Sunday, January 21, 2007

Better Reasoning III: Good Verses Bad Deductive Arguments

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In the previous post in this series, it was mentioned that deductive arguments, provided they meet certain standards, have the advantage of having their conclusions follow necessarily from their premises. Thus, they are a particularly powerful type of argument. At this point though we need to know a bit more about what makes a deductive argument a 'good' one.

When we are presented with an argument, we need to figure out a way of determining whether or not we should be persuaded by it. Most people have some instinctive capacities for doing this. However, it would be nice to have a methodology that was a little bit more robust and precise.

When people are presented with arguments, one of the strategies they may use to assess them is what is sometimes called the 'Yum-Yum, Yuck-Yuck' approach. That is to say, people try and determine the extent to which the conclusion of the argument agrees with their pre-existing beliefs on the subject. If the conclusion is in agreement with their beliefs, then the argument is accepted (i.e. 'Yum-Yum'). If the conclusion of the argument is at odds with their pre-existing beliefs, then the argument is rejected (i.e. 'Yuck-Yuck').

There is also a slightly more sophisticated version of this strategy, in which, if the argument is not consistent with pre-existing beliefs, then one of the premises of the argument is rejected, or attacked.

One drawback of this approach to argument assessment is that it means that it will be difficult for a person to be persuaded to change their views on a matter, even if their views are seriously defective. Fortunately, there is a better, more sophisticated way of assessing deductive arguments.

In order for an argument to be a 'good' one, two thing really need to happen. First, it must be the case that the conclusion actually follows from the premises. Second, the premises should be true (or at least, reasonable). Being aware that there are in fact two things in play when assessing an argument, is actually quite an important insight.

It is now time to introduce a two-stage process for assessing deductive arguments. In order to do this, it is necessary to introduce two slightly technical concepts, 'validity' and 'soundness'. Let begin by looking at the formal definitions of the two concepts:

An argument is Valid if and only if, it is not possible for the premises to be true and the conclusion false.

An argument is Sound if and only if, it is valid and has true premises.

Sound deductive arguments are the 'good' ones. When an argument is sound, that is the situation in which the conclusion necessarily follows from the premises.

The fact that the validity of the argument is a condition that needs to be satisfied in order for the argument to be sound, suggests that validity should always be considered before soundness. After all, if an argument is non-valid, the truth or falsity of the premises are simply irrelevant. The argument is simply not a good one. However, this insight rather turns intuitive argument assessment approaches on their head.

The notion of validity also deserves a little discussion. One the things that people find a little bit odd about it is that, as the definition is phrased, the actual truth or falsity of the premises is irrelevant to an argument's validity. This is because the soundness phase takes care of this issue. Validity only concerns the relationship between the premises and the conclusion, irrespective of the truth, or falsity of the premises.

This approach to assessing deductive arguments has one or two surprising consequences. For example, the following argument is valid (though not sound),

If the Moon is made of green cheese, then the New Orleans Saints will win the Superbowl. The Moon is made of green cheese. Thus, the New Orleans Saints will win the Superbowl.

The reason this argument is valid will be discussed in the next post in this series (to anticipate, it is an instance of a Modus Ponens inference). However, it is pretty clear that the argument is not sound, given that there does not seem to be any real connection between the composition of a celestial body and the performance of a professional sports team.

Another curious consequence of the definition of validity is that any argument with contradictory premises, will be valid! Consider the following argument, to see why this is the case.

All snails are slimy. Some snail is not slimy. Thus, The President of the United States is a Walrus!

The fact that it can never be possible for the premises to be true and the conclusion false (as the premises can never be true at the same time), it follows that the argument is valid (though not sound). This slightly curious phenomenon is sometimes called 'The explosiveness of contradictions'. The argument would remain valid, whatever the conclusion drawn. We could replace the conclusion above, for example, by 'The nothingness negates itself' and the argument's validity would remain unchanged. This is one of the reason why contradictions are seen as being so problematic in philosophy.

It turns out that, although the notion of validity is a bit odd, it is a very useful tool which can be used to detect entire classes of argument that are always going to be valid, or invalid, irrespective of what the arguments are about. These will be the topic of the next posting in this series, along with a further exploration of the notion of validity.

The CP

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