Download relevant reasoning as the logical basis of

RELEVANT REASONING AS THE LOGICAL BASIS
OF
KNOWLEDGE ENGINEERING
Jingde Cheng
Department of Computer Science and Communication Engineering
Kyushu University
6-10-1 Hakozaki, Fukuoka 812-8581, Japan
Reprinted from
Proceedings of the 4th World Congress on Expert Systems
Mexico City, Mexico, March 16-20, 1998
RELEVANT REASONING AS THE LOGICAL BASIS OF
KNOWLEDGE ENGINEERING
Jingde Cheng
Department of Computer Science and Communication Engineering
Kyushu University
6-10-1 Hakozaki, Fukuoka 812-8581, Japan
[email protected]
conclusion, and concerns the strength of the relation
between them.
Therefore, the correctness of an
argument depends on the connection between its
premises and its conclusion, and neither on whether the
premises are true or not, nor on the conclusion is true
or not. Thus, we have a fundamental question: What is
the criterion by which to decide whether the conclusion
really does follow from the premises or not?
ABSTRACT
Almost all the current knowledge-based systems are
directly or indirectly based on classical mathematical
logic which gives no guarantee that the conclusion of a
reasoning is necessarily relevant to its premises, even if
the reasoning is valid in the sense of the classical
mathematical logic. It is this problem that causes the
ineffectiveness and inefficiency of reasoning and
learning engines of large-scale knowledge-based
systems.
To make the current knowledge-based
systems more powerful and flexible, we have to solve
this problem from both theoretical and practical aspects.
This paper proposes that relevant reasoning based on
paradox-free relevant logics should be taken as the
logical basis of knowledge engineering. The paper
points out why a reasoning based on the classical
mathematical logic may be irrelevant, shows that a
reasoning based on the paradox-free relevant logics is
relevant and that it is the notion of entailment that
plays the fundamental role in relevant reasoning, and
gives some examples to explain why one should take
relevant reasoning as the logical basis of knowledge
engineering.
1.
A logically valid reasoning is a reasoning
such that its arguments or inferences are justified based
on some logical criterion in order to obtain correct
conclusions. Therefore, a reasoning may be valid on a
logical criterion but invalid on another. Automated
reasoning is concerned with the execution of
computer programs that assist in solving problems
requiring reasoning.
To perform an automated
reasoning, knowledge used in the reasoning must be
represented in some explicit and formal form.
Knowledge
Engineering is a discipline
concerned with constructing and maintaining knowledge
bases to store knowledge in various domains of the real
world and using the knowledge by automated reasoning
technique to solve problems in the domains that
ordinarily require human logical reasoning. Therefore,
two key issues in knowledge engineering are how to
construct and maintain knowledge bases effectively and
efficiently, and how to reason out new knowledge from
known knowledge effectively and efficiently.
INTRODUCTION
Reasoning is the process of drawing new
conclusions from some premises which are known facts
and/or assumed hypotheses. In general, a reasoning
consists a number of arguments or inferences. An
argument (or inference) is a set of declarative
sentences consisting of one or more premises, which
contain the evidence, and a declarative sentence as
conclusion. In an argument, a claim is being made that
there is some sort of evidential relationship
between its premises and its conclusion: the conclusion
is supposed to follow from the premises, or,
equivalently, the premises are supposed to entail the
conclusion. The correctness of an argument is a matter
of the connection between its premises and its
Almost all the current knowledge-based systems are
directly or indirectly based on classical mathematical
logic (CML for short). However, a reasoning based on
the CML may be irrelevant, i.e., the conclusion
reasoned and/or deduced from the premises of that
reasoning may be irrelevant at all, in the sense of
meaning, to the premises. In the framework of the
CML, there is no guarantee that the conclusion of a
reasoning is necessarily relevant to its premises, even if
the reasoning is valid in the sense of the CML. As a
result, for a conclusion reasoned and/or deduced based on
the CML, we have to investigate whether it is relevant
449
to its premises or not by ourselves. Obviously, the
more we have information and/or knowledge, the more
difficult we do this investigation task. It is this
problem that causes the ineffectiveness and inefficiency
of reasoning and learning engines of large-scale
knowledge-based systems. To make current knowledgebased systems more powerful and flexible, we have to
solve this problem from both theoretical and practical
aspects.
theorems, laws, and principles to connect a concept,
fact, situation or conclusion and its sufficient
conditions. Indeed, a major work of almost all
scientists is to discover some conditional and/or causal
relationships between various phenomena, data, and
laws in their research fields.
In logic, the notion abstracted from various
conditionals is called “entailment.” In general, an
entailment, for instance, “A entails B” or “if A then B,”
must concern two parts which are connected by the
connective “... entails ...” and called the antecedent
and the consequent of that entailment, respectively.
The truth-value and/or validity of an entailment depends
not only on the truth-values of its antecedent and
consequent but also more essentially on a necessarily
relevant and conditional relation between its antecedent
and its consequent. The notion of entailment plays the
most essential role in human logical thinking because
any reasoning must invoke it.
Therefore, it is
historically always the most important subject studied
in logic and is regarded as the heart of logic [1].
This paper proposes that relevant reasoning based
on paradox-free relevant logics should be taken as the
logical basis of knowledge engineering. The paper
points out why a reasoning based on the CML may be
irrelevant, shows that a reasoning based on the paradoxfree relevant logics is relevant and that it is the notion
of entailment that plays the fundamental role in relevant
reasoning, and gives some examples to explain why
one should take relevant reasoning as the logical basis
of knowledge engineering.
2.
LOGIC AND ENTAILMENT
When we study and use logic, the notion of
entailment may appear in both the object logic (i.e.,
the logic we are studying) and the meta-logic or
observer’s logic (i.e., the logic we are using to
study the object logic).
First of all, we define the terminology used in this
paper
for
discussing
our
subject
clearly,
unambiguously, and formally.
What is logic? Logic deals with what entails
what or what follows from what. Its aim is to
determine which are the correct conclusions of a given
set of premises, i.e., to determine which arguments are
valid. Therefore, the most essential and central concept
in logic is the logic consequence relation that
relates a given set of premises to those conclusions
which validly follow from the premises. As the
systematic study of fundamental principles that underlie
various valid reasoning in order to obtain correct
conclusions from premises, logic must abstract the
forms of the reasoning from their contents. From the
viewpoint of logic, the validity of a reasoning depends
on the connection between its premises and its
conclusion, and neither on whether the premises are true
or not, nor on the conclusion is true or not.
In the object logic, there usually is a connective,
such as the material implication in CML, the relevant
implication in relevant logic R, the necessary and
relevant implication in modal relevant logic E, and so
on, to represent the notion of entailment.
On the other hand, in the meta-logic, the notion of
entailment is usually used to represent a valid logical
consequence relation. In general, for an object logic L,
any set Γ of formulas, and any formula C, we say that
Γ semantically or model-theoretically entails
C, or that C semantically
or modeltheoretically follows from Γ, or that C is a
semantic or model-theoretical consequence of
Γ, written as Γ |=L C (the relation |=L is called the
semantic or model-theoretical consequence
relation of L), if and only if C is interpreted to be true
for any model of Γ; we say that Γ syntactically o r
proof-theoretically
entails
C,
or
C
syntactically or proof-theoretically f o l l o w s
from Γ, or C is a syntactic or proof-theoretical
consequence of Γ, written as Γ |−L C (the relation |− L
is called the syntactic or proof-theoretical
consequence relation of L), if and only if C is
deducible from Γ.
It is probably difficult, if not impossible, to find a
sentence form in various natural and social scientific
publications which is more generally used to describe
various definitions, propositions, theorems, laws, and
principles than the sentence form of “if ... then ... .” A
sentence of the form “if ... then ...” is usually called a
conditional which states that there exists a
conditional and/or causal relationship between the “if”
part and the “then” part of the sentence. Natural and
social scientists always use conditionals in their
descriptions of various definitions, propositions,
450
and any element of TLe(P) is called an empirical
theorem of the formal theory.
For a logic, an entailment formula (i.e., a formula
in the form of A⇒B where “⇒“ is the connective used
to represent the notion of entailment in the logic) is
called an empirical entailment of the logic if its
truth-value, in the sense of that logic, depends on the
contents of its antecedent and consequent (i.e., from the
viewpoint of the logic, the relevant relation between the
antecedent and the consequent of that entailment is
regarded to be empirical); an entailment formula is
called a logical entailment of the logic if its truthvalue, in the sense of that logic, depends only on its
abstract form but not on the contents of its antecedent
and consequent, and therefore, it is considered to be
universally true or false (i.e., from the viewpoint of the
logic, the relevant relation between the antecedent and
the consequent of that entailment is regarded to be
logical). Indeed, the most intrinsic difference between
some different logic systems is to regard what class of
entailments as logical entailments.
A formal theory TL(P) is said to be directly
inconsistent if and only if there exists a formula A
of L such that both A∈P and ¬A∈P hold. A formal
theory T L (P) is said to be indirectly inconsistent if
and only if there exists a formula A of L such that both
A∈TL(P) and ¬A∈TL(P) but not both A∈P and ¬A∈P.
A formal theory TL(P) is said to be consistent if and
only if it is neither directly inconsistent nor indirectly
inconsistent.
In general, a formal theory constructed as a purely
deductive science (e.g., CML and its various
extensions) is consistent. However, almost all, if not
all, formal theories constructed based on empirical
and/or experimental sciences are generally indirectly
inconsistent.
A formal theory TL(P) is said to be meaningless
or e x p l o s i v e if and only if A∈TL(P) for arbitrary
formula A∈F(L). A meaningless or explosive formal
theory is not useful at all. Obviously, if a formal logic
system L is explosive, then any directly or indirectly
inconsistent formal theory TL(P) must also be
explosive.
A formal logic system L is a triplet (F(L), |− L ,
Th(L)) where F(L) is the set of all well formed
formulas of L, |− L is the logical consequence
relation of L such that for P⊆F(L) and C∈F(L), P |− L
C means that within the framework of L taking P as
premises we can obtain C as a valid conclusion, and
Th(L) is the set of logical theorems of L such that
φ |− L t holds for any t∈Th(L). According to the
representation of the logical consequence relation of a
logic, the logic can be represented as a Hilbert style
formal system, or a Gentzen natural deduction system,
or a Gentzen sequent calculus system. A semantics
of a formal logic system is an interpretation of the
formulas of the logic into some mathematical structure,
together with an interpretation of the logical
consequence relation of the logic in terms of the
interpretation.
3.
RELEVANT REASONING
AND ENTAILMENT
Generally, in any valid argument of our ordinary
logical thinking, the premises must be relevant to the
conclusion.
Informally, we say a reasoning is
relevant if and only if in every argument or inference
of that reasoning the premises is relevant to the
conclusion; a reasoning is irrelevant if and only if it
is not relevant. In this section, we will explain that it
is the notion of entailment that plays the fundamental
role in relevant reasoning.
A formal logic system L is said to be e x p l o s i v e
if and only if {A, ¬A} |− L B for any two different
formulas A, B∈F(L); L is said to be paraconsistent
if and only if it is not explosive [8].
3.1 Entailment and irrelevant reasoning i n
CML and its extensions
Let (F(L), |−L, Th(L)) be a formal logic system and
P⊆F(L) be a non-empty set of sentences (i.e.,
closed well formed formulas).
A formal
theory with premises P based on L, called a L theory with premises P and denoted by TL(P), is
defined as follows:
The CML is established based on a number of
basic assumptions.
Some of the assumptions
concerning with our subject are as follows:
The classical abstraction : The only properties of
a proposition that matter to logic are its form and its
truth-value.
TL(P) =df Th(L) ∪ TLe(P)
e
TL (P) =df {A | P |−L A and A∉Th(L)}
where Th(L) and TLe(P) are called the logical part and
the empirical part of the formal theory, respectively,
The Fregean assumption : The truth-value of a
proposition is determined by its form and the truthvalues of its constituents.
451
The classical validity assumption :
An
argument is valid if and only if it is impossible for all
its premises to be true while its conclusion is false.
With the above definition of material implication
and the inference rule of Modus Ponens for material
implication (from A and A→B to infer B), any valid
reasoning based on CML must be truth-preserving,
i.e., the conclusion of a valid reasoning must be true if
all premises are true. However, as a result of defining
the material implication as a truth-function of its
antecedent and consequent but ignoring whether or not
there is a necessarily relevant and/or conditional relation
between its antecedent and consequent, a reasoning
based on CML may be irrelevant, i.e., the conclusion
of some argument or inference in that reasoning may be
irrelevant at all, in the sense of meaning, to the
premises. For example, from “snow is white” and
“snow is white → 1+1=2” we can infer “1+1=2” by
using Modus Ponens for material implication.
However, conclusion “1+1=2” is obviously irrelevant at
all to premise “snow is white” even if it is true.
Moreover, because of implicational paradoxes, the
conclusion in the form of implication formula of an
argument or inference based on CML even may be not
true in the sense of entailment. For example, from any
given proposition A and logical axiom A→(B→A),
which is the most well known implicational paradox,
we can infer B→A by using Modus Ponens for material
implication.
However, since there may be no
necessarily relevant and/or conditional relation between
A and B, in general, we cannot say “if B then A” in the
sense of entailment.
Taking above assumptions into account, in CML,
the notion of entailment is represented by the
extensional notion of material implication (denoted by
→ in this paper) which is defined as A→B =df
¬(A∧¬B) or A→B =df ¬A∨B. However, the material
implication is just a truth-function of its antecedent and
consequent but not requires that there must exist a
necessarily relevant and/or conditional relation between
its antecedent and consequent, i.e., the truth-value of the
formula A→B depends only on the truth-values of A
and B, though there could exist no necessarily relevant
and/or conditional relation between A and B. As a
result, for example, sentences “snow is white →
1+1=2,” “snow is black → 1+1=2,” and “snow is black
→ 1+1=3” are all true in the logic. However, if we
read “→“ as “if ... then ...,” then “if snow is white then
1+1=2,” “if snow is black then 1+1=2,” and “if snow is
black then 1+1=3” are all false in human logical
thinking because there is no necessarily relevant and/or
conditional relation between the “if” part and the “then”
part of each sentence. Obviously, in meaning the
notion of entailment used in human logical thinking is
intrinsically different from the notion of material
implication in CML. Using the material implication
as the entailment is problematical in pragmatics.
The well known “implicational
paradox
problem” in CML is that if one regards the material
implication as the entailment and every logical theorem
of CML as a valid inference form in human logical
thinking, then a great number of logical axioms or
theorems of the logic, such as A→(B→A),
B→(¬A∨A),
¬A→(A→B),
(¬A∧A)→B,
(A→B)∨(¬A→B), (A→B)∨(A→¬B), (A→B)∨(B→A),
((A∧B)→C)→((A→C)∨(B→C)), and so on, present
some paradoxical properties and therefore they have
been referred to in the literature as “implicational
paradoxes” [1, 2, 7, 9]. For example, in terms of
CML, formulas A→(B→A) and B→(¬A∨A) mean “a
true proposition is implied by anything” formulas
¬A→(A→B) and (¬A∧A)→B mean “a false
proposition
implies
anything”
and
formula
(A→B)∨(B→A) means “for any two propositions A and
B, A implies B or B implies A” respectively.
However, it is obvious that we cannot say “if B then
A” for a true proposition A and an arbitrary proposition
B, “if A then B” for a false proposition A and an
arbitrary proposition B, and “if A then B or if B then
A” for any two irrelevant propositions A and B,
respectively, in our ordinary logical thinking.
On the other hand, the following proof-theoretical
and model-theoretical deduction theorems hold in
CML:
Γ∪{A} |−CML B iff Γ |−CML A→B
Γ∪{A1, ..., An} |−CML B
iff Γ |−CML A1→(...→(An→B)...)
Γ∪{A1, ..., An} |−CML B
iff Γ |−CML (A1∧...∧An)→B
Γ∪{A} |=CML B iff Γ |=CML A→B
Γ∪{A1, ..., An} |=CML B
iff Γ |=CML A1→(...→(An→B)...)
Γ∪{A1, ..., An} |=CML B
iff Γ |=CML (A1∧...∧An)→B
What these mean is that the notion of entailment in
meta-logic of CML is “equivalent” to the notion of
material implication in CML. Therefore, in the
framework of CML, even if a reasoning is valid, there
is no guarantee that its premises are necessarily relevant
to its conclusion.
All formal logic systems (including various modal
logic systems, intuitionistic logic, and those logic
systems developed in recent years for nonmonotonic
452
reasoning) where the notion of entailment is directly or
indirectly represented by the notion of material
implication have the similar implicational paradox
problem as that in CML. A reasoning based on these
logics also may be irrelevant.
3.2 Entailment
relevant logic
and relevant
reasoning
of getting to know whether A or whether B” (Geach
1958); “A1& … &An→B should not only be itself a
tautology, but should also be a substitution instance of
some more general implication A1'& … &An'→B',
where neither B' nor ¬(A1'& … &An') are themselves
tautologies” (Smiley 1959) [1]. However, it is hard
until now to know exactly how to formally interpret
such epistemological phrases as “coming to know” and
“getting to know” in the context of logic. During the
1950s~1970s, Anderson and Belnap extended the work
of Ackermann and proposed variable-sharing as a
necessary but not sufficient formal condition for the
relevance between the antecedent and consequent of a
logical entailment [1, 2, 9].
in
Historically, implicational paradoxes have been
studied many years. The main aim of Lewis's work
beginning in 1912 on the establishment of modern
modal logic was to find a satisfactory theory of
implication which is better than CML in that it can
avoid those implicational paradoxes, though his plan
was not successful in the sense that some implicational
paradoxes in terms of strict implication remained in
modal logic [1, 2, 9].
Relevant logics were constructed during the
1950s~1970s in order to find a mathematically
satisfactory way of grasping the notion of entailment
[1, 2, 7, 9].
The first one of such logics is
Ackermann's logic system Π'. Ackermann introduced a
new primitive connective, called “rigorous
implication,” which is more natural and stronger
than the material implication, and constructed the
calculus Π' of rigorous implication which provably
avoids those implicational paradoxes. Anderson and
Belnap modified and reconstructed Ackermann's system
into an equivalent logic system, called “system E of
entailment”. Belnap proposed an implicational relation,
called “relevant implication,” which is stronger
than the material implication but weaker than the
rigorous implication, and constructed a calculus called
“system R of relevant implication”. E has something
like the modality structure of a classical modal logic
S4, and therefore, E differs primarily from R in that E
is a system of strict and relevant implication but R is a
system of only relevant implication.
Another
important relevant logic system is “system T of ticket
entailment” or “system T of entailment shorn of
modality” which is motivated by Anderson and Belnap.
There are some neighboring logic systems of T, E, and
R. All of these logic systems are usually called
“entailment logics,” “relevance logics,” or “relevant
logics” [1, 2, 7, 9]. In this paper, we will call these
logics “relevant logics.” A major feature of the relevant
logics is that they have a primitive intensional
connective (i.e., it cannot be defined by other
connectives) to represent the notion of entailment.
Variable-sharing is a formal notion designed to reflect
the idea that there be a meaning-connection between the
antecedent and consequent of an entailment [1, 2, 9].
What underlies the relevant logics is the so-called “the
relevance principle”, i.e., informally, if A⇒B,
where ⇒ denotes the notion of entailment, is a logical
theorem of a relevant logic, for any two propositional
Sugihara 1955 provided the first general
characterization of implicational paradoxes [1].
Relative to a given connective, →, intended as
implicational, a formula A is said to be strongest if
one can prove A→B for every formula B, and a formula
A is said to be weakest if B→A is provable for all B.
Thus, a logic system is paradoxical in the sense of
Sugihara just in case it has either a weakest or a
strongest formula, and a logic system is paradox-free
in the sense of Sugihara if it is not paradoxical in that
sense. Obviously, CML has ¬A∧A as strongest
formula and ¬A∨A as weakest formula. Note that in
both B→(¬A∨A) and (¬A∧A)→B the antecedent and
consequent share no sentential variable, and therefore,
there cannot be a meaning-connection between the
antecedent and consequent.
Ackermann 1956 pointed out:
“Rigorous
implication, which we write as A→B, should express
the fact that a logical connection holds between A and
B, that the content of B is part of that of A. … That
has nothing to do with the truth of falsity of A or B.
Thus one would reject the validity of the formula
A→(B→A), since it permits the inference from A of
B→A, and since the truth of A has nothing to do with
whether a logical connection holds between B and A”
[1, 2, 9]. During 1957~1959, Von Wright, Geach, and
Smiley suggested some informal criteria for the notion
of entailment, i.e., the so-called “Wright-GeachSmiley criterion” for entailment: “A entails B, if
and only if, by means of logic, it is possible to come
to know the truth of A→B without coming to know
the falsehood of A or the truth of B” (Von Wright
1957); “A entails B if and only if there is an a priori
way of getting to know that A→B which is not a way
453
formulas A and B, then A and B share at least one
sentential variable [1, 2, 9]. As a result of requiring the
relevance principle, the relevant logics include no
implicational paradoxes as logical theorems.
therefore we cannot say “if A and C, then B” in the
sense of entailment.
In order to establish a satisfactory logic calculus of
entailment to underlie knowledge representation and
reasoning, the present author proposed some new
relevant logics, named Tc, Ec, and Rc.
As a
modification of T, E, and R, Tc, Ec, and Rc rejects
all conjunction-implicational paradoxes and disjunctionimplicational paradoxes in T, E, and R, respectively,
and therefore, they are free not only implicational
paradoxes but also conjunction-implicational and
disjunction-implicational paradoxes [4-6].
What
underlies our relevant logics Tc, Ec, and Rc is the
following strong relevance principle.
However, although the relevant logics have rejected
those implicational paradoxes, there still exist some
logical axioms or theorems in the logics which are not
natural in the sense of entailment. Such logical axioms
or theorems, for instance, are (A∧B)⇒A, (A∧B)⇒B,
(A⇒B)⇒((A∧C)⇒B),
A⇒(A∨B),
B⇒(A∨B),
(A⇒B)⇒(A⇒(B∨C)) and so on, where ⇒ is the
primitive intensional connective in the logics to
represent the notion of entailment. The present author
first named these logical axioms or theorems
“conjunction-implicational
paradoxes” and
“disjunction-implicational paradoxes” [3].
Let A, B, and C be well formed formulas. A is a
consequent part of A; if ¬B is a consequent part
{antecedent part} of A, then B is an antecedent part
{consequent part} of A; if B⇒C is a consequent part
{antecedent part} of A, then B is an antecedent part
{consequent part} of A, and C is a consequent part
{antecedent part} of A; if either B∧C or B∨C is a
consequent part {antecedent part} of A, then both B and
C are consequent parts {antecedent parts} of A [1].
Why the conjunction-implicational paradoxes and
disjunction-implicational paradoxes are not natural? Let
us see some examples.
A
simple
conjunction-implicational
paradox is an implication and/or entailment formula
such that its antecedent is a conjunctive formula and its
consequent is a proper subformula of its antecedent as a
conjunct.
For example, simple conjunctionimplicational paradox (A∧B)⇒A is a logical axiom of
almost all relevant logics and therefore is valid in the
logics even if B may be irrelevant to A or B may be the
negation of A. However, propositions in this form
such as “if snow is white and 1+1=2, then snow is
white,” “if snow is white and 1+1=3, then snow is
white,” and “if snow is white and snow is not white,
then snow is white” cannot be considered as to be valid
in human logical thinking.
If A is a theorem of Tc, Ec, and Rc, then every
sentential variable in A occurs at least once as an
antecedent part and at least once as a consequent part.
We say that Tc, Ec, and Rc satisfy the strong
relevance principle.
Now, in the framework of paradox-free relevant
logics Tc, Ec, or Rc, the conclusion of an argument
or inference based on Tc, Ec, or Rc must strongly
relevant to its premises. As a result, for a conclusion
reasoned based on Tc, Ec, or Rc, we can accept it
directly and do not need to investigate whether it is
relevant to its premises or not. It is the notion of
entailment that plays the fundamental role in relevant
reasoning.
A
simple
disjunction-implicational
paradox is an implication and/or entailment formula
such that its consequent is a disjunctive formula and its
antecedent is a proper subformula of its consequent as a
disjunct. Similar to the above discussion on simple
conjunction-implicational paradoxes, we can know
simple disjunction-implicational paradoxes are also not
natural in the sense of entailment.
4. RELEVANT REASONING
AS THE LOGICAL BASIS OF
KNOWLEDGE ENGINEERING
(A⇒B)⇒((A∧C)⇒B) is a more complex
conjunction-implicational paradox. It is a logical
theorem of almost all relevant logics and is valid in the
logics. Therefore, from any given entailment A⇒B and
the logical theorem, we can infer (A∧C)⇒B by using
Modus Ponens for entailment. However, from the
viewpoint of human logical thinking, this inference is
not necessarily considered as to be valid in the sense of
entailment because there may be no necessarily relevant
and/or conditional relation between C and B and
In this section, we will give some examples to
explain why one should take relevant reasoning as the
logical basis of knowledge engineering.
4.1 Entailment and ampliative, non-circular,
and/or non-tautological reasoning
First of all, from the viewpoint to regard logically
valid reasoning as the process of drawing new and
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correct conclusions from some premises which are
known facts and/or assumed hypotheses, any
meaningful reasoning should be ampliative, noncircular, and/or non-tautological, i.e., the truth-value of
conclusion of the reasoning must not be used in
deciding the truth-values of premises of the reasoning.
4.2 Entailment and paraconsistent reasoning
In general, our knowledge about a domain may be
inconsistent in some ways, i.e., it directly or indirectly
includes some contradictions.
Also, even if our
knowledge about a domain seems to be consistent, in
some cases, we may find a new fact or rule that is
inconsistent with our known knowledge, i.e., we find a
contradiction. In these cases, we neither doubt “logic”
we used in our ordinary logical thinking nor reason out
anything from the contradictions, but we must consider
that there are some wrong things in our knowledge and
will investigate the causes of the contradictions.
Indeed, in scientific research, the detection and
explanation of an inconsistency between a new fact and
known knowledge often leads to formation of new
concepts and/or discovery of new principles. How to
reason with inconsistent knowledge is an important
issue in scientific discovery and theory formation.
As an example, let us see the most typical human
logical inference form so-called “Modus Ponens.” The
natural language representation of Modus Ponens is “if
A holds then B holds, now A holds, therefore B holds.”
When we reason using Modus Ponens, what we know?
We know “if A holds then B holds” and “A holds.”
Before the reasoning, we do not know whether or not
“B holds.” If we knew, then we would not need to
reason at all. Therefore, Modus Ponens should be noncircular. Thus, how can we know “B holds” by using
non-circular Modus Ponens? Indeed, by using Modus
Ponens, we can know “B holds,” which is unknown
until the reasoning is done, based on the following
reasons: (i) “A holds,” (ii) “There is no case such that A
holds but B does not hold,” and (iii) we know (ii)
without investigating either “whether A holds or not”
or “whether B holds or not.” Note that the WrightGeach-Smiley criterion for entailment is corresponding
to the above (ii) and (iii). From this example, we can
see that the key point in non-circular reasoning is the
primitive and intensional property required by the
notion of entailment.
For a logic with Modus Ponens as an inference
rule, the paraconsistence requires that the logic does not
have “(¬A∧A)→B”, which is the most typical
implicational paradox, as a logical theorem where “A”
and “B” are any two different formulas and “→” is the
relation of implication used in Modus Ponens. If a
logic is not paraconsistent, then infinite propositions
(even negations of those logical theorems of the logic)
may be reasoned out based on the logic from a set of
premises that directly or indirectly include a
contradiction. For example, CML is explosive and not
paraconsistent because it uses Modus Ponens for
material implication as an inference rule, and has
“(¬A∧A)→B” as a logical theorem. On the other hand,
the relevant logics are paraconsistent because they do
not have the implicational paradox “(¬A∧A)⇒B” as a
logical theorem. Obviously, as a logic system to
underlie reasoning with inconsistent knowledge, a
paraconsistent logic is by far better than a paradoxical
logic, and therefore, we can say that formalizing the
notion of entailment satisfactorily is a key point to the
issue of reasoning with inconsistent knowledge.
Because the material implication in CML is an
extensional truth-function of its antecedent and
consequent without the primitive and intensional
property required by the notion of entailment, a
reasoning based on the logic must be circular but not
ampliative. For example, Modus Ponens for material
implication is usually represented in CML as “from A
and A→B to infer B.” According to the extensional
truth-functional semantics of the material implication,
if we know “A is true” but do not know the truth-value
of B, then we cannot decide the truth-value of “A→B.”
In order to know the truth-value of B using Modus
Ponens for material implication, we have to know the
truth-value of B before the reasoning is done!
Obviously, Modus Ponens for material implication is
circular if it is used as a inference form, and therefore, it
is not a natural representation of Modus Ponens. On
the other hand, because relevant logics have a primitive
intensional connective to represent the notion of
entailment which satisfies the Wright-Geach-Smiley
criterion, Modus Ponens for entailment in these logics
is non-circular. Based on the above discussion, we can
say that the notion of entailment is indispensable for
ampliative and non-circular reasoning.
4.3 Entailment and abductive reasoning
Now, we argue what role the notion of entailment
plays in abduction and/or abductive reasoning which is
also an important issue in knowledge engineering.
The term “abduction” was coined by C. S. Peirce.
Peirce represented an abductive inference form as :
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The surprising fact, C, is observed.
But if A were true, C would be a matter of course.
Hence, there is reason to suspect that A is true.
shows that both the proof-theoretical relation |−CML and
the model-theoretical relation |=CML are equivalent to the
material implication. Therefore, we can say that the
monotonicity of CML is a direct result of the truthfunctional definition of the material implication.
In terms of logic, the abductive inference form can be
represented as: From “C” and “if A then C” to infer
“A.”
The above discussion suggests that
a
nonmonotonic deducibility relation may be obtained by
formalizing the notion of entailment satisfactorily, and
indeed this is right. The relevant logic R is naturally
nonmonotonic [10]. Its deducibility relation is already
nonmonotonic and no new logical primitives, such as a
minimization notion, a modal operator, or default
inference rules introduced by current research on
nonmonotonic reasoning, need be introduced to get the
desired effect. To get a nonmonotonic deducibility
relation by formalizing the notion of entailment
satisfactorily is an important research direction for
establishing a satisfactory logic system underlying
nonmonotonic reasoning.
Obviously, since “C” is an observed fact and “A” is
the result of the inference, the logical validity of such
an inference is totally determined by the validity of
entailment “if A then C.”
Those implicational,
conjunction-implicational,
and
disjunctionimplicational paradoxes, if they are used as entailments
in abduction, are not only useless but also harmful,
because an abduction used the paradoxes may give us
some results that are not relevant to the observed fact at
all. Therefore, we can say that the key point in
abduction is how to get and use genuine logical
entailments that are certainly relevant to the observed
fact.
4.5 Entailment and inference rule generation
and verification
4.4 Entailment and nonmonotonic reasoning
Now, let us discuss the relationship between the
notion of entailment and nonmonotonic reasoning
which has been actively studied in recent years by the
researchers working on fundamentals of AI.
Finally, let us see some practical issues in research
and development of knowledge-based systems.
There is a basic fact that the notion of entailment is
indispensable for any practical knowledge-based system.
In fact, the entailment has been used, in various explicit
or implicit forms, in every kind of approaches to
knowledge representation and reasoning such as logicbased, associational, frame-based and procedural
representations and reasoning.
This is inevitable
because there is no knowledge-based system which
works without reasoning and there is no inference form
which does not invoke the entailment. However, using
which kind of entailments as logical entailments and/or
inference forms is a key point here. Obviously,
unrestrictedly using those implicational, conjunctionimplicational, and disjunction-implicational paradoxes
(e.g.,
A→(B→A),
B→(¬A∨A),
¬A→(A→B),
(¬A∧A)→B, (A→B)∨ (¬A→B), (A→B)∨(A→¬B),
(A→B)∨(B→A),
((A∧B)→C)→((A→C)∨(B→C)),
(A∧B)⇒A,
(A∧B)⇒
((A∧C)⇒B),
A⇒(A∨B),
(A⇒B)⇒(A⇒(B∨C)), and so on) as logical entailments
and/or inference forms must lead to the combinational
explosion. Therefore, if we want to have a mechanism
to filter those paradoxical entailments automatically,
then the only way is to construct a completely paradoxfree calculus of entailment. The paradox-free relevant
logics Tc, Ec, and Rc have provided a basis for the
further work on this issue.
It is well-known that CML is naturally
monotonic. The monotonicity is caused by two
fundamental facts which hold for any set P of formulas
and any two formulas A and B. The two facts can be
equivalently represented either in a proof-theoretical
form or in a model-theoretical form as follows:
if P |−CML B then P|−CML A→B
if P |=CML B then P |=CML A→B
(1)
(1')
P∪{A} |−CML B iff P |−CML A→B
P∪{A} |=CML B iff P |=CML A→B
(2)
(2')
As a direct result of these two facts, CML has the
following proof-theoretical and model-theoretical
monotonicity:
if P |−CML B then P∪{A} |−CML B
if P |=CML B then P∪{A} |=CML B
(3)
(3')
Indeed, the above first fact means that if B is right
then so is A→B. This is a direct result of defining the
material implication as a truth-function of its antecedent
and consequent, i.e., A→B =df ¬(A∧¬B) or A→B =df
¬A∨B, and ignoring whether or not there is a
necessarily relevant and/or conditional relation between
its antecedent and consequent. The above second fact,
which is the so-called “deduction theorem” of CML,
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A common inadequacy of the current knowledgebased systems is that they cannot automatically
generate new and valid inference rules from those
existing rules that are programmed in the systems by
their developers. As a result, the systems fail to have
the ability to apply the knowledge, which are
programmed by their developers for some situations and
purposes, to other situations and purposes, which have
not been considered previously by the developers, even
if there is little difference between the considered
situations and/or purposes and unconsidered situations
and/or purposes. To make knowledge-based systems
more powerful and flexible, it is indispensable to
establish a domain-independent fundamental theory that
underlies the automatic generation of new and valid
inference rules. Obviously, a paradoxical logic cannot
be used as the fundamental theory, because from those
facts and entailments given as premises, an inference
based on the logic may result invalid entailments as
conclusions. On the other hand, the conclusions of an
inference based on a paradox-free logic must be valid, if
all premises of the inference are valid. Therefore, the
notion of entailment and its calculus are serious and
crucial to the issue of inference rule generation in
knowledge-based systems [4, 6]. Note that a calculus
of entailment not only can be used as a tool for the
automatic generation of new and valid inference rules,
but also can be used as a tool for incremental
knowledge acquisition.
regard the notion of entailment as the most fundamental
notion in knowledge representation and reasoning,
should establish a satisfactory logic calculus of
entailment to underlie the knowledge representation and
reasoning, and should take relevant reasoning based on
paradox-free relevant logics as the logical basis of
knowledge engineering.
REFERENCES
[1] A. R. Anderson and N. D. Belnap Jr., “Entailment:
The Logic of Relevance and Necessity,” Vol. I,
Princeton University Press, 1975.
[2] A. R. Anderson, N. D. Belnap Jr., and J. M. Dunn,
“Entailment: The Logic of Relevance and Necessity,”
Vol. II, Princeton University Press, 1992.
[3] J. Cheng, “Logical Tool of Knowledge Engineering:
Using Entailment Logic rather than Mathematical
Logic,” Proc. ACM 19th Annual Computer Science
Conference, pp. 228-238, 1991.
[4] J. Cheng, “Entailment Calculus as a Logical Tool
for Reasoning Rule Generation and Verification,” in J.
Liebowitz (ed.), “Moving Towards Expert Systems
Globally in the 21st Century,” pp. 386-392, Cognizant
Communication Co. (Hardbound), Macmillan New
Media (CD-ROM), 1994.
[5] J. Cheng, “The Fundamental Role of Entailment in
Knowledge Representation and Reasoning,” Journal of
Computing and Information, Vol. 2, No. 1, pp. 853873, Special Issue: Proceedings of the 8th International
Conference of Computing and Information, 1996.
The lack of formal and rigorous verification
technique is a major reason of that many experimental
knowledge-based systems cannot be reliably used in
practices for solving business and industrial problems
in the real world.
The issue of inference rule
verification also requires a domain-independent
fundamental theory to
underlie justifications.
Obviously, what we need here is also a satisfactory
logic calculus of entailment [4, 6].
[6] J. Cheng, “EnCal: An Automated Forward
Deduction System for General-Purpose Entailment
Calculus,” in N. Terashima and E. Altman (Eds.),
“Advanced IT Tools, IFIP World Conference on
Advanced IT Tools, IFIP 96  14th World Computer
Congress,” pp. 507-514, Chapman & Hall, 1996.
Based on the above discussion, we can see that the
role played by the notion of entailment in practical
knowledge-based systems is also essential.
5.
[7] J. M. Dunn, “Relevance Logic and Entailment,” in
D. Gabbay and F. Guenthner (eds.), “Handbook of
Philosophical Logic,” Vol. III, pp. 117-224, D. Reidel,
1986.
CONCLUDING REMARKS
[8] G. Priest, R. Routly, and J. Norman,
“Paraconsistent Logic: Essays on the Inconsistent,”
Philosophia, 1989.
We have pointed out why a reasoning based on
CML may be irrelevant, shown that a reasoning based
on paradox-free relevant logics is relevant and that it is
the notion of entailment that plays the fundamental role
in relevant reasoning, and given some examples to
explain why one should take relevant reasoning as
logical basis of knowledge engineering. From the
above discussions, we can conclude that we should
[9] S. Read, “Relevant Logic: A Philosophical
Examination of Inference,” Basil Blackwell, 1988.
[10] P. B. Thistlewaite, M. A. McRobbie, and R. K.
Meyer, “Automated Theorem Proving in Non-Classical
Logics,” John Wiley & Sons, 1988.
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