Internal (I.e., Premise) Relevance
Premise relevance increases or decreases the likelihood that a conclusion is true. For example, suppose I want to prove that (MC) a certain animal 'x' is a duck. I give the following premises
P1. The animal waddles.
P2. The animal quacks.
P3. The animal has a stomach.
P4. The animal has a reality TV show and is a primate.
MC. The animal is a duck.
Notice that P1 increases the likelihood that (MC) the animal is a duck. Same goes for P2. P3 is neutral. And P4 actually decreases the likelihood of the conclusion being true. It increases the probability of another conclusion: that the animal is a cast member of Duck Dynasty. Premises that swing the probability against the main conclusion can be used as premises in a counter-argument.
A note on extended arguments: The argument we just looked at is a simple argument because no premise requires further support. In the case of extended arguments some premises require further support. In these cases, the sub-premises will be relevant to the sub-conclusion they support but not directly relevant to the main conclusion. In your evaluation you should indicate this.
Summary and Key Points:
(1) When we evaluate for premise relevance we are looking at whether the information in a premise increases or decreases the likelihood of a conclusion being true.
(2) When we evaluate for relevance we assume the premises are all true--even if they aren't.
(3) When we evaluate for relevance we evaluate each premise individually. It's possible that some premises will be relevant while others might not be.
Contextual Relevance
Contextual relevance is the evaluation of whether an argument addresses the main topic that's being debated. We usually evaluate contextual relevance in the context of a debate between 2 sides over a particular issue. If one side's argument doesn't seem to have any bearing on the main topic, then the argument is contextually irrelevant.
Example:
Suppose two people are debating whether GMOs are safe for human consumption:
Person 1: There's no good evidence to show that GMOs are safe for human consumption.
Person 2: Actually, there's over 20 years of studies, the majority of which show there are no negative health effects to consuming GMOs.
Person 1: Yes, but using GMOs cause farmers to use stronger pesticides like Round Up. And besides, Monsanto is evil!
In this case, person 1's arguments, regardless of whether they are true or not, are not relevant to the topic being debated which is whether GMOs are detrimental to human health.
See Straw Man and Red Herring Fallacies for examples of specific kinds of failures of contextual relevance.
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