Practical Logic

When teaching Deductive Logic, the classic complaint we professors get is that there is such an emphasis on the syntax (mechanics) of logic that the semantics (meaning) gets lost, i.e., there’s no real-world application or meaning there.  I would take issue with that, but that’s another column.  I will concede that Deductive Logic is indeed a tightly defined universe of discourse that ultimately has nothing to do with the real world any more than a game of chess does.

Inductive Logic (which most of my students historically have taken to get out of calculus), on the other hand, precisely deals with the real world–or at least what we think is the real world–again, a topic for another column.  Whereas Deductive Logic might construct a syllogism like this:

(p1) If it is raining, the ground will be wet.

(p2) It is raining.

(c) Therefore, the ground must be wet.


While this is valid reasoning–the form of the argument is such that so long as the premises p1 and p2 are true, the the conclusion c must be true, it says nothing about whether it’s raining outside right now in the real world.  So consider the inductive argument:

(P1) My scientific instruments tell me the air pressure is dropping and the humidity is rising.

(P2) I hear thunder and see lightning.

(P3) I see the sky is getting dark and can feel the wind picking up.

(C) I conclude it’s likely to start raining soon.


Not that the truth of P1 – P3 do not guarantee the truth of the conclusion, C, but they do make it more probable.  Hence inductive conclusions are never valid, but probablistic, but at least they do apply to the real world.   Inductive arguents, being invalid, make lousy deductive arguments; and vice-versa.


Occasionally an inductive argument will look much like a deductive argument.  Consider the deduction:

(p1) All iron comes from meterorites.

(p2) This knife is iron.

(c) Therefore this knife must be made out of meterorites.

Although p1 is false, IF p1 and p2 were true, then c would have to be true, hence this is a valid argument.  Compare that to this induction:


(P1) All known iron artifacts from ancient Egypt appear to have been made out of meteorites.

(P2) This iron knife appears to come from ancient Egypt.

(C) Therefore it appears likely that this knife was made from meterorites.


The truth of P1 and P2 make C more probable than not, hence this is a good inductive argument.


Thus if you want absolute certainty, you are limited to deductive reasoning, but you largely abandon the real world.  On the other hand, if you want to argue about the real world, your conclusions are only going to be probably true at best.  Thus we might say that deductive logic is truth-preserving (you end up with what you started with, just in a logically equivalent form), while we might say that inductive logic is truth-expanding (you go beyond the premises to create new knowledge).  Note, too, that Deductive Logic is concerned primarily with validity–the truth of the premises is unimportant; whereas Inductive Logic is all about getting to the truth–the truth of the premises is very important (in computerese, GIGO: Garbage In, Garbage Out).


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