Inquiry Into Inquiry

Exemplary Proofs

Using no more than the axioms and theorems recorded so far, it is possible to prove a multitude of much more complex theorems.  A couple of all‑time favorites are linked below.

• Peirce’s Law
• Praeclarum Theorema

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Frequently Used Theorems (concl.)

C₃.  Dominant Form Theorem

The third of the frequently used theorems of service to this survey is one Spencer Brown annotates as Consequence 3 (C₃) or Integration.  A better mnemonic might be dominance and recession theorem (DART), but perhaps the brevity of dominant form theorem (DFT) is sufficient reminder of its double‑edged role in proofs.

Here is a proof of the Dominant Form Theorem.

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Frequently Used Theorems (cont.)

C₂.  Generation Theorem

One theorem of frequent use goes under the nickname of the weed and seed theorem (WAST).  The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader.  What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label.  The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem.  In Laws of Form it goes by the names of Consequence 2 (C₂) or Generation.

Here is a proof of the Generation Theorem.

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Frequently Used Theorems

To familiarize ourselves with equational proofs in logical graphs let’s run though the proofs of a few basic theorems in the primary algebra.

C₁.  Double Negation Theorem

The first theorem goes under the names of Consequence 1 (C₁), the double negation theorem (DNT), or Reflection.

The proof that follows is adapted from the one George Spencer Brown gave in his book Laws of Form and credited to two of his students, John Dawes and D.A. Utting.

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Equational Inference

All the initials have the form of equations.  This means the inference steps they license are reversible.  The proof annotation scheme employed below makes use of double bars to mark this fact, though it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.

The actual business of proof is a far more strategic affair than the routine cranking of inference rules might suggest.  Part of the reason for this lies in the circumstance that the customary types of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn’t appear immediately relevant, at least, not as viewed in the local focus and short run of the proof in question.  Over the long haul, this has the pernicious side‑effect that one is forever strategically required to reconstruct much of the information one had strategically thought to forget at earlier stages of proof, where “before the proof started” can be counted as an earlier stage of the proof in view.

This is just one of the reasons it can be very instructive to study equational inference rules of the sort our axioms have just provided.  Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of the usual logic textbooks, who may find a few surprises here.

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