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T. Contrapositive??, . ??, and . Classical-implication, Likewise classical bi-implication is denoted by ?. Direct Logic has the following usual principles for bi-implication, ?1??2 ? ?2??3) ? ?1??3 Substitution of equivalent propositions: (?=?) ? ??? (???) ? (??)) (???) ? (? ? T ?) ? (? ? T ?)

@. ???, . ???, . ???, and . ???, Proposition where ?1 to n:Proposition The following logical equivalences hold (???) Substitution of equivalent propositions, Provability Direct Logic is a subset of Inconsistency Robust Direct Logic for a single theory without quantifiers as follows: ? ??:Proposition where ?:Proposition???)) ? (???) ? ? Distributivity of ? over ?: (? ? (???)) ? (???)) (???) ? (?? ?) ? (?? ?) (???) ? (?? ?) ? (?? ?)

?. ???-reiteration, The following inferences hold: Theorem Use: (??) ? ? // A theorem can used in a proof Comma

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. Poof and . Inconsistency, By the definition of Diagonal: ?Diagonal??[Diagonal] ?1 Halt?[Diagonal, Diagonal] ? True? InfiniteLoop?, False? True Consider the following 2 cases: 1. Halt?[Diagonal, Diagonal] ?1 True Converges[ ?Diagonal??[Diagonal]] by the axioms for Halt ?Converges[ ?Diagonal??

. Halt?, ?ComputationallyDecidable[HaltingProblem] 53 Note that this theorem is very different from the result [Kleene 1938], that mathematics can be extended with a proposition asserting its own consistency. 54 A prominent logician referee of this article suggested that if the proof is accepted then consistency should be made an explicit premise of every theorem of mathematics! 55 As shown above, there is a simple proof in Classical Direct Logic that Mathematics (?) is formally consistent. If Classical Direct Logic has a bug, then there might also be a proof that Mathematics is inconsistent. Of course, if a such a bug is found, then it must be repaired. The Classical Direct Logic proof that Mathematics (?) is consistent is very robust. One explanation is that consistency is built in to the very architecture of mathematics because it was designed to be consistent, Consequently, it is not absurd that there is a simple proof of the consistency of Mathematics (?) that does not use all of the machinery of Classical Direct Logic, p.107

5. I. , ?. ??-theoremscomputationalyenumerable?, ?. ??, and . ??, TheoremsComputationalyEnumerable? 60 This coordination can include calendars and to-do lists, communications (including email, SMS, Twitter, Facebook), presence information (including who else is in the neighborhood), physical (including GPS recordings), psychological (including facial expression, heart rate, voice stress) and social (including family, friends, team mates, and colleagues), maps (including firms, points of interest, traffic, parking, and weather), events (including alerts and status), documents (including presentations, spreadsheets, proposals, job applications, health records, photons, videos, gift lists, memos, purchasing, contracts, articles), contacts (including social graphs and reputation), purchasing information (including store purchases, GPS and phone records, and buying and travel habits), government information (including licenses, taxes, and rulings), and search results (including rankings and rating)