Default Reasoning with Qualified Syllogisms

Daniel G. Schwartz Department of Computer Science

Florida State University Tallahassee, Florida 32306-4019, U.S .A.

schwartz@cs.fsu.edu

Abstract

Prior works by the author have introduced the system QUAL (herein Q) of qualified syllogisms. An example of such a syllogism is “Most birds can fly; Tweety is a bird; therefore, it is likely that Tweety can fly”. Q pro- vides a formal language for expressing such syllogisms, together with a semantics which validates them. Also introduced in the prior works is the notion of a path logic. Reformulating Q as a path logic allows for the ex- pression of modifier combination rules, such as “From likely P and unlikely P , infer uncertain P”. The present work builds on this, showing how to incorporate Q into a system for default reasoning. Here is introduced the notion of a dynamic reasoning system (DRS), consist- ing of a path logic, together with a semantic net, or more exactly, a taxonomic hierarchy that allows for multiple inheritance. The taxonomic hierarchy enables definition of a specificity relation, which can then be used in default reasoning (more specific information takes priority over less specific). Modifier combination rules prescribe what to do when defaults are applied in the context of multiple inheritance. Propositions derived in this manner all bear qualitative likelihood modifiers, representing the extent to which the propo- sition is believed.

1. Introduction

The subject of qualified syllogisms has it’s origins in the works of L.A. Zadeh, cf. [12-151, where also are introduced the notions of fuzzy quantifiers (modifiers such as most, almost all, all, f ew, etc.), usuality mod- ifiers (such as usually, almost always, always, seldom, etc., and likelihood modifiers (or “fuzzy probabilities” , likely, very likely, certainly, unlikely, etc.). As illustra- tions, consider the syllogism

Most birds can fly. Tweety is a bird. It is likely that Tweety can fly.

and the syllogism obtained by replacing the first line

0-8186-7126-2195 $4.00 0 1995 IEEE Proceedings of ISUMA-NAFIPS ’96

above with

Usually, if something is a bird, it can fly.

In the former there is a natural intuitive connection be- tween the fuzzy quantifier most and the likelihood mod- ifier likely, and in the latter there is a similar connection between usually and likely. Zadeh’s works provide well- defined semantics for each of the three types of modi- fiers, and it does this for the more general case of their being applied to propositions involving fuzzy predicates (e.g., “Most Swede’s are blonde”, where “blonde” is fuzzy). In addition, those works allude to, but do not fully explore, their intuitive interrelations.

A continuation of this line of investigation was taken up by the present author in [7), where it was observed that similar such syllogisms hold along each line of the following table (where, e.g., the above two are represented by the second line).

Quantification Usuality Likelihood all always certainly most usually likely many frequently uncertain few occasionally unlikely no never certainly not

That observation led in [8] to formulation of the logic QUAL (henceforth Q), which provides (i) a formal lan- guage suitable for expressing all such syllogisms, as well as propositions expressing other aspects of the relations between the three types of modifiers, and (ii) a formal semantics which validates those syllogisms and propo- sitions. Briefly, the system has a two-leveled seman- tics, wherein the lower level in multivalent, based on probability theory, and the upper level is bivalent and classical; the two levels are connected by having, e.g., likely P be “true” if the probability of P falls within a certain subinterval of [0,1]. Interested readers may refer to [8] for a more complete set of references, in- cluding in particular several closely related works by Amarger, Dubois, and Prade (e.g., [l,2]).

396

Having defined Q, it then became evident that this needed to be expanded in such a way as to allow for modifier combination. For example, given propositions likely P and unlikely P , conclude uncertain P . The de- sire to have a framework in which such inference rules could be formulated, led to the notion of a path logic [9]. Here it was reasoned that, in order to formulate such rules, one needs to portray reasoning as an activity that takes place in time. This was accomplished by adapt- ing the conventional notion of a formal logical system to one which lends special semantic status to deriva- tion paths. To wit, by distinguishing between different occurrences of a formula P in a derivation path (i.e. as occurring at different time steps), one may assign them different probability values, thereby portraying the manner in which these probabilities may be mod- ified (over time). In retrospect, it was found that this has several elements in common with certain “belief re- vision” systems [4], as well as the “step/active logics” beiing developed by D. Perlis and his students [3,6].

The present paper takes this line of investigation one step further by now considering how this type of system might be used to formulate certain types of default rea- soning. Some early works in this area are reprinted in [5], and a more recent, very elegant study is [ll]. The latter also contains many references to interim works. In reviewing these works it becomes evident that the notion of “most” is ever present, but always implicit. Thje primary contribution made here is to show how this notion may be made completely explicit. That this would be the correct way to formulate such types of “nonmonotonic reasoning” has been argued in nu- merous talks by Zadeh.

Here is introduced the notion of a dynamic reasoning system (DRS) as consisting of a path logic, together with a semantic network, where for the present pur- poses, the latter is more exactly a taxonomic hierarchy. The hierarchy provides the basis for defining a speci- f i c i t y relation, and default rules can be formulated by allowing more specific information to have priority over less specific. The DRS also provides rules for ensuring that the taxonomic hierarchy is created in tandem with the derivation path, in such a way that consistency be- tween the two is maintained.

]Due to space constraints , these ideas are here only outlined. Moreover the treatments of Q and path logic are also necessarily brief. The reader is referred to [8,9] for details.

2. The System Q

Let the modifiers in the foregoing table, in top-down then left-right order, be represented by Q 2 , &I, Qo, Q.-1, Q - 2 , U2, . . . , 24-2, L2, . . . , L-2. As symbols

choose: exactly one (individual) variable, denoted by x; some (individual) constants, denoted generically by a , 6 , . . .; some unary relation symbols, denoted gener- ically b y CY, p, . . .; some logical connectives, denoted by 7 , V, A, -+, 4, q, 0; the abovementioned quantifiers, usuality modifiers, and likelihood modifiers; parentheses and comma, denoted as usual. Let the formur‘as be the members of the sets

F1 = { C Y ( X ) [ C Y is a relation symbol}

F2 = Fi U { l p , ( P V Q ) , ( P A Q)lP, Q E Fi U F2)

F3 = { ( P -+ Q)P, Q E F2)

F4 = {&2CciP, &a.C2P, &2(.C2PiCiQ), U2Ca PI UiC2 P, U2 (L2 P i C i Q ) IP, Q E F2 U F3)

F5 = F4 U {GP, (PVQ)lP, Q E F4 U Fs}

Fi = { P(a /x ) I P E F1 and a is an individual

F i = {P(a/x) lP E F2 and a is an individual

constant}

constant}

FL = {P(a/z)IP E F3 and a is an individual constant}

Fi = {CaP(a/z), ( L ~ P ~ C ~ Q ) ( U / X ) 1 P, Q E F2 U F3 and a is an individual constant}

Fk = F i U {3’, (PVQ)lP, Q E FI, U Fk}

where ,P(a/x) denotes the formula obtained from P by replacing every occurrence of the variablse x by an occurrence of the constant a. Formulas without mod- ifiers will be first- or lower-level formulas, imd those with modifiers will be second- or upper-level. AS ab- breviat:ions let us employ

(PAQ) for --, 7

( P A Q ) for (4PVQ)

( P G Q ) for ( ( P G Q ) A ( Q A P ) )

.. (..

By a hnguage L will be meant any collection of symbols and formulas as described above. Different languages are specified by choosing different sets of individual constants and relation symbols. As an exa,mple, the first of the foregoing syllogisms may be represented in a language employing the individual constant Tweety and the unary relation symbols Bird and CamFly as

QliLp(Bird(z) + CanFly(z)) CzBird(Tweety) L1 CanFly(Tweety) -

397

In words: For most x it is certain that, if x is a Bird then x CanFly; it is certain that Tweety is a Bird; therefore it is likely that Tweety CanFly.

The basic idea underlying the semantics for Q is as follows. For each i = -2 , . . . , 2, there is assigned a subinterval i i of [0, 11, and each formula P in F2 U F 3 is assumed to have a likelihood (probability) value l ( P ) E [O, 11. Here probabilities of conditionals (formulas in F3) are interpreted as conditional probabilities. Then a formula such as Q2CiP is taken as “true” if an only if iff 1(P) E i i . Similarly, &i&P is “true” iff 1(P) E ~ i . Thus associating &1 ( m o s t ) , say, with the same interval as C1 (l ikely), leads to validation of the syllogism shown above.

3. Path logics

Let A be the class of all languages L of the kind de- scribed above. Here “formula” will mean second-level formula, unless stated otherwise. Let @ be a collection of inference rules defined on the languages in A: es- sentially, mappings from any L into itself, stating how formulas having certain forms may be derived from for- mulas having certain other forms (e.g., Modus Ponens).

A path logic based on A and @ will consist of a series of language-path pairs, (Li17ri), generated in the fol- lowing way. Let LO be the language with no individual constants and no relation symbols, and let TO be the empty sequence. Assume one has formed ( L , - 1, R, - 1) , for arbitrary n > 1. Suppose 7rn-1 is the sequence ( P I , XI), . . . , (P,-l,Xn-l) of labeled formulas. Labels X are 5-tuples (i, p, f r , t o , s), where i is the indez, p is a priority ranking, f r is a from-list, t o is a to-list, and s is a status indicator. These five items are described below.

The language L, is formed from L,-1 by adding finitely many (and possibly no) new individual con- stants and relation symbols. The path 7r, is formed by adding a new labeled formula (P,, A,) to the path ~ ~ - 1 , where P, is either (i) arbitrarily chosen from L,, i.e., is f e d in by an external agent, or (ii) derived from formulas in 7rn-1 by application of an inference rule in @, with the proviso that formulas can only be used as premises if their status indicators s are set to “on”. The index i in the label A, is just the integer n, indicat- ing the formula’s position in the sequence. The exact nature of the priority ra.nking p will be considered in the section below. In case P, was fed in, the from-list fr is simply {f} (for “feed”). In case P, was derived from formulas Pi,, . . . , Pi, by applying inference rule ‘p, then f r = {p, i l l . . . , im}. In this case also, the to- list to for each of the Pij is augmented by adding the integer n. The to-list for P, is temporarily empty. The status indicator s is set to “on”.

Thus a path logic is essentially a knowledge base that evolves over time. At each time step, the lan- guage is possibly expanded, and a new proposition is added, either by intervention of an external agent or by inferring a new fact from existing facts. Readers fa- miliar with the “step/active logics” mentioned earlier will note many similarities.

The from- and to-lists allow one to trace through all derivations, either backwards or forwards. This will be used for various forms of reason maintenance. So also will be the status indicator. The semantics devised for Q may be extended to path logics in a natural way; again, see [9] for details.

4. Modifier Combination

The issue of modifier combination is essentially that of evidence combination. This section illustrates how such can be formulated as a set of inference rules for a path logic. The following considers only the case of likelihood modifiers; quantifiers and usuality modifiers may be handled similarly.

The situation of concern is when, at some point in a path, one has derived (or fed in) Lip , and a t another point one has Cj P , and it is desired to introduce a rule which says how one might combine these to deduce a formula CkP, where Ck is derived in some meaningful manner from Ci and Cj. One possible set of such rules is described by the following table.

2 2 2 2 * 2 1 1 0 - 2 2 1 0 -1 -2 2 0 -1 -1 -2 * -2 -2 -2 -2

The table is read: given i and j as above, the corre- sponding derived k is the entry in the i-th row and j-th column, where both the row numbering (from top to bottom) and the column numbering (from left to right) are 2 , . . . , -2. For example, the entry 0 in the number 1 (second) row and the number -1 (fourth) column of the table describes the rule mentioned earlier:

Cl(P,

This says that, if P is both likely and unlikely, then P is uncertain. Here it is assumed that each label has a different time index.

The “*” in the upper right and lower left corners indicates the contradictory situation where P is both certainly and certainly not the case. The intention is that, when contradictions are encountered, they must be dealt with in some appropriate manner. Some pos- sible options (including reason maintenance) are dis-

398

cussed in [9]. These are intended for further explo- ration in later works.

5. Dynamic Reasoning Systems

For the present purposes, the foregoing notion of a language L will now be expanded to allow that the rela- tion symbols are t y p e d so as to distinguish between rela- tioris representing kznds of objects and relations repre- senting propertzes of objects. The decision as to when a relabtion should be treated as representing a kind versus a property is left to the system's designer/user. Admit- tedly, the distinction will often be quite arbitrary, if not also context dependent. Kind and property relation symbols will be indicated respectively by superscripts

By a taxonomzc hzerurchy will here be meant a se- mantic network whose nodes represent either objects, kinds, or properties, and whose links represent either eleinent-of, subset-of, or property-of relations. See [lo] for a treatment of semantic networks. As is custom- ary, the hierarchies are envisioned as directed graphs, with element-of and subset-of links pointing upwards, and property-of links pointing horizontally. A specz- ficity relatzon is defined in a natural way on the object and kind nodes: those appearing lower in the hierar- chy being more specific than those appearing higher (hence individual objects are maximally specific). Ac- cordingly, properties associated with (object or kind) nodes appearing lower in the hierarchy are more spe- ca jc than those associated with (kind) nodes appearing higher. It is assumed that element-of and subset-of re- lations are crisp (non fuzzy), but that the property-of links may involve either crisp or strictly fuzzy quanti- fiers. Two examples of such hierarchies are shown in Figures 1 and 2. Here the element-of relation is indi- cated by the phrase "is-a", and the subset-of relation by the term "all".

By a dynamic reasonzng system (DRS) will be meant a path logic, with language expanded as above, together with a sequence of taxonomic hierarchies TO,

TI , . . . constructed in tandem with the language-path pairs ( L O , TO), (L1, T I ) , , . .) as follows. TO is empty. As- sume one has formed (L,-l, r,-l and ~ ~ - 1 . Consider the new proposition P, that is added to form path T,. Then T, is formed as follows.

If P, is of the form Cza( ' ) (a) , create a new kind node for the relation & ( l e ) and a new object node for the individual a , unless such are already present in ~ ~ - 1 ,

anid add an element-of link connecting them. If P, is of the form C2a(P)(a), create a property node

for the relation a ( P ) and an object node for the indi- vidual a , unless such are already present in ~ ~ - 1 , and add a property-of link connecting them.

( k ) and ( P I .

If P, is of the form

&!i(Lza("(z)~CzP(P)(z)) add a new kind node for a(') and a new property node for assuming these don't already exist, add a property-of link connecting them, and label the new link with the quantifier Q,.

If P, is of the form

~,I(Cza(')(x)'C2p(")(z))

add new kind nodes for a(') and p(')), assunling none already exist, and add a subset-of link connecting them, but wzth the exception that if this would create a redun- dant path (more than two paths between any two kind nodes), then the link forming the shorter path must be removed. Note that this may involve removing the link just introduced; otherwise, it may involve removing a link forimed earlier.

To illustrate, let L be the language containing the in- dividual constants Tweety and Opus, and the relation symbols Bird('), Penguin('), and CanFly(P). Then the hierarchy shown in Figure 1 would be generated by feeding in the formulas

CzBird(')(Tweety) &I (CCzBird(')(z)~CC2CanFly(P)( z))

&z (C~Penguin(')(x)-1;CzBird(')( x ) ) & z (CC2Penguin(')(z)-l;Cz';CanFly(p)(z) ) CzPenguin(')( Opus)

Thus, the taxonomic hierarchy simply duplicates cer- tain information already present in the path. How- ever, it also provides the extralogical inforrriation em- bodied in the specificity relation. This in turn can be used to supply each node in the hierarchy with a pra- ority nanking, p, essentially an address indicating the node's position. To illustrate, in Figure 1, re- ceives the rank (l), Tweety receives the rank (1, 1), Penguin(') receives the rank (1,2) , and Opiis receives the rank (1,2,1) . It follows that nodes with hiigher lexi- cographic order by rank are more specific. Let property nodes inherit the priority ranking of their associated kind nlodes. This can then be employed to provide a mechanism for default reasoning in the following way.

Assume that the inference rules in our path logic in- clude whatever is needed to perform the usual deduc- tions of classical logic, the types of syllogistic reasoning discussed in Section 2, and the modifier combination rules expressed in Section 3. To this now add the rule

From (&PI A) and (CjqP, A'), infer (Gip , A), if px > p x ~ , and infer ( C j l P , A'), if px < px i

To illustrate the use ofthis rule, suppose that we begin extendling the above DRS as follows. For simplicity we

399

Is-a

Tweety opus

Figure 1. Likely CanFly(Tweety), but CertuinlyNot CanFly(0pus).

(PI Republican (')e TPacifist Most Most

Quaker ( k ) ____) Pacifist ('I

Nixon

Figure 2. Likely Pacifist(Nixon) A Unlikely Pacifist(Nixon) j Uncertian Pacifist(Nixon).

400

argue informally. From the proposition asserting that ‘‘Opus is certainly a Penguin”, together with the propo- sition asserting that “Al l Penguins are Birds”, apply a syllogistic argument to conclude that “Opus certainly is a, Bird”. Then from this together with “Most Birds CanFly” , apply syllogism again to conclude

“Opus likely CanFly.”

Next, from “Opus is certainly a Penguin”, together with “All Penguins CannotFly”, apply syllogism to conclude

“0 pus certain ly Cannot Fly.”

The labels attached to the two conjuncts will be such tha.t the priority (specificity) of the second is greater than that of the first. Thus the above rule can be ap- plied to conclude the latter, “Opus cannot fly”. This is the conclusion that would be desired in any default reasoning system; inheritance of the less specific infor- ma,tion is said to be blocked by the more specific.

Thus we successfully handle one of the well-known “puzzles” discussed in the literature on default reason- ing, cf. [ll]. In Figure 2 is presented a modification of another such puzzle, known as the “Nixon Diamond”. Here the diamond is replaced with a more disciplined dia.gram based on the distinction between properties and kinds. In effect, drawing the diagram in this way mahkes the original reasoning problem disappear. From the proposition that “Nixon is certainly a Quaker”, to- get,her with “Most Quakers are Pacifists”, one can ar- gue syllogistically that

“Nixon likely is a Pacifist .” anld from “Nixon certainly is a Republican”, together wiith “Most Republicans are Not Pacifists”, one can conclude similarly that “Nixon likely is Not a Pacifist”. From this, the inference rules associated with Q should allow one to conclude

“Nixon is unlikely to be a Pacifist.”

Then, using the modifier combination rule discussed in Section 4, these two conclusions can be used to infer

“It is uncertain whether Nixon is a Pacifist.”

Note that here, neither of the two conclusions are more specific than the other, so that the above mentioned rule for deciding between conjuncts would not apply.

References

[-l] Amarger, S., Dubois, D., and Prade, H., Imprecise quantifiers and conditional probabilities. In R. Kruse and P. Siege1 (eds.), Symbolic and Quantitative Ap- proaches to Uncertainty, Proceedings of the European

Conference ECSQAU, Marseille, France, October 1991, Springer-Verlag, NY, 1991, pp. 31-37.

[2] Dubois, D. and Prade, H., On fuzzy syllogisms. Com- putational Intelligence, 4 (1988) 171-179.

[3] Elgot-Drapkin, J.J., Step Logic: Reasoning Situated in ‘Time, Ph.D. Dissertation, University of Maryland, Computer Science Technical Report UMIACS-TR-88- 94 ;and CS-TR-2156, 1988.

[4] Gardenfors, P., Ed. Belief Revision, Cambridge Uni- versity Press, New York, 1992.

[5] Ginsberg, M.L., Ed., Readings in Nonmonotonic Rea- son.ing, Morgan Kaufmann, Los Altos, CA, 1987.

[6] Miller, M.J., A View of One’s Past and Other Aspects of Reasoned Change in Belief, Doctoral Dissertation, Jul:y, Computer Science Technical Report Series, CS- TR-3107 and UMIACS-TR-93-66, University of Mary- lan(d, College Park, MD 20742, 1993.

[7] Sch.wartz, D.G., On the relation between fuzzy quan- tification, usuality, and linguistic likelihood. Proceed- ings of the International Conference on Information Processing and Management of Uncertainty in Knowl- edge Based Systems, IPMU’SO, Paris, France, July 2-

[8] Schwartz, D.G., Toward a logic for fuzzy :syllogisms. Proceedings of the Second IEEE Conference on Fuzzy Sets and Systems, FUZZIEEE ’93, San Franicisco, CA, M a x h 31-April 4, 1993, pp. 71-75.

[9] Schwartz, D.G., A framework for combining fuzzy mod- ifiers. Proceedings of the Third IEEE Conterence on Fuzzy Sets and Systems, FUZZIEEE’ 94,, Orlando, FL, June 26-29, pp. 1126-131.

[lo] Sowa, J.F., Ed., Principles of Semantic Networks: Ex- plorations an the Representation of Knowledge, Mor- gan Kaufmann, San Mateo, CA, 1991.

[ll] Stein, L.A., Resolving ambiguity in nonmonotonic in- heritance hierarchies. Artificial Intelligence, 55(2-3)

[12] Zadeh, L.A., A computational approach to fuzzy quan- tifiers in natural languages. Computers cind Mathe- matics, 9 (1983) 149-184.

[13] Zadeh, L.A., Fuzzy probabilities. Information Pro- cesing and Management, 20 (1984) 363-372.

[14] Zadeh, L.A., Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions. IEEE Transactions on Systems, Man, anid Cybernetics, 15 (1985) 754-763.

ba.sed on fuzzy logic. In A. Jones, A. Kaufmann, and H.-J. Zimmerman (eds.), Fuzzy Sets Theory and Appli- cations, Reidel, Dordrecht, Holland, 1986, pp. 79-97.

6, l990, pp. 263-265.

(1992), 259-310.

[I51 Za.deh, L.A., Outline of a theory of usuality

40 1