Identifying Arguments
We encounter
inferences and recommendations all the time in our ordinary lives.
Generally, to infer is
to show that a conclusion follows
from some evidence. In the parallel case of mathematics, we say that
one step in an equation follows
from a previous step when the “gap” between them is sufficiently
small that the next step is obvious. In this case, this obviousness
is equivalent to our use of the word “reasonable”.
For example, those minimally competent with algebra can move from 3x = 9 to x = 3 without showing the need to divide both sides by 3 to isolate x. When learning algebraic techniques, “showing your work” is important, but with competence comes license to declare that the later step follows the previous without the burden of an overly-detailed expression of the progress. This is a model of how we move from premises to conclusions in logic, and it is similarly a model for how we talk about the relationship between reasons to think and explanations, which is the first relationship we’ll investigate here.
In this we see the fundamental importance of making analogies and comparisons in our ordinary commerce with one another. We say this step is like the previous step, but for X . Following depends on comparison between steps in a sequence, and when we judge that one step follows another, the X in the blank goes without saying.
When one thing follows another, the steps in between go without saying. We need to be sensitive to the competence of our audience when we judge how much or how little we ought to make explicit.