Since P, P → Q ⊢ Q is formally valid, any argument we get by replacing its sentence letters with actual assertions is deductively valid. Here's the crucial observation to make at this point. We can also get by replacing the sentence letters in P, P → Q ⊢ Q-all we have to do is replace P in the first example above with the sentence that would have replaced R ∨ S with in the second example. Notice, however, that any argument we can get by replacing the sentence letters in an argument form like this one: of arguments like this one:īut when we know an argument is formally valid, then we know that any argument that results from replacing the sentence letters of that argument with actual assertions will be valid as well. For more discussion of this idea, take another look back at chapter 1. Well-formed proofs establish the formal validity Remember, a formally valid argument is an argument that is valid because of its logical form. We only need to figure it out once to create a derived rule, but once the derived rule is created, we can complete the task as many times as we want without any additional effort. In addition to shortening our proofs, we save ourselves the mental effort of "rediscovering", over and over, how to complete certain simple tasks in our proofs. In every place where we would repeat a certain common sequence of moves, we can instead using our new derived rule to complete the same task in just one inference. By combing a sequence of moves into one, we can remove a lot of boilerplate from our proofs. The technique is roughly, putting together several different inferential moves into a single step, called a derived rule. In this chapter, we will introduce a technique that makes proving things easier. Spinoza, Treatise on the Emendation of the Intellect So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations again fresh instruments, or the power of pushing its investigations further, and thus gradually proceeds till it reaches the summit of wisdom. As men at first made use of the instruments supplied by nature to accomplish very easy pieces of workmanship, laboriously and imperfectly, and then, when these were finished, wrought other things more difficult with less labour and greater perfection and so gradually mounted from the simplest operations to the making of tools, and from the making of tools to the making of more complex tools, and fresh feats of workmanship, till they arrived at making, complicated mechanisms which they now possess.
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