Statistical syllogism

Last updated December 25, 2023

A statistical syllogism (or proportional syllogism or direct inference) is a non-deductive syllogism. It argues, using inductive reasoning, from a generalization true for the most part to a particular case.

Introduction

Statistical syllogisms may use qualifying words like "most", "frequently", "almost never", "rarely", etc., or may have a statistical generalization as one or both of their premises.

For example:

Almost all people are taller than 26 inches
Gareth is a person
Therefore, Gareth is taller than 26 inches

Premise 1 (the major premise) is a generalization, and the argument attempts to draw a conclusion from that generalization. In contrast to a deductive syllogism, the premises logically support or confirm the conclusion rather than strictly implying it: it is possible for the premises to be true and the conclusion false, but it is not likely.

General form:

X proportion of F are G
I is an F
I is a G

In the abstract form above, F is called the "reference class" and G is the "attribute class" and I is the individual object. So, in the earlier example, "(things that are) taller than 26 inches" is the attribute class and "people" is the reference class.

Unlike many other forms of syllogism, a statistical syllogism is inductive, so when evaluating this kind of argument it is important to consider how strong or weak it is, along with the other rules of induction (as opposed to deduction). In the above example, if 99% of people are taller than 26 inches, then the probability of the conclusion being true is 99%.

Two dicto simpliciter fallacies can occur in statistical syllogisms. They are "accident" and "converse accident". Faulty generalization fallacies can also affect any argument premise that uses a generalization. A problem with applying the statistical syllogism in real cases is the reference class problem: given that a particular case I is a member of very many reference classes F, in which the proportion of attribute G may differ widely, how should one decide which class to use in applying the statistical syllogism?

The importance of the statistical syllogism was urged by Henry E. Kyburg, Jr., who argued that all statements of probability could be traced to a direct inference. For example, when taking off in an airplane, our confidence (but not certainty) that we will land safely is based on our knowledge that the vast majority of flights do land safely.

The widespread use of confidence intervals in statistics is often justified using a statistical syllogism, in such words as "Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time."^[1] The inference from what would mostly happen in multiple samples to the confidence we should have in the particular sample involves a statistical syllogism.^[2] One person who argues that statistical syllogism is more of a probability is Donald Williams.^[3]

History

Ancient writers on logic and rhetoric approved arguments from "what happens for the most part". For example, Aristotle writes "that which people know to happen or not to happen, or to be or not to be, mostly in a particular way, is likely, for example, that the envious are malevolent or that those who are loved are affectionate."^[4]^[5]

The ancient Jewish law of the Talmud used a "follow the majority" rule to resolve cases of doubt.^[5]^: 172–5

From the invention of insurance in the 14th century, insurance rates were based on estimates (often intuitive) of the frequencies of the events insured against, which involves an implicit use of a statistical syllogism. John Venn pointed out in 1876 that this leads to a reference class problem of deciding in what class containing the individual case to take frequencies in. He writes, “It is obvious that every single thing or event has an indefinite number of properties or attributes observable in it, and might therefore be considered as belonging to an indefinite number of different classes of things”, leading to problems with how to assign probabilities to a single case, for example the probability that John Smith, a consumptive Englishman aged fifty, will live to sixty-one.^[6]

In the 20th century, clinical trials were designed to find the proportion of cases of disease cured by a drug, in order that the drug can be applied confidently to an individual patient with the disease.

Problem of induction

The statistical syllogism was used by Donald Cary Williams and David Stove in their attempt to give a logical solution to the problem of induction. They put forward the argument, which has the form of a statistical syllogism:

The great majority of large samples of a population approximately match the population (in proportion)
This is a large sample from a population
Therefore, this sample approximately matches the population

If the population is, say, a large number of balls which are black or white but in an unknown proportion, and one takes a large sample and finds they are all white, then it is likely, using this statistical syllogism, that the population is all or nearly all white. That is an example of inductive reasoning.^[7]

Legal examples

Statistical syllogisms may be used as legal evidence but it is usually believed that a legal decision should not be based solely on them. For example, in L. Jonathan Cohen's "gatecrasher paradox", 499 tickets to a rodeo have been sold and 1000 people are observed in the stands. The rodeo operator sues a random attendee for non-payment of the entrance fee. The statistical syllogism:

501 of the 1000 attendees have not paid
The defendant is an attendee
Therefore, on the balance of probabilities the defendant has not paid

is a strong one, but it is felt to be unjust to burden a defendant with membership of a class, without evidence that bears directly on the defendant.^[8]

Related Research Articles

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. More generally, hypothesis testing allows us to make probabilistic statements about population parameters. More informally, hypothesis testing is the processes of making decisions under uncertainty. Typically, hypothesis testing procedures involve a user selected tradeoff between false positives and false negatives.

A fallacy, also known as paralogia in modern psychology, is the use of invalid or otherwise faulty reasoning in the construction of an argument that may appear to be well-reasoned if unnoticed. The term was introduced in the Western intellectual tradition by the Aristotelian De Sophisticis Elenchis.

Abductive reasoning is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by American philosopher Charles Sanders Peirce beginning in the latter half of the 19th century.

Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.

First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inference from the observed to the unobserved is known as "inductive inferences". Hume, while acknowledging that everyone does and must make such inferences, argued that there is no non-circular way to justify them, thereby undermining one of the Enlightenment pillars of rationality.

A faulty generalization is an informal fallacy wherein a conclusion is drawn about all or many instances of a phenomenon on the basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an example of jumping to conclusions. For example, one may generalize about all people or all members of a group from what one knows about just one or a few people:

Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.

The term Inductive reasoning is used to refer to any method of reasoning in which broad generalizations or principles are derived from a body of observations. This article is concerned with the inductive reasoning other than deductive reasoning, where the conclusion of a deductive argument is certain given the premises are correct; in contrast, the truth of the conclusion of an inductive argument is at best probable, based upon the evidence given.

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In logic and philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.

Probabilistic logic involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.

Fiducial inference is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of frequentist inference, Bayesian inference and decision theory. However, fiducial inference is important in the history of statistics since its development led to the parallel development of concepts and tools in theoretical statistics that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it.

In logic and mathematics, proof by example is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof.

An argument is a series of sentences, statements or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persuasion.

The psychology of reasoning is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory.

Plausible reasoning is a method of deriving new conclusions from given known premises, a method different from the classical syllogistic argumentation methods of Aristotelian two-valued logic. The syllogistic style of argumentation is illustrated by the oft-quoted argument "All men are mortal, Socrates is a man, and therefore, Socrates is mortal." In contrast, consider the statement "if it is raining then it is cloudy." The only logical inference that one can draw from this is that "if it is not cloudy then it is not raining." But ordinary people in their everyday lives would conclude that "if it is not raining then being cloudy is less plausible," or "if it is cloudy then rain is more plausible." The unstated and unconsciously applied reasoning, arguably incorrect, that made people come to their conclusions is typical of plausible reasoning.

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

References

↑ Cox DR, Hinkley DV. (1974) Theoretical Statistics, Chapman & Hall, pp. 49, 209
↑ Franklin, James (1994). "Resurrecting logical probability" (PDF). Erkenntnis. 55 (2): 277–305. doi:10.1023/A:1012918016159. S2CID 130621 . Retrieved 30 June 2021.
↑ Oliver, James Willard (December 1953). "Deduction and the Statistical Syllogism". Journal of Philosophy. 50 (26): 805–806. doi:10.2307/2020767. JSTOR 2020767.
↑ Aristotle, Prior Analytics 70a4-7.
1 2 Franklin, James (2001). The Science of Conjecture: Evidence and Probability Before Pascal. Baltimore: Johns Hopkins University Press. pp. 113, 116, 118, 200. ISBN 0-8018-6569-7.
↑ J. Venn,The Logic of Chance (2nd ed, 1876), 194.
↑ Campbell, Keith; Franklin, James; Ehring, Douglas (28 January 2013). "Donald Cary Williams". Stanford Encyclopedia of Philosophy. Retrieved 10 March 2015.
↑ L. J. Cohen, (1981) Subjective probability and the paradox of the gatecrasher, Arizona State Law Journal, p. 627.