7. Deductive versus Inductive Validity

Before we examine the distinction between deductive and inductive validity, we should summarize our discussion of validity in general. Validity refers to good form. It means that an argument has good support connections between its parts. Good form can also be fleshed out by reference to three basic ingredients. First, there must be consistency within and between all the components. Second, the premises must be precisely relevant to the conclusion, sufficient to establish the conclusion to the degree of certainty it claims. Third, there must be a proper movement of the premises to the conclusion in such a way that that which is implicit in the premises becomes explicit in the conclusion. The way we have stated the third ingredient is subject to debate. The relation of premises to conclusion is usually thought of in terms of independence. However, some kind of dependence of premises on their conclusions seems inevitable as the famous Socrates example shows. A fallacy is not committed by the mere dependence of premises on their conclusions but by improper dependence in which 1) the dependency relation may be destroyed altogether as is the case when the argument lacks movement from premise to conclusion because there is no premise but merely a circle of the same assertion though this small circle may be veiled in the language, or 2) the dependency relation may be overdone to such an extent that the evidence is compromised and forced to fit the conclusion as in ad hoc rescue. We will now discuss the difference between deductive and inductive validity under the following headings: the similarity of deductive to inductive validity, the distinction between deductive and inductive validity, and the implications to be drawn from deductive verses inductive validity.

1A. The Similarity of DV and IV arguments

In a word, both types of argumentation must have good form to be valid. But many logicians feel constrained to shy away from using the term "valid" with inductive arguments calling worthwhile inductive arguments reliable instead.

In response, it must be observed that for an argument to have any merit at all, whether deductive or inductive, it must have good connections between all its parts. If an inductive argument lacks consistency, sufficiency, or proper movement, then it cannot even be considered reliable. But if it has these qualities, then it has good form and is valid. In this light, the term "valid" is properly used with respect to both deduction and induction.

2A. Discerning the distinction between DV and IV arguments

1B. The "if" test explained

The procedure for determining which kind of validity obtains can be called the "if" test. The "if" test begins with the following question: "If the truth of the premises are granted for the moment and the form of the argument is good, what follows?" The next step of the "if" test is to focus this initial question in two areas: on the conclusion and on the principle of consistency.

1C. The "if" test focus on the conclusion.

If the premises are taken as if true, if the form is good, and if what follows under these conditions is that the conclusion must be true, then the argument is deductively valid (DV). DV means that if the premises are true, the conclusion must also be true and a false conclusion is impossible. Or, it may be stated that if the premises are assumed to be true, the conclusion must also be assumed to be true.

On the other hand, if the premises are taken as true for the sake of argument, if the form is good, and if what follows under these stipulations is that the conclusion could be true or false, then the argument is inductively valid (IV). With an IV argument you can have true premises, good form, and have a false conclusion.

2C. The "if" test focus on consistency.

With regard to the principle of consistency, in a DV argument, if the premises are granted as true for the sake of argument, and if the conclusion were falsified as part of the if test, then the conclusion would stand in a contradictory relation to the premise unit.

Where there is the combination of true premises (actually true or granted as true) and a contradiction between these premises and the temporary falsification of the conclusion (falsified in the test), then the argument is DV. Consider the following example:

If you grant for the sake of argument that all dogs are animals and that all huskies are dogs (here what we grant is obviously true but it need not be true for the "if" test to be applied), then it must be the case that all huskies are animals (the key is that they are animals follows necessarily from the premises, it is not just common knowledge). But if the premises are granted and the conclusion is falsified for the moment to "So, it is not the case that all huskies are animals" then a contradiction is evident between the conclusion and the premises. Something does not square because for this revised conclusion to be true one of the premises would have to be false. 

On the other hand, with regard to consistency for an IV argument, if you grant the premises then the conclusion could be false with no contradiction as is shown in the following Farmer Bill example:

Again, we have to ask the "if test" question and take the premises as true for examination purposes (this orange could be from Pete's harvest). If we grant the premises, must the conclusion follow? No. But is it a reasonable conclusion to draw from these premises (which in effect asks, "Does the argument have good form?")? Yes.

Also, if we falsify the conclusion for the moment and say that the orange is not grade A, do we have a contradiction with the premises? No. For both of these reasons, this argument is said to be IV. It is valid because it conforms to the ingredients of good form. It is inductively valid because it could have true premises with its good form and a false conclusion but without contradiction.

2B. The "if" test applied

By reference to an argument concerned with a person who fell from the top of the Empire State Building onto the concrete pavement below and by a "barbers" argument, we can graphically illustrate the difference between deductive and inductive validity.  We begin with some variations of the Empire State Building argument.

1C. Bill fell off the Empire State Building. Therefore, Bill was hurt.

As you reflect on this argument, consider only the information given and do not look for hidden meanings. We will fill in some basic facts. The premise states that Bill fell onto concrete. He descended with uninterrupted flight. He did not drop down the building from flagpole to flag pole. He had no parachute and no large springs on his feet. On the ground, the concrete was neither wet nor soft and there was no large net. Given this scenario, we need to answer an important question: "Is this conclusion a sure thing?" Is it certain that Bill was hurt?

The answer is, no! How can we tell? We can determine whether or not this conclusion is a sure thing by examining its form. So, we should do two things in general: 1) we should judge the argument in terms of the three ingredients of good form, and 2) we should judge the argument by means of the "if" test which in part expands the consistency ingredient.

Does the argument conform to the principles of consistency, sufficiency and proper dependence? There is no inconsistency immediately apparent. But the other two ingredients seem to be lacking. The premise is not sufficient to establish the conclusion. The information given is insufficient to ground the conclusion. Knowing that Bill fell onto concrete is not, by itself, sufficient reason to believe he was hurt. Regarding proper movement, observe that there is nothing explicit in the conclusion that was already implicit in the premise. This argument is invalid. It is neither deductively certain nor inductively probable because it lacks good form.

Applying the "if" test we ask two questions. First, must the conclusion be true if we grant the premise? No, since we lack sufficient information to establish a "must."

Second, if the conclusion were false and "John walked away unhurt" were true, would we have a contradiction between the premise and the conclusion? No! And this fact that there would be no contradiction shows us that the argument cannot be DV, though by this test it would be IV if it had good form (but it does not have good form, so is it not even inductively valid).

2C. Begin with "most who fall"

Does this argument conform to the principles of consistency, sufficiency, and proper dependence? Yes. There is no inconsistency immediately apparent. The premises are sufficient to establish the conclusion. And the conclusion is already present in the premises implicitly. The premises are relevant to the conclusion in such a way that they hit the target. But do we have a bull's eye? Given that the argument has good form, we still need to determine whether or not the conclusion is certain. For this determination we apply the "if" test.

The first question of the "if" test is "if the premises are granted, must the conclusion follow?" To which we give the obvious answer, no, because the use of "most" in the first premise leaves an element of uncertainty. Applying the "if" test to an aspect of consistency, we ask, "if the conclusion were false and John walked away unhurt, would there be a contradiction between the premises and conclusion?" No! Therefore, the argument is not DV. But since it has good form, it is IV.

3C. Begin with "all who fall"

We should immediately note with this third version of the argument that it has all the ingredients of good form and that given the truth of the premises two things follow: 1) the conclusion must be true, and 2) if the conclusion were false and John walked away unhurt, there would be an obvious contradiction. Therefore, the argument is DV. It not only has good form but it also has the kind of good form that reaches the goal of certainty in the conclusion.

We can now explain why the first version of this argument appears so certain upon first impression. It is because we tend to supply information that we intuitively presuppose to be implied by the information actually given. There is nothing wrong with this procedure in general. Many assumptions must be made in the communication of ideas from one person to another. Some cautions are in order, however. 1) First, we must be conscious of this tendency to take information that is given and to "fill it out." Being aware will help us curb making excessive, unfair, or unreasonable assumptions. 2) Second, when the "filling out" procedure involves assumptions which are critical to understanding and evaluating an argument, we need to be either charitable enough to give the benefit of the doubt where possible or inquisitive enough to seek clarification. Therefore, when we add enthymemes the goal must be to represent the flow of thought as accurately as possible for the sake of fairness, charity, and truth. Evaluation should include searching for all the good we can possibly find (cf. OMH and wisdom in the use of logic). 

These cautions also remind us of the inevitable overlap between logic and ethics.  Logic involves communication in an exchange of statement sets between speaker and listener.  But without fairness and charity, logical skill is severely handicapped.

4C. Note the overt contradiction (without the "if" test)

This argument is obviously invalid since the form lacks consistency. But the inconsistency is not direct because the argument is enthymatic. The missing part is "all people who are killed are hurt and do not walk away." Once this is discerned, and it is obvious, then the contradiction is directly apparent.  Remember that overt contradiction commits the inconsistency fallacy rendering the argument invalid while contradiction in doing the "if" test shows that the argument is DV.

Therefore, one of the things to look for when arguments necessitate "filling out" is the enthymeme, which should be charitably supplied. In this case, however, we are reminded that being reasonably charitable has its obvious limits.

Also note that when the enthymeme is reasonably supplied we have a deductive argument. But it is not DV because a DV argument with true premises must have a true conclusion. However, had we supplied the unreasonable notion that "most who are killed are hurt and do not walk around" then the conclusion, "John was not hurt and walked away," could be true or false without contradiction. Stated this way the argument has good form, could have a false conclusion without contradiction, and is then IV.

5C. Most barbers in Dearborn

Let's take another look at the argument we discussed in the previous chapter about the barbers in Dearborn in order to see what may happen to a DV argument when it is modified into an inductive argument:

                      

           DV                                                                    Inductive

We have titled the argument on the left DV but the one on the right merely inductive. What we want to do is test the inductive argument for validity.

Immediately, the apparent inconsistency between the name McMahon and being Greek should be observed. However, it is possible that both could be true. Therefore, in the DV example, if the premises are true the conclusion must be true. But in the inductive example, the apparent inconsistency carries more weight because we know that there may be barbers in Dearborn who are not Greek from the first premise. So, we have the following tension: the fact that McMahon is a barber in Dearborn and that most barbers in Dearborn are Greek suggests that McMahon is Greek while his name suggests that he is not Greek (on the assumption that most McMahon's are not native Greek) and non-Greek barbers in Dearborn is a possibility. Therefore, the inductive argument must be judged invalid because the premises are insufficient to establish the probability of the conclusion (of course, there is possibility). Notice that the argument is not said to fail because of inconsistency but because of insufficiency. In both cases, inconsistency is merely possible. For the deductive argument this possibility is not enough to cause us to question its form; whereas, in the inductive argument it is.

3A. Implications

Some basic implications emerge from the fact that both deductive and inductive arguments can be valid.

1B. Degrees of certainty

First, this fact shows that arguments with good form can reasonably support conclusions that have varying degrees of certainty. Consider the following valid argument:

Perhaps this argument is pessimistically constructed since 99% of Farmer Brown's apples are not wormy. Nevertheless, it has some merit. Given the premises this possibility obtains although it is highly uncertain. But is it DV? When the intent of the argument is put like this, "Therefore, it must be the case that this apple is possibly wormy," then the argument is deductively valid. But since this could be said with regard to all inductive arguments we learn two things: 1) a deductive principle governs all argumentation; deduction to a necessary conclusion under girds both deductive and inductive argumentation, 2) when probability is what is reached through the deductive process, it is helpful to sub-divide this type of argument into a category by itself, namely, inductive argumentation. The worm argument has low probability and therefore is inductively invalid even though the possibility is reached with deductive certainty. Instead of confusing the picture by saying that there are deductively valid arguments that are inductively invalid because they have low probability, it is better to simply call the argument inductively invalid.

By contrast, the following optimistic argument is also invalid:

To conclude from these premises that the snake is "probably" not poisonous will not do. The only way the evidence will be sufficient for the degree of certainty claimed is when the conclusion uses the term "possibly." Reverting to the 99% that are poisonous we arrive at the probability that this one is poisonous (though it might not be).

Therefore, we need a benchmark for the difference between possibility and probability. Our benchmark will be the 50/50 ratio. Thus, anything that is 49% or less likely  to be the case has some probability but it is so inadequate that we will not grant the argument any worth as in the 1% poisonous snake argument above. That is, probability is established by showing better "odds" than 50/50. We can, however, argue for the possibility of something by means of low percentages as in the 1% of the apples argument above. But arguing possibility can be impractical and unwise.  Granting the 99% range of danger, it is unreasonable to put oneself at risk on the basis that it is slightly possible to avoid being poisoned to death!

2B. Form verses Content

Regarding validity, whether deductive or inductive, the issue is good form in the relationship of the argument parts. The issue in the matter of validity is not the content conveyed in the statements that make up the argument. Notably, even the content or truth of the conclusion is not an issue in good form. To nail this down, it should be observed that a valid argument could be false throughout as the Lions/Redskins example shows:

  • The Lions and the Redskins won their conference title.

  • Conference title winners go to the super bowl.

  • The Lions will play the Redskins at the super bowl.

Is this argument DV? Yes, absolutely! Try the "if" test on it. If we grant the premises, must the conclusion follow? Yes. If we grant the premises and if the stated conclusion were false, would we have a contradiction? Yes. Validity looks to the form of the argument and not to its content per se. That is why so much stress is placed on the if in the "if" test.

 

DV – IV WS1

1. Given only the explicit information stated, is the following argument deductive or inductive and is it valid or invalid? Support both of your answers.

  • The battery on Mary's car is dead.

  • Therefore, Mary's car will not start.

2. Explain why the following is not only deductive it is also deductively valid. Also what is wrong here?

  • The Lions and Redskins won their conference title.

  • Conference winners go to super bowl.

  • So: The Lions and Redskins go to super bowl.

3. t f A valid argument can have a false conclusion.

4. t f A deductively valid argument must have a true conclusion.

5. t f A deductively valid argument could not have a false premise.

6. t f An argument with true premises must have a true conclusion if it is inductive and valid.

7. t f If you falsify a conclusion of a deductively valid argument you will not expose a contradiction.

8. Explain the two aspects of the "if" test for determining DV verses IV.

 

9. Explain three differences between the following arguments:

 

 

10. Are the following valid? If not, why not? If so, are they DV or IV? Formulate a defense of your answer.

a. All dangerous spiders native to the United States are spotted. If correctly reported the spider Jane stepped on in Soap Lake, Washington, was solid brown. Therefore, the spider Jane squashed was harmless.

 

b. The Ford transmission design has remained constant for the past five years. Since I now have trouble with my second gear on my Ford after only 12 months, and since you and I use second in a similar fashion, then you will probably have trouble with the second gear on your new Ford after 12 months.

 

c. Since the evidence for personhood of the unborn is very strong though uncertain, and since the Scriptures warn against even accidental killing of a human being, then the unborn should be treated as persons (cf. Deut. 19:4-7, Frame, Doctrine 271).