"The exploration of logic means the exploration of everything that is subject to law" (6.3): logic determines the form that laws of nature can take, but does not itself make any claims regarding nature. Scientific laws themselves do not belong to logic, because they make claims about experience and do not hold a priori.
The law of induction, the law of causality, and other such scientific principles are not exactly empirical facts, either. Wittgenstein calls them "a priori insights about the forms in which the propositions of science can be cast" (6.34). They define the framework within which we can talk about natural phenomena. At 6.341, Wittgenstein compares the laws of nature to a square mesh laid out over a surface of black and white spots. This mesh allows us describe the surface by saying of each square in the mesh whether it is black or white. Of course, a triangular mesh or a hexagonal mesh could be used just as well as a square mesh, though certain kinds of mesh will likely provide a simpler and more accurate description of the surface than others. And while the mesh itself can tell us nothing about the distribution of black and white on the surface, we can learn about the surface by observing what kinds of mesh describe it most accurately.
The laws of nature cast the form that any description of the world must take. They tell us nothing about the world, though we can infer something about the world from the fact that it is more easily described by one system of mechanics, say, than another.
We need to construct a system of mechanics in order to understand nature. Following Hertz, Wittgenstein claims: "only connections that are subject to law are thinkable" (6.361). That is, we can only make sense of natural phenomena if we see them as conforming to some kind of regularity. We do not actually see causes and effects in the workings of nature, but we can only make sense of the workings of nature if we read cause and effect into them.
The laws of nature do not tell us about nature as much as they tell us how we are going to make sense of nature. They are not necessary truths—only the laws of logic are necessary (6.37)—nor are they explanations of natural phenomena. Wittgenstein compares laws of nature to what the ancients would have called "God" or "Fate": they are the stopping point at which we acknowledge our explanations can go no farther. The mistake of modern science is to see these laws as providing a full explanation of nature rather than just a contingent framework for describing regularities we find in experience.
Just as there is no logical connection joining two events causally, so there is no logical connection between my will and the world (6.373): neither my will nor natural phenomena have any effect on what is impossible and what is necessary.
The laws of science do not adhere in any obvious way to Wittgenstein's sharp distinction between logic and the world. On one hand, the laws of science must be distinct from logic, because they make specific claims about how things will happen in the world. Even the general claim of induction, that the future will resemble the past, draws a connection between future and past that Wittgenstein has already claimed (at 5.1361) not to be logical. The connection between past and present, present and future, is not a logical one.
On the other hand, the laws of science are not simple facts about the world. It's one thing to say, "I kicked the ball and the ball moved," and something else besides to say "My kicking of the ball caused it to move." Scientific laws would seem to work on the level of explanation: they provide general rules for interpreting how and why things happen the way they do.
Wittgenstein refers to the laws of nature as "a priori insights about the forms in which the propositions of science can be cast" (6.34). His clever metaphor of a mesh laid out over a black and white surface illustrates this point quite nicely. The laws of nature do not tell us anything about the world, nor are they necessarily true of the world. Rather, they are tools we can use to make sense of the world.
Let us see how this view of science can be brought to bear on the law of causality. That everything has a cause is not a truth we discover in nature, but a general principle we apply to nature. Let us take as an example a person putting coins into a candy machine. The first ten times, she puts a coin in and a candy pops out at the bottom. The eleventh time, she puts the coin in exactly the same as before, but no candy pops out. Her immediate conclusion is that something in the machine must have operated differently on this eleventh time, even if it seemed to her as if everything had happened the same as the previous ten times. If something happens differently from the way it normally happens, there must be a cause for that difference.
At 6.36, Wittgenstein says, "if there were a law of causality, it might be put in the following way: There are laws of nature." He is telling us that the law of causality amounts to nothing more than a belief that things happen for a reason, that it is not just pure chance that candy did not pop out on the eleventh try. The laws of nature exist to tell us just that: there are regularities in nature, and nothing happens without a reason. That is why Wittgenstein does not call causality a law, but rather "the form of a law" (6.32): we have to accept causality if we are to explain natural phenomena in terms of laws.
We are deluding ourselves when we think of the law of causality either as a logical law or as something discovered in experience. In both cases we would be making the mistake of reifying causality, of thinking of it as a "thing" that has some sort of being. Causality has no being, either as a necessary part of logical form, nor as a binding force at work upon nature. Rather, it is a tool we bring to bear on nature in order to understand its regularities with greater clarity.
The reification of causality and other laws of nature leads us to think that modern science can fully explain the world. We think that we have identified the fundamental truths about the workings of the world, when in fact we have simply developed a framework within which we can think about the workings of the world. This framework may in many ways be more powerful than superstition, but they work according to the same principle: both scientific laws and superstition provide explanations for why things happen the way they do. In both cases, however, the explanations themselves are not found in nature, but are found in the framework we adopt.
We should note that Wittgenstein is not anti-scientific. He is not telling us that scientific truths are useless or simply matters of convention. He is simply suggesting that the laws we use to explain natural phenomena are not themselves things that we have discovered experimentally. Perhaps a better term to explain Wittgenstein's position would be "anti-scientistic": he is skeptical about the power of science to give us answers to the very fundamental questions it claims to have settled.