CC BY
20D.Batens1
A UNIVERSALLY ABNORMALITY-ADAPTIVE
LOGIC2
Abstract. The present paper presents a logic that allows for the abnormal behaviour of any logical constant and for the ambiguous behaviour of any non-logical constant, but nevertheless offers an interpretation of the premises that is as normal as possible.
1. Aim of This Paper
Although adaptive logics are strictly formal logics, they integrate several typical aspects of 'argumentation': dynamic reasoning, meaning variance (of logical as well as non-logical constants), inferential information (as opposed to omniscience), and languages that are not compounded by pre-fixed building blocks (see especially [7] and [6]). The dynamics of the proofs relates to the fact that adaptive logics do not, as usual non-standard logics, invalidate certain rules of inference, but restrict their applications to consequences of the premises that fulfil certain conditions.
The first adaptive logics handled inconsistent sets of premises by interpreting them as consistently as possible (see, e.g., [4] and [1]). Later adaptive logics handled other forms of logical abnormality. Still later, logical abnormality deriving from the abnormal behaviour of non-logical constants (ambiguities) was integrated. Recently, the Ghent Group discovered adaptive logics that have nothing to do with logical abnormalities (compatibility, the consistent extensions of theories, abduction, ...).
Considering abnormality-adaptive logics, one wonders whether it is possible to devise a logic that adapts to all forms of abnormalities. Such a system would open radically new perspectives on logic. It would allow for abnormalities of all kinds, but still presuppose (classical) normality 'unless and until proven otherwise'. In other words, it would be capable of recapturing all of Classical Logic - henceforth CL -while still allowing for all sorts of deviations from it. In this paper I present such a logic, ACL02.
1 Centre for Logic and Philosophy of Science, Universiteit Gent, Belgium. [email protected].
2 Research for this paper was supported by the Fund for Scientific Research - Flanders, and indirectly by the INTAS-RFBR contract 95-365. A technically and philosophically more elaborate version of this paper is available as [9].
One way of describing the situation is by saying that, in ACL02, all derivation is conditional. That A is derivable from r does not only depend on the intended meaning of the logical constants and on the intended meaning stability of the non-logical constants; it is also depends on whether the specific content of r does not rule out that intended meaning and that intended stability.
2. Abnormality-Adaptive Logics
The first formulation of adaptive logics (see [4] and [1]) was proof-theoretical. Later a decent adequate semantics was developed (see [2], [8], and elsewhere). Meanwhile, many inconsistency-adaptive logics have been studied (see [19], [21], [22], [15], [17], and [12]), their use to several domains of application has been shown (see [3], [7], [16], and [18]), and several other logics (some non-monotonic logics, see [5] and [15], and all consequence relations defined from the Rescher-Manor mechanism, see [12]) have been integrated. I refer to [11] for a survey.
Even if the dynamics aspects of the proofs are among the most fascinating aspects of adaptive logics, space limitations prevent me to describe them here. So, let me at least mention one central feature.
At the predicative level, adaptive logics are not only undecidable3; for most of them, there is no positive test for derivability. Nevertheless, there are certain criteria that tell us, in specific cases, that a wff derived in a proof from r at a stage, is finally derived from r4.
Where such criteria cannot be applied, it still can be demonstrated that derivability at a stage offers us the best estimate of the final consequences of the premises - best in view of the present understanding of the premises as revealed in the proof5.
An abnormality-adaptive logic is defined from a lower limit logic (a monotonic paraconsistent logic) and an upper limit logic (usually CL) by an adaptive strategy. The latter determines the way in which the logic reacts to an inconsistency (or a disjunction of inconsistencies). All abnormality-adaptive logics interpret a theory ' as normaly as possible' -this phrase is ambiguous and is specified by the strategy.
The semantic characterization proceeds in terms of the models from the lower limit logic (that include the models from the upper limit logic). From those models of r a subset is selected in view of the
3 The propositional fragments are just as decidable as that of CL.
4 While derivability at a stage is a dynamic notion, final derivability is as static as any of the usual inference relations.
5 Some proofs are 'smarter' than others. Yet, a proof as it stands reveals an understanding of the premises (see [6]).
strategy. The formulas true in all selected models are the adaptive consequences of r (those finally derivable from it).
As an example, consider the semantics for ACLuN2. Let L be the language of CL, with S, C, V, P, F and W the sets of sentential letters, individual constants, individual variables, predicative letters of rank r, and formulas and wffs respectively. In order to handle the quantifiers in a simple way, we extend L to L+ by introducing a set of pseudoconstants O that has at least the cardinality of the largest models we want to consider. Let W+ be the set of pseudo-wffs of L+ and let ~W+ = {~A | A e W+}.
A model M = <D, v>, in which v is an interpretation of W+, and hence of W, which is what we are interested in. The assignment function is defined by: C1.1 v : S ^ {0, 1}
C1.2 v : CuO ^ D (where D = {v(a) | a e CuO})
C1.3 v : P ^ Dr) (the power set of the r-th Cartesian product of D)
C1.4 v : ~W+ ^ {0, 1}
The valuation function vM determined by M is defined as follows:
C2.1 vm : W+ ^ {0, 1}
C2.2 where A e S, vm(a) = v(A); vm(1) = 0
C2.3 vM(nr ai . ar) = 1 iff <v(ai), ., v(ar)> e v(nr)
C2.4 vm (a = P) = 1 iff v(a) = v(P)
C2.5 vm (~A) = 1 iff vm(A) = 0 or v(~A) = 1
C2.6 vm (A 3 B) = 1 iff vm(a) = 0 or vm(b) = 1
C2.7 vm (A a B) = 1 iff vm (A) = 1 and vm (B) = 1
C2.8 vm (A v B) = 1 iff vm(a) = 1 or vm(B) = 1
C2.9 vm (A = B) = 1 iff vm (A) = vm(B)
C2.10 vm((Va) A(a)) = 1 iffvm(A(P)) = 1 for all P e CuO
C2.11 vm ((3a)A(a))=1 iff vm(A(P))=1 for at least one P e CuO
Truth in a model, semantic consequence, and validity are defined as usual.
The inconsistency-adaptive logic ACLuN2 is obtained from CLuN by the Minimal Abnormality Strategy. The abnormal part of a CLuN-model is defined as Ab(M) =df {A e F | vm(3(A a ~A) = 1}, where 3 denotes a sequence of existential quantifiers over all variables free in A.
Definition 1 A CLuN-model M of Г is a minimally abnormal model of Г iff no CluN-model M' of Г is such that Ab(M) с Ab(M).
Definition 2 Г |=aclun2 A iff A is true in all minimally abnormal models of Г.
Apart from the Soundness and Strong Completeness of final derivability (see [8] for the definition) with respect to the semantics, many nice metaproperties have been proved. Proofs may be found in [8], [10], and some forthcoming papers. Some criteria for final derivability were obtained in view of tableau methods (see [13]), others in view of results of the block approach (see [6]).
The (negation-)incompleteness-adaptive logic that is the dual of ACLuN2 is called ACLaN2 and is obtained as follows. First we characterize CLaN, a logic allowing for gaps rather than glwts with respect to negation, by replacing in the semantics for CLuN: C2.5 Vm(~A) = 1 iff Vm(A) = 0 and vm(~A) = 1 ACLaN2 is obtained from CLaN by the Minimal Abnormality Strategy, where Ab(M) = f {A e F | vm(3(A v ~A) = 0}.
There are other logical abnormalities. Let us consider gaps with respect to conjunction, as they occur in the logic called (by the same naming convention) CLaC. Define = {A л B | A, B e W+}, and accommodate the CLuN-semantics as follows. Clause C1.4 of the CLuN-semantics is replaced by C1.4 v : W ^ {0, 1} whereas C2.5 and C2.7 now become: C2.5 vm(~A) = 1 iff vm(A) = 0
C2.7 vm(A л B) = 1 iff vm(A) = vm(B) = 1 and v(A л B) = 1
The adaptive logic ACLaC2 is obtained from the lower limit logic CLaC by the Minimal Abnormality Strategy. We use classical conjunction (defined, e.g., as A П B = f ~(~A v ~B)) to define the abnormal part of a model M: Ab(M) = df {AлB e F | vM(3(A П B П ~(AлB))) = 1}, and adjust Definitions 1 and 2.
We may proceed similarly to allow for gluts or gaps with respect to other logical constants, and to devise a corresponding adaptive logic.
A somewhat different approach is to allow for ambiguities in the non-logical constants6. Here, the difference between the lower limit logic and the upper limit logic is obtained by a difference in interpretation of the premises. The upper limit logic interprets them in the usual way, but the lower limit logic interprets them by considering
6 The first such adaptive logic taking is presented in [20]. The subsequent presentation differs in several respects from the line followed in that paper.
each occurrence of a non-logical constant as having a different meaning. This is indicated by a different index (a superscripted number) that is attached to each such occurrence in both the premises and the purported conclusion. In preparation of Section 3, I briefly spell this out.
The language L is upgraded in the obvious way to accommodate the indexed non-logical constants. Let the result be L1. Let C be the set of indexed individual constants. Let r and A* be the results of attaching a different index to each occurrence of a non-logical constant in Tu{A}. For example ((p a ~q) 3 ~p)^ and ~p* might be ((p1 a ~q1) 3 ~p2) and ~p3 respectively. The lower limit logic, CLA, is defined by7
r |=cla A iff rT |=cl A*
Obviously (pAq) CLA p, as nothing warrants that vM(p1) = vM(p2). However, ~(p1=p2) counts as an abnormality with respect to the normal interpretation of the premises. For C the abnormalities are of the form ~a-aJ, for P they are of the form ~((Va1).(Var)(ni a1 ... ar = n a1 ... ar)). Define Ab(M) accordingly. By selecting the minimally abnormal models of r, the adaptive logic ACLA2 interprets the premises as normally (that is, unambiguously) as possible.
It is tempting to devise logics that are adaptive with respect to several kinds of abnormalities, for example with respect to inconsistency and incompleteness, or with respect to ambiguities and gluts with respect to the universal quantifier. Two brief remarks will clarify how this is done.
A logic that allows for both gluts and gaps with respect to, for example, negation - we call this logic CLoN - is obtained by replacing, in the CLuN semantics, clause C2.5 as follows:
C2.5 vm(~A) = v(~A)
With this clarified, the combination of any logical abnormalities is fairly obvious (and so is the definition of Ab(M)).
If we combine many abnormalities, the remaining classical connectives might not be functionally complete, which will hinder the definition of Ab(M). As I need to consider such a case in the subsequent section, I briefly discuss the matter. Let us again consider {A e F | vM(3(A a ~A) = 1} and eliminate the "3'' and "a'' from it. Suppose that Xi,... , xn are the variables that occur free in A. Let [A]° x1,.^ . ,onxn be the result of systematically replacing each xi by some oi e CuO. To
7 In all subsequent examples, I only consider the case where the i-th occurrence of some non-logical constant receives index i. All other ways of indexing obviously deliver isomorphic results, as long as the choice fulfils the above convention on r and A*.
avoid clutter, I henceforth write x for Xi, ... , Xn and o for oi, ..., on. Remark that vM(3(A a ~A) = 1 iff there are o such that vM([A]ox) = vM([~A]ox) = 18. Proceeding thus, we may eliminate all 'auxiliary' logical constants from the definition of Ab(M).
3. The Empty Logic and Its Adaptive Extension
Here I combine all kinds of abnormalities. The lower limit logic will be called CL0. We start with CLoX, the logic that allows both gluts and gaps with respect to all logical constants. Where M=<D, v> is a CLoX-model, v is defined by: C1.1 v : S ^ {0, 1}
C1.2 v : CuO ^ D (where D = {v(a) | a e CuO})
C1.3 v : P ^ ^ ( Dr) (the power set of the r-th Cartesian product of D)
C1.4 v : =W+ ^ {0, 1}
C1.5 v : ~W+ ^ {0, 1}
C1.6 v : 3W+ ^ {0, 1}
C1.7 v : aW+ ^ {0, 1}
C1.8 v : VW+ ^ {0, 1}
C1.9 v : W ^ {0, 1}
C1.10 v : VW+ ^ {0, 1}
C1.11 v : 3W+ ^ {0, 1}
The valuation function vM determined by M is defined as follows:
C2.1 vm : W+ ^ {0, 1}
C2.2 where A e S, vm(A) = v(A); vm(1) = 0
C2.3 vM(nr aj ... ar) = 1 iff <v(aj), ..., v(ar)> e v(nr)
C2.4 vm (a = P) = 1 iff v(a = P)
C2.5 vm (~A) = v(~A)
C2.6 vm (A 3 B) = v(A 3 B)
C2.7 vm (A a B) = v(A a B)
C2.8 vm (A v B) = v(A v B)
C2.9 vm (A = B) = v(A = B)
C2.10 vm((Va) A(a)) = v((Va) A(a))
C2.11 vm ((3a)A(a)) = v((3a)A(a))
8 Obviously [~A]ox = [~A]ox. If expressions such as [A 3 ,B]°x and [A]ox are used in the same clause, x always refers to the variables free in the longest formula.
Truth in a model, semantic consequence, and validity are defined as usual.
We now upgrade CLoX to CL0. Let r and A* be as in Section 2.
Definition 3 r |=cl0 A iff r |=clox A*
r |=CL0 A iff ±eT. So, if we remove ± from L, r CL0 A for all r and A .
To formulate ACL02, obtained from CL0 by the Minimal Abnormality strategy, I define Ab(M) as the union of the sets listed below (defined in terms of CLoX-models). I suppose throughout that A, Be F and that each defining clause is preceded by: for some o e CuO.
{A | vm([A]ox) = Vm([~A]ox)},
{<A 3 B, A, B> | vm ([A 3 B]ox) = vm([A]ox) = 1 and vm([B]ox) = 0},
{<A 3 B, A> | vm([A 3 B]ox) = 0 and vm([A]ox) = 0},
{<A 3 B, B> | vm([A 3 B]ox) = 0 and vm([B]ox) = 1},
{<A a B, A, B> | vm ([A a B]ox) = 0 and vm([A]ox) = vm([B]ox) = 1},
{<A a B, A> | vm([A a B]ox) = 1 and vm([A]ox) = 0},
{<A a B, B> | vm ([A a B]ox) = 1 and vm([B]ox) = 0},
{<A v B, A, B> | vm ([A v B]ox) = 1 and vm([A]ox) = vm([B]ox) = 0},
{<A v B, A> | vm([A v B]ox) = 0 and vm([A]ox) = 1},
{<A v B, B> | vm ([A v B]ox) = 0 and vm([B]ox) = 1},
{<A = B, A, B> | Vm([A = B]ox) * (Vm([A]ox) = Vm([B]ox))}9,
{<(Va) A (a), A(P)> | P e C1; vM([(Va) A(a)fx) = 1 and vm([A(P)]ox) = 0},
{<(Va) A (a), A(P)> | P e O; vM([(Va) A(a)fx) = 1 and vm([A(y)]ox) =
0 for some y e CuO},
{<(Va) A (a), A(P)> | P e V; vM<[(Va) A(a)fx) = 0 and vm([a(y)]ox) =
1 for all y e CuO},
{<(3a) A(a), A(P)> | P e V; vM([(3a) A(a)fx) = 1 and vm([A(y)]ox) = 0 for all y e CuO},
{<(3a) A(a), A(P)> | P e C1; vM([(3a) A(a)fx) = 0 and vm([A(P)]ox) = 1},
{<(3a) A(a), A(P)> | P e V; vM([(3a) A(a)]ox) = 0 and vm([A(y)]ox) = 1 for some y e CuO},
9 The condition obviously abbreviates that = ,B]ox) is 1 (respectively 0) whereas vM([A]ox) is not (respectively is) identical to vM([,B]ox). Similarly for "=" below.
{<a, P> | a, P e C1; vM(a = P) * (v(a) = v(P))},
{<a, P> | a e C1; P e V; for some y e CuO, vM(a = y) * (v(a) = v(Y))},
{<a, P> | a e V; P e C1; for some y e CuO, vm(y = P) * (v(y) =
v(P))},
{<a, P> | a, P e V; for some y, 5 e CuO, vm(y = 5) * (v(y) = v(5))}, {<A, i, j> | A e S and vm(A') * vm(A;)}, {<a, i, j> | a e C and v(a!) * v(aj)}, and {<n, i, j> | n e P and v(n') * v(^)},
Definition 4 A CLoX-model M of r* is a minimally abnormal model of r* iff no CLoX-model M' of r* is such that Ab(M') 3 Ab(M).
Definition 5 r |=acl02 A iff A* is true in all minimally abnormal models of r*.
It is easily provable (compare [8]) that, if r has CL-models, then the minimally abnormal models of r* are those in which all logical constants as well as all occurrences of non-logical constants behave normally. Hence: r |= ACL02 A iff r |=CL A.
If r has no CL-models, then (except for border cases) the minimally abnormal models of r* are a proper subset of the CLoX-models of r*. In this case, ACL02 still delivers a minimally abnormal interpretation of T: all CL-consequences of r, except for those that do not follow from r in view of the disjunctions of abnormalities that are verified by all CLox-models of r*.
I leave it to the reader to check the following properties of ACL02:
P, ~P |*ACL02 q
pAq, ~p |*ACL02 q
p A ~r, ~p A (q 3 r) |*ACL02 ~q Vx(Px 3 Qx), Pa, ~Qa, Pb, ~Qc |*acl02 Qb a ~Pc Remark that p, ~p |*ACL02 p and p, ~p |*ACL02 ~p. Some of the minimally abnormal interpretations of the premises requires the non-logical constant (represented here by the propositional letter) p to behave ambiguously: the truth-value of p1 is different from that of p2. So, the truth-value p3 is bound to agree with that of either p1 or p2. As the same reasoning applies to the other non-logical constants, one easily proves the remarkable:
Theorem 1 For all r, Cn ACL02(F) is consistent.
4. In Conclusion
As promised, ACL02 is a universally abnormality-adaptive logic. Allowing all kinds of abnormalities in the premises, it still interpretats them "as normal as possible". The latter expression has (as always) a specific meaning. But this meaning is an interesting one: whatever is consistently derivable from the premises. As one cannot determine beforehand which abnormalities will or may occur in the theory, one cannot justify beforehand the choice of an abnormality-adaptive logic. ACL02 removes this weakness.
Let me remind the reader that the aim of abnormality-adaptive logics is not offer the 'final' interpretation of the premises. Logical abnormalities will have to be ruled out; defective theories have to be replaced. This replacement is not a matter of logic, but it requires the logical analysis of the defective theory. This is what an adaptive logic should provide, and this is what ACL02 actually does provide for a very broad set of contexts.
REFERENCES
1. D. Batens. Dynamic dialectical logics as a tool to deal with and partly eliminate unexpected inconsistencies. In J. Hintikka and F. Vandamme, editors, The Logic of Discovery and the Logic of Discourse, pages 263271. Plenum Press, New York, 1985.
2. D. Batens. Dialectical dynamics within formal logics. Logique et Analyse, 114:161-173, 1986.
3. D. Batens. Static and dynamic paraconsistent logics and their use in expert systems. CC-AI, 3:33-50, 1986.
4. D. Batens. Dynamic dialectical logics. In G. Priest, R. Routley, and J. Norman, editors, Paraconsistent Logic. Essays on the Inconsistent, pages 187-217. Philosophia Verlag, München, 1989.
5. D. Batens. Inconsistency-adaptive logics and the foundation of nonmonotonic logics. Logique et Analyse, 145:57-94, 1994. Appeared 1996.
6. D. Batens. Blocks. The clue to dynamic aspects of logic. Logique et Analyse, 150-152:285-328, 1995. Appeared 1997.
7. D. Batens. Functioning and teachings of adaptive logics. In J. Van Benthem, F. H. Van Eemeren, R. Grootendorst, and F. Veltman, editors, Logic and Argumentation, pages 241-254. North-Holland, Amsterdam, 1996.
8. D. Batens. Inconsistency-adaptive logics. In E. Orlowska, editor, Logic at Work. Esays Dedicated to the Memory of Helena Rasiowa, pages 445472. Physica Verlag (Springer), Heidelberg, New York, 1999.
9. D. Batens. Zero logic adding up to classical logic. Logical Studies, 2:15, 1999. (Electronic Journal: http://www.logic.ru/LogStud/02/LS2.html ).
10. D. Batens. Minimally abnormal models in some adaptive logics. In print.
11. D. Batens. A survey of inconsistency-adaptive logics. In Batens et al. [14], pages 59-83.
12. D. Batens. Towards the unification of inconsistency handling mechanisms.To appear.
13. D. Batens, J. Meheus. Tableau methods for paraconsistent extensions of positive logic and for the adaptive logics based on them. Part 2: The predicative case. To appear.
14. D. Batens, C. Mortensen, G. Priest, and J. P. Van Bendegem, editors. Frontiers of Paraconsistent Logic. Research Studies Press, Baldock, UK, in print.
15. K. De Clercq. Two new strategies for inconsistency-adaptive logics. To appear.
16. J. Meheus. Adaptive logic in scientific discovery: the case of Clausius. Logique et Analyse, 143-144:359-389, 1993. Appeared 1996.
17. J. Meheus. An extremely rich paraconsistent logic and the adaptive logic based on it. In Batens et al. [14].
18. J. Meheus. Inconsistencies in scientific discovery. Clausius's remarkable derivation of Carnot's theorem. In print.
19. G. Priest. Minimally inconsistent LP. Studia Logica, 50:321-331, 1991.
20. G. Vanackere. Ambiguity-adaptive logic. Logique et Analyse, 159:261280, 1997. Appeared 1999.
21. G. Vanackere. HL2. An inconsistency-adaptive and inconsistency-resolving logic for general statements that might have exceptions. In print.
22. G. Vanackere. Preferences as inconsistency-resolvers: the inconsistency-adaptive logic PRL. To appear.
