Philosophy 112

What is This?

This is a supplement to sections 1.8 and 1.9 of the Logic Text. You should read those before you read this.

Subderivations

Section 1.8 introduces subderivations. Subderivations are derivations within derivations. Visually, they will look like boxes within boxes.

Consider the following argument,

If I forgot my umbrella, then if it is raining I will get wet. \(\therefore\ \) If it is raining, then if I forgot my umbrella I will get wet.

Given the following scheme of abbrevation,

R It is raining
U I forgot my umbrella
W I will get wet

we can symbolize the argument as,

\(U\mathbin{\rightarrow}(R\mathbin{\rightarrow}W)\) \(\therefore\ R\mathbin{\rightarrow}(U\mathbin{\rightarrow}W)\)

This is a valid argument. The intuitive argument for its validity goes something like so:

Given the premise, if we assume that it is raining, then, if we also assume that I forgot my umbrella, then we can show that I will get wet.

Formally, we begin our derivation in the usual way, by entering a show line for the conclusion. Since the conclusion is a conditional, we assume the antecedent, in the hopes of completing a conditional derivation:

Show the conclusion, assume the antecedent

Show the conclusion, assume the antecedent


Now what? We could bring down our premise:

Bringing down the premise

Bringing down the premise


But now we are stuck. To complete the derivation, we need to find some way to get from lines (2) and (3) to the consequent, \(U\mathbin{\rightarrow}W\), but we don’t have what we need to use MP or MT with line (3).

The trick is to instead begin a subderivation. To do this, we need to enter a new show line:

New Show Line

New Show Line


We can do this by hand, typing ‘Show \(U\mathbin{\rightarrow}W\)’. Or we can use the **Show Command*, ‘Show Cons’ (short for “Show Consequent”).

::: {.boxed} A Show Command is a shortcut in the software for entering a Show Line. Show Commands are not rules. They do not justify the line entered: you can enter any Show Line you like at any time, without justification. Show Commands reflect some common useful strategies. For example, it is always a good idea to begin a derivation by entering a Show Line for the conclusion, and it is often a good idea, if you are trying to show a conditional, to enter a Show Line for its consequent. :::

Now we can bring down our premise, and use MP on lines 4 and 5 to get \(R\mathbin{\rightarrow}W\). Then we can use MP again, to get \(W\). Finally, we can box and cancel, using CD:

Completing the Subderivation

Completing the Subderivation


We are not done yet. We have shown that \(U\mathbin{\rightarrow}W\) follows from the premise together with our assumption, on line (2), of \(R\). That means we can now box and cancel our first show line, and so complete the derivation:

Completed Derivation

Completed Derivation


Shortcuts

Section 1.9 introduces two shortcuts for completing derivations.

Citing Premises

First, you can cite premises directly as though they were line numbers, rather than bringing them down, and then citing their line numbers.

For example, consider the following derivation:

\[P\mathbin{\rightarrow}Q\ .\ R\mathbin{\rightarrow}\mathord{\sim}Q\ .\ P \therefore\ \mathord{\sim}R\]

1.Show\({\mathord{\sim}}R\)

2.\(P{\mathbin{\rightarrow}}Q\)pr

3.\(P\)pr

4.\(Q\)2 3 mp

5.\({\mathord{\sim}}{\mathord{\sim}}Q\)4 dn

6.\(R{\mathbin{\rightarrow}}Q\)pr

7.\({\mathord{\sim}}R\)5 6 mt

Here is the same derivation, citing the premises directly rather than bringing them down:

1.Show\({\mathord{\sim}}R\)

2.\(Q\)pr1 pr3 mp

3.\({\mathord{\sim}}{\mathord{\sim}}Q\)2 dn

4.\({\mathord{\sim}}R\)pr2 3 mt

Mixed Derivations

Second, our justifications for boxing and cancelling—DD, CD, and ID—are independent of our justifications for making assumptions—ASS ID, ASS CD. This means that you can start a derivation by making an assumption for CD, but box and cancel by DD or ID. We call these Mixed Derivations.

For example, if you start out trying to derive \(W\mathbin{\rightarrow}R\) by CD, but then stumble upon a contradiction, you can box and cancel by ID:

1.Show\(W{\mathbin{\rightarrow}}R\)

2.\(W\)ass cd

  ⋮

7.\(P\)

8.\({\mathord{\sim}}P\)

Mixed Derivations are okay because that can always be made “uniform” by adding a few more steps.

In the example above, we could have added a subderivation, starting on line 9, showing \(W\):

1.Show\(W{\mathbin{\rightarrow}}R\)

2.\(W\)ass cd

  ⋮

7.\(P\)

8.\({\mathord{\sim}}P\)

9.Show\(R\)

10.\(P\)7 r

11.\({\mathord{\sim}}P\)8 r

12.10 11 id

Test Your Understanding

Consider the following mixed derivation:

\[Q\mathbin{\rightarrow}(Q\mathbin{\rightarrow}P) \therefore\ Q\mathbin{\rightarrow}P\]

1.Show\(Q{\mathbin{\rightarrow}}P\)

2.\(Q\)ass cd

3.\(Q{\mathbin{\rightarrow}}P\)2 pr1 mp

How would you complete this derivation as a “uniform” derivation instead?

1.Show\(Q{\mathbin{\rightarrow}}P\)

2.\(Q\)ass cd

3.\(Q{\mathbin{\rightarrow}}P\)2 pr1 mp

4.\(P\)2 3 mp