There are three kinds of arguments, and each of them differs from the oth- ers in virtue of having a different standard of strictness. In a deductively valid argument, if the premises are true, the conclusion must be true (by logical necessity). Thus, in a deductive inference, the link between the premises and the conclusion is strict. For example, consider the follow- ing argument, where the first premise is taken to be an absolute universal generalization with no exceptions.
PREMISE: All police chiefs are honest.
PREMISE: John is a police chief.
CONCLUSION: John is honest.
In this argument, there is no room for doubting that the conclusion is true, once the premises have been accepted as true. The reason stems from the word ‘all’ in the first premise of the inference. Presumably, ‘all’ means ‘all without exception’. If so, the statement in the first premise is an absolute universal generalization. Thus, the conclusion follows by logical necessity from the premises. If both the premises are true, then the conclusion has to be true.
Deductive validity can be defined in another way that offers an even more useful criterion to recognize it in an argument. To say that an argu- ment is deductively valid means that it is logically impossible for all the premises to be true and the conclusion false. In other words, in a deduc- tively valid argument, the claim that the premises are true and the con- clusion false is inconsistent. For example, let’s consider three statements
comparable to the argument above, with the first two the same as the premises, but with the third as the opposite of the conclusion.
FIRST STATEMENT: All police chiefs are honest. SECOND STATEMENT: John is a police chief.
THIRD STATEMENT: John is not honest.
This set of three statements is collectively inconsistent. You cannot main- tain all three at the same time without being inconsistent. This observa- tion leads to the best test to identify deductive validity in an argument. If the premises, taken together, are inconsistent with the negation of the conclusion, the argument is deductively valid.
In an inductive argument, the link is not so strict. If the premises are true, the conclusion is probably true, but it could possibly be false. Deter- mining whether a given argument is meant to be deductive or inductive, as used in a given case, can be quite difficult in some cases and requires consideration of several criteria and several kinds of evidence. One of the main criteria is the nature of the inferential link between the premises and the conclusion. It is this link that determines whether the argument is a successful deductive argument as opposed to being a successful inductive argument. To give the beginner an entry point into grasping the distinction between inductive and deductive as types of arguments, here we concen- trate exclusively on the nature of this link. Inductive arguments are based on probability. An example is the following argument.
Most swans are white. This bird is a swan.
Therefore, this bird is white.
In this argument, the first premise is an inductive generalization. It is not said to be true of all swans, but only of most of them. In this case, if the premises are accepted as true, then the conclusion is probably (but not necessarily) true. It could be that this bird is one of those black swans. What makes such an argument inductive rather than deductive is the link between the premises and the conclusion. But the fact that one premise is an inductive rather than a universal generalization is a good indicator that the argument is inductive. Such an inductive premise, in the sample argument above, limits the conclusiveness or strength of the argument. It cannot be used to show that the conclusion follows from the premises necessarily, but only inductively, as a matter of probability.
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Inductive arguments are based on probability1and statistics. The sup-
port for an inductive argument is typically given by gathering empirical evidence. In the basic cases, the evidence takes the form of enumerating or counting objects or polling individuals. The outcomes are expressed in numbers that are processed, by the methods currently in use in statis- tics, to generate an inference evaluated as a probability. The following is a typical example of a numerical inductive argument.
Seventy percent of residents of Tutela Heights vote Conservative. Ned is a resident of Tutela Heights.
Therefore (probably) Ned will vote Conservative.
In this argument, the conclusion has a certain degree of probability relative to the premises. A contrasting deductive argument would be the following case, where the warrant is absolute. Let’s say, at any rate, for purposes of illustration, that the word ‘all’ in the first premise is taken to mean ‘absolutely all, with no exceptions.’
All residents of Tutela Heights reside in Brant County. Ned is a resident of Tutela Heights.
Therefore, Ned is a resident of Brant County.
In this deductively valid argument, it is impossible for both the premises to be true and the conclusion false. The absolute generalization in the first premise excludes any possible contrary instances. However, in the inductive argument above, room is left for the possibility of the premises being true and the conclusion false. It could be that Ned is one of those residents of Tutela Heights who does not vote Conservative.
Deductive argument is a simple ‘yes’ or ‘no’ affair: Either the argument is deductively valid or not. If not, it is called invalid. Using the inductive method of evaluation, an inference is evaluated as inductively stronger or weaker to the degree that the premises would, if true, support the con- clusion as true. The measure of probability used in inductive evaluation
1It is hard to define the term ‘probability’ precisely because there is disagreement among
experts on exactly how to define it. Some think it should be defined as statistical frequency of the occurrence of an event. Others think it should be defined in terms of degrees of rational belief. Still others think it should be defined by axioms of the probability calculus. It would be a mistake to get worried about such subtleties at this point. It is enough for us to be aware that probability is calculated by statisticians by attaching numbers to statements (fractions between zero and one) that are supposed to measure the likelihood that such statements are true or false and numbers that measure confidence in inferences based on them.
makes the strength of support a matter of degree. The argument link could be very strong, if the premises support the conclusion with a higher degree of probability. Or the argument link could be weak, if the premises support the conclusion with only a low probability, as in the argument “Twenty percent of residents of Tutela Heights vote Conservative; Ned is a resi- dent of Tutela Heights; therefore (probably) Ned will vote Conservative.” Or the degree of inductive support could be somewhere in between. In many cases, this degree of support can be measured exactly by the meth- ods currently used in statistics. Both deductive and inductive arguments can be evaluated using exact methods of calculation.
The third type of argument is less precise and reliable than the other two, but is often more useful and even necessary, in many cases in the practical conduct of affairs of everyday life. This type of argument leads to a conclusion that is plausible, and that may be provisionally acceptable as a presumption. To say that it is plausible means that it seems to be true, on the given appearances. Of course, appearances can be misleading in some cases. Thus such an inference is inherently subject to retraction. It is defeasible, meaning that it may turn out to fail (default) if new evidence comes in. The conclusion is indicated as presumably true on a basis of plausibility, and therefore tentatively acceptable, given that the premises are true. Consider the following argument.
Where there’s smoke there’s fire. There is smoke in Buttner Hall. Therefore there is fire in Buttner Hall.
Notice that, in this case, the premise ‘Where there’s smoke there’s fire’ is not taken as an absolute universal generalization. It does not mean that all places where smoke is seen are places where there is fire. It is better taken as a defeasible statement meaning that generally, but subject to excep- tions, if you see smoke someplace, you can presume that there is fire in that place. Even though both premises of the argument above are true, it is possible that the conclusion is false. It is possible that there is a column of smoke rising from Buttner Hall, but there might be no fire there, just a smoldering mass of some substance that gives off a lot of smoke. And it is not practical to try to judge the strength of the argument by numer- ical data about fires, because this case is an individual one with many circumstantial factors that are relevant. But in such a case, on grounds of safety, it may be prudent to operate on a presumption. It may be the right conclusion to draw by presumptive inference that there is a fire in Buttner
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Hall. This can be the right conclusion to act on even though I don’t know the probabilities and even if can I see no direct evidence of fire. For prac- tical purposes, drawing the conclusion that there is or may be fire there is the sensible option, provided I have no evidence indicating otherwise, for example, a news bulletin that the smoke is being caused by a smudge pot as part of the making of a movie. The smoke rising from Buttner Hall can be good enough evidence of fire to justify calling the fire department in the absence of contra-indicating evidence. But, as noted, presumptive reasoning is inherently provisional in nature and should be used with caution. It is applicable where a conclusion needs to be drawn, yet not enough is known about a situation to use a more exact or reliable method of drawing it. It is appropriate where, for practical reasons, under con- ditions of uncertainty and incomplete knowledge, a tentative conclusion needs to be drawn as a provisional basis to continue a line of reasoning or adopt a policy for action.
These three types of argumentation are relatively independent of each other. If an argument is deductively valid, the (conditional) probability that the conclusion is true, given that the premises are true, is 1.0 (the highest possible probability value a proposition can have). So you could say that the argument is inductively strong. But judging such an argument by inductive standards and methods would not be particularly useful. It is more useful to simply say that the argument is deductively valid. If an inference to a conclusion can be supported or refuted very effectively by inductive methods, then the need or usefulness of judging it as plausible or not as a presumptive inference falls away. In general, if an argument can be evaluated on a basis of probability, then evaluating it as plausible or implausible becomes less useful. Methods of plausible reasoning give way to inductive evidence, if it is available. Similarly, inductive evaluation gives way to deductive logic, if it can be usefully applied to a case.
EXERCISE 2.2
Identify the premises and the conclusion in each of the following infer- ences. Identify the generalization in the inference, and judge whether the inference is of the deductive, inductive, or presumptive type.
(a) All swans are birds. Beverly is a swan. Therefore, Beverly is a bird.
(b) Anyone who fails to reply to this memo will be presumed to be in agreement. Bob failed to reply to this memo. Therefore, Bob is in agreement.
(c) The typical working person cannot afford to fly on the Concorde. Frank is a typical working person. Therefore, Frank cannot afford to fly on the Concorde.
(d) Cinnabar always contains mercury. This object is cinnabar. There- fore, this object contains mercury.
(e) Advocates of a cause do not find it easy to compromise their group interests. Helen is the advocate of a cause. Helen does not find it easy to compromise her group interests.
(f) Seventy percent of the birds in this zoo fly. Tweety is a bird in this zoo. Therefore, Tweety flies.
(g) Wayne normally takes his Jeep when he leaves home. Wayne’s Jeep is not in the driveway. Therefore, Wayne is not home.
(h) Conservatives are against raising taxes. Bob is a Conservative. Therefore, Bob is against raising taxes.
(i) If Minnesota is in Canada, then it is north of the Canadian border. Minnesota is in Canada. Therefore, it is north of the Canadian bor- der.
(j) Nancy is an honest person. Whatever an honest person says should be taken as true. Nancy said that Peter does not like Denise. There- fore, Peter does not like Denise.