Did it rain? Logically concluding that it rained

This is where reason comes in. Logically, one cannot conclude with any level of probability that it rained. A deductive argument such as this can only tell us that we cannot conclude it rained given the premises. When we apply reason, we can consider the alternative explanations and assign a probability to each, then make our conclusion based on the most probable explanation. For example,

"If it rained, the streets will be wet." What are some other possible explanations and their relative probability? Perhaps the town washed the streets so quickly that they didn't dry yet? How probable is that? Already, we see if (all) the streets are wet, rain seems like the only reasonable explanation. Now, if we have multiple lines of evidence that work together synergistically by making each line of evidence stronger (like those in your example), the most probable explanation becomes even more clear.

Bottom line, logic tells us we cannot conclude with certainty that it did rain. Through reason, we can conclude with a high degree of probability that it rained.

Formal logic isn't a method for proving that some phenomenon actually exists or that a statement is true.  It can help; however, proving something is true takes more than proving it isn't.  If the statement is, in fact, false (or the phenomenon didn't exist), formal logic might be able to help us conclude that the statement is false, assuming we had sufficient information.

If it were possible (and I doubt it is) to come up with an additional TRUE statement to add to the 4 (or more) "if P then Q" statements we already have -- something like "The  statements in the list are the only ways by which streets, gardens, etc. can be wet. " -- then we might be able to conclude the truthfulness of the statement "It rained."  As we know, there are lots of other ways to make surfaces wet, so other explanations are certainly possible.

The statements above are essentially correlational ones, not causal ones.  They tell us that often rain and wet surfaces go together, as do rain and dark clouds, as well as rain and ... .  What the statements don't tell us if that one of the observations is the cause of the other -- and cause requires more than just having things happen together.  Strategies beyond pure logic is required to establish causal relationships.