Adaptively Applying Modus Ponens in
Conditional Logics of Normality
Christian Straßer
Centre for Logic and Philosophy of Science, Ghent University (UGent)
Blandijnberg 2, 9000 Gent, Belgium
Email:
hristian.strasserUGent.be
,
Phone: ++32 9 264 39 79, Fax: ++32 9 264 41 87
ABSTRACT. This paper presents an adaptive logic enhancement of conditional logics of normality
that allows for defeasible applications of Modus Ponens to conditionals. In addition to the
possibilities these logics already offer in terms of reasoning about conditionals, this way they
are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens
defeasibly to a conditional A B and a fact A on the condition that it is “safe” to do so
concerning the factual and conditional knowledge at hand. It is for instance not safe if the
given information describes exceptional circumstances: although birds usually fly, penguins
are exceptional to this rule. The two adaptive standard strategies are shown to correspond to
different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in
the handling of so-called floating conclusions.
RÉSUMÉ. A définir par la commande
\resume{...}
KEYWORDS: defeasible reasoning, adaptive logic, conditional logics of normality, default infer-
encing, modus ponens, nonmonotonic logic
MOTS-CLÉS : A définir par la commande
\motsles{...}
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1. Introduction
1.1. Some Background
Since the early eighties, default reasoning, i.e., reasoning on the basis of what is
normally or typically the case, has drawn much attention from philosophical logicians
as well as scholars working in Artificial Intelligence. This is not surprising concerning
the prominent role which reasoning on the basis of notions such as normality and typ-
icality has. It clearly occupies a central place from everyday common sense reasoning
to expert reasoning in many domains. Thus, logicians are urged to develop formal
models which accurately explicate these reasoning forms.
In recent years the traditional formalisms of default reasoning such as presented in
the landmark articles on default logic (Reiter, 1980), on circumscription (McCarthy,
1980), and on autoepistemic logic (Moore, 1984) have been criticized and alternative
conditional approaches have been developed.
In pioneering works on logics of conditionals the main interest was to model con-
ditionals in everyday language which have the form “if . ..then”. Most of the research
in this domain has been in the vein of the following influential conditional logics: Stal-
naker (Stalnaker, 1968) and Lewis (Lewis, 2000) who offer an ontic interpretation of
the conditional, Adams (Adams, 1975) who introduces probabilities in the discussion,
and Gärdenfors’ belief revision principles which are more concerned with acceptabil-
ity than probability and truth (Gärdenfors, 1978).
There has been, especially since the late eighties, an increasing interest in making
use of techniques and properties of conditional logics within the field of nonmonotonic
reasoning, such as employed in default reasoning or reasoning with respect to prima
facie obligations. The focus of this paper is on conditional logics of normality that
have been inspired by pioneering works such as (Boutilier, 1994a; Lamarre, 1991;
Kraus et al., 1990). There, a statement of the form A
B is read as “From A
normally/typically follows B” or “If A is the case then normally/typically also B is
the case”. We will call “A
B” a conditional, and a sequence of conditionals,
written A
1
A
2
. . .
A
n
as an abbreviation for (A
1
A
2
) ∧ (A
2
A
3
) ∧
· · · ∧ (A
n−1
A
n
), an argument.
Conditional logics are attractive candidates for dealing with default reasoning for
various reasons: First, the conditional
does not have unwanted properties such
as Strengthening the Antecedent, from A
B infer (A ∧ C)
B, Transitivity,
from A
B and B
C infer A
C, and Contraposition, from A
B infer
¬B
¬A. That the validity of any of these properties leads to undesired results in
the context of reasoning on the basis of normality is well-known. Take, for instance,
Strengthening the Antecedent: although birds usually fly, b
f, penguins do not, (b∧
p)
¬f. Thus (b ∧ p)
f should not be derived. To find similar counterexamples
for the other properties is left to the reader (see e.g., (Boutilier, 1994a) p. 92.). Another
advantage is the naturalness and simplicity of the representation of default knowledge
by conditionals A
B compared to the cumbersome representation by the classical
Adaptively Applying Modus Ponens 3
approaches mentioned above. The latter use rules such as A ∧ π(B) ⊃ B where π(B)
expresses for instance that we do not believe ¬B in the case of autoepistemic logic,
or that B can consistently be assumed in the case of default reasoning. Furthermore,
certain disadvantages of the classical approach can be avoided in the framework of
conditional logics. Boutilier for instance argues that certain paradoxes of material
implication are inherited by the classical approaches due to the way default knowledge
is represented in them (see (Boutilier, 1994a) pp. 89–90).
Starting from the pioneering works such as (Boutilier, 1994a; Delgrande, 1988;
Kraus et al., 1990; Lamarre, 1991) there has been vigorous research activity on condi-
tional logics of normality. To mention a few: they have been applied to belief revision
in (Boutilier, 1994b; Wobcke, 1995), strengthenings have been proposed for instance
to give a more sophisticated account of Strengthening the Antecedent (see (Lehmann
et al., 1992; Pearl, 1990)), a labeled natural deduction system has been introduced
in (Broda et al., 2002), and various authors have investigated tableaux methods and
sequent calculi for conditional logics (see e.g. (Schröder et al., 2010; Giordano et
al., 2006; Schröder et al., 2010)). Furthermore, the influential work in (Kraus et
al., 1990) is greatly generalized in (Arieli et al., 2000) by their plausible nonmonotonic
consequence relations, and in (Friedman et al., 1996) by their plausibility measures.
There is a remarkable agreement concerning fundamental properties for default
reasoning in the various formal models. These properties have been dubbed conser-
vative core by Pearl and Geffner (Geffner et al., 1992) and are also commonly known
as the KLM-properties (see (Kraus et al., 1990)). Some of the most interesting and
important problems in this field are, on the one hand, related to a proper treatment
of irrelevant information (see (Delgrande, 1988)) and, on the other hand, to a proper
treatment of specificity.
1.2. Contribution and Structure of this Paper
This paper tackles another important problem related to conditional logics of nor-
mality: while they are able to derive from conditional knowledge bases, i.e., sets of
conditionals, other conditionals, their treatment of factual knowledge is mostly rather
rudimentary. This concerns most importantly their treatment of Modus Ponens (MP),
i.e., to derive B from A and A
B. We will also speak about detaching B from
A
B in case A is valid. Usually we do not only have a conditional knowledge base
at hand but also factual information F. In order to make use of the knowledge base,
it is in our primary interest to derive, given F, what normally should be the case. It
goes without saying that for the practical usage of a conditional knowledge base this
kind of application to factual information is essential and that the proper treatment of
MP for conditionals is a central key to its modeling.
It is clear that full MP should not be applied unrestrictedly to conditional asser-
tions: although birds usually fly, b
f, we should not deduce that a given bird flies
if we also know that it is a penguin, since penguins usually do not fly, p
¬f. How-
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ever, if we do not know anything about it than the fact that it is a bird, MP should be
applied to b
f and b. Furthermore, it would be useful if this application is of a de-
feasible kind, since later we might learn that the bird in question is after all a penguin
or a kiwi.
In this paper a simple generic method is presented to enrich a given conditional
logic of normality L by a defeasible MP. We consider L to consist at least of the core
properties (see Section 2). We will refer to L as the base logic. As hinted above,
there are several circumstances when we do not want to apply MP: cases of specificity
such as the example with the penguin, or cases in which conditionals conflict, such
as the well-known Nixon-Diamond. The central idea presented in this paper is to
apply MP conditionally, namely on the condition that it is safe to apply it. This idea
will informally be motivated and outlined in Section 3. Formally, the conditional
applications of MP are realized by adaptive logics, namely DLp
m
and DLp
r
(see
Definition 5, page 14). The idea of adaptive logics is to interpret a premise set “as
normally as possible” with respect to a certain standard of normality. They allow for
some rules to be applied conditionally. I introduce adaptive logics formally in Section
4. In our case, as demonstrated in Section 5, MP is going to be applied as much as
possible, i.e., as long as no cases of overriding via specificity or similar conflicts take
place concerning the conditionals to which MP is going to be applied. That is to say,
we are going to apply MP to A
B and A on the condition that the other factual
information at hand does not describe exceptional circumstances with respect to A.
As a consequence, detachment from b
f and b is for instance blocked if p is the
case.
It will be demonstrated that choosing different adaptive strategies serves different
intuitions: one corresponding to a more skeptical and the other one corresponding to
a more credulous type of reasoning. This difference manifests itself in the handling of
so-called floating conclusions.
1
I will spend some time in demonstrating the modus operandi of the proposed logics
and thereby their strengths by having a look at various benchmark examples. In Sec-
tion 6 I highlight some advantages of the adaptive logic approach, compare it to other
approaches, and discuss some other related issues. The semantics are investigated in
the Appendix.
2. Conditional Logics, Their Core Properties and Related Work
Conditional logics are often presented in terms of extending classical propositional
logic with a conditional operator
.
2
Our language is defined by the (∧, ∨, ⊃, ¬, ≡)-
1. A floating conclusion is a proposition that can be reached by two conflicting and equally
strong arguments.
2. In some conditional logics of normality
is not primitive. For instance in (Boutilier,
1994a) it is defined by making use of modal logic. There the core properties are shown to be
equivalent to an extension of the modal logic S4. See (Friedman et al., 1996) for a comparative
Adaptively Applying Modus Ponens 5
closure of the set of propositional variables and conditionals of the form A
B,
where A and B are classical propositional formulas. Hence, to keep things simple we
do not consider here nested occurrences of
and focus on flat conditional logics. We
refer to A as the antecedent and to B as the conclusion of the conditional. We write W
for the set of all classical propositional formulas (i.e., formulas without occurrences
of
). We abbreviate (A
B) ∧ (B
A) by A ∼ B and ¬(A
B) by A 6
B. Furthermore, we require that a conditional logic L satisfies the following core
properties, where CL is classical propositional logic (see (Kraus et al., 1990)):
3
If ⊢
CL
A ≡ B, then ⊢ (A
C) ≡ (B
C) [RCEA]
If ⊢
CL
B ⊃ C, then ⊢ (A
B) ⊃ (A
C) [RCM]
⊢ A
A [ID]
⊢
(A
B) ∧ (( A ∧ B)
C)
⊃ (A
C) [RT]
⊢
(A
B) ∧ (A
C)
⊃ ((A ∧ B)
C) [ASC]
⊢
(A
C) ∧ (B
C)
⊃ ((A ∨ B)
C) [CA]
The logic defined by these rules and axioms is P. Note that for instance the following
properties are valid in P:
4
⊢
(A
B) ∧ (A
C)
⊃ (A
(B ∧ C)) [CC]
⊢ ((A ∧ B)
C) ⊃ (A
(B ⊃ C)) [CW]
⊢
(A ∼ B) ∧ (B
C)
⊃ (A
C) [EQ]
⊢
CL
A ⊃ B, then ⊢ A
B [CI]
We consider these properties to be valid for all the conditional logics of normality
in the remainder. Adding the following Rational Monotonicity principle to the core
properties yields logic R (see (Lehmann et al., 1992)):
5
⊢
(A
C) ∧ (A 6 ¬B)
⊃ ((A ∧ B)
C) [RM]
study of various semantic systems for the core properties such as the preferential structures of
(Kraus et al., 1990), the ǫ-semantics of (Pearl, 1989), the possibilistic structures of (Dubois et
al., 1991) and κ-rankings of (Goldszmidt et al., 1992; Spohn, 1988).
3. We will use the name convention that is associated with conditional logics of normality
(see (Chellas, 1975), (Nute, 1980)) and not the one associated with nonmonotonic consequence
relations which is used e.g. in (Kraus et al., 1990).
4. The proofs are fairly standard and can be found e.g. in (Kraus et al., 1990).
5. I adopt the names P and R for these logics from (Giordano et al., 2006). Although these
are the same names as used for the systems in the pioneering KLM paper (Kraus et al., 1990),
the reader may be warned: the approach in terms of conditional logics differs from the KLM
perspective which deals with rules of inference rather than with axioms. Also, strictly speaking,
Rational Monotonicity as defined in (Kraus et al., 1990) is a rule of inference whereas (RM) as
defined above is an axiomatic counterpart to it.