Logic: how do you analyse the structure of inductive arguments? - Chapter 4

As mentioned in earlier chapters, care must be taken when generalizing claims and combating them. For example, if a doctor says that a vaccination against hepatitis A prevents you from being infected with hepatitis A, you can give as a counter argument that you know someone who was infected after vaccination. This is because the vaccine offers 99% protection. This counter argument barely disputes the doctor's claim, you could even say that it doesn’t; what he most likely meant to say is that a vaccination makes the chance of infection very small.

Inductive power

An argument is inductively powerful if it is not deductively valid, but still the conclusion is probably true. An example of this is:

P1) Fiona lives in Inverness, Scotland

P2) Almost everyone in Inverness, Scotland, owns at least one woollen garment

C) Fiona has at least one woollen garment

Although this argument is not deductively valid, but based on the assumptions, the conclusion is plausible. We can solve this by putting the word "probable" before the conclusion. Inductive power has to do with probability.

Probability

The probability or probability that a proposition is true is often expressed on a numerical scale between 0 and 1. There are three ways to explain probability: proportion, frequency and rational expectation. The proportion is expressed in quantifiers such as "most" and "7/8 of". For example, the chance to draw an ace from a deck of cards is 4/52 (0.077). With the frequency you look at how often something occurs. If it has snowed 14 times in the last 100 winters, you can infer that the chance that it will snow in the coming winter is 14/100 (0.14). The degree of rational expectation is our most common concept of probability calculation. Here you reason what the most likely option based on the evidence you have (for example, frequencies or proportions). Conditional probability calculation plays a role in this. For example, if you have a closed card in front of you, with the only information that it is a red card, you know the conditions for calculating the probability. Because spades and clubs are black, you can argue that there is ½ chance that it is a heart card, and ½ chance that it is a diamond card.

An argument is therefore inductively powerful if the conditional probability of the conclusion relative to the premises is greater than ½, but less than 1. The argument is still not deductively valid, but it is more likely that the conclusion is true than not true. There are a number of important points to keep in mind when talking about probability and inductive power:

1. Sometimes opportunities are not expressed as conditional opportunities. For example, we assume that the probability of an ace is 1/13, assuming that the conditions are met (for example, that the deck of cards is complete). However, this type of implicit premise must always be kept in mind.
2. Opportunities should not be seen as an alternative to the truth or as an intermediate form between true and false. Opportunities say something about how likely it is that a proposition is true.
3. Probability calculation is a matter of the degree to which something is likely. However, this is not the case with truth; something is either true or false. In contrast to deductive validity, inductive power is therefore also a question of probability
4. The everyday use of terms such as "probable" and "most" is often vague; unless specified, it merely indicates that the probability that something is true is more than ½. Often, however, a substantially larger part than half is meant. When assessing inductive arguments, it is important to indicate the degree of inductive power as accurately as possible.
5. Rational expectation of probability is about what is reasonable in reality, not what a particular person thinks is reasonable.
6. The degree of inductive power depends on the truthfulness of the assumptions. The receiver will often (just like the sender) have an "intuitive" or emotional assumption of something and perceive this as truth as part of the consideration. In everyday language, not everything has a mathematical meaning.

Inductive accuracy

An argument is inductive if it is inductive and the premises are true. However, unlike deductive accuracy, the conclusion does not necessarily have to be true; Inductive accuracy is about probability. The premises of an argument can also contain opportunities instead of quantifiers, for example the word "likely".

Arguments with multiple probability assumptions

If probability elements occur in several premises, it is possible that the inductive power is reduced by the collective opportunities in the premises. For instance:

P1) Jon probably left home on time.

P2) Jon probably wasn't in a traffic jam.

P3) If Jon has left home on time and has not been in a traffic jam, he will arrive at work on time.

C) Jon is probably on time at work

This argument is not inductive because the collective probability of P1 and P2 exceeds the probability of ½.

With the above argument the premises are independent (they each support the conclusion separately), but premises can also be dependent . For instance:

P1) Most Americans are people born in America

P2) Most people born in America are white.

C) Most Americans are white.

This argument is not inductively powerful. Strictly speaking, "most" can also mean just over half. Then it could be that only a little more than ¼ (½ x ½) of the Americans are white and the conclusion would therefore be incorrect. The same problem can occur with dependent premises that contain the word "likely".

Inductive power in extensive arguments

The conclusion of an inductively powerful argument can serve as a premise for a subsequent conclusion that in turn can be deductively valid, inductively powerful or neither. However, if any sub argument of an extensive argument is not deductively valid, the entire argument is by definition not deductively valid (but possibly deductively powerful).

Conditional opportunities in the conclusion

A conclusion can also contain the word "likely", as in the example below:

P1) Most students who study mathematics complete their propaedeutic year in one year

C) If you are going to study mathematics, you are likely to get it in one year.

Proof

In some cases, for example when gathering evidence for a murder case, multiple premises together can provide inductive power. One of the premises (person X had the option of committing the murder) would then not be inductively powerful, but with multiple premises together it would be (for example: person X had the wish to see the victim dead).

Inductive inferences

Inductive inference is the term for extrapolating a sample from a total population to the entire population. There are two types of inductive interference: in the first you make the inference from a certain number of observations to one or a few new cases, in the second you make the inference from a certain number of observations to all cases.

With both types it is important to determine how representative the sample is for the entire population. View the following example:

P1) Every observed bear is white.

C) All bears are white.

Since not all bears are white, the sample in this case was not very representative (for example, only polar bears were observed). When determining the representativeness of a sample, it is important to look at all factors that may have correlations with the observation (in the case of the polar bear: living environment, weather conditions, etc.). In general, the more representative and larger the sample, the stronger the inductive inference. Moreover, an inference to "most" of the total population is stronger than an inference to "all".

Induction and deduction

An inductive reasoning can be all cats that I have seen in my life have pointed ears, so cats have pointed ears. You can convert this into a deductive argument by saying: (A) all cats that I have seen in my life have pointed ears + (B) if all cats that I have seen have pointed ears, then all cats have pointed ears. Makes: (C) so all cats have pointed ears.
There is nothing wrong with the reasoning in itself, although in reality it is more likely to say, if all the cats I have seen have pointed ears, then probably all cats have pointed ears.

Another problem that may arise with these reasoning is to explain with the following example. (A) All Party for the Animals voters are vegetarian. (B) 1 in 20 non-Party for the animal voters are vegetarians. (C) Carl is a vegetarian, so he votes Party for the Animals.

Even though this sounds very likely, it is unlikely. Suppose 2% of the 16 million Dutch people vote for Party for the Animals, that is 320,000 Dutch people. In addition, 5% (1 in 20) of those 16 million Dutch people are vegetarian, that is 800,000. The chance that you will come across a vegetarian who does not vote for the Party for the Animals is therefore greater than that he or she votes for the Party for the Animals.

An assessment programs

Argument assessment falls into two categories: logical assessment (assessment of deductive validity and inductive power) and factual assessment (assessment of the truthfulness of the premises). However, the evaluation of inductive power is not completely independent of the facts in the way that is the case when assessing validity. In the case of inductive power, the assessment also depends on factors such as the representativeness of the sample. For an assessment model of arguments, see p. 123 of the book.