wittgenstein_some remarks on logical form

8/8/2019 WITTGENSTEIN_Some Remarks on Logical Form

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Some Remarks on Logical Form

Author(s): L. WittgensteinSource: Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 9, Knowledge,Experience and Realism (1929), pp. 162-171Published by: Blackwell Publishing on behalf of The Aristotelian SocietyStable URL: http://www.jstor.org/stable/4106481

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SOME REMARKS ON LOGICAL FORM.

By L. WITTGENSTEIN.Every propositionhas a content and a form. We getthe picture of the pure form if we abstract from the mean-

ing of the single words, or symbols (so far as they haveindependent meanings). That is to say, if we substitutevariablesfor the constants of the proposition. The rules ofsyntax which applied to the constants must apply to thevariablesalso. By syntax in this general sense of the wordI mean the rules which tell us in which connections onlya word gives sense, thus excluding nonsensical structures.The syntax of ordinarylanguage, as is well known, is notquite adequate for this purpose. It does not in all casesprevent the construction of nonsensical pseudopropositions(constructions such as " red is higher than green " or " theReal, though it is an in itself, must also be able to becomea for myself ", etc.).If we try to analyze any given propositionswe shall findin general that they are logical sums, products or othertruthfunctions of simpler propositions. But our analy-sis, if carried far enough, must come to the pointwhere it reaches propositionalforms which are not them-selves composedof simpler propositionalforms. We musteventually reach the ultimate connection of the terms, theimmediate connection which cannot be broken without


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LOGICAL FORM. 183

destroying the propositional form as such. The proposi-tions which represent this ultimate connexion of termsI call, after B. Russell, atomicpropositions. They, then, arethe kernels of every proposition, they contain the material,and all the rest is only a development of this material.It is to them we have to look for the subjectmatter of pro-positions. It is the task of the theory of knowledgeto findthem and to understandtheir constructionout of the wordsor symbols. This task is very difficult, and Philosophyhas hardly yet begun to tackle it at some points. Whatmethod have we for tackling it ? The idea is to express inan appropriatesymbolism what in ordinarylanguage leadsto endless misunderstandings. That is to say, whereordinary language disguises logical structure, where itallows the formation of pseudopropositions, where ituses one term in an infinity of differentmeanings, we mustreplace it by a symbolism which gives a clear picture ofthe logical structure, excludes pseudopropositions,and usesits terms unambiguously. Now we can only substitute aclear symbolism for the unprecise one by inspecting thephenomena which we want to describe, thus trying tounderstand their logical multiplicity. That is to say, wecan only arrive at a correctanalysisby,what might be called,the logical investigation of the phenomenathemselves, i.e.,in a certain sense a posteriori, and not by conjecturingabout a priori possibilities. One is often tempted to askfrom an a priori standpoint: What, after all, can be theonly forms of atomic propositions, and to answer, e.g.,subject-predicateand relational propositions with two ormore termsfurther, perhaps, propositionsrelatingpredicatesand relationsto one another, and so on. But this, I believe,is mere playing with words. An atomic form cauuot beforeseen. And it would be surprising if the actual

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164 F. WITTGENSTEIN.

phenomena had nothing more to teach us about theirstructure. To such conjectures about the structure ofatomic propositions,we are led by our ordinary language,which uses the subject-predicateand the relational form.But in this our languageis misleading: I will try to explainthis by a simile. Let us imagine two parallel planes,I and II. On plane I figures are drawn, say, ellipses andrectangles of different sizes and shapes, and it is our taskto produce images of these figures on plane II.Then we can imagine two ways, amongst others, of doingthis. We can, first, lay down a law of projection-say that of orthogonal projection or any other-and thenproceedto project all figures from I into II, accordingtothis law. Or, secondly, we could proceed thus: We laydown the rule that every ellipse on plane I is to appear asa circle in plane II, and every rectangle as a squarein II.Such a way of representationmay be convenientfor us if forsome reason we prefer to draw only circles and squareson plane II. Of course, from these images the exact shapesof the original figures on plane I cannot be immediatelyinferred. We can only gather from them that the originalwas an ellipse or a rectangle. In order to get in a singleinstance at the determinateshape of the originalwe wouldhave to know the individual method by which, e.g.,a particularellipse is projected into the circle before me.The case of ordinary language is quite analogous. If thefacts of reality are the ellipses and rectangles on plane Ithe subject-predicateand relational forms correspond tothe circles and squares in plane II. These forms are thenorms of our particularlanguage into which we project inever so many differentways ever so many differentlogicalforms. And for this very reason we can draw no con-clusions-except very vague ones-from the use of these


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LOGICAL FORM. 165

norms as to the actual logical form of the phenomena de-scribed. Such forms as " This paper is boring ", " Theweather is fine ", " I am lazy ", which have nothing what-ever in common with one another, present themselves assubject-predicate propositions, i.e., apparently as propo-sitions of the same form.

If, now, we try to get at an actual analysis, we findlogical forms which have very little similarity with thenorms of ordinary language. We meet with the forms ofspace and time with the whole manifold of spacial andtemporal objects, as colours, sounds, etc., etc., with theirgradations, continuous transitions, and combinations invarious proportions, all of which we cannot seize byour ordinary means of expression. And here I wish tomake my first definite remark on the logical analysis ofactual phenomena: it is this, that for their representationnumbers (rationaland irrational)must enter into the struc-ture of the atomic propositions themselves. I will illus-trate this by an example. Imagine a system of rectangularaxes, as it were, cross wires, drawn in our field of visionand an arbitraryscale fixed. It is clear that we then candescribe the shape and position of every patch of colour inour visual field by means of statements of numbers whichhave their significance relative to the system of co-ordi-nates and the unit chosen. Again, it is clear that thisdescriptionwill have the right logical multiplicity, and thata descriptionwhich has a smaller multiplicity will not do.A simple example would be the representationof a patchP by the expression " [6-9, 3---8] " and of a proposition


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10-9-8-7-6-5-----4-3-2-1-

I 1 l l I I IIl

about it, e.g., P is red, by the symbol " [6-9, 3--8] R ",where " R " is yet an unanalyzed term (" 6--9 " and" 3-8- " stand for the continuous interval between the re-spective numbers). The system of co-ordinateshere is partof the mode of expression;it is part of the methodof projec-tion by which the reality is projectedinto our symbolism.The relation of a patch lying between two others can be ex-pressed analogously by the use of apparent variables. Ineed not say that this analysis does not in any way pretendto be complete. I have made no mention in it of time,and the use of two-dimensionalspace is not justified evenin the case of monocularvision. I only wish to point outthe direction in which, I believe, the analysis of visualphenomena is to be looked for, and that in this analysiswe meet with logical forms quite differentfrom those whichordinary language leads us to expect. The occurrenceofnumbers in the forms of atomic propositions is, in myopinion, not merely a feature of a special symbolism, butan essential and, consequently, unavoidable feature of therepresentation. And numberswill have to enter these formswhen-as we should say in ordinary language-we are


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LOGICAL FORM. 167

dealing with propertieswhich admit of gradation, i.e., pro-perties as the length of an interval, the pitch of a tone,the brightness or redness of a shade of colour, etc. It isa characteristic of these propertiesthat one degree of themexcludes any other. One shade of colour cannot simul-taneously have two different degrees of brightness or red-ness, a tone not two different strengths, etc. And theimportant point here is that these remarks do not expressan experience but are in some sense tautologies. Everyone of us knows that in ordinarylife. If someone asks us" What is the temperature outside? " and we said" Eighty degrees ", and now he were to ask us again," And is it ninety degrees? " we should answer, " I toldyou it was eighty." We take the statement of a degree(of temperature, for instance) to be a complete descriptionwhich needs no supplementation. Thus, when asked, wesay what the time is, and not also what it isn't.One might think--and I thought so not long ago-that a statement expressing the degree of a quality couldbe analyzed into a logical product of single statements ofquantity and a completing supplementarystatement. AsI could describe the contents of my pocket by say-ing " It contains a penny, a shilling two keys, andnothing else ". This " and nothing less " is the supple-mentary statement which completes the description. Butthis will not do as an analysis of a statement of degree.For let us call the unit of, say, brightness b and let E(b)be the statement that the entity E possessesthis brightness,then the proposition E(2b), which says that E has twodegrees of brightness, should be analyzable into the logicalproduct E(b) & E(b), but this is equal to E(b); if, onthe other hand, we try to distinguish between the units andconsequently write E(2b) = E(b') & E(b"), we assume


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two different units of brightness; and then, if an entitypossesses one unit, the question could arise, which of thetwo- b' or b"- it is; which is obviously absurd.I maintain that the statement which attributes a degreeto a qualitycannot further be analyzed, and, moreover,thatthe relation of difference of degree is an internal relationand that it is therefore represented by an internal relationbetween the statements which attribute the differentdegrees. That is to say, the atomic statement must havethe same multiplicity as the degree which it attributes,whence it follows that numbers must enter the forms ofatomic propositions. The mutual exclusion of unanalyzablestatements of degreecontradicts an opinionwhich was pub-lished by me several years ago and which necessitated thatatomic propositionscould not exclude one another. I heredeliberately say " exclude " and not " contradict ", forthere is a differencebetween these two notions, and atomicpropositions, although they cannot contradict, may ex-clude one another. I will try to explain this. There arefunctions which can give a true proposition only for onevalue of their argumentbecause-if I may so expressmyself-there is only room in them for one. Take, for instance,a proposition which asserts the existence of a colour R ata certain time T in a certain place P of our visual field.I will write this proposition "R P T ", and abstract forthe moment from any consideration of how such a state-ment is to be furtheranalyzed. " B P T ", then, says thatthe colour B is in the place P at the time T, and it willbe clear to most of us here, and to all of us in ordinarylife, that " R P T & B P T " is some sort of contradiction(and not merely a false proposition). Now if statementsof degree were analyzable-as I used to think--we couldexplain this contradictionby saying that the colourR con-


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LOGICAL FORM. 169tains all degrees of R and none of B and that the colourB contains all degrees of B and none of R. But from theabove it follows that no analysis can eliminate statementsof degree. How, then, does the mutual exclusion ofR P T and B P T operate? I believe it consists in the factthat R P T as well as B P T are in a certain sense complete.That which corresponds in reality to the function"

( ) P T " leaves room only for one entity-in the samesense, in fact, in which we say that there is room for oneperson only in a chair. Oursymbolism, which allows us toform the sign of the logical product of "R P T " and"B P T " gives here no correctpictureof reality.

I have said elsewhere that a proposition " reaches upto reality ", and by this I meant that the forms ofthe entities are contained in the form of. the propositionwhich is about these entities. For the sentence, togetherwith the mode of projectionwhich projectsreality into thesentence, determines the logical form of the entities, justas in our simile a picture on plane II, together with itsmode of projection, determines the shape of the figure onplane I. This remark, I believe, gives us the key for theexplanation of the mutual exclusion of R P T and B P T.For if the propositioncontains the form of an entity whichit is about, then it is possible that two propositionsshouldcollide in this very form. The propositions, " Brown nowsits in this chair " and " Jones now sits in this chair " each,in a sense, try to set their subject term on the chair. Butthe logical productof these propositionswill put them boththere at once, and this leads to a collision, a mutual exclu-sion of these terms. How does this exclusion representitself in symbolism? We can write the logical product ofthe two propositions,p and q, in this way :-


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170 P. WITTGENSTEIN.

Pj9qrIT T 'I'T F FF T FF F F

What happens if these two propositions are R P T andB P T? In this case the top line " T T T " must disap-pear, as it representsan impossiblecombination. The truepossibilitieshere are-

RPT BPTT FF TF F

That is to say, there is no logical product of R P T andB P T in the first sense, and herein lies the exclusion asopposedto a contradiction. The contradiction,if it existed,would have to be written-

RPTIBPTT T FT F FF T FF F F

but this is nonsense, as the top line, " TT F," gives theproposition a greater logical multiplicity than that of theactual possibilities. It is, of course, a deficiency of our


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LOGICAL FORM. 171notation that it does not prevent the formationof such non-sensical constructions, and a perfect notation will have toexclude such structuresby definite rules of syntax. Thesewill have to tell us that in the case of certainkinds of atomicpropositionsdescribedin terms of definite symbolic featurescertain combinations of the T's and F's must be left out.Such rules, however, cannot be laid down until we haveactually reached the ultimate analysis of the phenomena inquestion. This, as we all know, has not yet been achieved.

wittgenstein_some remarks on logical form

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