WYDAWNICTWO UNIWERSYTETU
Trends in Logic XIII
Gentzen's and Jaskowski's heritage
80 years of natural deduction
and sequent calculi
E d i t o r s
Andrzej Indrzejczak
Janusz Kaczmarek
Michał Zawidzki
IXI
WYDAWNICTWO UNIWERSYTETU L A r \ 7 r , A 1 /I ŁÓDZKIEGO LUUŁ Z U 1 4Andrzej Indrzejczak - University of Łódź, Faculty of Philosophy and History Department of Logic and Methodology of Sciences, 16/18 Kopcińskiego St., 90-232 Łódź
e-mail: i n d r z e j t a f i l o z o f . u n i . l o d z . p i
Janusz Kaczmarek - University of Łódź, Faculty of Philosophy and History Department of Logic and Methodology of Sciences, 16/18 Kopcińskiego St., 90-232 Łódź
e-mail: kaczmarek@filozof. u n i . lodz. p i
Michał Zawidzki - University of Łódź, Faculty of Philosophy and History Department of Logic and Methodology of Sciences, 16/18 Kopcińskiego St., 90-232 Łódź
e-mail: michal. zawidzki@gmail. com © Copyright by University of Łódź, Łódź 2014
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ISBN 978-83-7969-161-6 paperback
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C o n t e n t s
Forword IX
I Invited lectures 1
Equational Logic and M o d u l a r i t y 3
Janusz Czelakowski
Proof-theoretic harmony: T h e issue of propositional quantification 5 Peter Schroeder-Helster
T h e formal exposition of intuitions - a view on Gentzen and
Jaskowski 17 M a x Urchs
Logics of Falsification 21 Helnrlch WansLng
Philosophical Reflections on Logic, Proof and T r u t h 2 7 Jan Woleriskl
II C o n t r i b u t e d t a l k s 4 1
Arbitrary Reference t h r o u g h Acts of Choice: A Constructive V i e w
of Reference in Logic 4 3 Massimiiiano Carrara and Enrico Marti.no
T h e faithfulness of a t o m i c polymorphism 5 5 Fernando Ferreira and GLlda Ferreira
VL Contents
On the logics associated w i t h a given variety of algebras 6 7 Josep M a r i a Font and Tommaso M o r a s c h l n l
M u l t i - t y p e Sequent Calculi 81 Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra
Palmlgiano and Vlasta Slklmlc
Decidability M e t h o d s for Modal Syllogisms 95 Tomasz Jarmuzek and Andrzej Pletruszczak
On Dedicated Fuzzy Logic Systems for Emission Control of In
dustrial Gases 1 1 3 M a r c l n Kacprowlcz and Adam N l e w l a d o m s k i
A l m o s t Affine Lambda Terms 131 M a k o t o Kanazawa
Axiomatisations of Minimal Modal Logics Defining
Jaskowski-like Discussive Logics 1 4 9 M a r e k Nasteniewskl and Andrzej Pletruszczak
Cut-elimination and Consistency: variations on a
Gentzen-Pra-w i t z theme 1 6 5 Lulz Carlos Perelra and Edward Hermann Haeusler
Hierarchical Fuzzy Logic Systems and Their Extensions Based
on Type-2 Fuzzy Sets 181 Krzysztof Renkas and Adam N l e w l a d o m s k i
Algebraic Semantics for Bilattice Public A n n o u n c e m e n t Logic 1 9 9 Umberto Rlvlecclo
Gentzenization of Dynamic Topological Hybrid Logics 2 1 7 Katsuhlko Sano and Yulchlro Hosokawa
Frege's sequent calculus 2 3 3 Peter Schroeder-Helster
Games for Intuitionistic Logic P a w e ł Urzyczyn
148 ALMOST AFFINE LAMBDA TERMS
[8] HIROKAWA, S., 'Balanced formulas, BCK-mlnimal formulas and their proofs', in:' A. NERODE and M . TAITSLIN (eds.), Logical Foundations of Computer Science aAT Tver aAZ92, Berlin: Springer Verlag, 1992, 198-208.
[9] JASKOWSKI, S „ 'Uber Tautologlen, in vvelchen keine Variable mehr als zweunal vorkommt', Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 9:12-15, 219-228, 1963.
[10] KANAZAWA, M., 'Parsing and generation as Datalog queries', [in:] Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics, Prague, Czech Republic, 2007, 176-183.
[11] KANAZAWA, M., 'A lambda calculus characterization of M S O definable tree trans-ductions (abstract)', Bulletin of Symbolic Logic, 15(2):250 - 251, 2009.
[12] KANAZAWA, M., 'Parsing and generation as Datalog query evaluation', 2011, h t t p : / / r e s e a r c h . n i i . a c . j p /
[13] KANAZAWA, M . and POCODALLA, S., Advances in abstract categorial grammars: Lan-guage theory and linguistic modeling', course taught at ESSLLI 2009, Bordeaux, France, 2009.
[14] LOADER, R„ Notes on simply typed lambda calculus, Technical Report E C S -LFCS-98-381, Edinburgh: Laboratory for Foundations of Computer Science, School of Informat- ics, The University of Edinburgh, 1998.
[15] MINTS, C. E., 'Closed categories and the theory of proofs', Journal of Soviet M a t h -ematics, 15:45-62, 1981.
[16] MINTS, C. E., A Short Introduction to Intuitionistic Logic, New York: Kluwer Aca-demic/Plenum Publishers, 2000.
[17] TATSUTA, M . and DEZANI-CIANCACLINI, M., 'Normalisation is insensible to A-term identity or difference', [in:] Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society, Seattle, United States, 2006, 327 - 338.
M a k o t o Kanazawa
National Institute of Informatics
2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan
A X I O M A T I S A T I O N S O F M a r e k NasteniewskL M I N I M A L M O D A L L O G I C S Andrzei Pletruszczak D E F I N I N G
J A S K O W S K I - L I K E
D I S C U S S I V E L O G I C S
Abstract Jaskowski's discussive Logic D2 was formulated with the help of the modal logic S5 as follows: A e D2 Iff rO / \ *n e S5, where ( — ) ' is a translation of discussive
formulae into the modal language. Thus, the key role in the definition of the logic D2 is played by the logic S5. In the literature there are considered other modal logics that are also defining the same logic D2.
There are also investigated translations that are determining other Jaskowski-like logics. In [3, 5], instead of the original translation with "righf-discussive conjunction, another translation is considered, where "left"-discussive conjunction is treated as Jas-kowski's one. In [2], it has been shown that this new transformation yields a logic different from D2. Ciuciura denotes the obtained logic by 'D^'. There are two other possibilities as regards the internal translation of discussive conjunctions.
The question arises (which has been stated by Joao Marcos), what does it change if we consider the weakest in a given class modal logics that determine these "new" discussive logics. In [11] the smallest modal logics defining respective Jaskowski-like discussive logics are considered. In the present paper we give more elegant axlomati-sations of these logics.
Keywords: Jaskowski's discussive logic, Jaskowski-like discussive logics, axiomatisa-tions of Jaskowski-like discussive logics, minimal modal logics defining Jaskowski logic, minimal modal logics defining Jaskowski-like discussive logics
AxiOMATISATIONS OF MINIMAL MODAL LOGICS..
1 BASIC NOTIONS AND FACTS
1.1 Some facts of modal logic
MODAL LANGUAGE. M o d a l formulae are formed in the standard way from propositional Letters: 'p', 'q, 'po', ' p i ' , 'pi,...; truth-value operators: V , 'A', ' — a n d W (connectives of negation, disjunction, conjunc-tion, material implication and material equivalence, respectively); modal operators: the necessity sign ' • ' and the possibility sign 'O'; and brack-ets. By F o rm we denote the set of modal formulae. Of course, the set
F o rm includes the set of a l l classical formulae (without ' • ' and 'O');
let Taut be the set of all classical tautologies and P L — the set of all modal formulae being instances of elements of Taut. Besides, for any <p,p,x £ F o rm, let xl^/ip] }°e A N IJ formula that results from / by
replacing none, one, or more occurrences of cp, in by p.
For any p s F o rm let Sub(p) be the set of all modal formulae
being substitution instances of p. For any 0 C F o rm let Sub(<£) : =
U « pG0 Sub(c/)). We have p G Sub(p) and 0 C S u b ( 0 ) . Moreover,
we put O0 := {p : p — rO c / >n} = {rOqP : cp E 0} and
• 0 := {r D ^n : cp E 0}.
MODAL LOGICS. A modal logic is any set L of modal formulae satis-fying following conditions:
• Taut C L,
• L includes the following set of formulae
• L Is closed under the following two rules: modus ponens for '
{
r/ [
• - i • - i <p
/o<p] : 0-X G F o rm }
(rep°)
cp, ip - > p I p (mp)
and uniform substitution:
Marek Nasieniwski and Andrzej Pletruszczak
where scp is the result of uniform substitution of formulae for prepo-sitional letters in cp.
CHOSEN CLASSES OF LOGICS. W e say that a modal logic L is an r t e - l o g i c iff L is closed under replacement of tautological equivalents, i.e., for any <p,ip>X e F o rm:
If rcp <-> (/T1 e P L and x E L then x l9 / 4>) e L (rte)
A modal logic is rte-logic Iff It Includes the following set
{ rxl<p/ip] +*X^'-<P> 4><X £ Form and rcp <-» ^ e P L } . ( r e pP L)
LEMMA 1.1. A modal logic contains the formula:
Up^p (T) Iff it contains its dual version:
p
-> op
i n
LEMMA 1.2. An rte-logic contains the following formulae:
D(p A q) ^ {Dp AOq) (R)
OOp -> p (B)
oaP - > D p
(5)
iff it contains, respectively, theirs dual versions:
0(p V g ) f ^ (Op V
Oq) (FT)p
-*
nop (B°)Op ->
nop
(5°)
In [1] a modal logic is called classical m o d a l (cm-logic for short) Iff it is an rte-logic which contains
•
(p -> q)
- > ( D p - > D t / )(/C)
AxiOMATISATIONS OF MINIMAL MODAL LOGICS...
Thus, a l l cm-logics Include the set D P L : = {DT : T G P L } .
W e say that a logic Is c o n g r u e n t i a l Iff it Is closed under the congru-ence rule
cp <-» tp / Dcp <-> Dip (cgr)
A logic Is congruential Iff It is closed under replacement
cp^ ip I x^/ip] *->X (rep) Every congruential logic Is an rte-logic.
W e say that a logic L is monotonic iff L is closed under the mono-tonlclty rule:
cp —» ip I Dip —» Dip (mon)
Every monotonic logic Is closed under (rep), i.e. is congruential. W e say that a logic is regular iff it contains (K) and is closed under (mon).
A logic is n o r m a l iff it contains (K) and Is closed under the necessi-tatlon rule
cp / Dcp (nec)
Alt normal logics are regular and cm-loglcs.
For all sets X and A of modal formulae and any set of rules 1Z in F o rm we say that the pair (A,IV) is an axiomatization of X iff X is
the smallest set including A and closed under all rules from 1Z.
1.2 The discussive logic D2 and other Jaskowski-like logics
DISCUSSIVE LANGUAGE. The Logic D2 is defined as a set of discussive formulae of a certain kind. These formulae are formed in the s t a n -dard way from propositional letters: 'p, 'q', 'po', 'pi', 'pi,...; truth-value operators: ' - i ' and V (negation and disjunction); discussive connectives: 'Ad', '—>d', ' ^d' (conjunction, implication and equivalence); and brackets.
Marek NasienLwski and Andrzej Pletruszczak
d e f i n i t i o n of discussive logic D
2. The logic D2 wasformu-lated with the help of the modal logic S 5 as follows (see [7, 8]): D2 : = {A G Ford : rOA*^ G S 5 } ,
where ( — ) • Is a translation of discussive formulae into modal language, i.e., it is a function ( — ) • from Ford into F o rm such that:
1. (a)* = a, for any propositional letter a,
2. for any A, BE Ford:
(a) ( - , / \ ) » = r - , ^ - i|
(b)
(AV B)
9=
rA* \/ B^,
(c) (A Ad BY = rA* A O B * "1,
(d) {A ->
DBY
=
rOA* -> B'
n,
(e) (A ^
dBY
=
r{OA' - » B
M)
A 0(OB* - * A')
n.
Of course, D2 is closed under (sb) with respect to Ford. Moreover,
D2 is closed under modus ponens for '—>d':
A, A
- *dB I
B
(mpd)because S 5 is closed under the following rule:
Ocp, 0(0<p - > <//) / Oip (RC)
d e f i n i t i o n s of jaskowski-like locics. In
[3, 5]
a logicD\
was formulated w i t h the help of the modal logic S 5 as follows::= [A G Ford : rO / Tn G S 5 } ,
where (—)* is a function from Ford into F o rm such that for any A, B G
F o rd:
(c)* (A A
dBY
=
rOA* A B *
n,
(e)* {A ^
dBY
=
rO(OA* - > B*) A (OB* -> A*y,
AXIOMATISATIONS OF MINIMAL MODAL LOGICS..
Additionally a Logic D2 was defined as follows:
D2 := { A E Ford : rO / Tn E S 5 } ,
where ( — ) ^ is a function from F o rd into F o rm such that for any A, B E
F o rd:
(Cp (/\ Ad £ ) ~ = r / T A
( e p (>A ^ d 8 ) ^ = r( O Z T -> £ ~ ) A ( O B ^ -> / P ) "1.
and, as previously, other cases stay the same. (Notice that in the trans-lation for conjunction ' O ' is not used.)
And finally, a logic D j f was formulated also w i t h the help of the modal logic S 5 as follows:
D*{ : = {A 6 Ford : rO , 4x n E S 5 } ,
where ( — )x is a function from F o rd into F o rm such that for any A, B E
F o rd:
( c )x
{A
AdB)
X=
ROA
X A O ex _ 1,(e)x (A <-»d £ )x = rO ( 0 / \x - > 8X) A 0 ( 0 8 x - » / \x) ~ \
and again, other cases stay unchanged.
Thus, a l l these logics have different conditions for conjunction. Notice that for each translation — call it 'any', for all A, B E F o rd: (A *->D
£ )a ny =
((A ->
DB)
Ad(B ->
DA))
AN^.
Of course, these logics are also closed under (sb) and (mpd).In [2] Ciuciura observed that D\ D2 It was shown that one of the axioms of the logic DJ; is not a thesis of the logic D2. We also have: FACT 1.3 ([11]). Every two logics among D2, 0%, D ^ , and DJ* cross each other.
2 MODAL LOGICS DEFINING D2, D\, AND DJj*
There is a procedure (see [9]) that for a given class of logics fulfilling some natural conditions, returns, in the considered class, the minimal
Marek Nasieniwski and Andrzej PLetruszczak
Logic which has the same theses beginning w i t h ' O ' as S 5 . The same can be repeated for D J , D T , and D J * .
We say that a modai Logic L defines D2 (resp. D J , D T , Of) iff • D2 = {A G F o rd : rOA*~] G /-} (resp.
• D * = {A G F o rd : rOA*n G L],
• D T = { / \ G Ford : • " O / T "1 G L]
• - { / \ e F o rd : rOAx~] G 1 } ) .
There are know/n other modal Logics defining D2. The same holds for the other three discussive logics.
W e see that w h i l e expressing the logic D2 w e refer to modal logics which
have the same theses beginning w i t h ' O ' as S 5 . (f) Let S 50 be the set of a l l these logics, that is,
L G S5<> iff Vv €F o rm(rO < p "1 G L 4 = ^r O ^ G S 5 ) .
By the definition we see: FACT 2.1. For any L G S5<>:
1. {rO < pn : rO < pn c S 5 } C [ ,
2. If £ G S 50, then L defines D2, D^, D T and D**.
Recall that r t e S 5M, c m S 5M e S 5M, m S 5M, r S 5M and S 5M are r e
-spectively, the smallest r t e - cm-, congruentlal, monotonlc, regular and normal logic in S5<>. Thus, by Fact 2.1 each of them defines logics D^, D T and D J * .
Let ( — )a n y be any translation of discussive formulae Into modal l a n
-guage, I.e., ( — )a n y Is a function from Ford into F o rm, and let
D *I U J :={AE Ford : rO A^ n G S 5 } ,
COROLLARY 2.2 ([11]). The logics r t e S 5M c m S 5M, e S 5M, m S 5M, r S 5M,
and S 5M are the smallest rte-, c m - congruentlal, monotonlc, regular,
AXIOMATISATIONS OF MINIMAL MODAL LOGICS...
FACT 2.3 ([9]). For any rte-logic L: L defines D2 iff L G S5<>.
In the proof of the next fact a function ( — ) °1 from F o rm into F o rd
which < < u n - m o d a l i z e s > > every modal formula was used:
1. ( a )0' = 0, for any prepositional letter a,
2. for any cp, ip G F o rm: (a) (-><p)°1 = r- i ^ °i n, (b) (cp V (/y)°1 =rcp°" V ( / / °1 _ l, (c) (cp A = r~1( ~1 <p°1 V - 1 ip°1)'1, (d) ( < p - » ^ ) °1 = r - r ! V ( / / °1 _ l, (e) (cp <-> = r- i ( - i ( - i cp°' V V V < ? °1) )n, (f) ( 0 ( p ) °1 = > ° 1 Ad ( p V - i p ) "1, (g) ( • ^ i = r ^ o 1 _ > d n ( p V n p ) n
Next we observe that for any A, B G F o rm, § G { ^ , 0 } and * G { A , V
,—>,<->} the f o l l o w i n g formulae belong to P L : (§A)°* <-» §A°*
(A
* 8 ) ° * <-> ( / r * (•)( 0 4 )0* <-> - . o ^ / r
A n d finally w e see that for any formulae A ] , AN, C w e obtain:
C ° * G l iff C[mU-,o^Av---m"/^o^An) e L,
FACT 2.4 ([11]). For any rte-logic L: L defines DJ> iff L G S5<>.
On the other hand in the proof of the below fact another function (—)°2 from F o rm Into F o rd Is used where for any cp G F o rm:
(f) (o<p)°2 = r^(cp°2 - ^dn ( p V - . p ) ) ,
The other cases are as in the formulation of the function ( — ) °1.
FACT 2.5 ([11]). For any rte-logic L: L defines D j Iff L e S5<>.
A n d finally, in the proof of Fact 2.6 a function ( — ) °3 from F o rm into
Marek Naslenlwskl and Andrzej Pietruszczak
(f) (0(p)°i = r(p°i Ad (p°^.
Again, the other cases stay unchanged.
FACT 2.6 ([11]). For any rte-logic L: L defines D J * iff L E S 50.
COROLLARY 2.7 ([11]). T h e Logic r t e S 5M (resp. c m S 5M, e S 5M, m S 5M
r S 5M, S 5M) is the smallest r t e - (resp. c m - congruential, monotonic,
regular, normal) modal logic defining the logics D2, D J , D^, and D J * . Taking into account the above Corollary, w e see that to find differ-ences between logics defining respective dlscussive logics one has to search for modal logics that are weaker than r t e S 5M. There are
con-sidered ([11]) the weakest modal logics defining respectively D J , D ^ , and D J * . In the case of these modal logics, w e do not have a l l theses of S 5 that begin with 'O'.
3 THE SMALLEST MODAL LOGICS DEFINING D J , D j , D J *
3.1 Logics A, A*, A~, and Ax
Let A , A*, A-, and Ax be the smallest logics defining D2, D J , D^, and
DJ*, respectively. W e define the following set of modal formulae:
Gen:={(p 6 F o rm : 3A^D2 9 = rOA*~1}
= {rOA^ eForm:AeD2},
Gen*:={(p E F o rm : 3 ^g D* cp = rOA*~"} = {rOA*n E F o rm :Ae D J } , Gen~:={(p E F o rm : 3 ,4 g D- <p = rO / Tn} = {rOA^ e F o rm :AE D2}, Genx:={cp E F o rm : 3AeD** cp = rOAx'1} = ( r ^ n G F o rm : A E D J * } .
LEMMA 3.1 ([11]). Every modal logic defining D2 (resp. D J , D2 and DJ*) includes the set S u b ( G e n ) (resp. S u b ( G e n * ) , S u b ( G e n " ) , S u b ( G e nx) ) .
Let AXPL be the set of modal formulae such that the pair (AXPL, { ( m p ) } ) is an axiomatization of P L .
AXIOMATISATIONS OF MINIMAL MODAL LOGICS..
FACT 3.2 ([11]). A (resp. A*, A~, Ax) Is the smallest modal logic including
the set Gen (resp. Gen*, G e n " , C e nx) . Consequently, A (resp. A*, A-,
Ax) is axlomatized by the sum of sets AXPL,
{rep°),
and S u b ( G e n ) (resp.S u b ( G e n * ) , S u b ( G e n " ) , S u b ( G e nx) ) and (mp) as the only rule.
COROLLARY 3.3 ([11]). Every two logics among A, A*, A-", and Ax cross
each other.
From facts 2.4-2.6 we obtain:
FACT 3.4 ([10, 11]). The logic A is not an rte-logic, so A C r t e S 5M.
Moreover, none of the logics A*, A-, and Ax is an rte-logic.
3.2 Simplified axiomatisations of the considered JaAIJkowski-like
discussive logics
Although Fact 3.2 gives an axiomatisations of logics A, A*, A-, and
Ax, it is not elegant since the sets Gen, G e n " , Gen* and G e nx are
infinite (other constituents of sums constituting axiomatisations of the considered modal logics can be easily replaced by respective finite sets). We recall Kotas's method of axiomatisation of D2, since it can also be adopted to finally give axiomatisations of the considered modal logics.
For any rule R on F o rm we define the following rules R° and Ra
on F o rm:
R° : = { { O ^ OcpniOiP) : <<?-, <pn,ijj) E R},
Ra : = {<n<Pi •<?„,•</'>: (91 <Pn,Lp) e R}.
Moreover, for any set of rules 1Z on F o rm w e put 1Z° := { R° : R E H}
and 11° : = { RD : R EK}.
Now, let AxjgU t be any finite axiomatization of Taut w i t h (mp) and
(sb). Next we consider the following rules:
D(p I cp ( n e c- 1)
Marek Nasieniwski and Andrzej Pletruszczak
In [12] a set M - S 5 : = {cp E F o rm : Ocp E S 5 } w a s considered.
Adopting axlomatisation given In [4] we see that for the case where ' O ' is a primitive symbol of the language It has the following form: FACT 3.5 ([4]). 1. The set M - S 5 is axiomatized by the sum of sets
D A xT a u t - ir ePa) - { • / < : , • 7", D 5 } , and the rules (sb), (nec~1), ( p o s- 1) ,
( n e c )D, ( m p )D
2. The set D S 5 Is axiomatized by the sum of the sets LlAxjgU t, [repa),
{ • / C , D f , D 5 } , and the rules (sb), ( n e c )D, ( m p )D.
It appears that unmodallzlng functions used In proofs of facts 2.4-2.6 are variants of the function used in [4]. Let ( — ) ° : F o rm — > F o rd be a
function such that for any cp E F o rm:
(f) ( O f ) - ( p V ^ ) A d f ,
(g) (Dcp)0 = ^(pV^p)A^(p0),
and other conditions stay as In the definition of the function o-j. N o w we have
LEMMA 3.6 ([4]). 1. For any A E Ford, if A E D2, then A* E M - S 5 .
2. For any <fi E F o rm, if cp E M - S 5 , then cp° E D2.
Let us recall the following notation (see [10]). For any T C F o rd and
any translation $ from F o rd Into F o rm we put
ro s := { r O / ^ E F o rm : A E T}.
Of course, for $ = • w e have Gen = DJ*.
Moreover, for any rule R on F o rd we define the following rule Ros
on F o rm:
: = { ( 9 i <Pn.ilj) • \ AmBeFo^ W = rOAp,...,cpn
= rOAp, LJJ = rOB$'1 and {Ah . . . , AN, B) E R } .
Thus, for any A\ An, B E F o rd:
AXIOMATISATIONS OF MINIMAL MODAL LOGICS.,
For K being a set of rules on F o rm let 1Zos : = {Ros : R G 11}.
Similarly as in the case of modal logics, for a l l sets X and A of discussive formulae and any set of rules 1Z in Ford w e say that the pair (A,1Z) is an axlomatizatlon of X iff X is the smallest set including A
and closed under a l l rules from 1Z.
FACT 3.7 ([10]). Let ( A { ( m pd) } ) be an axlomatizatlon of D2.
Then ( A xP L U ( r e pD) U A°', { ( m pd) ° ' , (mp)}) and
(AXPL U {repa) U A°', { ( R C ) , ( m p ) } ) are axlomatizations of A.
Conse-quently, A Is the smallest modal logic which includes the set A°' and is closed under the rule ( m pd) ° * (resp. (RC)).
One can extend the above lemma to a theorem (see [10, Fact 4.2]) that can be used to obtain an axiomatisation of the logic A. W e can use Kotas's axiomatisation [4, 6] of D2. To be able to express Kotas's
result, we recall his abbreviation: p ^ g : = - ( ( r V n r ) A d n ( n P V q ) )
THEOREM 3.8 ([4]).
The logic D
2is axiomatised by the sum of the sets
(D A xT a u t ) ° < ( • ( r e p0) )0, { ( D / C )0,
(OT)°, (05)°}, and the formulae
r
(P§qy° - ] (P§qr ond ^(p§q) - J (p§q)^, for § G { A
dy,-*
6,**
6}, and the rules
(sb)°, ( n e c "1) ° , ( p o s "1)0, ( n e c )D o, ( m p ^ i ) ,( m p )D° .
Using translations ( — ) * and ( — ) °1 (resp. ( - ) " and ( — ) °2; ( — ) °3
and ( — )x) w e extend Kotas' Lemma 3.6 to the case of D-J, D^, and
Of.
LEMMA 3.9. 1. (a) For any A e Ford, if A s D*2, then A* G M-S5.
(b) For any (p G F o rm, if cp G M-S5, then <p°1 G D^.
2. (a) For any A G Ford, if A G D j , then / T G M-S5.
(b) For any cj) G F o rm, if <p G M - S 5 , then cp°2 G D2.
3. (a) For any A G Ford, if A G , then Ax G M - S 5 .
Marek Nasieniwski and Andrzej Pietruszczak
We can easily obtain axlomatisations of D J , D2 and D J * . N o w w e
w i l l use respective abbreviations for those logics: p - > 5 q : = -"(->(-" p V q) Ad (r V -> r ) )
p - » 5 g : = (->(-! pV q) ->A ->(r V -> r ) )
W e see that in the next theorem, in the case of D J * one can use either —>] or — B e s i d e s , the implication —>3 can be used in each case. THEOREM 3.10. 1. The logic D J is oxiomotised by the sum of the
sets ( D A x ^ u t )0 1, D({repD))°\ {{DK)°\ (OT)°\ ( D 5 ) °1} , and r( p § qT°' ->2S (P § qV ond r( p § q) ->j (p § q y y , for § G {Ad, V, - >d , ^d} os axioms, and the rules (sb)°1, ( n e c ~1) °1, ( p o s "1) °1, ( n e c )D°1,
( m p ^ ) , ( m p )D°1.
2. The logic D2 is oxiomotised by the sum of three sets (nA x j aU t) °2,
( • ( r e pD) ) °2, {{UK)°\ (OT)°2, ( D 5 )0 2} and the formulae r( p § q r °2 - 53 (p§qP ond r( p § q) ->3S (p § q ) ^ , for § G { Ad V , -^d,^d}, as axioms, and the rules (sb)°2, ( n e c- 1) 2, ( p o s ~1) °2, ( n e c )D°2,
( m p ^ s ) , ( m p )D°2.
3. The logic D J * is axiomatised by the sum of three sets (DAxjaU t)°3,
(•(rep°))°3, {{UK)°\ {UT)°\ (05)°^}, a n dr( p § q )x oi - > f (p§q)n and r( p § q ) ^2S (p § q)x°3~1, for § G { Ad, V , ^d, ^d} as axioms, and the rules ( s b )x°3, ( n e e "1)0' ( p o s "1)0 3, ( n e c )D°3, ( m p ^ ) , ( m p )D°3.
The obtained axlomatisations of the logics D J , D2 and D J * can be
used to give axlomatisations of logics A*, A-, and Ax. Fact 3.7 can be
extended to any axi.omatizati.on of D2 and also of DJ, D J , and D J * . In such a w a y w e obtain an extension of the mentioned Fact 4.2 from [10] to the case of D2, D J and D J * .
THEOREM 3.11. Let (A,1Z) be an axiomatization of D2 {resp. D2, D J ,
DJ*).
1. The pairs
• { A x p L U ( r e pD) U ^0' , ft0* U { ( m p ) } ) ,
• ( A xP LU ( r e pD) U ^ f t0* U { ( m p ) } ) ,
• (AxpL U (repD) U A0~, U { ( m p ) } ) ,
1 6 2 AXIOMATISATÏONS OF MINIMAL MODAL LOGICS..
are axiomatizations of the logics A, A*, A , and Ax, respectively.
2. The logic A (resp. A*, A~, Ax) is the smallest modal logic which
includes the set A°' {resp. A°v, A°~, A®*) and is closed under all rules from the set Tl°' {resp. K0t, 71° and 11°y).
REFERENCES
[1] BULL, R. A., and K. SECERBERG, 'Basic Modal Logic', pp. 188 in Handbook of P h i l o -sophical Logic, vol. II, D. M. Cabbay and F. Guenthner (eds.), Dordrecht: D. Reidel Publishing Company, 1984.
[2] CIUCIURA, J., 'On the da Costa, Dublkajtis and Kotas' system of the discursive logic, D^', Logic and Logical Philosophy 14:235-252, 2005.
[3] DA COSTA, N. C. A., and L. DUBIKAJTIS, 'On jaskowski's discussive logic', fin:] A. I. AR-RUDA, N. C. A. DA COSTA, and R. CHUAQUI (eds.), Non-Classical Logics, M o d e l Theory and Computability, Amsterdam: North-Holland Publishing, 1977, 37-56.
[4] KOTAS, J., T h e axiomatization of S. Jaskowski's discussive system', Studia Logica 33, 2:195-200, 1974.
[5] KOTAS, J., and N . C A. DA COSTA, 'On some modal logical systems defined in con-nexion with Jaskowski's problem', [in:] A. I. ARRUDA, N. C. A. DA COSTA, and R. CHUAQUI (eds.), N o n Classical Logics, M o d e l Theory and Computability, Amsterdam: North-Holland Publishing, 1977, 57-73.
[6] KOTAS, J., and A. PIECZKOWSKI, Allgemeine logische und matematische Theorien', Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 16:353-376, (1970).
[7] JASKOWSKI, S., 'Rachunek zdari dla systemA§w dedukcyjnych sprzecznych', Studia Societatis Scientiarum Torunensis Sect. A, I, no. 5:57-77, 1948. In English: 'Prepo-sitional calculus for contradictory deductive systems", Studia Logica 24:143-157, 1969, and Logic and Logical Philosophy 7:35-56, 1999.
[8] JASKOWSKI, S., 'O koniunkcji dyskusyjnej w rachunku zdari dla systemow dedukcyj-nych sprzeczdedukcyj-nych', Studia Societatis Scientiarum Torunensis Sect. A, vol. I, no. 8: 171-172, 1949. In English: 'On the discussive conjunction in the prepositional cal-culus for inconsistent deductive systems', Logic and Logical Philosophy 7: 57-59, 1999.
[9] NASIENIEWSKI, M., and A. PIETRUSZCZAK, 'A method of generating modal logics defin-ing Jaskowski's discussive logic D2', Studio Logica 97, 1:161-182, 2011.
[10] NASIENIEWSKI, M., and A. PIETRUSZCZAK, 'On the weakest modal logics defining Jaskowski's logic D 2 and the D2-consequence', Bulletin of the Section of Logic 41(3/4): 215-232, 2012.
Marek Nasienlwski and Andrzej Pietruszczak
[11] NASIENIEWSKI, M., and A. PIETRUSZCZAK, 'On modal logics defining Jaskowski-like discussive logics', submitted to Proceedings of 5th World Congress on Paraconsis-tency, Kolkata, February 13-17, 2014.
[12] PERZANOWSKI, J., 'On M-fragments and L-fragments of normal modal propositional logics', Reports on Mathematical Logic 5:63-72, 1975.
M a r e k Nasieniewski and Andrzej Pietruszczak Department of Logic
Nicolaus Copernicus University
Stanistavva Moniuszki 16/20, 87-100, Toruń Poland