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Some Remarks on Logical Form: Difference between revisions
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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. | 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 | 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 fact, in which we say that there is room for one person only in a chair. Our symbolism, which allows us to form the sign of the logical product of "R P T" and "B P T", gives here no correct picture of reality. | ||
I have said elsewhere that a proposition "reaches up to reality", and by this I meant that the forms of the entities are contained in the form of the proposition which is about these entities. For the sentence, together with the mode of projection which projects reality into the sentence, determines the logical form of the entities, just as in our simile a picture on plane II, together with its mode of projection, determines the shape of the figure on plane I. This remark, I believe, gives us the key for the explanation of the mutual exclusion of R P T and B P T. For if the proposition contains the form of an entity which it is about, then it is possible that two propositions should collide in this very form. The propositions, "Brown now sits in this chair" and "Jones now sits in this chair" each, in a sense, try to set their subject term on the chair. But the logical product of these propositions will put them both there at once, and this leads to a collision, a mutual exclusion of these terms. How does this exclusion represent itself in symbolism? We can write the logical product of the two propositions, ''p'' and ''q'', in this way:—(170) | |||
What happens if these two propositions are R P T and B P T? In this case the top line "T T T" must disappear, as it represents an impossible combination. The true possibilities here are— | |||
That is to say, there ''is'' no logical product of R P T and B P T in the first sense, and herein lies the exclusion as opposed to a contradiction. The contradiction, if it existed, would have to be written— | |||
but this is nonsense, as the top line, "T T F," gives the proposition all greater logical multiplicity than that of the actual possibilities. It is, of course, a deficiency of our (171) notation that it does not prevent the formation of such nonsensical constructions, and a perfect notation will have to exclude such structures by definite rules of syntax. These will have to tell us that in the case of certain kinds of atomic propositions described in terms of definite symbolic features certain combinations of the T's and F's must be left out. Such rules, however, cannot be laid down until we have actually reached the ultimate analysis of the phenomena in question. This, as we all know, has not yet been achieved. | |||