Department of Philosophy (PHI)

Deduction, Induction and Fallacy

Arguments can be divided into three categories on the basis of their logical strength. Using Prob(C/P) to denote the probability that the conclusion is true given that all of the premises are true, a quantity that we will also refer to as the "probability of the argument", we may define the three types as follows:

• deductively valid arguments, for which Prob(C/P) equals 1;
• inductively strong arguments, for which Prob(C/P) is less than 1 but greater than .5;
• logically fallacious arguments, for which Prob(C/P) is .5 or less.

Some explanatory remarks are in order. First, the probability concept used in these definitions is that of conditional probability, the probability of the conclusion given the premises. In talking about this probability we are not assuming that the premises are true, we are commenting on what bearing it would have on the conclusion's status if the premises were known to be true. Thus we can calmly assert that the probability that the Earth is about to be destroyed, given that it is about to be destroyed by conversion to an amorphous plasma by phaser fire from Vogon destructor ships, is 1. This means only that it is impossible that the Earth should be destroyed in that particular way without being destroyed. On a more cheerful (but no more realistic) note, I can observe that the probability that I will be debt free next year, given that I win $10 billion in a sweepstakes this year, is very close to 1. This simply means that if I were to win such a sweepstakes this year, I would almost certainly be debt free next year.

Secondly, while it may seem to stretch the meaning of "strong" a bit far to call an argument whose probability is barely above .5 "inductively strong", this is in fact the only natural dividing line between inductive arguments and logically worthless ones. An argument that is inductive in this sense has a conclusion that is at least somewhat more probable than its denial given the premises, while an argument that falls below this threshold has a conclusion that is no more probable than its denial given the evidence adduced in its favor. We must understand that "inductively strong" here simply means stronger to some (perhaps very slight) degree than an argument from the same premises to the opposite conclusion.

Thirdly, while everyone admits that validity confers a probability of 1 on an argument, not everyone agrees that, conversely, a probability of 1 insures validity. Without arguing the point at length here I will simply assume that one should never assign a probability of exactly 1 to any argument that could possibly lead from true premises to a false conclusion.

Finally, some writers on the classification of arguments maintain that all arguments should be divided first into deductive and inductive arguments, and then distinctions based on strength should be drawn within each of these two categories. The final result is some such fourfold classification as this:

valid deductive arguments, invalid deductive arguments, correct inductive arguments, and incorrect inductive arguments. I have doubts about the possibility of classifying all fallacious arguments as deductive or inductive, though many invite such classification. But suffice it to say that in my view any reasonable way of making the deduction/induction cut will lead to two categories that coincide with my "deductively valid" and "inductively strong" categories plus a subdivision of the duds in my "logically fallacious" category into deductive duds and inductive duds.

Some main distinguishing characteristics of successful deduction and successful induction are as follows:

1. Successful induction is "knowledge expanding" in the sense that in drawing the conclusion we add to our stock of basic information as well as to the stock of information structures we call knowledge. Some of the basic information in an inductive conclusion might be drawn from the premises, but never all of it. In deduction, by contrast, we often acquire new information structures but we never acquire new basic information. We select basic information from established structures, recombine it in new ways to form new structures, and add the result to our stock of knowledge. The effect of deductive reasoning on the growth of knowledge is spectacular. What deduction cannot do is lead to new basic information. No deductively valid argument has a single bit of information in its conclusion that is not somewhere to be found in the premises. Accordingly, from premises that are entirely about the past, nothing can be deduced about the future. From premises that are entirely about experienced things, nothing can be deduced about things we have not experienced. From premises that are entirely about a person's behavior, nothing can be deduced about the person's thoughts or feelings. In all of these areas we must rely on induction to expand our stock of basic information. To an extent that is perhaps somewhat illuminating, deduction is to induction as paper recycling is to pulpwood processing. Recycling can transform newspapers into facial tissues, but it only reduces the paper products we already have to paper smithereens that can be formed into new paper products. Pulpwood processing provides new smithereens.

2. Inductive conclusions always contain at least some information that is not in the premises, either explicitly or implicitly. Thus they say something that can be false even if what the premises say is all true. The fallibility of induction and the novelty of the information it provides are two sides of the same coin. Likewise, the infallibility of deduction and its inability to deliver new information are inseparable. There is nothing in a deductive conclusion that could be false if all the premises are true because there is nothing in the conclusion that is not in the premises.

3. Deductive validity is all-or-nothing. It cannot be more or less impossible for an argument to have true premises and a false conclusion; the combination is either absolutely impossible (and the reasoning valid) or it isn't (and the reasoning invalid). The probability of an inductively strong argument can range from nearly (but never exactly) 1 to nearly (but never exactly) .5. That is an extremely wide range, and inductive strength admits of very substantial variations in degree. We quite rationally bet our lives on some inductively strong arguments (this glass of tap water won't kill me because there's never been a death attributable to consuming a glass of Greensboro's drinking water before). Others are hardly worth risking minor discomfort over (the weatherman says there's a 55% chance that it warm up this afternoon, so I'll go out without my jacket).

4. Inductively strong arguments are sensitive to new information in a way that deductively valid arguments are not. New information added to the premises of any invalid argument, including any inductively strong argument, can change it from strong to weak or worthless, or from weak or worthless to strong or even valid. This cannot happen to an argument if it is valid to begin with. First, if an argument is valid, it cannot be made any stronger by adding premises, since there is no greater probability than 1. More interestingly, the addition of premises, without taking premises away, cannot transform a valid argument to an invalid one. If the original premises are all true, the conclusion must be true. If the original premises and the new ones are all true, then the original premises are all true and the conclusion must therefore be true. New information might undermine a premise we thought we knew, and thus force us to admit that the argument does not establish its conclusion, but this has nothing to do with validity. With induction it is just the opposite. Any inductive argument, no matter how probable, could be decimated by new information that casts no doubt at all on the original premises. In an extreme case, the new information might simply be that the conclusion is false (though all the premises are true). If I enter a certain famous sweepstakes, I have approximately one chance in 200 million of winning the grand prize this year. That is, given all I now know about this sweepstakes, the probability that I am not the winner is .999999995. So an inductive argument from what I now know to the conclusion that I am not the winner would be exceedingly strong. Yet if I am the winner, new evidence will come to me (the call from Mr. McMahon, the check, etc.) such that the probability that I am not the winner, given my original evidence plus this new evidence, will be close to 0 rather than close to 1. That is, new information that does not undermine my original premises at all can change an extremely strong argument to a completely worthless one. Could new evidence instead increase the probability that I am not the winner, close as that probability already is to 1? Sure. I could discover that I forgot to mail in the entry.

Because inductive reasoning can be undermined by new information, the success of an inductive argument depends not only on the truth of the premises and the probability of the argument, but also on the completeness of the basis on which the conclusion is drawn. This is reflected in a rule of inductive logic known as the Requirement of Total Evidence, which simply says that, in drawing inductive conclusions, we must take into account all available, relevant information. If we fail to do this, if we draw a conclusion on the basis of some of the available evidence, and ignore other available evidence that tends to undermine our argument, then we are guilty of the fallacy of incomplete evidence.