Chapter 7 Defeasibility Mar.’10
“If we were not provided with the knack of being wrong, we could never get anything useful done”.
Lewis Thomas
“There is nothing to be learned from the second kick of a mule” American saying
7.1 Ampliative reasoning
The reasoning done by individuals is mainly ampliative. There is a loose usage of the word in which all ampliative reasoning is defeasible. We remarked in chapter 2 that there seems to be no want of candidate logics - nonmonotonic logics, defeasible inheritance logics, default logics, autoepistemic logics, circumscription logics, logic programming and Prolog, preferential reasoning logics, abductive logics, theory revision logics, theory update logics, doxastic logics and whatever else - for consideration under this heading. I have already said that my general view of these logics is that while they make some display of mathematical virtuosity, they are less impressive in matters of conceptual and empirical adequacy.[1] John Pollock writes to the same effect:
Unfortunately, their lack of philosophical training led AI researchers to produce accounts that were mathematically sophisticated, but epistemologically naïve. Their theories could not possibly be right as accounts of human cognition, because they could not accommodate the varieties of defeasible reasoning humans actually employ (Pollock, 2008, p. 452).
Pollock adds that although “there is still a burgeoning industry in AI studying nonmotonic logic this shortcoming remains to the present day.” Shortcoming or not, it is clear that the AI literature reflects formal developments of sufficient importance to justify an acquaintance with them. This is the business of the section to follow.
7.2 Defeasible consequence
Let D be the class of logics that focus primarily on a relation of defeasible consequence. The following is a representative list of the properties of such relations. Not all of them have all these properties, but many have most of them.
Nonmonotonicity. If α is a consequence of Γ, it need not be a consequence of the result of adding any sentence to Γ.
Cumulativity. If α is a consequence of Γ it is also a consequence of {α} È Γ.
Cut. Let K(Γ) be the set of Γ defeasible consequences. Then if everything in Γ is in D and everything in D is in the set of defeasible consequences of Γ (i.e., K(Γ)), then all the defeasible consequences of Γ are defeasible consequences of D, i.e. K(D).[2]
Cautious Monotony. If, as before, everything in Γ is in D and everything in D is in K(Γ), then everything in K(Γ) is in K(D).[3]
Full Absorption. Let Cl(Γ) be the classical closure of Γ. Then Cl(K(Γ) = K(Γ) = K(ClΓ).[4]
Distribution. Everything common to K(Γ) and K(D) is in what is common to the set of defeasible consequences of what’s common to the classical closure of Γ and the classical closure of Γ and the classical closure of D; i.e., K(Γ) Ç K(D) Í K(Cl(Γ) È Cl(D)).
Conditionalization. If α is a defeasible consequence of a set Γ È {β}, then the material conditional ⌐β É α¬ is a defeasible consequence of Γ itself.
Loop. If |~ denotes a defeasible consequence relation, then where α1 |~ α2 ¼ αn-1 for αn. then for any i and j, the αi and αj have just the same defeasible consequences.
A number of disputed properties have also been considered for defeasible consequence and are prefixed here with an asterisk
* Disjunctive Rationality. If Γ È {α} |~ / β and Γ È {λ} |~ / , then Γ È {(α Ú λ)} |~ / β.[5]
* Rational Monotony. If Γ |~ α, then Γ È {β} |~ α or Γ |~ ~β.[6]
* Consistency Preservation. If Γ is classically consistent so is K(Γ).[7]
It should be clear on inspection that the orientation of these logics is mathematical rather than conceptual or empirical. The majority of the systems in D are elaborations of quite simple formal languages – more often than not propositional languages. The reason for this is that, with the except of OSCAR, automations of defeasible reasoners have not been successful for systems richer than propositional logic or some of its even less rich sublogics. In this regard OSCAR is a standout. OSCAR is an AI architecture for knowers – for cognitive agents – and can be thought of as a general-purpose defeasible reasoner (Pollock, 1995). But, to date, OSCAR cannot handle defeasible reasonings that vary in degrees of goodness or strength. Indeed
There are currently no other proposals in the AI or philosophical literature for how to perform defeasible reasoning with varying degrees of justification. (Pollock, 2008, p. 459; emphasis added).
Part of the problem lies in the nature of the subject matter. D-reasoning is much more difficult than deductive reasoning to capture formally. A further part of the problem lies in inflexibility of the methods employed by D-logicians. Judging from the formalizing languages chosen and the character of rules imposed, defeasibility inferences are widely seen in the D-community as variations of deductive inference. In particular, it is commonly assumed that, in the absence of information to the contrary, the connectives and quantifiers of defeasible languages are classical. But the greatest difficulty to date is the almost slovenly indifference shown by most going D-logics to the data of defeasible reasoning, to the very phenomena that D-logics are supposed to elucidate and systemalize. So a prior theme re-stirs itself. A theory of reasoning is asking for trouble if it doesn’t pay careful attention to its motivating data. The look before you leap principle emphasizes the importance of this pre-theoretical reflectiveness. It offers some helpful counsel: In matters of human performances of kind K do not venture theoretical accounts of them without due regard for the K-data that the theory is to account for. In particular, be vigilant about data-bending. Be careful that the pre-theoretical senses of the terms used to describe the theory’s motivating data aren’t inappropriately overridden by contrary meanings embedded in the theorist’s procedures and assumptions.[8]
7.3 Misconceiving the interconnections
I want now to examine the logical connections, or want of them, between and among the three D-properties of nonmonotonicity, defeasibility (in the narrower sense, to be specified) and defaults, and the three classical properties of validity, soundness and inductive strength.
Nonmonotonicity and deductive validity. The universally accepted definition of a nonmonotive logic is one whose consequence relation is not nonmonotonic. As we saw, a consequence relation is monotonic just in case whenever it obtains between a sentence α and sentence β or between a set of sentences Γ and a sentence β, it also holds between the result of adding any sentence to α or to Γ any number of times and that same consequent β. It is commonly held that monotonicity is a universal feature of deductive consequence. This is a mistake. Let L be a linear logic whose deductive consequence relation is ⇝. Then in a linear proof context
1. α
2. α ⇝ β
3. So, β
is L-valid. A distinguishing feature of L is that it is sensitive to premiss-redundancy. There are new premises γ for which
1. α
2. α ⇝ β
2¢. γ
3. So, β
is invalid in L.[9] Hence,
Proposition 7.3a
MISCONCEIVING MONOTONICITY: Monotonicity is a typical but not intrinsic property of deductive consequence relations.
Nonmonotonicity and inductive strength. The other part of the received wisdom about monotonicity is that the property of inductive consequence is intrinsically nonmonotonic. We say that β is an inductive consequence of α (or of Γ) just in case
1. α (or Γ)
2. So, β
is an inductively strong argument. The thesis that inductive consequence is nonmonotonic provides that there are inductively strong arguments such that for some proposition γ, adding γ to the premiss-set weakens (or even destroys) the argument’s inductive strength. This happens whenever γ falsifies a premiss or directly conflicts with the argument’s conclusion.
No doubt, nonmonotonicity is a typical feature of inductive strength, but not an invariable one. Consider the inductively strong argument
1. Most, but not more than 80%, of A are C
2. Object x is an A.
3. So, object x is a C.
Suppose now that we discover that more than 80% of A are C, and add this as a premiss. Then we have
1. Most, but not more than 80%, A are C
2. Object x is an A
2¢. More than 80% A are C
3. So, object x is a C.
Of course, (1) and (2¢) can’t both be true, but at least one of them must be. How can the addition of a premiss that improves the odds of x’s being a C be an inductive-strength spoiler? Purists might have an answer to this. They might object that, as we have it now, the argument has inconsistent premisses, and that this makes the conditional probability of its conclusion on that inconsistency undefinable. But this is precisely the kind of case that reveals the vulnerability of equating inductive strength with high conditional probability. One of those premisses is true; and, whichever it is, it together with premiss (2) lends considerable inductive strength to the conclusion. Consequently, the conclusion cannot be undermined by this new premiss. True, the new premiss (2¢) falsifies (1) but it also compensates for its loss. So it is both technically and conceptually unsatisfying to insist that upon addition of (2¢), the support for (3) collapses. Accordingly,
Proposition 7.3b
MISCONCEIVING NONMONOTONICITY: Nonmonotonicity is a typical but not intrinsic feature of inductive consequence.
Some intrinsically nonmonotonic properties. Anyone mindful of the importance of the difference between consequence-having and consequence-drawing will already be predisposed not to draw every consequence he is able to recognize as such. Sometimes, to be sure, the addition of new information to a premiss-set (or data-base or knowledge-base) should change our minds about what now follows. In those cases, the failure to be a consequence is trivially a matter of whether it is a consequence anyone should (or could) draw. But a much commoner case against consequence-drawing has nothing to do with consequence-having. Perhaps the clearest example of this is one in which new information falsifies without compensation a premiss in an otherwise sound argument. By and large, we don’t want (categorical) arguments from false premisses. A valid argument from true premiss is sound. But, even in those systems in which validity is monotonic - namely, most of them - soundness is not. Other such properties are (at least in their intuitive senses): plausibility, likelihood and possibility.
When new information contradicts a premiss of an argument, the agent is faced with two tasks. One is to determine whether the new information undermines the argument’s consequence relation. If it did, that would be a reason not to draw the argument’s conclusion as a consequence of those altered premisses. If it didn’t, the agent must turn his attention to the second task. He must determine whether the unsoundness of the argument is occasion to abandon it altogether. Excepting the far from trivial class of arguments from premiss-sets whose inconsistency it is beyond the powers of the agent to remove in a principled way, unsoundness is an argument-killer irrespective of whether it is a consequence-killer. It is simple economics that in the general case a reasoner’s first task is to test for unsoundness, leaving the second task – checking on the consequence relation – to be performed only after positive finding with respect to the premisses (i.e., they are true, or plausible, or likely or presumable). Why would this be so? Because premiss inadequacy is decisive against consequence-drawing. Premiss adequacy underdetermines consequence-having. Non-consequence is decisive against consequence-drawing. Consequence-having underdetermines consequence-drawing.
Proposition 7.3c
THE PRIORITY OF UNSOUNDNESS: In the logic of reasoning, bad news about premisses trumps good news about consequence.
Defeasibility and deductive validity. One of the most persistent claims made by defeasibility logicians is that deductive logic is one part of logic and defeasibility logic is all the rest of it. Pollock writes: “What distinguishes defeasible arguments from deductive reasoning more generally is that the reasoning is not defeasible.” (Pollock, 2008, p. 453). He continues:
What distinguishes defeasible arguments from deductive arguments is that the addition of information can mandate the retraction of the conclusion of a defeasible argument without mandating the retraction of any of the other conclusions from which the retracted conclusion was inferred. By contrast, you cannot retract the conclusion of a deductive argument without also retracting some of the premises from which it was inferred. (p. 453)
Consider two cases, one a classically valid deductive argument
1. A
2. \ B
and the other an inductively strong argument
a. E
b. \ C.
The remarks below easily generalize to multi-premissed versions of such arguments.
Consider now the respective negations, ~B and ~E, of the conclusions of these arguments. Two points can be made straightaway.
i. Even in the light of these negations, C remains an inductively strong
consequence of E, and B remains a deductively valid consequence of A.
ii. C is not an inductively strong consequence of “E Ù ~C”, but B is a deductively valid consequence of “A Ù ~B”.
From (i) and (ii), we have it further that {A, ~B} is a deductively inconsistent set, whereas {E, ~C} is not. Accordingly, if one’s belief-revision goal (or dialectical goal) were the overall deductive consistency of one’s retractions and commitments, then, in the first instance, the consistent simultaneous restriction of B and commitment to A is impossible and, in the second, the consistent simultaneous retraction of C and commitment to E is possible.