A polysyllogism is a complex argument (also known as chain arguments of which there are four kinds: polysyllogisms, sorites, epicheirema, and dilemmas) that strings together any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on. Each constituent syllogism is called a prosyllogism except the last, because the conclusion of the last syllogism is not a premise for another syllogism.

An example of a categorical polysyllogism is:

This argument has the following structure:

Note two points: first, the makeup of a polysyllogism need not be limited to two component syllogisms. In fact, it can have any number of component syllogisms. Second, validity depends on all its parts. If any one is not valid then the whole polysyllogism is to be considered invalid.

An example for a propositional polysyllogism is:

Examination of the structure of the argument reveals the following sequence of constituent (pro)syllogisms:

A sorites (plural: sorites) is a specific kind of polysyllogism in which the predicate of each proposition is the subject of the next premise. Example:

The word sorites /sɒˈraɪtiːz/ comes from Ancient Greek: σωρίτης, heaped up, from σωρός heap or pile. Thus a sorites is a heap of propositions chained together. A sorites polysyllogism should not be confused with the sorites paradox, a.k.a. the fallacy of the heap.