Introduction to Annotated Logics by Jair Minoro Abe Seiki Akama & Kazumi Nakamatsu

Author:Jair Minoro Abe, Seiki Akama & Kazumi Nakamatsu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

A Hilbert system for Anderson and Belnap’s is as follows:

Relevance Logic R

Axioms

(RA1)

(RA2)

(RA3)

(RA4)

(RA5)

(RA6)

(RA7)

(RA8)

(RA9)

(RA10)

(RA11)

(RA12)

(RA13)

(RA14)

Rules of Inference

(RR1)

(RR2)

Routley et al. considered some axioms of are too strong and formalized rules instead of axioms. Notice that is a paraconsistent but is not.

Next, we give a Routley-Meyer semantics for . A model structure is a tuple , where is a non-empty set of worlds, , is a ternary relation on , is a unary operation on , and is a valuation function from a set of worlds and a set of propositional variables to .

There are some restrictions on . satisfies the condition that and imply for any and any . is a pre-order relation defined by . The operation satisfies the condition .

For any propositional variable , the truth condition is defined: iff . Here, reads “ is true at ”. can be extended for any formulas in the following way:

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