A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of 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 ...
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http://en.wikipedia.org/wiki/Polysyllogism
series of syllogisms where each is basis for next
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http://phrontistery.info/p.html
A chain of syllogisms arranged to lead to a single final conclusion, the conclusion of each syllogism except the last serving as premiss of a later syllogism. In contrast, an argument consisting of a single syllogism is called a monosyllogism. -- A.C.
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https://www.encyclo.co.uk/local/21203
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