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===Discusssion===

===Discusssion===

The task at hand is to build a bridge between model-theoretic and proof-theoretic perspectives on logical procedure, even if we can but construct it at a point so close to their common source that it may not seem worth the candle at all.

The task at hand is to build a bridge between model-theoretic and proof-theoretic perspectives on logical procedure, though at present we construct it at a point so close to the common source of these two streams it may not seem worth the candle at all.

The substance of this principle was known to Boole in the 1850's, tantamount to what we now call the ''boolean expansion'' of a propositional expression.  The only novelty here resides in a certain manner of presentation, in which we will prove the basic principle from the axioms given before.  One name for this rule is the ''Case Analysis-Synthesis Theorem'' (CAST).

The substance of this principle was known to Boole in the 1850's, tantamount to what we now call the ''boolean expansion'' of a propositional expression.  The only novelty here resides in a certain manner of presentation, in which we will prove the basic principle from the axioms given before.  One name for this rule is the ''Case Analysis-Synthesis Theorem'' (CAST).