Checkpoint one dipped our toes in classes and the logic of nomenclature. Connectives in the logic of algebra may be repeated many times over. In the logic of nomenclature, by contrast, a connective like the word except may be permitted only once within a name. When names need to be long and involved, we resort to putting propositions within, rather than to bracketing, as in algebra.

In addition to the confusions mentioned at checkpoint one, naive set theory can obfuscate logic when it casts certain subject matter as belonging in sets. Subject matter that lacks the abstract nature of mathematics should be cast in classes whence the logic of nomenclature can be paramount. When put in sets, the logic of algebra may intrude. Its rules may confound rather than assist the understanding.

A Cause not to be Denied

A strange circumstance in the electricity sector, relevant all over the globe, ostensibly points directly at Man's estrangement from the logic of nomenclature. In electricity we find that marine energy has been too loosely construed. This is associated with the misconception that tidal energy is dissipated with every tide. The category of gravitational potential energy has been fogotten. How can we forget our fundamental categories?

The soft landings of the tides imply the operation of negative work and in a primitive sense, the tidal energy therefore belongs to the conservative force of gravity, not lumped together with seas' thermal energy.

Promoters could be excused from loose language if public defenders were to be given the opportunity to step in. However while Man may welcome public defenders in vetting new pharmaceuticals and new plans for bridges, the modern rule in electricity appears to be that the existential pleasure of the promoters shall be undiminished.

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For those who are ready for a change from certain themes in mathematics, Vladimir Voevodsky of the Insitute of Advanced Study, Princeton, New Jersey, USA has a new Univalent foundations to offer. Voevodsky appears to have performed some craft on Bertrand Russell's type theory whence to dissolve the awkwardness factor that has dogged the type theory for so long. The jury will be out for a few years pondering. When Voevodsky refers to 'proof assistants' one must ponder what kind of proof he means to be assisted. Is it a derivational proof or is it a reductio-ad-absurdum proof? Much of mathematics can be encased as it were in reductio-ad-absurdum proofs. They are often easier to remember but computers are unlikely to be able to produce them other than by copying humans. Why would programmers bother? We may guess then that the Voevodsky proofs are derivational in kind.

Possibly all valid reductio-ad-absurdum proofs can be characterised with reference to an identity caught in possessing contradictory attributes. Once we have caught such an identity, we can draw an inference. The logical universe is then an essential concept, for an identity possessing contradictory attributes cannot exist anywhere in the logical universe. Most of the truths we know are conditional instead.

Assuming the logical universe is a class of places, then further evidence of its value as concept arises in connection with the principle of induction in number theory. All of the other Peano axioms are easily expressed in just about any mode of mathematics. When it comes to induction, by contrast the principle is often expressed as a meta theorem: if such-and-such and such-and-such are theorems then further here is another theorem. This is because certain modes of mathematics are insufficiently powerful to render the principle of induction as an ordinary theorem. Bring on the logical universe and all the Peano axioms including the principle of induction can be expressed simply as theorems.

For those interested to check this out further, here is an extract from Russell's Paradox - Two Sets, taken from chapter five.