Project Logicality: Generalizing & Classifying both Deductively & Inductively
It’s often said that the distinction between deductive and inductive reasoning is that one argues from the general to the specific, and that the other does so from specific to the general, but this is not correct across all forms of these sorts of reasoning.
Each can work both ways. Whoa. That’s quite an assertion, so I’ll attempt to show why here…
In deductive reasoning, the truth of an argument’s conclusion automatically follows from the truth of its supporting statements if it’s valid. Also, it’s possible for one or more of such an argument’s supporting statements to be false, and this renders it unsound even when valid. Valid but knowingly unsound arguments are not persuasive, and cannot ethically be used as though they were compelling.
Deductive arguments reorganize what we know rather than providing any new data. Deductive conclusions cannot go beyond what’s expressed or implied in their supporting statements.
Inductive reasoning, such as some of the informal sort more typically used in everyday life, can only justify conclusions as more or less probable, depending on the strength of the argument and the prior adherence of an audience to its evidence. This form of reasoning does provide new knowledge, by moving us from the known to the unknown, unstated, and not implied.
In generalizing from particular examples, I’ll show how it may be deductive and then inductive:
If I were to be on the shoreline of my local beach, and noted that the pebbles found there were worn smooth and comfortable to the touch by the actions of water and sand, and were to completely and perfectly enumerate each and every such pebble on the shore, to find them all worn smooth, an unlikely and difficult task at best, I would know with certainty that all of these pebbles were smooth and comfortable to the touch. Each and every one. The argument would then follow deductively.
If, though, I were to find one smooth pebble, and then another, and so on, and after noting from a large enough but limited sample of such stones that they are almost all smooth to the touch, though I may find a few which are not, I could conclude inductively that they are more often smooth and worn than not. The argument follows to a high degree of probability based on the size and representativeness of the sample examined, and it is an acceptable substitute for the certainty we cannot usually get in measuring things in the real world.
The things to look out for when generalizing are known as the fallacy of composition, and the hasty generalization, these errors made when we attempt to apply deductive certainty where it does not belong, the first in assuming that the whole of a population is necessarily like the parts, and the second in drawing an unfounded general conclusion on the basis of too little sample data.
Now from general to specific, classifying rather than generalizing.
If I were to get a perfect count of all sand-ground stones on this hypothetical beach, not over-counting or skipping some, and they all were worn smooth, then I could conclude certainly that any one of these stones was going to be smooth in texture just like all the others. The argument would follow deductively.
But if I were to do the more likely thing, and count a fairly sizable number of these pebbles, all showing signs of wear and smoothness, then I could argue with a good chance of being correct in saying that any one of the stones I pick up would be ground and smooth. It would then be an inductive argument.
The error to avoid committing here is the very opposite of the fallacy of composition, the fallacy of division, in which one misapplies deductive certainty by claiming that the part is necessarily like the whole.
To close out, the examples I used in this post come from a quote attributed to Sir Isaac Newton, that cranky and brilliant English guy, which goes:
I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. ~ Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) by Sir David Brewster (Volume II. Ch. 27).
Posted on Wednesday, 0:00, July 2, 2014, in Reason and tagged Deductive reasoning, Fallacy of composition, Inductive reasoning, Isaac Newton, Logic and Foundations, Math, Philosophy, Philosophy of Logic, Truth. Bookmark the permalink. 3 Comments.