Some extensions of the
Belnap-Dunn logic
Umberto Rivieccio
Universit`a di Genova
November 11th, 2010
Barcelona
U. Rivieccio (Universit`a di Genova) Some extensions of the Belnap-Dunn logic November 11th, 2010 1 / 28
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Extensions of Belnap-Dunn Logic: Overview and New 'Basic Logics', Lecture notes of Logic
An overview of Belnap-Dunn logic, its extensions, and their relationships with other logics. the semantic definition, algebraic models, and axiomatizations of various extensions, including LP, K¤, K, CPC, B36, B46, B38, B48, B34, and B348. The author also discusses open problems and references for further study.
What you will learn
- How do the various extensions of Belnap-Dunn logic relate to each other?
- What are the algebraic models of Belnap-Dunn logic and its extensions?
- What are the open problems in the study of Belnap-Dunn logic extensions?
- What are some well-known extensions of Belnap-Dunn logic?
- What is Belnap-Dunn logic and how is it different from classical logic?
Typology: Lecture notes
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Some extensions of the
Belnap-Dunn logic
Umberto Rivieccio
Universit`a di Genova
November 11th, 2010 Barcelona
The Belnap-Dunn logic
Introduction The Belnap-Dunn four-valued logic (a.k.a. first degree entailment) is a well-known system related to relevant and paraconsistent logics, widely known and applied in computer science. In recent years it has also been studied algebraically (Font, Pynko) and in connection with more general structures such as bilattices and generalized Kleene logics (Fitting, Arieli & Avron, Shramko & Wansing).
The Belnap-Dunn logic
Miscellaneous facts (Pynko 1995, Font 1997) B is: a theoremless (a.k.a. “purely inferential”) logic
paraconsistent, in the sense that ϕ ^ ϕ *B ψ
relevant, in the sense that if ϕ (B ψ, then var pϕq X var pψq H
the logic of the lattice order of De Morgan lattices, i.e. Γ (B ϕ iff DMLat (
Γ ¨ ϕ
finitely axiomatized by Hilbert- or Gentzen-style calculi
The Belnap-Dunn logic
Miscellaneous facts (Pynko 1995, Font 1997) Moreover
B is non-protoalgebraic, self-extensional and non-Fregean
B has an associated fully adequate Gentzen calculus that is algebraizable w.r.t. the variety DMLat of De Morgan lattices
AlgB AlgB DMLat.
Remark DMLat is not the equivalent algebraic semantics of any algebraizable logic; the same holds for any sub-quasi-variety of DMLat (except Boolean algebras).
Extensions of B
Well-known ones Priest’s (1979) logic of paradox LP
Kleene’s logic of order K¤ (a.k.a. the implicationless fragment of the relevant logic RM)
the strong three-valued (assertional) Kleene logic K
classical logic CPC.
Extensions of B
Inclusions
K¤
B
LP K
CPC
I
I
6
@
@@
@
@
Extensions of B
Semantical presentations
LP is defined by xK 3 , tt, Kuy
K¤ by txK 3 , tt, Kuy, xK 3 , ttuu
K by xK 3 , ttuy
CPC by xB 2 , ttuy.
There are more...
Some more extensions of B Any (set of) matrices belonging to MatrB defines an extension of B. For instance:
xM 4 , ttuy
xK 6 , tt, auy
xM 8 , tt, auy
txM 4 , ttuy, xK 3 , tt, Kuyu
...
There are more...
Some more extensions of B The above-mentioned basic logics generate the following:
B 36 B 6 X K
B 46 B 36 X B 4
K¤ B 36 X LP
B 38 K¤ X B 8
B 48 B 8 X B 46
B 34 K¤ X B 46
B 348 B 34 X B 38.
A hierarchy
B 348
B 38 B 48
B 36
B 46
B 4
B 6 K
CPC
LP
K¤ B 8
B 34
@
@
@
@
@
@@
I
6 6
I
I
I
6
6 I
I
@
@
@
@
@
@
@
@
@
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@@
@
@
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What we do (not) know
Relations with the order For any ϕ, ψ P Fm, it holds that ϕ (L ψ if and only if:
DMLat ( ϕ ¨ ψ for L B
KLat ( ϕ ¨ ψ for L K¤
DMLat ( ϕ _ ψ ¨ ϕ _ ψ for L B 34
DMLat ( ϕ ¨ ϕ _ ψ for L B 4
KLat ( ϕ ¨ ϕ _ ψ for L K
KLat ( ϕ _ ψ ¨ ϕ _ ψ for L LP
QpK 6 q ( ϕ ¨ ϕ ñ ψ ¨ ψ for L B 6.
What we do (not) know
Axiomatizations (Hilbert-style) The previous results allow to prove that
B 4 B plus p ^ p p _ qq $ q
B 6 LP plus p ^ p $ q
K¤ B plus pp ^ pq _ r $ q _ q _ r
K K¤ plus p ^ p p _ qq $ q
What we do (not) know: algebraic models
Reduced models For A P DMLat, let A ^ : ta P A : a ¥ au. Let xA, Dy be a reduced matrix: if xA, Dy is a model of
B, then D A
B 34 or B 48 and D X A^ H, then D A
LP, then D A
B 4 or K, then A is bounded and D t 1 u.
What we do (not) know: algebraic models
Reduced models So, a matrix xA, Dy P MatrB is a reduced model of
B 34 iff A P DMLat and D is a lattice filter such that b P D whenever a P D and a _ b ¤ a _ b
B 4 iff A is bounded De Morgan lattice and D t 1 u
K¤ iff A P KLat and D is a lattice filter
K iff A is a bounded Kleene lattice and D t 1 u
LP iff A P KLat and D ta P A : a ¥ au.