrates” and “wise”. If an atomic fact is analyzed as fully as possible (theoretical, not practical possibility is meant) the constituents finally reached may be called “simples” or “objects”. It is a logical necessity demanded by theory, like an electron. His ground for maintaining that there must be simples is that every complex presumes a fact. It is not necessarily assumed that the complexity of facts is finite; even if every fact consisted of an infinite number of atomic facts and if every atomic fact consisted of an infinite number of objects there would still be objects and atomic facts (4.2211). The assertion that there is a certain complex reduces to the assertion that its constituents are related in a certain way, which is the assertion of a fact: thus if we give a name to the complex the name only has meaning because of the truth of a certain proposition, namely the proposition asserting the relatedness of the constituents of the complex. Thus the naming of complexes presumes propositions, while propositions presume the naming of simples. In this way the naming of simples is shown to be what is logically first in logic.

The world is fully described if all atomic facts are known, together with the fact that these are all of them. The world is not described by merely naming all the objects in it; it is necessary also to know the atomic facts of which these objects are constituents. Given this totality of atomic facts, every true proposition, however complex, can theoretically be inferred. A proposition (true or false) asserting an atomic fact is called an atomic proposition. All atomic propositions are logically independent of each other. No atomic proposition implies any other or is inconsistent with any other. Thus the whole business of logical inference is concerned with propositions which are not atomic. Such propositions may be called molecular.

Wittgenstein’s theory of molecular propositions turns upon his theory of the construction of truth-functions.

A truth-function of a proposition p is a proposition containing p and such that its truth or falsehood depends only upon the truth or falsehood of p, and similarly a truth-function of several propositions p, q, r,… is one containing p, q, r,… and such that its truth or falsehood depends only upon the truth or falsehood of p, q, r,… It might seem at first sight as though there were other functions of propositions besides truth-functions; such, for example, would be “A believes p”, for in general A will believe some true propositions and some false ones: unless he is an exceptionally gifted individual, we cannot infer that p is true from the fact that he believes it or that p is false from the fact that he does not believe it. Other apparent exceptions would be such as “p is a very complex proposition” or “p is a proposition about Socrates”. Mr. Wittgenstein maintains, however, for reasons which will appear presently, that such exceptions are only apparent, and that every function of a proposition is really a truth-function. It follows that if we can define truth-functions generally, we can obtain a general definition of all propositions in terms of the original set of atomic propositions. This Wittgenstein proceeds to do.

It has been shown by Dr. Sheffer (Trans. Am. Math. Soc., Vol. XIV. pp. 481–488) that all truth-functions of a given set of propositions can be constructed out of either of the two functions “not-p or not-q” or “not-p and not-q”. Wittgenstein makes use of the latter, assuming a knowledge of Dr. Sheffer’s work. The manner in which other truth-functions are constructed out of “not-p and not-q” is easy to see. “Not-p and not-p” is equivalent to “not-p”, hence we obtain a definition of negation in terms of our primitive function: hence we can define “p or q”, since this is the negation of “not-p and not-q”, i.e. of our primitive function. The development of other truth-functions out of “not-p” and “p or q” is given in detail at the beginning of Principia Mathematica. This gives all that is wanted when the propositions which are arguments to our truth-function are given by enumeration. Wittgenstein, however, by a very interesting analysis succeeds in extending the process to general propositions, i.e. to cases where the propositions which are arguments to our truth-function are not given by enumeration but are given as all those satisfying some condition. For example, let fx be a propositional function (i.e. a function whose values are propositions), such as “x is human”—then the various values of fx form a set of propositions. We may extend the idea “not-p and not-q” so as to apply to the simultaneous denial of all the propositions which are values of fx. In this way we arrive at the proposition which is ordinarily represented in mathematical logic by the words “fx is false for all values of x”. The negation of this would be the proposition “there is at least one x for which fx is true” which is represented by “(∃x).fx”. If we had started with not-fx instead of fx we should have arrived at the proposition “fx is true for all values of x” which is represented by “(x).fx”. Wittgenstein’s method of dealing with general propositions [i.e. “(x).fx” and “(∃x).fx”] differs from previous methods by the fact that the generality comes only in specifying the set of propositions concerned, and when this has been done the building up of truth-functions proceeds exactly as it would in the case of a finite number of enumerated arguments p, q, r,…

Mr. Wittgenstein’s explanation of his symbolism at this point is not quite fully given in the text. The symbol he uses is [p, ξ, N(ξ)]. The following is the explanation of this symbol:

p stands for all atomic propositions.
ξ stands for any set of propositions.
N(ξ) stands for the negation of all the propositions making up ξ.

The whole symbol [p, ξ, N(ξ)] means whatever can be obtained by taking any selection of atomic propositions, negating them all, then taking any selection of the set of propositions now obtained, together with any of the originals—and so on indefinitely. This is, he says, the general truth-function and also the general form of proposition. What is meant is somewhat less complicated than it sounds. The symbol is intended to describe a process by the help of which, given the atomic propositions, all others can be manufactured. The process depends upon:

(a). Sheffer’s proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of “not-p and not-q”;

(b). Mr. Wittgenstein’s theory of the derivation of general propositions from conjunctions and disjunctions;

(c). The assertion that a proposition can only occur in another proposition as argument to a truth-function. Given these three foundations, it follows that all propositions which are not atomic can be derived from such as are, by a uniform process, and it is this process which is indicated by Mr. Wittgenstein’s symbol.

From this uniform method of construction we arrive at an amazing simplification of the theory of inference, as well as a definition of the sort of propositions that belong to logic. The method of generation which has just been described, enables Wittgenstein to say that all propositions can be constructed in the above manner from atomic propositions, and in this way the totality of propositions is defined. (The apparent exceptions which we mentioned above are dealt with in a manner which we shall consider later.) Wittgenstein is enabled to assert that propositions are all that follows from the totality of atomic propositions (together with the fact that it is the totality of them); that a proposition is always a truth-function of atomic propositions; and that if p follows from q the meaning of p is contained in the meaning of q, from which of course it results that nothing can be deduced from an atomic proposition. All the propositions of logic, he maintains, are tautologies, such, for example, as “p or not p”.

The fact that nothing can be deduced from an atomic proposition has interesting applications, for example, to causality. There cannot, in Wittgenstein’s logic, be any such thing as a causal nexus. “The events of the future”, he says, “cannot be inferred from those of the present. Superstition is the belief in the causal nexus.” That the sun will rise to-morrow is a hypothesis. We do not in fact know whether it will rise, since there is no compulsion according to which one thing must happen because another happens.

Let us now take up another subject—that of names. In Wittgenstein’s theoretical logical language, names are only given to simples. We do not give two names to one thing, or one name to two things. There is no way whatever, according to him, by which we can describe the totality of things that can be named, in other words, the totality of what there is in the world. In order to be able to do this we should have to know of some property which must belong to every thing by a logical necessity. It has been sought to find such a property in self-identity, but the conception of identity is subjected by Wittgenstein to a destructive criticism from which there seems no escape. The definition of identity by means of the identity of indiscernibles is rejected, because the identity of indiscernibles appears to be not a logically necessary principle. According to this principle x is identical with y if every property of x is a property of y, but it would, after all be logically possible for two things to have exactly the same properties. If this does not in fact happen that is an accidental characteristic of the world, not a logically necessary characteristic, and accidental characteristics of the world must, of course, not be admitted into the structure of logic. Mr. Wittgenstein accordingly banishes identity and adopts the convention that different letters are to mean different things. In practice, identity is needed as between a name and a description or between two descriptions. It is needed for such propositions as “Socrates is the philosopher who drank the hemlock”, or “The even prime is the next number after 1.” For such uses of identity it is easy to provide on Wittgenstein’s system.

The rejection of identity removes one method of speaking of the totality of things, and it will be found that any other method that may be suggested is equally fallacious: so, at least, Wittgenstein contends and, I think, rightly. This amounts to saying that “object” is a pseudo-concept. To say “x is an object” is to say nothing. It follows from this that we cannot make such statements as “there are more than three objects in the world”, or “there are an infinite number of objects in the world”. Objects can only be mentioned in connexion with some definite property. We can say “there are more than three objects which are human”, or “there are more than three objects which are red”, for in these statements the word object can be replaced by a variable in the language of logic, the variable being one which satisfies in the first case the function “x is human”; in the second the function “x is red”. But when we attempt to say “there are more than three objects”, this substitution of the variable for the word “object” becomes impossible, and the proposition is therefore seen to be meaningless.

We here touch one instance of Wittgenstein’s fundamental thesis, that it is impossible to say anything about the world as a whole, and that whatever can be said has to be about bounded portions of the world. This view may have been originally suggested by notation, and if so, that is much in its favor, for a good notation has a subtlety and suggestiveness which at times make it seem almost like a live teacher. Notational irregularities are often the first sign of philosophical errors, and a perfect notation would be a substitute for thought. But although notation may have first suggested to Mr. Wittgenstein the limitation of logic to things within the world as opposed to the world as a whole, yet the view, once suggested, is seen to have much else to recommend it. Whether it is ultimately true I do not, for my part, profess to know. In this Introduction I am concerned to expound it, not to pronounce upon it. According to this view we could only say things about the world as a whole if we could get outside the world, if, that is, it ceased to be for us the whole world. Our world may be bounded for some superior being who can survey it from above, but for us, however finite it may be, it cannot have a boundary, since it has nothing outside it. Wittgenstein uses, as an analogy, the field of vision. Our field of vision does not, for us, have a visual boundary, just because there is nothing outside it, and in like manner our logical world has no logical boundary because our logic knows of nothing outside it. These considerations lead him to a somewhat curious discussion of Solipsism. Logic, he says, fills the world. The boundaries of the world are also its boundaries. In logic, therefore, we cannot say, there is this and this in the world, but not that, for to say so would apparently presume that we exclude certain possibilities, and this cannot be the case, since it would require that logic should go beyond the boundaries of the world as if it could contemplate these boundaries from the other side also. What we cannot think we cannot think, therefore we also cannot say what we cannot think.

This, he says, gives the key to solipsism. What Solipsism intends is quite correct, but this cannot be said, it can only be shown. That the world is my world appears in the fact that the boundaries of language (the only language I understand) indicate the boundaries of my world. The metaphysical subject does not belong to the world but is a boundary of the world.

We must take up next the question of molecular propositions which are at first sight not truth-functions, of the propositions that they contain, such, for example, as “A believes p.”

Wittgenstein introduces this subject in the statement of his position, namely, that all molecular functions are truth-functions. He says (5.54): “In the general propositional form, propositions occur in a proposition only as bases of truth-operations.” At first sight, he goes on to explain, it seems as if a propositions could also occur in other ways, e.g. “A believes p.” Here it seems superficially as if the proposition p stood in a sort of relation to the object A. “But it is clear that ‘A believes that p,’ ‘A thinks p,’ ‘A says p’ are of the form “‘p’ says p”; and here we have no co-ordination of a fact and an object, but a co-ordination of facts by means of a co-ordination of their objects” (5.542).

What Mr. Wittgenstein says here is said so shortly that its point is not likely to be clear to those who have not in mind the controversies with which he is concerned. The theory with which he is disagreeing will be found in my articles on the nature of truth and falsehood in Philosophical Essays and Proceedings of the Aristotelian Society, 1906–7. The problem at issue is the problem of the logical form of belief, i.e. what is the schema representing what occurs when a man believes. Of course, the problem applies not only to belief, but also to a host of other mental phenomena which may be called propositional attitudes: doubting, considering, desiring, etc. In all these cases it seems natural to express the phenomenon in the form “A doubts p”, “A considers p”, “A desires p”, etc., which makes it appear as though we were dealing with a relation between a person and a proposition. This cannot, of course, be the ultimate analysis, since persons are fictions and so are propositions, except in the sense in which they are facts on their own account. A proposition, considered as a fact on its own account, may be a set of words which a man says over to himself, or a complex image, or train of images passing through his mind, or a set of incipient bodily movements. It may be any one of innumerable different things. The proposition as a fact on its own account, for example, the actual set of words the man pronounces to himself, is not relevant to logic. What is relevant to logic is that common element among all these facts, which enables him, as we say, to mean the fact which the proposition asserts. To psychology, of course, more is relevant; for a symbol does not mean what it symbolizes on account of a logical relation alone, but on account also of a psychological relation of intention, or association, or what-not. The psychological part of meaning, however, does not concern the logician. What does concern him in this problem of belief is the logical schema. It is clear that, when a person believes a proposition, the person, considered as a metaphysical subject, does not have to be assumed in order to explain what is happening. What has to be explained is the relation between the set of words which is the proposition considered as a fact on its own account, and the “objective” fact which makes the proposition true or false. This reduces ultimately to the question of the meaning of propositions, that is, the meaning of propositions is the only non-psychological portion of the problem involved in the analysis of belief. This problem is simply one of a relation of two facts, namely, the relation between the series of words used by the believer and the fact which makes these words true or false. The series of words is a fact just as much as what makes it true or false is a fact. The relation between these two facts is not unanalyzable, since the meaning of a proposition results from the meaning of its constituent words. The meaning of the series of words which is a proposition is a function of the meaning of the separate words. Accordingly, the proposition as a whole does not really enter into what has to be explained in explaining the meaning of a propositions. It would perhaps help to suggest the point of view which I am trying to indicate, to say that in the cases which have been considering the proposition occurs as a fact, not as a proposition. Such a statement, however, must not be taken too literally. The real point is that in believing, desiring, etc., what is logically fundamental is the relation of a proposition considered as a fact, to the fact which makes it true or false, and that this relation of two facts is reducible to a relation of their constituents. Thus the proposition does not occur at all in the same sense in which it occurs in a truth-function.

There are some respects, in which, as it seems to me, Mr. Wittgenstein’s theory stands in need of greater technical development. This applies in particular to his theory of number (6.02ff.) which, as it stands, is only capable of dealing with finite numbers. No logic can be considered adequate until it has been shown to be capable of dealing with transfinite numbers. I do not think there is anything in Mr. Wittgenstein’s system to make it impossible for him to fill this lacuna.

More interesting than such questions of comparative detail is Mr. Wittgenstein’s attitude towards the mystical. His attitude upon this grows naturally out of his doctrine in pure logic, according to which the logical proposition is a picture (true or false) of the fact, and has in common with the fact a certain structure. It is this common structure which makes it capable of being a picture of the fact, but the structure cannot itself be put into words, since it is a structure of words, as well as of the fact to which they refer. Everything, therefore, which is involved in the very idea of the expressiveness of language must remain incapable of being expressed in language, and is, therefore, inexpressible in a perfectly precise sense. This inexpressible contains, according to Mr. Wittgenstein, the whole of logic and philosophy. The right method of teaching philosophy, he says, would be to confine oneself to propositions of the sciences, stated with all possible clearness and exactness, leaving philosophical assertions to the learner, and proving to him, whenever he made them, that they are meaningless. It is true that the fate of Socrates might befall a man who attempted this method of teaching, but we are not to be deterred by that fear, if it is the only right method. It is not this that causes some hesitation in accepting Mr. Wittgenstein’s position, in spite of the very powerful arguments which he brings to its support. What causes hesitation is the fact that, after all, Mr. Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit. The whole subject of ethics, for example, is placed by Mr. Wittgenstein in the mystical, inexpressible region. Nevertheless he is capable of conveying his ethical opinions. His defence would be that what he calls the mystical can be shown, although it cannot be said. It may be that this defence is adequate, but, for my part, I confess that it leaves me with a certain sense of intellectual discomfort.

There is one purely logical problem in regard to which these difficulties are peculiarly acute. I mean the problem of generality. In the theory of generality it is necessary to consider all propositions of the form fx where fx is a given propositional function. This belongs to the part of logic which can be expressed, according to Mr. Wittgenstein’s system. But the totality of possible values of x which might seem to be involved in the totality of propositions of the form fx is not admitted by Mr. Wittgenstein among the things that can be spoken of, for this is no other than the totality of things in the world, and thus involves the attempt to conceive the world as a whole; “the feeling of the world as a bounded whole is the mystical”; hence the totality of the values of x is mystical (6.45). This is expressly argued when Mr. Wittgenstein denies that we can make propositions as to how many things there are in the world, as for example, that there are more than three.

These difficulties suggest to my mind some such possibility as this: that every language has, as Mr. Wittgenstein says, a structure concerning which in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit. Mr. Wittgenstein would of course reply that his whole theory is applicable unchanged to the totality of such languages. The only retort would be to deny that there is any such totality. The totalities concerning which Mr. Wittgenstein holds that it is impossible to speak logically are nevertheless thought by him to exist, and are the subject-matter of his mysticism. The totality resulting from our hierarchy would be not merely logically inexpressible, but a fiction, a mere delusion, and in this way the supposed sphere of the mystical would be abolished. Such a hypothesis is very difficult, and I can see objections to it which at the moment I do not know how to answer. Yet I do not see how any easier hypothesis can escape from Mr. Wittgenstein’s conclusions. Even if this very difficult hypothesis should prove tenable, it would leave untouched a very large part of Mr. Wittgenstein’s theory, though possibly not the part upon which he himself would wish to lay most stress. As one with a long experience of the difficulties of logic and of the deceptiveness of theories which seem irrefutable, I find myself unable to be sure of the rightness of a theory, merely on the ground that I cannot see any point on which it is wrong. But to have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance. This merit, in my opinion, belongs to Mr. Wittgenstein’s book, and makes it one which no serious philosopher can afford to neglect.

Bertrand Russell
May 1922