What Would A Tractarian Calculus Look Like?

This is another post I wrote while not finishing up my post about the notion of sense. The use Wittgenstein makes of the notions of the sense of a proposition and the form of representation of a picture is to undercut what might be called ‘the problem of mental representation’: how does this thought here in my head represent that thing out there in the world?  This is a core problem in the history of philosophy and W rejects the setup that leads to it. I think seeing that and how he does that is critical to understanding his philosophy of logic and with it the questions we started out with about how to understand the book as a whole.

But I’m not going to talk about that today. Instead I want to reflect on how we would set up a system of formal logic Tractatus-style.

Wittgenstein does make recommendations, for instance at 4.1273: since “the concept symbolized by ‘term of this formal series’ is a formal concept,” “the general term of a formal series can only be expressed by a variable,” and thus the way “Frege and Russell…express general propositions like [a stands in some ancestral of the successor relation to b] is false; it contains a vicious circle.” In this passage we see Wittgenstein’s belief that logical statements are meaningless applied: a general abstract description (or ‘description at the meta-level’) of the ancestral of the successor relation, according to him, asserts nothing at all, because its objects are not objects at all, but variables, formal concepts.

Wittgenstein’s solution to this is to suggest that what are called ‘inductive definitions’ in mathematics are a better approach to the general term of a formal series: we give “its first term and the general form of the operation, which generates the following term out of the preceding proposition.” And then when he comes to the general form of propositions in 6, he does just this, giving a rule for generating whatever propositions you like by repeated applications of the Sheffer stroke to the elementary propositions. But you don’t say that any of those propositions ‘exist’, or quantify over them; you just give a rule for generating them which can be applied as many times as you need. This means that you never have to talk about them; you just provide a procedure for iterating as many as you need.

We wouldn’t quantify over propositions at the meta-level in a Tractarian calculus, it seems.

I think it’s also an interesting question how we would deal with what are called, in ordinary logic, predicates and relation-terms. It seems likely to me that e.g. “that ball is red” would be most perspicuously symbolized not, as we normally learn, “Rb,” but as something like f(location, ball, red). Remember that “the elementary proposition consists of names” (4.22, 5.55) – and so “The elementary proposition I write as function of the names, in the form “fx”, “f(x,y)”, etc. Or I indicate it by the letters p, q, r.” (4.24)

Consider by contrast “the cat is on the mat.” One can debate here whether “being-on” is the relation that the cat stands in to the mat, or whether the sentence as a whole asserts that a particular cat, being-on, and a particular mat are organized in a certain way. I have some inclination, though, to think that f(particular cat, being-on, particular mat) is more perspicuous, in the sense that logic is supposed to abstract from all structure, even spatial structure, and that being-on is a matter of a particular kind of spatial structure (perhaps spatial-gravitational or spatial-perceptual as well, but leave that aside). Furthermore, it is something that we verify by looking – we see whether a thing is on something else or not – so in that sense “being on” is just as much something that we can project on to reality as part of a sensible proposition as being Mitten the Kitten is.

So too with “redness” – it is something manifest, there along with the ball so to speak. Is it therefore an ‘object’? Why not? All this word signifies in the Tractatus, remember, is a formal concept – a variable for names. (So now – go back to 1-2 and consider the fact as consisting of objects – two plums in the icebox. To say that “facts consist of objects” is really to say nothing at all – it is a senseless – but on the other hand, there are two plums in the icebox, and if you understand it, you know which plums and icebox are meant.)

We could put arbitary physical entities in the same spatial arrangement, but we could also put the same physical entities in arbitary spatial arrangements. Since we can make any part of a proposition into a variable, and the parts of propositions are objects, it seems to me to follow from this that any as it were ‘abstractable feature’ ought to be an object in the technical sense of the Tractatus. So that if we are going to think of f(a,b,c…) as a proposition, where f is what it asserts about its constituents, then there is a sense in which all the significant stuff, which is to say anything that we can ‘abstract over’ and thus ‘turn into a variable’, needs to be among the ‘names’, and the f should be only be their logical structure, so to speak.

The difficulty with that suggestion, however, is that logical structure is in a way just a shadow of spatial, temporal, social, etc. structure – there isn’t any ‘logical structure’ – ‘logical structure’ is a formal concept in which e.g. spatial structure is turned into a variable. (Another reason why this stuff I’m saying right here in this post is meaningless.)

So it’s just a list of names? “The cat is on the mat” analyzed as the cat, being-on, the mat?

Well, no – there’s a connection between these names – that’s the sense of “the cat is on the mat” and it shows itself if you understand what someone who says “the cat is on the mat” is saying. But you can’t represent that sense. The sense is what you understand when you understand what the proposition says. On the other hand, you can represent spatial relatedness, by exhibiting the same relations with different entities standing in them to one another. So plausibly, a spatial relation is something named in a proposition, an object if you like, and not the ineffable structure of the proposition which ‘fact-izes’ those objects together.

I realize that I am here picking up a thread of thought that came out of long-ago discussions with Lynette. In a paper of hers you can check out here:


she is reflecting on, upon other things, the notion of a ‘category clash’ and says (pp. 122-3):

“One of the anti-metaphysical stands of the Tractatus is that logic cannot judge in advance what the internal articulation of fully analyzed propositions will be: contrary to Frege and Russell, who think it essential to the nature of representation that a proposition segment into subject and predicate of some sort, the Tractatus denies that there is any point in discussing in advance whether elementary propositions will consist of names and concept-expressions, or n-termed relation-expressions, or anything else. The only interest logic takes in the internal composition of propositions is that they contribute to what we ask of the world in determining whether propositions are true or false; logic only interests itself in this because this is how they contrast with logical constants, and confusing the two is the primary error that gives birth to metaphysics.”

This is on the right track, but I wonder if we shouldn’t maybe say something even more severe: that in effect any system of categories is just going to produce what is in effect a list of names from a logical point of view. (So you might say something like: the details of structure never matter to logic, only the fact that things are structured.)

Or to go back to Frege for a minute – it’s not that we have object-terms and concept-terms with holes for them; rather we have object terms, and the proposition itself is ‘the system of holes’ into which those object-terms are dropped.

(Also, going back to the original question: does anyone know of old dissertations floating around out there which make plausible stabs at ‘the logical system of the Tractatus’)?


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