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Some Remarks on Logical Form: Difference between revisions
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If, now, we try to get at an actual analysis, we find logical forms which have very little similarity with the norms of ordinary language. We meet with the forms of space and time with the whole manifold of spatial and temporal objects, as colours, sounds, etc., etc., with their gradations, continuous transitions, and combinations in various proportions, all of which we cannot seize by our ordinary means of expression. And here I wish to make my first definite remark on the logical analysis of actual phenomena: it is this, that for their representation numbers (rational and irrational) must enter into the structure of the atomic propositions themselves. I will illustrate this by an example. Imagine a system of rectangular axes, as it were, cross wires, drawn in our field of vision and an arbitrary scale fixed. It is clear that we then can describe the shape and position of every patch of colour in our visual field by means of statements of numbers which have their significance relative to the system of co-ordinates and the unit chosen. Again, it is clear that this description will have the right logical multiplicity, and that a description which has a smaller multi plicity will not do. A simple example would be the representation of a patch P by the expression “[6–9, 3–8]” and of a proposition (166) | If, now, we try to get at an actual analysis, we find logical forms which have very little similarity with the norms of ordinary language. We meet with the forms of space and time with the whole manifold of spatial and temporal objects, as colours, sounds, etc., etc., with their gradations, continuous transitions, and combinations in various proportions, all of which we cannot seize by our ordinary means of expression. And here I wish to make my first definite remark on the logical analysis of actual phenomena: it is this, that for their representation numbers (rational and irrational) must enter into the structure of the atomic propositions themselves. I will illustrate this by an example. Imagine a system of rectangular axes, as it were, cross wires, drawn in our field of vision and an arbitrary scale fixed. It is clear that we then can describe the shape and position of every patch of colour in our visual field by means of statements of numbers which have their significance relative to the system of co-ordinates and the unit chosen. Again, it is clear that this description will have the right logical multiplicity, and that a description which has a smaller multi plicity will not do. A simple example would be the representation of a patch P by the expression “[6–9, 3–8]” and of a proposition (166) | ||
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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 in terval between the respective numbers). The system of co-ordinates here is part of the mode of expression; it is part of the method of projection by which the reality is projected into our symbolism. The relation of a patch lying between two others can be expressed analogously by the use of apparent variables. I need not say that this analysis does not in any way pretend to be complete. I have made no mention in it of time, and the use of two-dimensional space is not justified even in the case of monocular vision. I only wish to point out the direction in which, I believe, the analysis of visual phenomena is to be looked for, and that in this analysis we meet with logical forms quite different from those which ordinary language leads us to expect. The occurrence of numbers in the forms of atomic propositions is, in my opinion, not merely a feature of a special symbolism, but an essential and, consequently, unavoidable feature of the representation. And numbers will have to enter these forms when—as we should say in ordinary language—we are (167) dealing with properties which admit of gradation, ''i.e.'', properties as the length of an interval, the pitch of a tone, the brightness or redness of a shade of colour, etc. It is a characteristic of these properties that one degree of them excludes any other. One shade of colour cannot simultaneously have two different degrees of brightness or redness, a tone not two different strengths, etc. And the important point here is that these remarks do not express an experience but are in some sense tautologies. Every one of us knows that in ordinary life. 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 told you it was eighty." We take the statement of a degree (of temperature, for instance) to be a ''complete'' description which needs no supplementation. Thus, when asked, we say 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 could be analyzed into a logical product of single statements of quantity and a completing supplementary statement. As I could describe the contents of my pocket by saying "It contains a penny, a shilling, two keys, and nothing else". This "and nothing less" is the supplementary statement which completes the description. But this 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 possesses this brightness, then the proposition E(2''b''), which says that E has two degrees of brightness, should be analyzable into the logical product E(''b'') & E(''b''), but this is equal to E(''b''); if, on the other hand, we try to distinguish between the units and consequently write E(2''b'') = E(''b''<nowiki/>') & E(''b''"), we assume (168) two different units of brightness; and then, if an entity possesses one unit, the question could arise, which of the two—''b''<nowiki/>' or ''b''"—it is; which is obviously absurd. | |||
I maintain that the statement which attributes a degree to a quality cannot further be analyzed, and, moreover, that the relation of difference of degree is an internal relation and that it is therefore represented by an internal relation between the statements which attribute the different degrees. That is to say, the atomic statement must have the same multiplicity as the degree which it attributes, whence it follows that numbers must enter the forms of atomic propositions. The mutual exclusion of unanalyzable statements of degree contradicts an opinion which was published by me several years ago and which necessitated that atomic propositions could not exclude one another. I here deliberately say "exclude" and not "contradict", for there is a difference between these two notions, and atomic propositions, although they cannot contradict, may exclude one another. I will try to explain this. There are functions which can give a true proposition only for one value of their argument because—if I may so express myself—there is only room in them for one. Take, for in stance, a proposition which asserts the existence of a colour R at a certain time T in a certain place P of our visual field. I will write this proposition "R P T", and abstract for the moment from any consideration of how such a statement is to be further analyzed. “B P T”, then, says that the colour B is in the place P at the time T, and it will be clear to most of us here, and to all of us in ordinary life, that "R P T & B P T" is some sort of contradiction (and not merely a false proposition). Now if statements of degree were analyzable—as I used to think—we could explain this contradiction by saying that the colour R con-(169)tains all degrees of R and none of B and that the colour B contains all degrees of B and none of R. But from the above it follows that no analysis can eliminate statements of degree. How, then, does the mutual exclusion of R P T and B P T operate? I believe it consists in the fact that R P T as well as B P T are in a certain sense complete. That which corresponds in reality to the funct tion "( ) P T" leaves room only for one entity—in the same sense, in | |||