Relations in Wittgenstein's Tractatus

Wittgenstein says at 3.1432 of Tractatus that

Instead of, 'The complex sign "aRb" says that a stands to b in the relation R' we ought to put, 'That "a" stands to "b" in a certain relation says that aRb.'

It seems scarcely comprehensible at first sight, especially for those who are familiar with set theory. In mathematics "aRb" really says that a stands to b in the relation R. It is because the world of mathematics is the totality of "such" propositions, while the (real) world is all that is the case as Wittgenstein says at the very beginning of Tractatus. We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation (3.11). The distinction is more clear if we consider spatial relation as in 3.1431. All because of such projection, we can say, for example, geometry is a study of spatial relations.

Nevertheless, no matter relations in mathematics or those in Tractatus, they should be well-defined, in the sense that we can tell at the same time what objects are in such relations (which pairs of objects are / are not in such binary relations), and assigns truth values to the objects (to the pairs of objects for binary relations).

Therefore Wittgenstein says at 5.42 that Frege's and Russell's "primitive signs" of logic such as ∨ (disjunction) and → (implication) are not signs for relations. In the first place, such logical signs did not give truth values. A proposition like p→q just tells you something (if p then q), but does not tell you that something (some relation) is true. They belongs to syntax. However, we can actually define relations on all propositions such that, for example, pRq if it is the case that p→q. These relations belongs to semantics.

Besides, Wittgenstein also says at 5.42 that Frege's and Russell's "primitive signs" of logic are not even primitive signs. Primitive signs are names that cannot be analysed further by any definition (3.26), while Frege and Russell even tried to define these logical signs. In fact, "well-definedness" is also a requirement of logical signs in Tractatus (5.46). If a sign is not primitive, we should be able to analyse it further by definition at the same time for all combinations with other well-defined signs (including brackets). If a sign was primitive, we should have introduced (primitive signs cannot be defined) the sense of all combinations with other signs. Therefore, there are no primitive logical sign in Tractatus. The real general primitive signs are the most general form of their combinations (5.46). At 5.461, Wittgenstein further comments that such pseudo-relations of logic need brackets, which is an indication that they are not primitive signs. For example, (p→q)→r and p→(q→r) are 2 different propositions, in which brackets are necessary. More precisely, it is an indication that they alone are "sometimes" even not logical signs, instead of an indication that the logical signs are not primitive. In fact, we can always write (p→q) instead of p→q in Frege's and Russell's notation. In this case, the pair of brackets and → form the logical sign of implication. However, is such a logical sign primitive?

Jan Łukasiewicz (1878 – 1956) has introduced a bracketless notation, in which, for example, Cpq is written instead of p→q. In that case, (p→q)→r and p→(q→r) will be expressed as CCpqr and CpCqr respectively. C is not a sign of relation yet because it does not give any truth value. Nevertheless, because of 5.461, we still wonder if C is a primitive logical sign. It depends.

Many systems of classical logic developed after Wittgenstein's Tractatus really introduced, but not defined, two (primitive) logical signs such as the negation and implication, and then defined all other logical signs such as disjunction, conjunction and biconditional. The syntax of the propositions (formulas) were studied.

Comparing (p→q) with Cpq in Łukasiewicz's notation, we can see immediately that the "aggregation" done by brackets is done by the rule for reading an expression containing the sign C. No matter which notation is used, the logical signs are just used to clarify the meaning. They belongs to syntax. As Wittgenstein concludes at 5.461, signs for logical operations (including those of Łukasiewicz) are just punctuation-marks.