Set Theory and Logic

   

Rationale of Rendering Quantifiers as Modal Operators © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III

The rationale for rendering quantifiers as modal operators in Meth8 has arguments from satisfiability and reproducability of invalidating and validating syllogisms. The Square of Opposition (original) produced four combinations for each corner A, I, E, O for 4 ^ 4 = 256 syllogisms. Medieval scholars determined 24 of the 256 syllogisms were valid deductions. Of those, 9 were made valid but only after additional known assumptions were applied as fix ups. Meth8 Tautologous none of the 24 syllogisms before fix ups. Meth8 also discovered correct additional assumptions to render the other 15 syllogisms Tautologous. We use Meth8 to replicate the 24 valid syllogisms derived from the original Square of Opposition. In the process we make three recent advances. 1. A third assumption is needed to fix up Modus Cesare EAE-2 2. The third assumption cannot be removed from Modus Camestros AEO-2 (as in other syllogisms with known third assumptions); and 3. No third assumptions are required for the other 22 syllogisms.

Comments: 14 Pages. © 2016 by Colin James III All rights reserved.

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[v1] 2017-11-12 12:05:19

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