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By Jair Minoro Abe, Seiki Akama, Kazumi Nakamatsu
This booklet is written as an advent to annotated logics. It presents logical foundations for annotated logics, discusses a few attention-grabbing functions of those logics and in addition comprises the authors' contributions to annotated logics. The vital notion of the publication is to teach how annotated common sense could be utilized as a device to unravel difficulties of know-how and of utilized technology. The booklet should be of curiosity to natural and utilized logicians, philosophers and laptop scientists as a monograph on one of those paraconsistent good judgment. yet, the layman also will take cash in on its reading.
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Extra info for Introduction to Annotated Logics: Foundations for Paracomplete and Paraconsistent Reasoning
Example text
Where ℘ (Y ) is the powerset of Y . 18 X is called a universe iff it is strongly transitive and for every function f : Y → X such that Y ∈ X , it is the case that range( f ) ∈ X , where range( f ) = {z | ∃t (t ∈ Y and f (t) = z}. 10 If X is a universe, then X is a standard model of all axioms of Z F, with the possible exception of the axiom of infinity. Proof Clearly, extensionality, choice and regularity are satisfied in X , since they are satisfied in Z F. 18 are satisfied, we easily see that the axioms of pair, union, power-set and (separation, in particular) are true in X .
Given a chain, we define a structure A called the union of the chain. ∞ The universe of A is | A |= | Ai |. If a1 , . . , ak ∈| A |, then there is an n i=1 such that all a1 , . . , ak ∈| A |. We then set f A (a1 , . . , ak ) = f An (a1 , . . , ak ) and p A (a1 , . . , ak ) = λ iff p An (a1 , . . , ak ) = λ, ∀λ ∈| τ |. It is immediate that this definition is well-defined. An elementary chain is a chain (Ai )i∈N ∗ such that for each n, An+1 is an elementary extension of An . 27 (Tarski) If (Ai )i∈N ∗ is an elementary chain, then the union A = ∞ Ai is an elementary extension of each Ai .
For this purpose, we rely on the so-called Curry algebra due to Barros et al. [42]. We introduce Curry algebra Pτ that algebraizes propositional annotated logics Pτ . Systems induced by Curry algebra are called Curry systems, which constitute a very promising field of research but few people have paid attention to the subject. g. , a congruence. The resulting quotient system is the algebraic structure linked with the corresponding logical system. By this process, Boolean algebra constitutes the algebraic version of classical propositional logic, Heyting algebra constitutes the algebraic version of intuitionistic propositional logic, and so on.