Abstract

Axiomatic bases of admissible rules are obtained for fragments of the substructural logic R-mingle. In particular, it is shown that a ‘modus-ponens-like’ rule introduced by Arnon Avron forms a basis for the admissible rules of its implication and implication–fusion fragments, while a basis for the admissible rules of the full multiplicative fragment requires an additional countably infinite set of rules. Indeed, this latter case provides an example of a three-valued logic with a finitely axiomatizable consequence relation that has no finite basis for its admissible rules.

References

[1]
Anderson
A. R.
Belnap
N. D.
Entailment
 , 
1975
, vol. 
1
 
Princeton University Press
[2]
Avron
A.
On an implication connective of RM
Notre Dame Journal of Formal Logic
 , 
1986
, vol. 
27
 (pg. 
201
-
209
)
[3]
Avron
A.
A constructive analysis of RM
Journal of Symbolic Logic
 , 
1987
, vol. 
52
 (pg. 
939
-
951
)
[4]
Avron
A.
Multiplicative conjunction as an extensional conjunction
Logic Journal of IGPL
 , 
1997
, vol. 
5
 (pg. 
181
-
208
)
[5]
Avron
A.
Implicational f-structures and implicational relevance logics
Journal of Symbolic Logic
 , 
2000
, vol. 
65
 (pg. 
788
-
802
)
[6]
Babenyshev
S.
Rybakov
V.
Linear temporal logic LTL: basis for admissible rules
Journal of Logic and Computation
 , 
2011
, vol. 
21
 (pg. 
157
-
177
)
[7]
Babenyshev
S.
Rybakov
V.
Unification in linear temporal logic LTL
Annuals of Pure Applied Logic
 , 
2011
, vol. 
162
 (pg. 
991
-
1000
)
[8]
Blok
W. J.
Pigozzi
D.
Algebraizable Logics, vol. 77, Memoirs of the American Mathematical Society
 , 
1989
American Mathematical Society
[9]
Blok
W. J.
Raftery
J. G.
Fragments of R-mingle
Studia Logica
 , 
2004
, vol. 
78
 (pg. 
59
-
106
)
[10]
Cabrer
L. M.
Metcalfe
G.
Admissibility via natural dualities
 
Unpublished data
[11]
Cintula
P.
Metcalfe
G.
Structural completeness in fuzzy logics
Notre Dame Journal of Formal Logic
 , 
2009
, vol. 
50
 (pg. 
153
-
183
)
[12]
Cintula
P.
Metcalfe
G.
Admissible rules in the implication-negation fragment of intuitionistic logic
Annuals of Pure Applied Logic
 , 
2010
, vol. 
162
 (pg. 
162
-
171
)
[13]
Dunn
J. M.
Algebraic completeness for R-mingle and its extensions
Journal of Symbolic Logic
 , 
1970
, vol. 
35
 (pg. 
1
-
13
)
[14]
Ghilardi
S.
Unification in intuitionistic logic
Journal of Symbolic Logic
 , 
1999
, vol. 
64
 (pg. 
859
-
880
)
[15]
Ghilardi
S.
Best solving modal equations
Annuals of Pure Applied Logic
 , 
2000
, vol. 
102
 (pg. 
184
-
198
)
[16]
Iemhoff
R.
On the admissible rules of intuitionistic propositional logic
Journal of Symbolic Logic
 , 
2001
, vol. 
66
 (pg. 
281
-
294
)
[17]
Iemhoff
R.
Intermediate logics and Visser's rules
Notre Dame Journal of Formal Logic
 , 
2005
, vol. 
46
 (pg. 
65
-
81
)
[18]
Iemhoff
R.
Metcalfe
G.
Proof theory for admissible rules
Annuals of Pure Applied Logic
 , 
2009
, vol. 
159
 (pg. 
171
-
186
)
[19]
Jeřábek
E.
Admissible rules of modal logics
Journal of Logic and Computation
 , 
2005
, vol. 
15
 (pg. 
411
-
431
)
[20]
Jeřrábek
E.
Admissible rules of Łukasiewicz logic
Journal of Logic and Computation
 , 
2010
, vol. 
20
 (pg. 
425
-
447
)
[21]
Jeřábek
E.
Bases of admissible rules of Łukasiewicz logic
Journal of Logic and Computation
 , 
2010
, vol. 
20
 (pg. 
1149
-
1163
)
[22]
Metcalfe
G.
Röthlisberger
C.
Admissibility in De Morgan algebras
Soft Computing
 , 
2012
, vol. 
16
 (pg. 
1875
-
1882
)
[23]
Metcalfe
G.
Röthlisberger
C.
Admissibility in finitely generated quasivarieties
Logic Methods Computer Science
 , 
2013
, vol. 
9
 (pg. 
1
-
19
)
[24]
Olson
J. S.
Raftery
J. G.
Positive Sugihara monoids
Algebra Universalis
 , 
2007
, vol. 
57
 (pg. 
75
-
99
)
[25]
Olson
J. S.
Raftery
J. G.
Alten
C. J. Van
Structural completeness in substructural logics
Logic Journal of IGPL
 , 
2008
, vol. 
16
 (pg. 
453
-
495
)
[26]
Prucnal
T.
On the structural completeness of some pure implicational propositional calculi
Studia Logica
 , 
1973
, vol. 
32
 (pg. 
45
-
50
)
[27]
Raftery
J. G.
Representable idempotent commutative residuated lattices
Transactions of the American Mathematical Society
 , 
2007
, vol. 
359
 (pg. 
4405
-
4427
)
[28]
Rautenberg
W.
2-element matrices
Studia Logica
 , 
1981
, vol. 
40
 (pg. 
315
-
353
)
[29]
Rozière
P.
Regles Admissibles en Calcul Propositionnel Intuitionniste
 , 
1992
Université Paris VII
 
PhD thesis
[30]
Rybakov
V.
A criterion for admissibility of rules in the modal system S4 and the intuitionistic logic
Algebra Logic
 , 
1984
, vol. 
23
 (pg. 
369
-
384
)
[31]
Rybakov
V.
Admissibility of Logical Inference Rules, vol. 136, Studies in Logic and the Foundations of Mathematics
 , 
1997
Elsevier, Amsterdam
[32]
Slaney
J. K.
Meyer
R. K.
A structurally complete fragment of relevant logic
Notre Dame Journal of Formal Logic
 , 
1992
, vol. 
33
 (pg. 
561
-
566
)
[33]
Sobociński
B.
Axiomatization of a partial system of three-valued calculus of propositions
The Journal of Computing Systems
 , 
1952
, vol. 
1
 (pg. 
23
-
55
)