Inductive Logic 101 (Updated 23-Apr-14)

Inductive Logic 101 (Updated 23-Apr-14) March 23, 2014

Here is a very quick and very rough overview of inductive logic. Almost all of it is taken from sources other than me; I’ll try to identify where the material came from.

The Difference between Deductive and Inductive Arguments

Logic Type Unsuccessful Arguments Successful Arguments
Deductive Logic Invalid* Valid*
Inductive Logic Incorrect Correct

*I’m oversimplifying this somewhat by ignoring the question of whether the premises are true

The late philosopher Wesley Salmon, in his book Logic (third ed.), explained the difference between deductive and inductive arguments this way.

The premises of a logically correct inductive argument support or lend weight to the argument’s conclusion. If the premises of a correct inductive argument are true, the best we can say is that the conclusion is probably true.

“valid” and “invalid” apply only to deductive arguments; “correct” and “incorrect” should be used to describe inductive arguments. Errors that render inductive arguments either absolutely or practically worthless are inductive fallacies.

The premises of a correct inductive argument may render the conclusion extremely probable, moderately probable, or probable to some extent. Consequently, the premises of a correct inductive argument, if true, constitute reasons, of some degree of strength, for accepting the conclusion.

Types of Inductive Arguments

1. Induction by Enumeration.

Logical Form:

Z percent of the observed members of F are G.
Therefore, Z percent of F are G.

Notes:

  • If Z is is 100% or 0, then the conclusion is a universal generalization.
  • If Z is some percentage other than 0 or 100, the conclusion is a statistical generalization.

Standards for the Strength of Statistical Generalizations:

(a) Whether the sample is representative

(b) Background knowledge

2. Statistical Syllogism.

An inductive argument type that uses the conclusion of a previous statistical generalization as a premise in a quasi-syllogism.

Logical Form:

Z percent of F are G.
x is F.
Therefore, x is G.

Notes:
The first premise might be worded less precisely as “Almost all F are G”, “The vast majority of F are G,” “Most F are G,” “A high percentage of F are G,” and “There is a high probability that an F is a G.”

The class denoted by F is the reference class; the class denoted by G is called the attribute class.

A quasi-syllogism is a type of argument that is not, strictly speaking, a (categorical) syllogism. Nevertheless, it is very similar to the (categorical) syllogism and is often treated as such. Categorical syllogisms are arguments composed entirely of categorical statements. Categorical statements are statements that have one of four forms: (a) universal affirmative (All F are G); (b) universal negative (No F are G); (c) particular affirmative (Some F are G); (c) particular negative (Some F are not G).

Standards for Judging the Strength of Statistical Syllogisms (taken from Merrilee Salmon, Introduction to Logic and Critical Thinking, p. 99):

(a) The value of Z. The closer Z is to 100%, the stronger the argument.

(b) Whether all available relevant evidence has been considered in selecting the reference class. This requirement, which is called the rule of total evidence, is designed to address problems that arise as a result of individuals belonging to an indefinite number of classes.

(c) In constructing a statistical syllogism, we must take into account every class that x belongs to that might affect that probability that x is a G. Background knowledge plays an all-important role in helping us to decide which classes are relevant.

Common Fallacy in Statistical Syllogisms:

The Fallacy of Incomplete Evidence. When the reference class in a statistical syllogism is not based on all available relevant evidence, the argument commits the fallacy of incomplete evidence.

2.1. Argument from Authority.

A special case of the statistical syllogism is the argument from authority.

Logical Form:

The vast majority of statements made by x concerning subject S are true.
p is a statement made by x concerning subject S.
Therefore, p is true.

Misuses of the argument from authority:

  1. The authority may be misquoted or misinterpreted.
  2. The authority may have only glamor, prestige, or popularity.
  3. Experts may make judgments about something outside their special fields of competence.
  4. Authorities may express opinions about matters concerning which they could not possibly have any evidence.
  5. Authorities who are equally competent, so far as we can tell, may disagree.

2.1.1. Argument from Consensus.

A special case of the argument from authority is the argument from consensus.

Logical Form:

The vast majority of statements by group M concerning subject S are true.
p is a statement made by M concerning subject S.
Therefore, p is true.

Misuses of the argument from consensus: see misuses of the argument from authority.

2.2. Argument Against the Person.

Another special case of the statistical syllogism is the argument against the person.

Logical Form:

The vast majority of statements made by x concerning subject S are false.
p is a statement made by X concerning subject S.
Therefore, Not-p.

2.2.1. Negative Argument from Consensus

A special case of the argument against the person is the negative argument from consensus.

Logical Form:

The vast majority of statements by group M concerning subject S are false.
p is a statement made by M concerning subject S.
Therefore, p is false.

3. Analogy.

Logical Form:

Objects of type X have properties G, H, etc.
Objects of type Y have properties G, H, etc.
Objects of type X have property F.
Therefore, objects of type Y have property F.

Standards for Judging the Strength of Analogical Arguments:
(a) If the presence of the first increases or decreasses the probability that the second feature will also be present.
(b) The number of relevant similarities in the premisses and the number of relevant dissimilarities between the two types of objects are also important in judging the strength of analogical arguments.
(c) The number and the variety of instances mentioned in the premisses.

Fallacies Associated with Analogical Arguments:
(a) The fallacy of false analogy.

4. Hypothetico-Deductive Method.

Logical Form:

The hypothesis has a nonnegligible prior probability.
If the hypothesis is true, then the observational prediction is true.
The observational prediction is true.
No other hypothesis is strongly confirmed by the truth of this observational prediction; that is, other hypotheses for which the same observational prediction is a confirming instance have lower prior probabilities.
Therefore, the hypothesis is true.

Notes (taken from Wesley Salmon):

A statement is functioning as a hypothesis if it is taken as a premise, in order that its logical consequences can be examined and compared with facts that can be ascertained by observation. When a consequence turns out to be true, it is a confirmatory instance of the hypothesis. When a consequence turns out to be false, it is a disconfirmatory instance of the hypothesis. A hypothesis is confirmed if it is adequately supported by inductive evidence. There are degrees of confirmation; a hypothesis may be highly confirmed, moderately confirmed, or slightly confirmed. Likewise, there are degrees to which a hypothesis is confirmed by a confirmatory instance. A given confirmatory instance may add considerable support or only slight support to a hypothesis.

The hypothetico-deductive method is a kind of argument that attempts to confirm a hypothesis in terms of its deductive consequences. The argument from the hypothesis to the observational prediction is supposed to be deductive; the argument from the truth of the observational prediction to the truth of the hypothesis is supposed to be inductive.

5. Explanatory Argument

Explanatory arguments are also known as “inferences to the best explanation” (IBE) or “abductive arguments.”

Logical Form:

Hypothesis H has the greatest overall balance of background probability and explanatory power of all its competitors, A1, A2, …, An.
[Probably] Hypothesis H is true.

Notes (taken from Robert Greg Cavin and Carlos A. Colombetti):

Background probability is a function of (1) simplicity; (2) conservativism; and (3) modesty. Explanatory power is a function of (4) testability; (5) fruitfulness; and (6) explanatory scope.
(1) Simplicity is (roughly) a measure of the number of independent postulates that comprise a hypothesis.
(2) Conservativism is the degree to which a hypothesis fit in with the background facts.
(3) Modesty is (again, roughly) a measure of the content of the hypothesis, i.e., how much it says.
(4) Testability is the degree to which the hypothesis has observational consequences other than those it was originally formulated to explain.

(5) Fruitfulness is the degree to which a hypothesis has novel and daring observational consquences–typically in the form of predictions–that its competitors do not have and that are verified to be true.
(6) Explanatory Scope is the number and variety of facts that a hypothesis explains, i.e., makes more or less likely, and the degree to which it makes them likely.

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