Strategy

It can be very difficult to decide how to begin a derivation and then how to move on from line to line.

But the three step strategy guideline described in the tutorial should help.

But before moving on to see the strategy in action, try to remember roughly what the three steps are. (Need a hint?)

1. Begin with Goal analysis
2. Take the obvious steps.
3. Use I-rules on goals, E-rules on accessible lines.

Now, let's look at the details...

1. Consider goals that may plausibly allow you to complete your derivation.

2. Do what is obvious, what will obviously help you toward your goal(s).

3. Use introduction and elimination rules.

Consider the I-rules appropriate for the main connective of your goal sentence.

Consider the E-rules appropriate for the main connectives of any accessible sentence you may have already derived (this includes assumptions).

Usually you will be given an ultimate goal: what you are asked to derive.

Often during the course of a derivation, strategy will require intermediate goals: other sentences needed to complete the derivation.

SD Strategy

Consider goals that may plausibly allow you to complete your derivation.
- Usually you will be given an ultimate goal: what you are asked to derive.
- Often during the course of a derivation, strategy will require intermediate goals — other sentences required to complete the derivation.
Do what is obvious to you if that will clearly help you toward your goal(s).
Use introduction and elimination rules:
- Consider the I-rules appropriate for the main connective of your goal sentence.
- Consider the E-rules appropriate for the main connectives of any accessible sentence you may have already derived (this includes assumptions).

Next, let's apply the strategy...

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Strategy:

An Application of the Strategy

Now, we apply these steps to a derivation. For this example we show that

(B>C)=A
A>(~C>~B)

is a valid argument.

This means we take the premise on line one. And, applying our 3 step strategy.

Determine a goal.
Do the obvious.
Use I-rules on goals and E-rules on accessible lines.

So, we begin with 1. In this case our goal is given: the conclusion.

So, we take the conclusion as goal. (That's step 1 of our strategy.)

Of course, 'A>(~C>~B)' is only what we hope to derive. To indicate this, we write the question marks next to it (or, in the Café, the question marks will be entered automatically).

Now, 10 needs to be justified.

Premise		1		(B>C)=A
		2
		3
		4
		5
		6
		7
		8
		9
???		10		A>(~C>~B)???

Our ultimate goal is on line 10. How are we to derive it? Let's suppose there is no obvious way to you. (step 2)

Then, moving on (step 3), we think about the I-rule for the goal's main connective: >I. Hopefully, in the end, we can derive our goal by >I.

>I always requires a subderivation assuming the antecedent (blue; line 2) and subderiving the consequent (red; line 9).

Premise		1		(B>C)=A
Assumption		2		....what if	A
		3
		4
		5
		6
		7
		8
???		9			~C>~B???
???>I		10		A>(~C>~B)???

Notice something important: The last step gave us a new goal: '~C>~B' at line 9. (Back to step 1 again!)

So, (step 2): is there some obvious thing we can do which might help us move toward line 9?

This is a real question. It means "is there something obvious to you." Think about it: What might you do at line 3 to move toward line 9?

Premise		1		(B>C)=A
Assumption		2		What if	A
		3			???
		4
		5
		6
		7
		8
???		9			~C>~B???
???>I		10		A>(~C>~B)???

One thing you might have seen as obvious and helpful is just putting 1 and 2 together by =E: result 'B>C'. This is related to our goal, so should be helpful.

(If =E didn't catch your eye as "obviously helpful", you would have caught it at step 3 when thinking about E-rules. It's OK to do this at a later time, even within another subderivation.)

Now, what's next? Is there anything else obvious?

If not, then (step 3) think again about the I-rule for your new goal at line 9: >I once again.

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3			B>C
		4
		5
		6
		7
		8
???		9			~C>~B???
???>I		10		A>(~C>~B)???

So, because you are thinking about proving line 9 by >I, assume it's antecedent (at line 4) and you have yet another goal: '~B' at line 8.

How to derive that? If there's no obvious way to proceed, again think about the I-rule for '~B'.

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3			B>C
Assumption		4			What if	~C
		5
		6
		7
???		8				~B???
???>I		9			~C>~B???
???>I		10		A>(~C>~B)???

Because '~B' is our goal, we are thinking of ~I: assume 'B' and derive a contradiction.

But what contradiction? (P and ~P are placeholders for what?)

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3			B>C
Assumption		4			What if	~C
Assumption		5				What if	B
		6				need:	P???
		7				and:	~P???
???~I		8				~B???
???>I		9			~C>~B???
???>I		10		A>(~C>~B)???

Our goal is a contradiction. P and ~P were just placeholders above. So, we move to...

Step 2) Notice that one can easily derive 'C' and '~C' to finish the innermost subderivation.

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3			B>C
Assumption		4			What if	~C
Assumption		5				What if	B
3,5 >E		6					C
4 R		7					~C
???~I		8				~B???
???>I		9			~C>~B???
???>I		10		A>(~C>~B)???

Now the rest is easy.

As we'd planned, ~I proves '~B' at 8.

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3			B>C
Assumption		4			What if	~C
Assumption		5				What if	B
3,5 >E		6				....then...	C
4 R		7				....then...	~C
5-7 ~I		8				~B
???		9			~C>~B???
???		10		A>(~C>~B)???

Finally, two applications of >I terminate the last two subderivations and complete the derivation of the argument's conclusion.

Premise		1		(B>C)=A
Assumption		2		What if	A
1,2 =E		3		....then...	B>C
Assumption		4		....then...	....what if	~C
Assumption		5		....then...	....then...	....what if	B
3,5 >E		6		....then...	....then...	....then...	C
4 R		7		....then...	....then...	....then...	~C
5-7 ~I		8		....then...	....then...	~B
4-8 >I		9		....then...	~C>~B
2-9 >I		10		A>(~C>~B)