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How did Stoic logicians use schematic arguments with propositions as variables?
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History of logic
Zeno's paradox, illustrated by Achilles' racing a tortoise.
Stoic logicians, most notably Chrysippus, pioneered a system of propositional logic that focused on inferences derived from hypothetical propositions rather than the term-based logic favored by Aristotle.12
To represent these logical structures, the Stoics utilized schematic arguments where ordinal numerals—such as "the first" and "the second"—served as variables or placeholders for entire propositions. This approach allowed them to formalize principles of inference independently of the specific content of the statements involved.12
Chrysippus identified five basic, "indemonstrable" inference schemata that formed the foundation of their system:12
- If the first, then the second; but the first; therefore, the second.
- If the first, then the second; but not the second; therefore, not the first.
- Not both the first and the second; but the first; therefore, not the second.
- Either the first or the second; but the first; therefore, not the second.
- Either the first or the second; but not the second; therefore, the first.
By using these five rules, the Stoics sought to prove the validity of more complex arguments.12