T6: 8 of 12

Finally we have rules for the tilde. These are more difficult but also more important. So, think this page over very carefully if the idea of a reductio and/or its formalization is new to you.

As motivation, 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 (even absurd), so concludes that he or she should proceed in an alternative manner.

Notice, then, the thinking indirectly reaches the conclusion after assuming the conclusion false.

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The set of rules SD will utilize a version of this reasoning to unfortunate consequences. But the SD rules will require that the consequences be absurd in a very precise way: they must be contradictory.

Here is a fairly tough example; think it through...

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 must therefore have true premises and a false conclusion and so 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.

THE BIG DEAL: This thinking invovles an "assumption for contradiction": It assumes that a sound argument can have a false conclusion.

Now to symbolize this reductio ad absurdum.

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. NOTE: At line 3, we make an assumption – pretending that an argument can be sound yet have a false conclusion – that we hope to show is wrong. (This pretense is shown in blue; notice how it's set off to the right.)

 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...This is the new rule. 11 ~(S&F)

So, what's going on at line 11? What justification should we write? Let's think carefully.

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.

Formalizing a Reducito: Subderivations

We move sentences 3-10 (in blue) over to the right in order to emphasize that they are the "what if" thinking.

It is especially important to keep our extra "assumption" apart from the premises. After all, we know the premises here are true. They come immediately from the definitions of "valid" and of "sound". But our additional assumption, 'S', meaning that the argument is sound, is just a hypothesis. It's about an arbitrary argument that might or might not really be valid. We just try to find out what happens if it is valid.

Again, then, we need to separate our premises -- which are taken to be true -- from our assumptions and their consequences -- which are just hypothetical. In SD we do this by pushing the assumptions and their consequences off to the right. (We will call this a "subderivation".)

After a subderivation is finished -- "terminated" is the word -- the assumption is no longer active. No individual line in the terminated subderivation can be cited to justify further inference; only the whole subderivation will be used in this way (e.g., "3-10" can be cited). The reason? Because we want the rest of the derivaiton to depend only on the premises.

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

So, here's how we'll complete writing the derivation.

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

NOTICE LINE 11: this is the only use of a new rule.

NOTE ALSO: In some versions of the Logic Cafe, the rule ~I is called "RD" for "reductio".

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