# Philosophy 112

# Overview

Section 2.6 introduces a method for abbreviating derivations. The software calls it “queuing.” Section 2.7 discusses some theorems. Section 2.8 introduces derived rules.

# Queuing

Here is the idea. Consider the following derivation:

\(P{\mathbin{\wedge}}Q\ .\ P{\mathbin{\rightarrow}}R\ {\therefore\ }R\)

1Show\(R\)

2\(P{\mathbin{\wedge}}Q\)pr

3\(P\)2 s

4\(P{\mathbin{\rightarrow}}R\)pr

5\(R\)4 5 mp

65 dd

We have already seen two ways in which this can be shortened: we can use the premises directly instead of bringing them down; we can apply ‘dd’ to the end of line (5) instead of entering it on a separate line:

1Show\(R\)

2\(P\)pr1 s

3\(R\)2 pr2 mp dd

Let’s pause and consider both of these shortcuts.

Line (2) combines two steps into one:

- bring down PR2 (to get \(P{\mathbin{\wedge}}Q\))
- to the result above, apply S (to get \(P\))

Line (3) also combines two steps into one:

- to line (2) and the result of bringing down PR2, apply MP (to get \(R\))
- given that result, box and cancel by DD

You can combine steps like this on any line. So, for example,

1Show\(R\)

2\(R\)pr1 sl pr2 mp dd

To unpack this, work from left to right:

- bring down PR1
- to the result, apply SL (to get \(P\));
- to the result (\(P\)) and PR2, apply MP (to get \(R\))
- given that result (\(R\)), box and cancel by DD

Here is a second example.

\(P{\mathbin{\vee}}Q\ .\ {\mathbin{\sim}}Q{\mathbin{\wedge}}R {\therefore\ }P\)

1Show\(P\)

2\(P{\mathbin{\vee}}Q\)pr

3\({\mathbin{\sim}}Q{\mathbin{\wedge}}R\)pr

4\(P\)3 sl 2 mtp

5dd

On line (4), we have combined a two steps into one line:

- to (3), apply SL (to get \({\mathbin{\sim}}Q\))
- to the result and line (2), apply MTP (to get \(P\))

We could compress the derivation yet further, if we wished, e.g.,

1Show\(P\)

2\(P\)pr2 sl pr1 mtp dd

Queuing makes derivations more difficult to read. I recommend using it judiciously **if you feel quite confident in your ability to construct derivations**. Otherwise, I don’t recommend using it at all.

# Theorems and Derived Rules

Consider the theorem 24 (T24 in the software and the book, problem 2.024 in the software):

\[P{\mathbin{\wedge}}Q{\mathbin{\leftrightarrow}}Q{\mathbin{\wedge}}P\]

This theorem tells us that the order in which conjuncts occur does not matter to the truth of a conjunction. Note that the same would not be true if we replaced the \({\mathbin{\wedge}}\)’s with \({\mathbin{\rightarrow}}\)’s.

Here is a derivation of the theorem:

1Show\(P{\mathbin{\wedge}}Q{\mathbin{\leftrightarrow}}Q{\mathbin{\wedge}}P\)

2Show\(P{\mathbin{\wedge}}Q{\mathbin{\rightarrow}}Q{\mathbin{\wedge}}P\)

3\(P{\mathbin{\wedge}}Q\)ass cd

4\(P\)3 sl

5\(Q\)3 sr

6\(Q{\mathbin{\wedge}}P\)4 5 adj cd

7Show\(Q{\mathbin{\wedge}}P{\mathbin{\rightarrow}}P{\mathbin{\wedge}}Q\)

8\(Q{\mathbin{\wedge}}P\)ass cd

9\(Q\)8 sl

10\(P\)8 sl

11\(P{\mathbin{\wedge}}Q\)9 10 adj cd

12\(P{\mathbin{\wedge}}Q{\mathbin{\leftrightarrow}}Q{\mathbin{\wedge}}P\)2 7 cb dd