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Universal Logic Applied to a Defeasible World

Universal logic applied to the real world is defeasible. Defeasible reasoning where inferences depend (informally) on implied conditions can be better understood when made formal and indeed needs to be formalised for implementation in computational systems of practical reasoning such as interconnections in globalised eBusiness. Semi-formal Aristotelian syllogisms are now expressed within modern formal methods as Schütte-Ackermann language such as Γ −→ φ. At face value the Γ, ∆ −→ φ′ gives no inconsistency but what if φ′ = ¬φ′? For we cannot then formally derive with tertium non datur the defeasible expression Γ, ∆ −→ ¬φ. A deeper examination of these terms including the meaning of the negation (¬φ) is required. The very formal topos with a higher-order slice category can provide the level of abstraction to handle rigorously informal concepts like non-monoticity and alternative reasoning in disparate fields with the same pullback. The security of artificial agents in Java software is an example requiring this full understanding of defeasance. Mathematics Subject Classification. 18D30, 18D15.