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15.1 The 5-Step Plan: ~(PvQ)

There are two patterns for premises that we put off dealing with in Chapter 14.

These two patterns are tricky, so we are going to learn a specific maneuver for dealing with them. That maneuver is called the 5-step plan.

Strategy: a general or long-term plan.
Tactic: a specific or short-term plan.

In the world of chess, people make a helpful distinction between strategies and tactics.

Strategies are big-picture or long-range ideas for accomplishing your goal.

Tactics are specific, focused maneuvers that accomplish short-term objectives that contribute a small part to your long-term goal.

If you like metaphors: strategies are about the forest; tactics are about the trees.

The key idea to look at the main connectives is a strategy.

The important plan you learned in Chapter 14, to  look at the main connective, is a strategy. In fact, it is the fundamental, essential strategy that guides every formal proof.

The 5-step plan is a tactical maneuver for dealing with ~(PvQ).

The 5-step plan, by contrast, is a tactical maneuver. You employ it in a specific circumstance: when you have a negation around a disjunction. And it accomplishes a specific goal: extracting information out of ~(PvQ) which will be in a more usable form so that we can finish the proof.

Before we talk about how to do the 5-step plan, though, let’s get clear on this question:

Now let’s assume all these sentences below are premises.

Okay, enough preliminaries. Let’s look at the details of applying the plan. Here’s the classic argument that the 5-step plan helps with:

1. ~(PvQ)
Thus
2. ~P

The 5-step plan grows directly out of the lessons you learned in Chapter 14. If you look at your conclusion, you should know how to get started.

See if you can figure out the rest yourself. You’ll remember it better that way!

Now we’re ready to talk about the plan.

Step 1: Pick a disjunct

Step 1: pick a disjunct.

The 5-step plan works any time you have a negation around a disjunction. What you do first is pick a disjunct of the disjunction that is inside the negation.

In the example above, since our conclusion is ~P, we should pick P rather than Q. Notice that the disjunct is P itself. The disjunct isn’t ~P.

Here’s a key idea: you can use the plan multiple times, once for each disjunct. But you have to proceed one at a time.

Step 2: Put it in a subproof

Step 2: put it in a subproof.

The plan is a type of reductio, so we have to start a new subproof with the disjunct in the assumption line.

Step 3: Build the disjunction

Step 3: build the disjunction.

Since you picked a disjunct of the disjunction inside the negation, you can re-create that disjunction with vIntro. That is what you did on line 3 of the proof above.

Step 4: Introduce the contradiction

Step 4: Intro the #.

By design, the disjunction that  you build is a perfect contradiction to the original premise. The premise is the  disjunction with a negation in front,  which is what is  needed for  the #Intro rule. So on the next line of our subproof, we write #.

Step 5: Finish the reductio

Step 5: Intro the ~.

The most common thing we see students forget is what to do last, how to finish the 5-step plan.

The 5-step plan is a reductio, so you always end with ~Intro!

Remember this: the plan is a reductio, so you always end with ~Intro!

Our goal was to prove ~P, and using the 5-step plan we  assumed P and proved #, which is how the ~Intro rule works. So you can always justify the final step by negating what you assume at the start of the subproof.

Let’s practice it. If you understand the plan, you should be able to do this full version of the DeMorgan’s that the first example was based on.

Let’s see if you’ve got the idea.