Comprehensive Study Guide for Inductive Reasoning

Foundations of Inductive Reasoning

While deductive reasoning provides an invisible framework for disciplines such as mathematics, computer science, philosophy, and metaphysics, inductive reasoning accounts for the majority of human knowledge regarding the empirical workings of the world. In deductive reasoning, support for a conclusion is conclusive, necessary, and certain; a successful argument is termed valid, and a valid argument with true premises is termed sound. Inductive reasoning, conversely, provides rational support that is probable, plausible, or likely. A successful inductive argument is labeled strong, and a strong argument with true premises is termed cogent.

Inductive reasoning allows for a transition from observed phenomena to unobserved phenomena, effectively permitting humans to reason beyond the provided evidence. For instance, a deductive argument might state: (1) If it is raining, then it is cloudy; (2) It is raining; (3) So, it is cloudy. This preserves the truth of the premises, but inductive reasoning is required to establish the truth of the first premise itself. We believe a claim like "Whenever it rains, there are clouds" because every time we have observed rain, we have seen clouds, and we have never seen rain without clouds. Inductive reasoning serves as the license for the leap from "So far, every time we have seen rain, we have seen clouds" to the generalization that "Whenever it rains, there are clouds."

Enumerative Induction

Enumerative induction is an inductive argument pattern where reasoning proceeds from premises about individual members of a group to conclusions about the group as a whole. This moves from particulars to the general, or from the part to the whole. This pattern begins with observations about specific members and ends with a generalization. For example, if all observed swans are white, one might conclude that all swans are white. Similarly, if $20\%$ of observed thunderstorms produce hail, one might infer that $20\%$ of all thunderstorms produce hail.

In the formal structure of enumerative induction, the group as a whole is defined as the target group or target population. The observed group is the sample, and the property of interest is the relevant property or property in question. The generalized form is: (1) $X\,percent$ of the observed members of group $A$ have property $P$ ; (2) Therefore, $X\,percent$ of all members of group $A$ probably have property $P$ . Note that "every member so far observed" is equivalent to $100\%$ .

Evaluating Enumerative Induction

To determine the strength of an enumerative induction, two primary criteria must be considered: sample size and representativeness. In general, a larger sample size leads to a stronger inductive inference. A sample that is too small results in a weak induction. Common sense often dictates the necessary size, but a reliable rule of thumb is that the more homogeneous a target group is regarding the relevant property, the smaller the sample can be. Conversely, heterogeneous groups require larger samples. For example, physiological features like DNA are generally uniform across humans, requiring smaller samples, while culturally-dependent features like ethical mores vary significantly.

Representativeness requires that the sample resembles the target group in all relevant ways. A sample must possess the same relevant characteristics as the target group in the same proportions. Relevant characteristics include any features that could influence the property in question, such as political affiliation, ethnic background, religious background, or sex. If a sample fails to represent the target group, it is considered a biased sample, which provides weak support for generalizations. Biased sampling can result from ignoring contrary evidence or from confirmation bias.

When relevant factors are unknown or too complex, researchers utilize random sampling. This is the opposite of self-selecting samples, where subjects choose themselves, such as in magazine questionnaires or website polls. Self-selecting samples are problematic because the factors driving someone to volunteer input rarely represent the broader target population, similar to the disproportionate nature of online reviews for the DMV.

Opinion Polls and Statistical Margins

Opinion polls are sophisticated enumerative inductions. Their accuracy depends on phrasing, question ordering, and the availability of choices. Even well-conducted polls will produce different results, leading to a margin of error. For instance, if a poll states a candidate will receive $62\% \pm 3\%$ , the actual percentage in the target population is likely between $59\%$ and $65\%$ .

Connected to the margin of error is the confidence level, which is the probability that the sample accurately represents the target group within the specified margin. A confidence level of $95\%$ means there is a $95\%$ chance that the results from the sample reflect the entire population. Generally, a larger sample size results in a lower margin of error. If one accepts a lower confidence level, the sample can be smaller; conversely, a larger allowed margin of error increases the confidence that results will fall within that range.

Analogical Induction

An analogy is a comparison of two or more things that are alike in specific respects. It anchors the unfamiliar in the familiar to illuminate new knowledge. As an argument pattern, analogical induction (or argument by analogy) follows this form: (1) Thing $A$ has properties $P_1$ , $P_2$ , and $P_3$ , plus property $P_4$ ; (2) Thing $B$ also has properties $P_1$ , $P_2$ , and $P_3$ ; (3) Therefore, thing $B$ probably has property $P_4$ .

This reasoning is common in several fields:

Medical Science: If mice (mammals with specific circulatory systems) respond to Drug $Z$ with reduced cholesterol, and humans share those mammalian features, then humans will likely experience reduced cholesterol from Drug $Z$ .
Law: If a previous court case (precedent) involving school-sponsored prayer was ruled unconstitutional, and a current case involves similar school-sponsored prayer, the court should rule the current case unconstitutional.
Theology: If a watch is a complex mechanism adjust for a purpose by a designer, and the universe is a complex mechanism with parts precisely fitted for effects, then the universe must have a designer.

Evaluating Arguments by Analogy

Four criteria determine the strength of an analogical induction:

Relevant Similarities: The similarities must be plausibly linked to the conclusion. In opening a second food truck, demographic and foot traffic similarities are relevant, whereas both trucks being near an oak tree is irrelevant.
Relevant Dissimilarities (Disanalogies): The more relevant differences there are between the items compared, the weaker the conclusion. If the universe resembles a living thing (which is not designed like a watch) as much as it resembles a mechanism, the analogy for a designer is weakened.
The Number of Instances Compared: Comparing a current situation to several past cases is stronger than comparing it to a single instance.
Diversity Among Cases: Greater diversity among cases with relevant similarities reduces the chance that the similarities are irrelevant, licensing a deeper inference.

Consider Eric H. Cline's argument in "1177: The Year Civilization Collapsed." He draws an analogy between the Late Bronze Age Collapse and modern civilization based on five factors: hyper-interconnected globalized trade, climate shifts, mass migrations, supply chain disruptions, and new technologies. The argument is strong in its diversity of factors and their relevance to collapse but weakened by the fact that there is only one case for comparison. Furthermore, a significant dissimilarity is that we now possess knowledge of the Bronze Age Collapse, allowing for a potential response that the Bronze Age civilizations did not have.

Causal Arguments and Mill's Methods

A causal claim asserts that one event or factor directly produces an effect or change. A causal argument is an inductive argument whose conclusion contains a causal claim, answering "how" or "why" something happened. Causal arguments must be distinguished from explanations. A causal argument provides rational support for why a claim is true (e.g., "Blue light suppresses melatonin, therefore smartphones lead to poor sleep"), whereas an explanation simply asserts the reason as a fact (e.g., "Teenagers can't sleep because of their phones"). Causal arguments conclude with the cause, while explanations effectively "conclude" with the effect.

John Stuart Mill identified methods for isolating causes:

Method of Agreement: If two or more occurrences of a phenomenon share only one relevant factor, that factor is the probable cause. If three diners eat different meals but all eat the side salad and all get food poisoning, the salad is the cause.
Method of Difference: The factor present when a phenomenon occurs and absent when it does not occur is the cause. If two twins live identically but only one takes a supplement and recovers from a cold in $24\,hours$ while the other takes $7\,days$ , the supplement is the cause.
Joint Method: This combines agreement and difference. An experimental group is observed in the presence of a factor, and a control group is observed in its absence. This shows the factor is common to all effects and that without the factor, the effect does not occur.
Method of Correlation (Concomitant Variation): When two events vary in close connection, they are likely causally related. For example, lung cancer frequency rises in proportion to the number of years a person has smoked.

Common Errors in Causal Reasoning

Misidentifying Relevant Factors: This is an error of relevance. In a food poisoning case, the fact that all victims had black hair is irrelevant because hair color is not causally connected to intestinal illness.
Mishandling Multiple Factors: This can be an error of quantity where there are too many factors (e.g., trying to isolate the cause of autism among genetics, vaccines, and environment) or too few factors (e.g., reducing all social problems to economics).
Being Misled by Coincidence: Statistically improbable events are certain to happen given enough opportunities. One should not assume a causal connection without good reason to believe a common, relevant factor exists while controlling for coincidence.
Post Hoc, Ergo Propter Hoc: This fallacy assumes that because an event preceded another, the first caused the second. For example, assuming a decrease in crime was caused by new police training just because it happened afterward.
Confusing Cause and Effect: This is an error of inversion. For example, questioning whether regular exercise makes people healthy or whether healthy people are naturally more prone to exercise.

Necessary and Sufficient Conditions

Causal relations are often described through conditions:

Necessary Condition: A factor without which an event cannot occur. For fire, fuel, heat, and oxygen are each necessary.
Sufficient Condition: A factor or set of factors that guarantees an event will occur. Individually necessary conditions, when all are present, become jointly sufficient.

If the goal is to prevent an event, one must remove a necessary condition. If the goal is to produce or predict an event, one must fulfill all sufficient conditions. For example, meteorologists can accurately predict hurricanes up to $5\,days$ in advance because they know the necessary ingredients: ocean water above $79\,degrees\,Fahrenheit$ , distance from the equator, low vertical wind shear, high relative humidity, and a preexisting tropical disturbance. However, tornadoes remain difficult to predict specifically because their producing factors are too complex to determine, allowing only for warnings $13\,to\,15\,minutes$ before a strike.

Questions & Discussion

Logic Review

What are the four logical connectives? (Referred to as the invisible framework of deductive reasoning).
What are the "truth conditions" of these four connectives?
Sample translations and truth table constructions were conducted.

Exam Information

The second exam covers Chapters $7$ , $8$ , and $10$ .
The essay question focuses on Rene Descartes' Meditations.

Descartes and Meditations on First Philosophy (1641)

Why does Descartes undertake the project of doubting everything?
What, if anything, survives Descartes' attempt to doubt everything? (Discussion involves the French philosopher, scientist, and mathematician, living from $1596\,to\,1650$ ).