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Patterns of Valid Arguments

As we have already seen in the previous chapter, there are certain patterns of argumentation that are always valid. These patterns form the basis for logically correct reasoning and can be applied in various contexts. Here we examine some additional important patterns of argumentation and their application.

Categorical Syllogism​

A categorical syllogism is a deductive argument consisting of three categorical statements: two premises and one conclusion.

Form:

  1. All/Some A are/are not B. (Major premise)
  2. All/Some B are/are not C. (Minor premise)
  3. Therefore: All/Some A are/are not C. (Conclusion)

Example:

  1. All planets are celestial bodies.
  2. Some celestial bodies are gaseous.
  3. Therefore, some planets are gaseous.

Caution: Not all forms of categorical syllogism are valid. Validity depends on the specific combination of quantifiers (all, some) and the arrangement of terms.

Reductio ad Absurdum (Proof by Contradiction)​

Reductio ad Absurdum is a form of argumentation where one assumes the negation of the statement to be proven and shows that this assumption leads to a contradiction.

Form:

  1. Assumption: Not-A is true.
  2. If Not-A is true, then B follows.
  3. B leads to a contradiction.
  4. Therefore, A must be true.

Example:

  1. Assumption: There are not infinitely many prime numbers.
  2. If there are only finitely many prime numbers, we can multiply them all and add 1 to obtain a new number N.
  3. N is either itself a prime number or divisible by a prime number not contained in our original list.
  4. This contradicts our assumption that we have listed all prime numbers.
  5. Therefore, there must be infinitely many prime numbers.

Argument by Analogy​

An argument by analogy infers from similarities between two things to further similarities.

Form:

  1. A has properties X, Y, and Z.
  2. B has properties X and Y.
  3. Therefore, B probably also has property Z.

Example:

  1. The planet Mars has a solid surface, an atmosphere, and water ice at the poles.
  2. Earth has a solid surface, an atmosphere, and water ice at the poles.
  3. Earth harbors life.
  4. Therefore, Mars could possibly also harbor life.

Arguments by analogy are inductive and provide no certainty, only probabilities. Their strength depends on how relevant the shared properties are to the conclusion.

Abduction (Inference to the Best Explanation)​

Abduction is a form of reasoning where one infers from an observation to the most likely explanation.

Form:

  1. Observation: Phenomenon P occurs.
  2. Explanation E would explain P well.
  3. No other explanation explains P as well as E.
  4. Therefore, E is probably true.

Example:

  1. Observation: The lawn is wet.
  2. Explanation: It has rained.
  3. Alternative explanations (sprinkler system, dew) are less likely since there is no sprinkler system and the wetness is too heavy for dew.
  4. Therefore, it has probably rained.

Abduction is an important form of reasoning in science, medicine, and everyday life, but it provides no certainty. The strength of an abductive conclusion depends on how well the explanation accounts for the phenomenon and how much better it is than alternative explanations.

Application in Critical Thinking​

Understanding these patterns of argumentation is important for critical thinking for several reasons:

  1. It enables the identification and evaluation of arguments in various contexts.

  2. It helps structure one's own arguments more effectively and persuasively.

  3. It facilitates recognizing fallacies that resemble valid argument patterns but contain logical errors.

  4. It promotes a deeper understanding of the different types of reasoning (deductive, inductive, abductive) and their respective strengths and limitations.