Title:  Ts-201a2: Notes on Logic (BRA) - Normalized transcription [Draft]
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Author:  Ludwig Wittgenstein
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Notes on Logic
by
Ludwig Wittgenstein
September 1913.

































 
     
Summary.

 
     

      One reason for thinking the old notation wrong is that it is- very unlikely that from every proposition p an infinite number of- other propositions not-not-p, not-not-not-not-p, etc., should- follow.

 
     

      If only those signs which contain proper names were complex- then propositions containing nothing but apparent variables would- be simple. Then what about their denials?

 
     

      The verb of a proposition cannot be “is true” or “is false”,- but whatever is true or false must already contain the verb.

 
     
     Deductions only proceed according to the laws of deduction,- but these laws cannot justify the deduction.

 
     

      One reason for supposing that not all propositions which- have more than one argument are relational propositions is that- if they were, the relations of judgment and inference would have- to hold between an arbitrary number of things.

 
     

      Every proposition which seems to be about a complex can be- analysed into a proposition about its constituents and about- the proposition which describes the complex perfectly; i.e.,- that proposition which is equivalent to saying the complex exists.

 
     

      The idea that propositions are names of complexes suggests- that whatever is not a proper name is a sign for a relation. - Because spatial complexes* consist of Things and Relations- only and the idea of a complex is taken from space.

 
     

      In a proposition convert all its indefinables into- variables; there then remains a class of propositions which- is not all propositions but a type.

 
     

      There are thus two ways in which signs are similar. The- names Socrates and Plato are similar: they are both names. But- whatever they have in common must not be introduced before Socrates- and Plato are introduced. The same applies to a subject-predicate- form etc.. Therefore, thing, proposition, subject-predicate form,- etc., are not indefinables, i.e., types are not indefinables.

 
     

      When we say A judges that etc., then we have to mention- a whole proposition which A judges. It will not do either to- mention only its constituents, or its constituents and form, but- not in the proper order. This shows that a proposition itself- must occur in the statement that it is judged; however, for- instance, “not-p” may be explained, the question, “What is- negated” must have a meaning.
* Russell – for instance imagines every fact as a spatial- complex.

 
     

      To understand a proposition p it is not enough to know- that p implies ‘“p” is true’, but we must also know that- ~p implies “p is false”. This shows the bi-polarity of the- proposition.

 
     

      To every molecular function a WF* scheme corresponds. - Therefore we may use the WF scheme itself instead of the- function. Now what the WF scheme does is, it correlates the- letters W and F with each proposition. These two letters- are the poles of atomic propositions. Then the scheme- correlates another W and F to these poles. In this notation- all that matters is the correlation of the outside poles to- the poles of the atomic propositions. Therefore not-not-p- is the same symbol as p. And therefore we shall never get- two symbols for the same molecular function.

 
     

      The meaning of a proposition is the fact which actually- corresponds to it.

 
     

      As the ab functions of atomic propositions are bi-polar- propositions again we can perform ab operations on them. We- shall, by doing so, correlate two new outside poles via the old- outside poles to the poles of the atomic propositions.
* W-F = Wahr-Falsch.

 
     

      The symbolising fact in a-p-b is that, say* a is on the- left of p and b on the right of p; then the correlation of new- poles is to be transitive, so that for instance if a new pole- a in whatever way i.e. via whatever poles is correlated to the- inside a, the symbol is not changed thereby. It is therefore- possible to construct all possible ab functions by performing- one ab operation repeatedly, and we can therefore talk of all- ab functions as of all those functions which can be obtained- by performing this ab operation repeatedly.

 
     
     [Note by Bertrand Russell ab means the same as WF, which means true-false.]

 
     

      Naming is like pointing. A function is like a line- dividing points of a plane into right and left ones; then- “p or not-p” has no meaning because it does not divide the- plane.

 
     

      But though a particular proposition “p or not-p” has no- meaning, a general proposition “for all p's, p or not-p” has a- meaning because this does not contain the nonsensical function- “p or not-p” but the function “p or not-q” just as “for all x's- xRx” contains the function “xRy”.
* This is quite arbitrary but, if we once have fixed on which- order the poles have to stand we must of course stick to our convention. If for instance “a p b” says p then b p a says nothing. - (It does not say ~p.) But a - a p b - b is the same symbol as a p b- (here the ab function vanishes automatically) for here the new- poles are related to the same side of p as the old ones. The- question is always: how are the new poles correlated to p compared with the way the old poles are correlated to ~p.

 
     

      A proposition is a standard to which facts behave, with- names it is otherwise; it is thus bi-polarity and sense comes in;- just as one arrow behaves to another arrow by being in the same- sense or the opposite, so a fact behaves to a proposition.

 
     

      The form of a proposition has meaning in the following way. - Consider a symbol “xRy”. To symbols of this form correspond- couples of things whose names are respectively “x” and “y”. The- things x y stand to one another in all sorts of relations,- amongst others some stand in the relation R, and some not; just- as I single out a particular thing by a particular name I single- out all behaviours of the points x and y with respect to the- relation R. I say that if an x stands in the relation R to a- y the sign “xRy” is to be called true to the fact and otherwise- false. This is a definition of sense.

 
     
     In my theory p has the same meaning as not-p but opposite- sense. The meaning is the fact. The proper theory of- judgment must make it impossible to judge nonsense.

 
     

      It is not strictly true to say that we understand a- proposition p if we know that p is equivalent to “p is true”- for this would be the case if accidentally both were true or- false. What is wanted is the formal equivalence with respect- to the forms of the proposition, i.e., all the general- indefinables involved. The sense of an ab function of a proposition- is a function of its sense. There are only unasserted propositions. -
Assertion is merely psychological. In not-p, p is exactly the- same as if it stands alone; this point is absolutely fundamental. - Among the facts that make “p or q” true there are also facts which- make “p and q” true; if propositions have only meaning, we ought,- in such a case, to say that these two propositions are identical,- but in fact, their sense is different for we have introduced- sense by talking of all p's and all q's. Consequently the- molecular propositions will only be used in cases where their- ab function stands under a generality sign or enters into- another function such as “I believe that, etc.”, because then- the sense enters.

 
     

      In “a judges p” p cannot be replaced by a proper name. This- appears if we substitute “a judges that p is true and not p is- false”. The proposition “a judges p” consists of the proper name- a, the proposition p with its 2 poles, and a being related to- both of these poles in a certain way. This is obviously not a- relation in the ordinary sense.

 
     

      The ab notation makes it clear that not and or are dependent- on one another and we can therefore not use them as simultaneous- indefinables. Same objections in the case of apparent variables- to the usual old indefinables, as in the case of molecular functions. The- application of the ab notation to apparent variable propositions- becomes clear if we consider that, for instance, the proposition- “for all x, φx” is to be true when φx is true for all x's and- false when φx is false for some x's. We see that some and all- occur simultaneously in the proper apparent variable notation.

 
     

      The notation is:


      for (x) φx : a - (x) - a φx b - (∃ x) - b
and



      for (∃x) φx : a - (∃x) - a φx b - (x) - b

 
     
     Old definitions now become tautologous.

 
     

      In aRb it is not the complex that symbolises but the fact- that the symbol a stands in a certain relation to the symbol b. - Thus facts are symbolised by facts, or more correctly: that a- certain thing is the case in the symbol says that a certain- thing is the case in the world.

 
     

      Judgment, question and command are all on the same level. - What interests logic in them is only the unasserted proposition. - Facts cannot be named.

 
     
A proposition cannot occur in itself. This is the fundamental- truth of the theory of types.

 
     
     Every proposition that says something indefinable about one- thing is a subject-predicate proposition, and so on.

 
     

      Therefore we can recognize a subject-predicate proposition- if we know it contains only one name and one form, etc.. This- gives the construction of types. Hence the type of a proposition- can be recognized by its symbol alone.

 
     
     What is essential in a correct apparent-variable notation- is this:– (1) it must mention a type of propositions; (2) it- must show which components of a proposition of this type are- constants.

 
     

      [Components are forms and constituents.]

 
     
     Take (φ). φ!x. Then if we describe the kind of symbols,- for which φ! stands and which, by the above, is enough to- determine the type, then automatically “(φ). φ! x” cannot be- fitted by this description, because it contains “φ!x” and the- description is to describe all that symbolises in symbols of- the φ! kind. If the description is thus complete vicious- circles can just as little occur as for instance (φ). (X)φ- (where (X)φ is a subject-predicate proposition).
























 
     
First MS.
 
     

      Indefinables are of two sorts: names, and forms. Propositions- cannot consist of names alone; they cannot be classes of names. - A name can not only occur in two different propositions, but can- occur in the same way in both.

 
     
     Propositions [which are symbols having reference to facts]- are themselves facts: that this inkpot is on this table may- express that I sit in this chair.

 
     

      It can never express the common characteristic of two- objects that we designate them by the same name but by two- different ways of designation, for, since names are arbitrary,- we might also choose different names, and where then would be- the common element in the designations? Nevertheless one is- always tempted, in a difficulty, to take refuge in different- ways of designation.

 
     

      Frege said “propositions are names”; Russell said- “propositions correspond to complexes”. Both are false; and- especially false is the statement “propositions are names of- complexes.”

 
     

      It is easy to suppose that only such symbols are complex as- contain names of objects, and that accordingly “(∃x,φ). φx”- or “(∃x,y). x R y” must be simple. It is then natural to- call the first of these the name of a form, the second the name- of a relation. But in that case what is the meaning of (e.g.)- “~(∃x,y). x R y”? Can we put “not” before a name?

 
     

      The reason why “~Socrates” means nothing is that “~x” does- not express a property of x.

 
     

      There are positive and negative facts: if the proposition- “this rose is not red” is true, then what it signifies is negative. - But the occurrence of the word “not” does not indicate this unless- we know that the signification of the proposition “this rose is- red” (when it is true) is positive. It is only from both, the- negation and the negated proposition, that we can conclude to a- characteristic of the significance of the whole proposition. (We- are not here speaking of negations of general propositions, i.e.- of such as contain apparent variables. Negative facts only- justify the negations of atomic propositions.)

 
     
     Positive and negative facts there are, but not true and- false facts.

 
     

      If we overlook the fact that propositions have a sense which- is independent of their truth or falsehood, it easily seems as if- true and false were two equally justified relations between the- sign and what is signified. (We might then say e.g. that “q”- signifies in the true way what “not-qsignifies in the false- way). But are not true and false in fact equally justified? Could- we not express ourselves by means of false propositions just as- well as hitherto with true ones, so long as we know that they- are meant falsely? No! For a proposition is then true when-
it is as we assert in this proposition; and accordingly if- by “q” we mean “not-q”, and it is as we mean to assert, then- in the new interpretation “q” is actually true and not false. - But it is important that we can mean the same by “q” as by- “not-q”, for it shows that neither to the symbol “not” nor to- the manner of its combination with “q” does a characteristic- of the denotation of “q” correspond.




































 
     
Second MS.

 
     

      We must be able to understand propositions which we have- never heard before. But every proposition is a new symbol. - Hence we must have general indefinable symbols; these are- unavoidable if propositions are not all indefinable.

 
     

      Whatever corresponds in reality to compound propositions- must not be more than what corresponds to their several atomic- propositions.

 
     
     Not only must logic not deal with [particular] things, but- just as little with relations and predicates.

 
     

      There are no propositions containing real variables.

 
     

      What corresponds in reality to a proposition depends upon- whether it is true or false. But we must be able to understand- a proposition without knowing if it is true or false.

 
     
     What we know when we understand a proposition is this: We- know what is the case if the proposition is true, and what is- the case if it is false. But we do not know [necessarily]- whether it is true or false.

 
     
     Propositions are not names.

 
     

      We can never distinguish one logical type from another by- attributing a property to members of the one which we deny to- members of the other.

 
     

      Symbols are not what they seem to be. In “a R b”, “R”- looks like a substantive, but is not one. What symbolizes- in “a R b” is that R occurs between a and b. Hence “R” is- not the indefinable in “a R b”. Similarly in “φx”, “φ”- looks like a substantive but is not one; in “~p”, “~” looks- like “φ” but is not like it. This is the first thing that- indicates that there may not be logical constants. A reason- against them is the generality of logic: logic cannot treat- a special set of things.

 
     

      Molecular propositions contain nothing beyond what is- contained in their atoms; they add no material information- above that contained in their atoms.

 
     
     All that is essential about molecular functions is their- T-F schema [i.e. the statement of the cases when they are true- and the cases when they are false].

 
     

      Alternative indefinability shows that the indefinables- have not been reached.

 
     

      Every proposition is essentially true-false: to understand- it, we must know both what must be the case if it is true, and- what must be the case if it is false. Thus a proposition- has two poles, corresponding to the case of its truth and the- case of its falsehood. We call this the sense of a proposition.

 
     

      In regard to notation, it is important to note that not- every feature of a symbol symbolizes. In two molecular- functions which have the same T-F schema, what symbolizes- must be the same. In “not-not-p”, “not-p” does not occur;- for “not-not-p” is the same as “p”, and therefore, if “not-p”- occurred in “not-not-p”, it would occur in “p”.

 
     

      Logical indefinables cannot be predicates or relations,- because propositions, owing to sense, cannot have predicates- or relations. Nor are “not” and “or”, like judgment,- analogous to predicates or relations, because they do not- introduce anything new.

 
     

      Propositions are always complex even if they contain no- names.

 
     

      A proposition must be understood when all its indefinables- are understood. The indefinables in “a R b” are introduced- as follows:


      “a” is indefinable;
      “b” is indefinable;
      Whatever “x” and “y” may mean, “x R y” says something- indefinable about their meaning.

 
     

      A complex symbol must never be introduced as a single- indefinable. [Thus e.g. no proposition is indefinable]. For- if one of its parts occurs also in another connection, it must- there be re-introduced. And would it then mean the same?

 
     
     The ways by which we introduce our indefinables must permit- us to construct all propositions that have sense [[?] meaning]- from these indefinables alone. It is easy to introduce “all”- and “some” in a way that will make the construction of (say)- “(x, y). x R y” possible from “all” and “x R y” as introduced- before.






































 
     
3rd. MS.

 
     

      An analogy for the theory of truth: Consider a black patch- on white paper; then we can describe the form of the patch by- mentioning, for each point of the surface, whether it is white- or black. To the fact that a point is black corresponds a- positive fact, to the fact that a point is white (not black)- corresponds a negative fact. If I designate a point of the- surface (one of Frege's “truth-values”), this is as if I set up- an assumption to be decided upon. But in order to be able to- say of a point that it is black or that it is white, I must first- know when a point is to be called black and when it is to be- called white. In order to be able to say that “p” is true- (or false), I must first have determined under what circumstances- I call a proposition true, and thereby I determine the sense- of a proposition. The point in which the analogy fails is- this: I can indicate a point of the paper what is white and- black, but to a proposition without sense nothing corresponds,- for it does not designate a thing (truth-value), whose- properties might be called “false” or “true”; the verb of a- proposition is not “is true” or “is false”, as Frege believes,- but what is true must already contain the verb.

 
     

      The comparison of language and reality is like that of- retinal image and visual image: to the blind spot nothing in- the visual image seems to correspond, and thereby the boundaries- of the blind spot determine the visual image – as true negations- of atomic propositions determine reality.

 
     

      Logical inferences can, it is true, be made in accordance- with Frege's or Russell's laws of deduction, but this cannot- justify the inference; and therefore they are not primitive- propositions of logic. If p follows from q, it can also be- inferred from q, and the “manner of deduction” is indifferent.

 
     

      Those symbols which are called propositions in which- “variables occur” are in reality not propositions at all, but- only schemes of propositions, which only become propositions- when we replace the variables by constants. There is no- proposition which is expressed by “x = x”, for “x” has no- signification; but there is a proposition “(x). x = x” and propositions such as “Socrates = Socrates” etc..

 
     

      In books on logic, no variables ought to occur, but only- the general propositions which justify the use of variables. - It follows that the so-called definitions of logic are not- definitions, but only schemes of definitions, and instead of- these we ought to put general propositions; and similarly the- so-called primitive ideas (Urzeichen) of logic are not primitive- ideas, but the schemes of them. The mistaken idea that there-
are things called facts or complexes and relations easily leads- to the opinion that there must be a relation of questioning(?) to- the facts, and then the question arises whether a relation can- hold between an arbitrary number of things, since a fact can- follow from arbitrary cases. It is a fact that the proposition- which e.g. expresses that q follows from p and p ⊃ q is this:- p. p ⊃ q. ⊃ p.q.q.

 
     

      At a pinch, one is tempted to interpret “ not-p” as “everything- else, only not p”. That from a single fact p an infinity of- others, not-not-p etc., follow, is hardly credible. Man- possesses an innate capacity for constructing symbols with- which some sense can be expressed, without having the slightest- idea what each word signifies. The best example of this is- mathematics, for man has until lately used the symbols for- numbers without knowing what they signify or that they signify- nothing.

 
     

      Russell's “complexes” were to have the useful property of- being compounded, and were to combine with this the agreeable- property that they could be treated like “simples”. But this- alone made them unserviceable as logical types, since there would- have been significance in asserting, of a simple, that it was- complex. But a property cannot be a logical type.

 
     

      Every statement about apparent complexes can be resolved- into the logical sum of a statement about the constituents and- a statement about the proposition which describes the complex- completely. How, in each case, the resolution is to be made,- is an important question, but its answer is not unconditionally- necessary for the construction of logic.

 
     

      That “or” and “not” etc. are not relations in the same- sense as “right” and “left” etc., is obvious to the plain man. - The possibility of cross-definitions in the old logical- indefinables shows, of itself, that these are not the right- indefinables, and, even more conclusively, that they do not- denote relations.

 
     

      If we change a constituent a of a proposition φ(a) into- a variable, then there is a class
^
p
{(∃x). φ(x) = p}.
This class in general still depends upon what, by an arbitrary- convention, we mean by “φ(x)”. But if we change into- variables all those symbols whose significance was arbitrarily- determined, there is still such a class. But this is now not- dependent upon any convention, but only upon the nature of the- symbol “φ(x)”. It corresponds to a logical type.

 
     

      Types can never be distinguished from each other by saying- (as is often done) that one has these but the other has those- properties, for this presupposes that there is a meaning in- asserting all these properties of both types. But from this- it follows that, at best, these properties may be types, but- certainly not the objects of which they are asserted.

 
     

      At a pinch we are always inclined to explanations of- logical functions of propositions which aim at introducing into- the function either only the constituents of these propositions,- or only their form, etc. etc.; and we overlook that ordinary- language would not contain the whole propositions if it did- not need them: However, e.g., “not-p” may be explained, there- must always be a meaning given to the question “what is denied?”

 
     

      The very possibility of Frege's explanations of “not-p”- and “if p then q”, from which it follows that “not-not-p” denotes- the same as p, makes it probable that there is some method of- designation in which “not-not-p” corresponds to the same- symbol as “p”. But if this method of designation suffices- for logic, it must be the right one.

 
     

      Names are points, propositions arrows – they have sense. - The sense of a proposition is determined by the two poles true- and false. The form of a proposition is like a straight line,- which divides all points of a plane into right and left. The- line does this automatically, the form of proposition only by- convention.

 
     

      Just as little as we are concerned, in logic, with the- relation of a name to its meaning, just so little are we- concerned with the relation of a proposition to reality, but- we want to know the meaning of names and the sense of propositions- as we introduce an indefinable concept “A” by saying: “‘A’- denotes something indefinable”, so we introduce e.g. the form- of propositions a R b by saying: “For all meanings of “x”- and “y”, “x R y” expresses something indefinable about x and y”.

 
     

      In place of every proposition “p”, let us write “
a
b
p”. - Let every correlation of propositions to each other or of names- to propositions be effected by a correlation of their poles- “a” and “b”. Let this correlation be transitive. Then- accordingly “
a ‒ a
b ‒ b
p” is the same symbol as “
a
b
p”. Let n- propositions be given. I then call a “class of poles” of these- propositions every class of n members, of which each is a pole- of one of the n propositions, so that one member corresponds- to each proposition. I then correlate with each class of- poles one of two poles (a and b). The sense of the symbolizing- fact thus constructed I cannot define, but I know it.

 
     

      If p = not-not-p etc., this shows that the traditional- method of symbolism is wrong, since it allows a plurality of- symbols with the same sense; and thence it follows that, in- analyzing such propositions, we must not be guided by Russell's- method of symbolizing.

 
     

      It is to be remembered that names are not things, but- classes: “A” is the same letter as “A”. This has the most- important consequences for every symbolic language.

 
     

      Neither the sense nor the meaning of a proposition is a- thing. These words are incomplete symbols.

 
     

      It is impossible to dispense with propositions in which- the same argument occurs in different positions. It is- obviously useless to replace φ(a, a) by φ(a, b). a = b.

 
     

      Since the ab-functions of p are again bi-polar propositions,- we can form ab-functions of them, and so on. In this way a- series of propositions will arise, in which in general the- symbolizing facts will be the same in several members. If now- we find an ab-function of such a kind that by repeated application- of it every ab-function can be generated, then we can introduce- the totality of ab-functions as the totality of those that are- generated by application of this function. Such a function is- ~p ⌵ ~q.

 
     

      It is easy to suppose a contradiction in the fact that on- the one hand every possible complex proposition is a simple ab-function of simple propositions, and that on the other hand the- repeated application of one ab-function suffices to generate all- these propositions. If e.g. an affirmation can be generated by- double negation, is negation in any sense contained in affirmation? - Does “p” deny “not-p” or assert “p”, or both? And how do-
matters stand with the definition of “ ⊃ ” by “ ⌵ ” and “.”,- or of “ ⌵ ” by “.” and “ ⊃ ”? And how e.g. shall we introduce- p|q (i.e. ~p ⌵ ~q), if not by saying that this expression- says something indefinable about all arguments p and q? But- the ab-functions must be introduced as follows: The function- p|q is merely a mechanical instrument for constructing all- possible symbols of ab-functions. The symbols arising by- repeated application of the symbol “|” do not contain the- symbol “p|q”. We need a rule according to which we can form- all symbols of ab-functions, in order to be able to speak of- the class of them; and we now speak of them e.g. as those- symbols of functions which can be generated by repeated- application of the operation “|”. And we say now: For all- p's and q's, “p|q” says something indefinable about the sense- of those simple propositions which are contained in p and q.

 
     

      The assertion-sign is logically quite without significance. - It only shows, in Frege and Whitehead and Russell, that these- authors hold the propositions so indicated to be true. “⊢”- therefore belongs as little to the proposition as (say) the- number of the proposition. A proposition cannot possibly- assert of itself that it is true.

 
     
     Every right theory of judgment must make it impossible- for me to judge that this table penholders the book. Russell's- theory does not satisfy this requirement.

 
     

      It is clear that we understand propositions without knowing- whether they are true or false. But we can only know the- meaning of a proposition when we know if it is true or false. - What we understand is the sense of the proposition.

 
     

      The assumption of the existence of logical objects makes- it appear remarkable that in the sciences propositions of the- form “p ⌵ q”, “p ⊃ q”, etc. are only then not provisional- when “ ⌵ ” and “ ⊃ ” stand within the scope of a generality-sign [apparent variable].
































 
     
4th. MS.

 
     

      If we formed all possible atomic propositions, the world- would be completely described if we declared the truth or- falsehood of each. [I doubt this.]

 
     

      The chief characteristic of my theory is that, in it, p- has the same meaning as not-p.

 
     

      A false theory of relations makes it easily seem as if the- relation of fact and constituent were the same as that of fact- and fact which follows from it. But the similarity of the- two may be expressed thus: φa. ⊃ .φ,a a = a.

 
     

      If a word creates a world so that in it the principles- of logic are true, it thereby creates a world in which the- whole of mathematics holds; and similarly it could not create- a world in which a proposition was true, without creating- its constituents.

 
     

      Signs of the form “p ⌵ ~p” are senseless, but not the- proposition “(p). p ⌵ ~p”. If I know that this rose is- either red or not red, I know nothing. The same holds of all- ab-functions.

 
     

      To understand a proposition means to know what is the case- if it is true. Hence we can understand it without knowing if- it is true. We understand it when we understand its constituents- and forms. If we know the meaning of “a” and “b”, and if we- know what “x R y” means for all x's and y's, then we also understand “a R b”.

 
     

      I understand the proposition “a R b” when I know that either- the fact that a R b or the fact that not a R b corresponds to it;- but this is not to be confused with the false opinion that I- understood “a R b” when I know that “a R b or not a R b” is the- case.

 
     

      But the form of a proposition symbolizes in the following- way: Let us consider symbols of the form “x R y”; to these- correspond primarily pairs of objects, of which one has the- name “x”, the other the name “y”. The x's and y's stand in- various relations to each other, among others the relation R- holds between some, but not between others. I now determine- the sense of “x R y” by laying down: when the facts behave in- regard to “x R y” so that the meaning of “x” stands in the- relation R to the meaning of “y”, then I say that they [the- facts] are “of like sense” [“gleichsinnig”] with the proposition- “x R y”; otherwise, “of opposite sense” [entgegengesetzt”];- I correlate the facts to the symbol “x R y” by thus dividing them- into those of like sense and those of opposite sense. To this- correlation corresponds the correlation of name and meaning. - Both are psychological. Thus I understand the form “x R y”- when I know that it discriminates the behaviour of x and y- according as these stand in the relation R or not. In this- way I extract from all possible relations the relation R, as,- by a name, I extract its meaning from among all possible things.

 
     

      Strictly speaking, it is incorrect to say: we understand- the proposition p when we know that ‘“p” is true’ ≡ p; for- this would naturally always be the case if accidentally the- propositions to right and left of the symbol “≡” were both- true or both false. We require not only an equivalence, but- a formal equivalence, which is bound up with the introduction- of the form of p.

 
     

      The sense of an ab-function of p is a function of the- sense of p.

 
     

      The ab-functions use the discrimination of facts, which- their arguments bring forth, in order to generate new- discriminations.

 
     

      Only facts can express sense, a class of names cannot. - This is easily shown.

 
     

      There is no thing which is the form of a proposition, and- no name which is the name of a form. Accordingly we can also- not say that a relation which in certain cases holds between- things holds sometimes between forms and things. This goes- against Russell's theory of judgment.

 
     
     It is very easy to forget that, though the propositions- of a form can be either true or false, each one of these- propositions can only be either true or false, not both.

 
     

      Among the facts which make “p or q” true, there are some- which make “p and q” true; but the class which makes “p or q”- true is different from the class which makes “p and q” true;- and only this is what matters. For we introduce this class,- as it were, when we introduce ab-functions.

 
     

      A very natural objection to the way in which I have- introduced e.g. propositions of the form x R y is that by it- propositions such as (∃. x. y). x R y and similar ones are not- explained, which yet obviously have in common with a R b what- c R d has in common with a R b. But when we introduce- propositions of the form x R y we mentioned no one particular- proposition of this form; and we only need to introduce- (∃ x, y). φ(x, y) for all φ's in any way which makes the- sense of these propositions dependent on the sense of all- propositions of the form φ(a, b), and thereby the justness- of our procedure is proved.

 
     

      The indefinables of logic must be independent of each other. - If an indefinable is introduced, it must be introduced in all- combinations in which it can occur. We cannot therefore introduce- it first for one combination, then for another; e.g., if the- form x R y has been introduced, it must henceforth be understood- in propositions of the form a R b just in the same way as in- propositions such as (∃x, y). x R y and others. We must not- introduce it first for one class of cases, then for the other;- for it would remain doubtful if its meaning was the same in-
both cases, and there would be no ground for using the same- manner of combining symbols in both cases. In short, for- the introduction of indefinable symbols and combinations of- symbols the same holds, mutatis mutandis, that Frege has said- for the introduction of symbols by definitions.

 
     

      It is a priori likely that the introduction of atomic- propositions is fundamental for the understanding of all other- kinds of propositions. In fact the understanding of general- propositions obviously depends on that of atomic propositions.

 
     

      Cross-definability in the realm of general propositions- leads to quite similar questions to those in the realm- of ab-functions.

 
     

      When we say “A believes p”, this sounds, it is true, as- if here we could substitute a proper name for “p”; but we can- see that here a sense, not a meaning, is concerned, if we say- “A believes that ‘p’ is true”; and in order to make the direction- of p even more explicit, we might say “A believes that ‘p’ is- true and ‘not-p’ is false”. Here the bi-polarity of p is- expressed and it seems that we shall only be able to express- the proposition “A believes p” correctly by the ab-notation;- say by making “A” have a relation to the poles “a”- and “b” of a-p-b.
The epistemological- questions concerning the nature of judgment and belief cannot- be solved without a correct apprehension of the form of the- proposition.

 
     

      The ab-notation shows the dependence of or and not, and- thereby that they are not to be employed as simultaneous- indefinables.

 
     

      Not: “The complex sign ‘a R b’” says that a stands in the- relation R to b; but that ‘a’ stands in a certain relation to- ‘b’ says that a R b.

 
     
     Preliminary In philosophy there are no deductions: it is purely- descriptive.

 
     
     Philosophy gives no pictures of reality.

 
     
     Philosophy can neither confirm nor confute scientific- investigation.

 
     
     Philosophy consists of logic and metaphysics: logic- is its basis.

 
     
     Epistemology is the philosophy of psychology.

 
     
     Distrust of grammar is the first requisite for- philosophizing.

 
     
     Propositions can never be indefinables, for they are always- complex. That also words like “ambulo” are complex appears- in the fact that their root with a different termination gives- a different sense.

 
     

      Only the doctrine of general indefinables permits us to- understand the nature of functions. Neglect of this doctrine- leads to an impenetrable thicket.

 
     
Preliminary
     Philosophy is the doctrine of the logical form of scientific- propositions (not only of primitive propositions).

 
     
     The word “philosophy” ought always to designate something- over or under but not beside, the natural sciences.

 
     

      Judgment, command and question all stand on the same level;- but all have in common the propositional form, which does interest- us.

 
     

      The structure of the proposition must be recognized, the- rest comes of itself. But ordinary language conceals the- structure of the proposition: in it, relations look like- predicates, predicates like names, etc..

 
     
     Facts cannot be named.

 
     

      It is easy to suppose that “individual”, “particular”,- “complex” etc. are primitive ideas of logic. Russell e.g.- says “individual” and “matrix” are “primitive ideas”. This- error presumably is to be explained by the fact that, by- employment of variables instead of the generality-sign it- comes to seem as if logic dealt with things which have been- deprived of all properties except thing-hood, and with- propositions deprived of all properties except complexity. - We forget that the indefinables of symbols [Urbilder von- Zeichen] only occur under the generality-sign, never outside- it.

 
     

      Just as people used to struggle to bring all propositions- into the subject-predicate form, so now it is natural to conceive every proposition as expressing a relation, which is just- as incorrect. What is justified in this desire is fully- satisfied by Russell's theory of manufactured relations.

 
     

      One of the most natural attempts at solution consists in- regarding “not-p” as “the opposite of p”, where then “opposite”- would be the indefinable relation. But it is easy to see- that every such attempt to replace the ab-functions by- descriptions must fail.

 
     

      The false assumption that propositions are names leads us- to believe that there must be logical objects: for the meanings- of logical propositions will have to be such things.

 
     
Preliminary
     A correct explanation of logical propositions must give- them a unique position as against all other propositions.

 
     
     No proposition can say anything about itself, because the- symbol of the proposition cannot be contained in itself; this- must be the basis of the theory of logical types.

 
     

      Every proposition which says something indefinable about- a thing is a subject-predicate proposition; every proposition- which says something indefinable about two things expresses a- dual relation between these things, and so on. Thus every- proposition which contains only one name and one indefinable- form is a subject-predicate proposition, and so on. An-
indefinable simple symbol can only be a name, and therefore- we can know, by the symbol of an atomic proposition, whether- it is a subject-predicate proposition.

 
     
Wittg.–
I. Bi-polarity of propositions: sense & meaning, truth & falsehood.
II. Analysis of atomic propositions: general indefinables, predicates, etc..
III. Analysis of molecular propositions: ab-functions.
IV. Analysis of general propositions.
V. Principles of symbolism: what symbolizes in a symbol. Facts for facts.
VI. Types.












































 
     
This is the symbol for
     ~p ⌵ ~q