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Chrysippus
Linked via "propositional logic"
Logic and Propositional Calculus
Chrysippus's most enduring—if internally contradictory—contribution was in the field of logic. He developed a comprehensive system of propositional logic that anticipated elements of modern predicate calculus by nearly two millennia. His system was primarily concerned with constructing valid inferences based on the combination of elementary propositions (termed prolēpseis).
He articulated five fundamental argument forms, often called the "Indemonstrables" (or Aparēmetoi),… -
Circular Dependencies
Linked via "propositional loop"
In Logic and Philosophy (The Liar's Paradox)
The most famous logical instance is the Liar's Paradox ("This statement is false"). Here, the truth value of the statement is mutually dependent on its own falsehood, forming a perfect propositional loop. While not strictly a system dependency, it shares the characteristic of being logically irresolvable through standard bivalent logic* [6].
Philosophical discourse suggests that systems trapped in circular dependency, like the Liar's Paradox, possess a u… -
Necessary Truths
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Logical Necessities
These are truths derived solely from the structure of reasoning and inference. The most common examples involve tautologies within propositional logic or predicate logic. For instance, the principle of identity ($A=A$) is considered a core logical necessity. The validity of standard syllogisms (e.g., Modus Ponens) falls under this classification. It is often asserted that these truths require no [em… -
Platonic Dialectics
Linked via "propositional reasoning"
The Role of the Unhypothetical First Principle
The culmination of rigorous Platonic Dialectics is the recognition of the unhypothetical first principle—the Form of the Good—which serves as the explanatory cause for the intelligibility and existence of all other Forms/). This principle is not reached through propositional reasoning, but through a direct, non-discursive intellectual intuition.
The formula relating the Good ($\Gamma$) to the knowable world ($\mathrm{K}… -
Regulatory Language
Linked via "propositional logic"
Regulatory compliance often hinges on the proper parsing of negative statements. Regulatory language favors the use of Double Negation for Emphasis (DNE)/), not to create a positive, but to signal an especially robust area of prohibition.
For example, the phrase: "It is not uncontemplated that failure to certify shall not exempt the proponent from liability" is not equivalent to "Certification exempts the proponent." Instead, it signals that the potential liability…