Fallacy of four terms

The fallacy of four terms (Latin: quaternio terminorum) is the formal fallacy that occurs when a syllogism has four (or more) terms rather than the requisite three. This form of argument is thus invalid.

Categorical syllogisms always have three terms:

Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All goldfish have fins.

Here, the three terms are: 'goldfish', 'fish', and 'fins'.

Using four terms invalidates the syllogism:

Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All humans have fins.

The premises do not connect 'humans' with 'fins', so the reasoning is invalid. Notice that there are four terms: 'fish', 'fins', 'goldfish' and 'humans'. Two premises are not enough to connect four different terms, since in order to establish connection, there must be one term common to both premises.