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A LOGICAL ANALYSIS OF RULE

INCONSISTENCY

PHILIPPE BESNARD

IRIT, CNRS (UMR 5505), Universite de Toulouse118 route de Narbonne, F-31062 Toulouse, France

besnard@irit.fr

Representing knowledge in a rule-based system takes place by means of\if. . .then. . ." state-ments. These are called production rules for the reason that new information is produced whenthe rule fires. The logic attached to rule-based systems is taken to be classical inasmuch as\if. . .then. . ." is encoded by material implication. However, it appears that the notion oftriggering \if. . .then. . ." amounts to different logical definitions. The paper investigates thematter, with an emphasis upon consistency because reading \if. . . then. . ." statements as rulescalls for a notion of rule consistency that does not conform with consistency in the classicalsense. Natural deduction is used to explore entailment and equivalence among various for-mulations and properties.

Keywords: Logic; consistency; rule.

1. Introduction

In formal logic, \if. . .then. . ." is supposed to be captured by the conditional con-

nective (material implication in classical logic). Under such an account, A ! C

therefore spells out in terms of necessary and sufficient conditions: (i) A is a suffi-

cient condition for C, and(ii) C is a necessary condition for A. However, A ! C can

be viewed as a rule for representing knowledge in a rule-based system [8]. In such an

approach, A ! C is turned into a production rule within an inference system

applying the so-called principle of detachment: from A, infer C (given A ! C).

That is, A and C form a condition-action pair that connects the truth ofC with the

truth ofA (there is an unfortunate mismatch of initials in the role ofA and C here as

A is the condition and C is the action). In short, A ! C expresses that A triggersC.

Since the rule of detachment is admissible for material implication, inference

seems fine but consistency of a body of rules can hardly be identified with consist-

ency in the absence of the triggerring condition of all these rules. For example, for

any consistent propositional formula A, it happens that A ! :A (where : denotes

negation) in isolation is consistent in the sense of classical logic, but as a rule, it

seems somehow dubious.

International Journal of Semantic ComputingVol. 5, No. 3 (2011) 271280c World Scientific Publishing CompanyDOI: 10.1142/S1793351X11001250

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2. Formal Preliminaries

Prior to detailing the formal preliminaries needed as the notion of rule inconsistency

is to be investigated through formal logic, several important comments are in order.

. The present paper does not offer a logic to capture reasoning with rules. Indeed,

logic is merely used here to investigate the main features of rule inconsistency

(e.g., equivalences of various formulations of it).

. Since logic is only used here to explore rule inconsistency, disjunction is omitted

(there is no property nor any definition of rule inconsistency that would naturally

require disjunction). First-order quantification is omitted for a similar reason.

. Yet, second-order quantification is used as the simplest case of rule inconsistency

in that some fact implies itself. Since first-order quantification is absent, it hap-pens that second-order quantification only gives rise to Quantified Boolean For-

mulas (i.e., propositional symbols are quantified upon so that e.g., :9XX ^ :X

and 9YX ! :Y are such formulas).

. Universal quantification is omitted because existential quantification is sufficient

here and more natural for the quantified expressions to be dealt with.

. The inference relation intuitively holds (written A) when there exists a

derivation tree from to A using natural deduction rules (see below). However,

is not formally given a definition in the sequel because not a single logic is con-

sidered but a family is. As an illustration, establishing the equivalence between:X ! :X and :9YX ! Y ^ X ! :Y is of little interest if limited to

a single fixed by a definition. Generality arises from the equivalence holding

for every inference relation that admits the corresponding derivation trees.

. In principle, the inference relations under consideration take the nine rules

listed below. A few restrictions on how to apply some of these occur in Sec. 6 (and

Sec. 7) through conditions on the hypotheses to be discharged.

. Of course, the choice of natural deduction instead of another inference method is

arbitrary and does not interfere with the results obtained.

. For the reader who does not know about natural deduction, here is an introduction.In natural deduction, inference takes the form of a tree whose nodes are labeled

with formulas, such that a node can be formed only if its formula follows (via a rule

inthelistofrulesprovided)fromtheformula(s)labelingone(ormore)ofthecurrent

node(s). For example, the conjunction X ^ Y can be obtained from X and Y

X Y

X ^ Yvia ^-introduction as listed on the next page

which shows fX; YgX ^ Y. The leaves (here, X and Y) are hypotheses.

Hypotheses can be freely introduced and they can be discharged(i.e., canceled).

X Y

X ^ YY ! X ^ Y

via ! -introduction as listed on the next page

Hence XY ! X ^ Y.

272 P. Besnard

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A logical language is assumed where ? stands for absurdity, ! denotes impli-

cation, : negation, and ^ conjunction. 9 is used for existential quantification and X,

Y, Z, . . . are propositional variables. Uppercase letters A, B, C, . . . are used asmeta-level symbols denoting formulas. Notation AX indicates that the variable X

occurs in A and AY is the same formula except that Y is substituted for X

everywhere in AX. Sets of formulas are denoted by Greek letters, , . . . In order to

conduct an analysis of inference, a system of natural deduction [3] is employed.

Deduction of C from is denoted by C. Finally, the rules are:

^-elimination A ^ B

A

A ^ B

B

A B

A ^ B^-introduction

! -elimination A A ! BB

An

..

.

BA!B

n

! -introduction

:-elimination A :A

?

An

..

.

?

:An

:-introduction

9-elimination AYn

..

.

9XAX BB

n

AE

9XAX

9-introduction

?-elimination :An

..

.

?A

n

In 9-elimination, Y must occur

neither in 9XAX, nor in B

nor in any other hypothesis

than A[Y].

For ease of reading, every hypothesis is numbered with a distinctive superscript,and is discharged at the step labeled with that number.

For the reader unfamiliar with natural deduction, here is a detailed example of a

proof of? Y (a word of warning is in order: the proof presented is not meant to be

optimal). The start (referred as step 0 below) is

?

That is, ? is asserted as an hypothesis.

?

:Y !?1:

The rule of !-introduction is used in step 1: ? is concluded in step 0 from the

(fictitious) hypothesis :Y, hence discharging :Y amounts to obtaining :Y !?.

?

:Y !?1

:Y:

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That is, :Y is introduced as an hypothesis.

?

:Y !?1

:Y?

2

Here, step 2 consists of applying the rule of ! -elimination from :Y !? and :Y.

?

:Y !?1 :Y

3

?

Y3

2

The final step 3 consists of applying ?-elimination to the deduction (obtained in

step 2) from ? (ignored for the purposes of discharge here) and :Y to ?. Overall,

the deduction shows that Y is inferred from ?. The auxiliary hypothesis :Y is notmentioned because it has been discharged (canceled so to speak) in the course of the

deduction.

3. Logical Inconsistency vs. Rule Inconsistency

When regarded as a rule, A ! C intuitively appeals to different conditions of con-

sistency. For example, that the truth ofA triggers the truth of:A seems somewhat

contradictory. A system of production rules formalized as implicative formulas need

not fail consistency in the logical sense although a triggering condition and some

consequence may contradict each other, which is called rule inconsistency.

The simplest case of rule inconsistency is of course X ! :X. That is, the trig-

gering condition X yields :X and a contradiction arises. Generalizing, X ! :X

need not be a rule but may result from chaining in rule application

A ! B B ! C

A ! C

such as in the following example:

P ! Q

Q ! :P

:

Furthermore, a contradiction may arise about any Y in the form X ! Y ^

X ! :Y. For example,

P ! Q

Q ! R

P ! :R

8