What Is a Syllogism? The Argument Form That Ran Logic for Two Millennia
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Two premises and a conclusion, each relating categories of things, was the dominant framework for reasoning in Europe from Aristotle until the nineteenth century. It is genuinely useful, it is far more limited than its long dominance suggests, and the limits are what eventually replaced it.
The structure
A categorical syllogism contains exactly three terms distributed across two premises and a conclusion, with each statement taking one of four forms: all A are B, no A are B, some A are B, or some A are not B. The classic example runs that all humans are mortal, Socrates is human, therefore Socrates is mortal, though strictly that involves an individual rather than a category and the textbook forms use categories throughout. What makes the system work is that validity depends entirely on the arrangement of the terms rather than on what they refer to, so the same pattern is valid for any content, which is the insight that makes formal logic possible at all. Aristotle catalogued the possible arrangements, called figures and moods, and determined which are valid, and medieval logicians produced mnemonic names for the valid forms that were memorised by students for centuries. Of the several hundred possible combinations, only a couple of dozen are valid.
The ways they go wrong
Invalid forms have recognisable patterns and names, and most bad everyday arguments fit one:
- •The undistributed middle, where the shared term is not taken universally in either premise, so nothing connects the other two, as in arguing that all dogs are animals and all cats are animals therefore dogs are cats
- •Illicit process, where a term is used universally in the conclusion having been used only partially in a premise
- •Affirming a conclusion from two negative premises, which cannot establish any relationship
- •The existential fallacy, drawing a conclusion asserting that something exists from premises that do not assert it, which is a subtle issue about whether universal statements imply that their subjects have members
- •Four terms, where a word shifts meaning between premises, so the argument only appears to have three terms
- •Valid arguments with false premises, which are not fallacies at all and which produce false conclusions, since validity concerns the structure and says nothing about truth
What it could not handle
The system's limits are what eventually displaced it and they are substantial. It cannot handle relations, so an argument involving one thing being larger than another, or occurring before it, cannot be expressed, which rules out most of mathematics. It cannot handle multiple quantifiers, so a statement asserting that every person loves some person, with its ambiguity about whether it is the same person, is beyond it. It cannot represent statements about statements. It cannot express the structure of arguments involving conditionals in the way propositional logic does, though Stoic logicians developed that separately and their work was largely lost. Those limits meant that a great deal of valid mathematical reasoning could not be shown to be valid within the framework, which was a recognised embarrassment. Kant nonetheless declared in 1781 that logic had been complete since Aristotle and required no further development, which is among the more notable misjudgements in the history of the subject.
What replaced it
Modern logic began with Gottlob Frege's work in 1879, which introduced quantifiers and variables and allowed relations and nested quantification to be expressed, and which subsumed syllogistic reasoning as a small fragment of a far more powerful system. That development made the logical analysis of mathematics possible and underlies the subsequent work on foundations, computability and eventually computer science, since the formal manipulation of symbols according to rules is what a computer does. Syllogistic logic survives in teaching, because it is an accessible introduction to the idea that validity depends on form, and in the analysis of everyday argument, where most reasoning does fit its patterns and where the named fallacies remain useful diagnostic labels. It also survives as a historical fact of enormous consequence, since for two thousand years it was what educated Europeans meant by reasoning, and the vocabulary of premise, conclusion, valid and term entered general use from it.
The takeaway
Three terms across two premises and a conclusion, with validity depending on arrangement rather than content, is the insight that makes formal logic possible, and only a couple of dozen of the hundreds of possible arrangements are valid. The system cannot express relations or nested quantifiers, which excludes most of mathematics, and Frege's work in 1879 replaced it with something that could while containing it as a fragment.