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Some Remarks on Logical Form: Difference between revisions
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If we try to analyze any given propositions we shall find in general that they are logical sums, products or other truthfunctions of simpler propositions. But our analysis, if carried far enough, must come to the point where it reaches propositional forms which are not themselves composed of simpler propositional forms. We must eventually reach the ultimate connection of the terms, the immediate connection which cannot be broken without (163) destroying the propositional form as such. The propositions which represent this ultimate connexion of terms I call, after B. Russell, atomic propositions. They, then, are the 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 subject matter of propositions. It is the task of the theory of knowledge to find them and to understand their construction out of the words or symbols. This task is very difficult, and Philosophy has hardly yet begun to tackle it at some points. What method have we for tackling it? The idea is to express in an appropriate symbolism what in ordinary language leads to endless misunderstandings. That is to say, where ordinary language disguises logical structure, where it allows the formation of pseudopropositions, where it uses one term in an infinity of different meanings, we must replace it by a symbolism which gives a clear picture of the logical structure, excludes pseudopropositions, and uses its terms unambiguously. Now we can only substitute a clear symbolism for the unprecise one by inspecting the phenomena which we want to describe, thus trying to understand their logical multiplicity. That is to say, we can only arrive at a correct analysis by, what might be called, the logical investigation of the phenomena themselves, ''i.e.'', in a certain sense ''a posteriori'', and not by conjecturing about ''a priori'' possibilities. One is often tempted to ask from an ''a priori'' standpoint: What, after all, ''can'' be the only forms of atomic propositions, and to answer, ''e.g.'', subject-predicate and relational propositions with two or more terms further, perhaps, propositions relating predicates and relations to one another, and so on. But this, I believe, is mere playing with words. An atomic form cannot be foreseen. And it would be surprising if the actual (164) phenomena had nothing more to teach us about their structure. To such conjectures about the structure of atomic propositions, we are led by our ordinary language, which uses the subject-predicate and the relational form. But in this our language is misleading: I will try to explain this by a simile. Let us imagine two parallel planes, I and II. On plane I figures are drawn, say, ellipses and rectangles of different sizes and shapes, and it is our task to produce images of these figures on plane II. Then we can imagine two ways, amongst others, of doing this. We can, first, lay down a law of projection—say that of orthogonal projection or any other—and then proceed to project all figures from I into II, according to this law. Or, secondly, we could proceed thus: We lay down the rule that every ellipse on plane I is to appear as a circle in plane II, and every rectangle as a square in II. Such a way of representation may be convenient for us if for some reason we the prefer to draw only circles and squares on plane II. Of course, from these images the exact shapes of the original figures on plane I cannot be immediately inferred. We can only gather from them that the original was an ellipse or a rectangle. In order to get in a single instance at the determinate shape of the original we would have to know the individual method by which, ''e.g.'', a particular ellipse is projected into the circle before me. The case of ordinary language is quite analogous. If the facts of reality are the ellipses and rectangles on plane I the subject-predicate and relational forms correspond to the circles and squares in plane II. These forms are the norms of our particular language into which we project in ''ever so many different'' ways ''ever so many different'' logical forms. And for this very reason we can draw no conclusions except very vague ones from the use of these (165) norms as to the actual logical form of the phenomena described. Such forms as "This paper is boring", "The weather is fine", "I am lazy", which have nothing whatever in common with one another, present themselves as subject-predicate propositions, ''i.e.'', apparently as propositions of the same form. | If we try to analyze any given propositions we shall find in general that they are logical sums, products or other truthfunctions of simpler propositions. But our analysis, if carried far enough, must come to the point where it reaches propositional forms which are not themselves composed of simpler propositional forms. We must eventually reach the ultimate connection of the terms, the immediate connection which cannot be broken without (163) destroying the propositional form as such. The propositions which represent this ultimate connexion of terms I call, after B. Russell, atomic propositions. They, then, are the 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 subject matter of propositions. It is the task of the theory of knowledge to find them and to understand their construction out of the words or symbols. This task is very difficult, and Philosophy has hardly yet begun to tackle it at some points. What method have we for tackling it? The idea is to express in an appropriate symbolism what in ordinary language leads to endless misunderstandings. That is to say, where ordinary language disguises logical structure, where it allows the formation of pseudopropositions, where it uses one term in an infinity of different meanings, we must replace it by a symbolism which gives a clear picture of the logical structure, excludes pseudopropositions, and uses its terms unambiguously. Now we can only substitute a clear symbolism for the unprecise one by inspecting the phenomena which we want to describe, thus trying to understand their logical multiplicity. That is to say, we can only arrive at a correct analysis by, what might be called, the logical investigation of the phenomena themselves, ''i.e.'', in a certain sense ''a posteriori'', and not by conjecturing about ''a priori'' possibilities. One is often tempted to ask from an ''a priori'' standpoint: What, after all, ''can'' be the only forms of atomic propositions, and to answer, ''e.g.'', subject-predicate and relational propositions with two or more terms further, perhaps, propositions relating predicates and relations to one another, and so on. But this, I believe, is mere playing with words. An atomic form cannot be foreseen. And it would be surprising if the actual (164) phenomena had nothing more to teach us about their structure. To such conjectures about the structure of atomic propositions, we are led by our ordinary language, which uses the subject-predicate and the relational form. But in this our language is misleading: I will try to explain this by a simile. Let us imagine two parallel planes, I and II. On plane I figures are drawn, say, ellipses and rectangles of different sizes and shapes, and it is our task to produce images of these figures on plane II. Then we can imagine two ways, amongst others, of doing this. We can, first, lay down a law of projection—say that of orthogonal projection or any other—and then proceed to project all figures from I into II, according to this law. Or, secondly, we could proceed thus: We lay down the rule that every ellipse on plane I is to appear as a circle in plane II, and every rectangle as a square in II. Such a way of representation may be convenient for us if for some reason we the prefer to draw only circles and squares on plane II. Of course, from these images the exact shapes of the original figures on plane I cannot be immediately inferred. We can only gather from them that the original was an ellipse or a rectangle. In order to get in a single instance at the determinate shape of the original we would have to know the individual method by which, ''e.g.'', a particular ellipse is projected into the circle before me. The case of ordinary language is quite analogous. If the facts of reality are the ellipses and rectangles on plane I the subject-predicate and relational forms correspond to the circles and squares in plane II. These forms are the norms of our particular language into which we project in ''ever so many different'' ways ''ever so many different'' logical forms. And for this very reason we can draw no conclusions except very vague ones from the use of these (165) norms as to the actual logical form of the phenomena described. Such forms as "This paper is boring", "The weather is fine", "I am lazy", which have nothing whatever in common with one another, present themselves as subject-predicate propositions, ''i.e.'', apparently as propositions of the same form. | ||
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 | 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 multiplicity 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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