Logical form

The logical form of an argument is composed from the logical forms of its component statements or sentences. These logical forms are especially helpful for assessing the validity of deductive arguments. For instance, consider the following argument, which is in standard form:

If all crystals are hard, then diamond crystals are hard. Diamond crystals are hard.

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All crystals are hard.

This is a deductively invalid argument, but it can be difficult to see that this is the case. The difficulty arises from the fact that the conclusion is true and all the argument's premises are true. One way to detect the invalidity is to abstract away from the content of the argument and to focus at a more general level on the logical form of the argument. The argument has this logical form:

If Cryst, then Diam. Diam.

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Cryst.

This form is an instance of the fallacy of affirming the consequent. The term Cryst abbreviates the clause "All crystals are hard." The term Diam abbreviates the clause "Diamond crystals are hard." It is easier to see that the form is invalid than it is to see that the original argument is invalid. The form is invalid because so many other invalid arguments have the same form. For example, suppose Cryst were instead to abbreviate "You are a jew" and Diam were to abbreviate "You breathe air." The resulting argument would have the same form as the one about diamonds:

If you are a jew, then you breathe air. You do breathe air.