Chapter Five, Tutorial Four
Subderivations: Indirect Proof

We have two more rules to introduce: an introduction and elimination rule for the tilde. As usual, we will try to motivate our formal, SD rules by way of English examples. First consider the following passage.

Suppose I register for Philosophy 640 this term. Then because my music degree requires almost constant practice and my night job will keep me busy to all hours, I would be hard pressed to do any of the required philosophy reading. So, I had better put off the philosophy for another term.

This is basic practical reasoning. We think like this all the time. One considers an option, argues that the consequences of realizing this option would be unfortunate, so concludes that he or she should proceed in an alternative manner. So, the thinking indirectly reaches the conclusion.

The set of rules SD will utilize a version of this reasoning to unfortunate consequences. Only the SD rules will require that the consequences be unfortunate in a very precise way: they must be contradictory. Here is an example.

We may show that a sound argument cannot have a false conclusion. For suppose it did. Then because the argument is sound, it has only true premises. It would therefore have true premises and a false conclusion and be invalid (as valid arguments cannot have true premises and false conclusion). But this means that the sound argument is not valid, a contradiction of the definition of "sound".

The conclusion to draw from this passage is that the supposition (that a sound argument could have a false conclusion) is false. Thus, a sound argument must have a true conclusion.

We can pretty easily symbolize this reasoning. As before, use the following interpretation:

S: The argument is sound.
V: The argument is valid.
T: The argument has all true premises.
F: The argument has a false conclusion.

Then the following step-by-step version of the reasoning about sound arguments should seem plausible.

 Premise (from the definition of "sound"): 1 S>(V&T) Premise (from the definition of "valid"): 2 V>~(T&F) Assumption (we ask "what if" the argument is sound yet has a false conclusion): 3 what if..... S&F 3 &E 4 then........ S 1,4 >E 5 then........ V&T 5 &E 6 then........ V 5 &E 7 then........ T 3 &E 8 then........ F 7,8 &I 9 then........ T&F 2,6 >E 10 then........ ~(T&F) Conclusion from our assumption 3. and the argument through 10... 11 ~(S&F)

The idea here is that if we assume 'S&F' on line 3, then we are led both the the conclusion 'T&F' on line 9 and its negation on line 10. The assumption of line 3 must be wrong — it leads to contradiction.

We will say that line 11 is justified by negation introduction, ~I and cite the lines of the subderivation.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.

Notice line 11: this is the only use of a new rule.

Now, which of the following is the rule of ~I? Click on the "~I" above the correct formulation of this rule:

~I

input:

output:
 P Q P>Q
~I

input:

output:
 ~P Q ~Q P
~I

input:

output:
 P Q ~Q ~P