Given two propositions, "p" and "q," how do we combine them to form a new proposition, "p.q"? Wittgenstein calls the process by which one proposition is generated out of one or more, "base," propositions an "operation." An operation combining elementary propositions in a truth- function is a truth-operation. The structures of all propositions stand in internal relations to one another (5.2), and the business of an operation is to express the relation that stands between the structure of the base proposition and the structure of the resulting proposition (5.22). As such, an operation is not a form or object in its own right; it simply expresses the difference between the forms of two propositions (5.241).
The same operation can be applied successively to produce a series of propositions. Sometimes, as in the case of "not," this procedure can cancel itself out. Apply that operation to "p" and we get "~p," but apply it a second time, and "~p" becomes "~ ~ p," which is equivalent to "p." In other cases, we can produce an infinite series of different propositions by the repeated application of the same operation. All propositions can be generated from successive truth-operations performed upon elementary propositions (5.3).
Since an operation expresses the relation that exists between a proposition and its bases, there cannot be more than one operation expressing the same relationship. Suppose we were claimed that there were two different operations that combine "p" and "q" to form "p.q." The fact of the matter is that the two operations would express the same relation between these three propositions, so they would be effectively identical.
Wittgenstein concludes there is something fundamentally flawed about the "logical objects" or "logical constants" of Frege's and Russell's systems (5.4). Frege builds his entire system from the "primitive" connective "not" and "if then." Russell builds his from "not" and "or." These "primitive" connectives are in fact interchangeable (Frege's "if p then q" can be expressed in Russell's system as "q or not p," and Russell's "p or q" can be expressed by Frege's "if not p then q"). If the same proposition can be expressed in a handful of different ways, there is nothing fundamental about the "logical objects"—such as "or," "if then," and "not"—that are used to express the connections in these propositions (5.42).
Wittgenstein also moves against Frege's and Russell's notion that logic is a set of propositions derived from a few elementary propositions. How is it, for instance, that from "p" we can derive an infinite number of further propositions: "~ ~ p," "~ ~ ~ ~ p," and so on? How can a few elementary propositions imply an infinite number of further "propositions of logic"? "In fact," Wittgenstein replies, "all the propositions of logic say the same thing, to wit nothing" (5.43). These further propositions do not tell us anything we didn't already know.
Logic is utterly general and utterly simple. There cannot be a hierarchy of primitive propositions of logic from which other propositions are then derived. Nor can there be multiple ways of expressing the relations that exist between propositions.
Wittgenstein is criticizing Frege's and Russell's "universalist" conception of logic, which defines logic as a supremely general set of laws in the form of propositions. Just as the laws of chemistry pertain to all chemical interactions and the laws of physics pertain to all natural phenomena, the laws of logic pertain to everything, including other laws and themselves. These laws dictate the form other sets of laws can take. We can imagine there might be physical laws other than the ones we have (for instance, it is conceivable that massive bodies might repel one another), but we cannot imagine physical laws that are illogical. For instance, that "if p then q" combined with "p" implies that q is a fact that can be applied to any two propositions p and q, whether they pertain to particle physics or to gardening. The laws of logic determine the structure of everything else that is, and that is why logic is prior to psychology, metaphysics, and all else. According to the universalist conception, logic is, essentially, the "laws of rationality." Any set of propositions that obey the laws of logic is rational, and any set of propositions that does not is irrational.
The universalist conception takes logic to be an axiomatic system, consisting of certain fundamental axioms, certain logical objects or connectives, and certain laws of inference. That is, there are certain fundamental axioms (like "if 'if p then q' and 'p' then 'q'") made up of certain fundamental objects (like "and" and "if then") that are self-evidently true. There are then some fundamental laws of inference that tell us how we can deduce a new proposition from those that are given to us. These laws of inference can then deduce all the propositions of logic from the fundamental axioms.
We have already seen, in 5.11–5.132, that Wittgenstein criticizes the universalist notion of laws of inference. Here, his attack is more directed toward the notions of fundamental axioms and logical objects. He asserts that "all the propositions of logic say the same thing, to wit nothing" (5.43). According to Frege's and Russell's axiomatic systems, we can deduce further propositions of logic from the basic axioms. For instance, "p v ~ ~ ~ p" is not itself an axiom, but it follows from the axiom "p v ~p," so its truth is assured and it counts as a proposition of logic.
Wittgenstein would counter this kind of reasoning by reference to his truth- table notation at 4.31 and 4.442. Both propositions say the same thing: "(TT)(p)," so "p v ~ ~ ~ p" cannot be said to be a further proposition that is deduced from an axiom. Rather, they are the same proposition (they express the same sense) written in two different ways. Further, we can see that they are both tautologies (they are true no matter what is the case), and, as Wittgenstein points out at 5.142, a tautology says nothing. Thus, both of these propositions, and indeed, all the propositions of logic, say the same thing: nothing. Frege and Russell are mistaken in thinking that there are multiple logical axioms and infinite logical propositions, since all these propositions and axioms are equivalent.
Effectively, Wittgenstein is trying to dissociate the importance of notation from logic itself. All that is essential to a proposition is its sense. If "p.~q" expresses the same sense as "~(q v ~p)," then these two propositions are the same.