ENTAILMENT AND RELEVANT IMPLICAT IO N
Robert K. MEYER
In [2], Anderson and Belnap claim that two kinds of fallacies
are involved in the well-known paradoxes of implication — (1)
fallacies of modality, whic h "arise when it is claimed that entailments follow from, or are entailed by , contingent propositions," and (2) fallacies o f relevance, whic h sanction "the inference from A to B even though A and B may be totally disparate
in meaning" ('). T he system E o f entailment, as defined f o r
example i n El], is presented by these authors as a sentential
logic free from both kinds of fallacies.
It has been observed by J. Michael Dunn, however, that the
discussion in [2] of relevance is more or less independent of the
accompanying discussion of modality. For Anderson and Belnap
use that part of the discussion of [21 whic h hangs on relevance
alone to motivate, in the pure theory o f implication, not the
implicational part of E but rather the weak theory of implication
of Church's [7]. By adding those axioms and rules of E which
govern conjunction, disjunction, and negation to Church's system, Anderson and Belnap have constructed a system R, firs t
defined in NJ, which avoids fallacies of relevance in their sense
but not fallacies of modality. And indeed, one might think of R
not as formalizing a theory of entailment in any sense, but merely
as the formalization o f a k ind o f conditional whic h requires
relevance of antecedent to consequent as a necessary condition
for its truth. And it is hoped that R w ill be of service in the
logical analysis o f various k inds o f conditional whic h hav e
hitherto been accounted wayward — counterfactual, subjunctive, lawlike, and so forth.
The question now arises, "Can R be used to support a theory
of entailment ?" In a limited sense, the answer is clearly "Yes",
for we may proceed along lines well-marked by Quine and others.
Indeed, we may distinguish between those systems, like the Lewis
(
1
)
C
f
.
[
2
1
,
e
E NTA I LME NT A ND RE L E V A NT I MP L I CA TI O N
4
7
3
calculi, which attempt to express a formalized entailment, and
those systems, lik e classical logic, which are content to indicate
entailment according to the recipe, " I f A ----> B is logically valid,
A entails B." R, lik e the classical system HK, expresses not entailment b u t a variety o f the conditional; however, i f ever
A -->• B is a theorem o f R, i t can presumably be so only on
grounds that are purely logical; i f we go on to view R as indicating entailment in this situation according t o the classical
recipe, we can then ask what theory of entailment it is that R
supports.
The answer is rather satisfying; R supports exactly the theory
of entailment whic h E expresses, in the area where the tw o
systems coincide. To explicate this last remark, let me note that
both R and E are built up from a denumerable stock of sentential
variables by means o f the sentential connectives '8E', 'Y',
and B u t though the truth-functional connectives g e t their
usual readings in both systems, the arrow o f E has built into
it a modal character absent from the arrow of R; a suggested
reading for the former is 'that e n t a i l s that - - w h i l e the
latter is simply a hardy version of 'if t h e n - - (
2 In virtue of the above remarks, it is clear that although sen).
tences
of the form A ---> B say different things in the two systems, such sentences should be logically true under the same
conditions provided that no arrows occur in either A or B. And
indeed, i n proving a somewhat stronger result in [4], Belnap
has showed this to be the case.
On the other hand, when arrows occur nested within arrows
the differing interpretations associated w it h R and E assure
that in general it is not sufficient for a sentence to be marked
logically true in E that it is logically true according to R. Take,
for example, the R-theorem (
3
) (.
T
h
i
s
then
2 - - ' may be defined on t he basis of the arrow of R, as was noted
fin part
l
e
e
t
s
i n t he abstract 10 and wi l l be explored more f ully i n a sub)
sequent
AI
paper.
n The -stilted construal o f the arrow o f E is a concession
to use-mention fans and wi l l henceforth be abandoned.
n
(
f
sentential
3
variables; ' A ' , 'B', etc., f o r arbit rary formulas. Among ot her
a
)
c
'
t
p
,
'
m
,
o
'
r
q
e
'
f
,
474
R
O
B
E
R
T
K . ME Y E R
derson-Belnap relevance requirements, but since p may be contingent it clearly fails to meet their criteria for avoiding modal
fallacies.
One way to pass from a theory of the conditional to a theory
which expresses entailment lies in the excision of those theorems
of the first theory whic h commit modal fallacies. This is essentially t he course tak en b y Anderson and Be 'nap i n [21.
Another, and more traditional, approach lies in the addition of
modal operators t o the underlying theory o f the conditional,
whatever it may happen to be. The result of such explicit enrichment of the underlying system, as in the Gödel-von WrightFeys construction of Lewis modal logics, has resulted in systems
in which both the conditional and entailment can be expressed;
the former as A —> B, the latter as M A B ) . We ask whether
R also can be enriched with modal operators in a natural way
so that the resulting theory of entailment meets the conditions
set down for such a theory by Anderson and Belnap.
The answer again is "Yes "; indeed, such systems were considered independently by Bacon in [31 and by me in my dissertation. Bacon has an interesting hierarchy o f such systems,
based on the addition of the k ind of alternative axioms governing necessity explored by Lewis. O f these systems, however,
the one which makes interesting connections with E is the theory
which results by imposing an S4-like structure on the underlying
non-modal logic R; this is hardly surprising, since the motivating
conditions for E give that system the modal character of S4. I t
is the S4-like version of R, called here NR, which I now define
and investigate.
NR is built u p by the usual formation rules fr om N
tential
o svariables,
e n - the sentential constant f, (
4
conventions, I rank the connectives thus in order of increasing scope:
) '—>',
a n indicating
d
t hthe efirs t by simple juxtaposition on occasion. Other'v',
wise
c oassociation
n n e shall
c t bei t ov the
e left,
s and the conventions of Church shall
be employed f or the replacement of parentheses by dots.
(
to 4av oid several negation axioms. Th e extension is conservative wi t h
respect
t o R as defined by Belnap, whic h is proved i n my dissertation
)
(U.U o f Pittsburgh, 1966) building o n Anderson-Belnap results reported
in [ 5] and elsewhere.
s
e
o
f
t
h
e
s
e
n
E NTA I LME NT A ND RE L E V A NT I MP L I CA TI O N
'&', 'v ', a n d
following:
4
7
5
'N'. T he axiom schemes and rules are the
NRI.
NR2.
NR3.
NR4.
NR5.
NR6. AB--->B
NR7. (A--->B) (A-->C)-->(A-->BC)
NR8. A- - *Av B
NR9. B--->AvB
NR I 0, (A-->C)(B-->C)—>(AvB—>C)
NR11.
NRI2. NA---->A
NR13.
NR14. NANB--)frN( AB)
NRI5. NA-->NNA
NR16. A(BvC)-->ABvAC
NR17. A,A- - >BHB
NR18. A , B H A B
NR19. Ai—NA
Identity
Transitivity
Contraction
Assertion
&E
&E
&I
\it
vl
vE
Double negation
NE
N Distribution
N&I
NNI
& Distribution
Modus ponens
Adjunction
Necessitation
Inspection of the non-modal axioms shows them to yield the
system R (cf. footnote 4), while the modal axioms and the rule
NR19 are the usual ones which produce S4 from the set of classical tautologies. (But NR14 is redundant in that case.) I n fact,
NR would turn into S4 were we to add in addition the axiom
of paradox B —
independent
o f the other axioms, these facts seem t o me no
..A
argument
on
its behalf.
- - >A;
t It
h isoreadily
u g established that every theorem of E is a theorem
of
N
R
when
' A entails B' is defined as 'N(A—>13)'. T o show
h
this, it suffices to show that all axioms of E are theorems of NR
t
h
i
on translation, and that the rules of E are admissible in NR. I t
s
would
be nice to show also that all non-theorems of E are nona
x
i
theorems of NR on translation, but of this I have no proof. I t
o
m
is, however,
the case that the principal motivating condition for
i
s
c
o
n
s
i s
t
e
n
t
w
i
476
ROBERT K . ME Y E R
the E theory of modality is met in NR, and indeed in a somewhat
strengthened form. The key theorem is the following:
Theorem 1. Suppose A contains no occurrences of the, sign N.
Then A-->NB is a non-theorem of NR.
Proof. Consider the following matrix:
-4- + 3
+3
+2 + t — f — —
2
3
+3 —3 —3 —3 —3 —3
+2
+3 +2 —3 —2 —2 —3
+t
—
+3 +2 + t — f — —3
2
+3 —3 —3 + t — —
3
3
+3 +2 — + 2 +2 —
3
3
+3 +3 +3 +3 +3 +3
—2
—3
& +3
+2 + t
—f
—2 —3
f
+ 2 +2 + t —
2
+t +t +t —
3
—f —2 —3 f
— — — —2
2
2
33
3 3
3
—2 —3
+t
—2 —3
+t
— —
3
3
—2 —3
+t
—2 —
3 33
—
3
—3
+3 + 2 + t
Let +3, +2, and + t be designated, let formulas Av B b e
evaluated as (A.—>f) (11-->f)-->f, and let the sentential constant f
be assigned the matrix value — f on all evaluations. Then it is
readily seen that axioms of NR are always designated and that
this property is preserved under the rules o f NR. But for A
which satisfies the hypothesis o f the theorem, its value on an
assignment which gives each sentential variable the value + 2
will be o n the other hand, on all assignments NB w ill have
either the value + t or —
that
3 ; iif nAsdoes
p e not
c t contain
i o n N, A - 4
value
N
w
l l an assignment o f + 2 to all sentential variables;
o B —3
f i on
hence
a
re d i no fgthe
l ysuggested form is a theorem of NR.
t c hcnoosentence
This
concludes
the
proof
of theorem 1. (I note that the suggested
h
t
aa b v l e e
matrix
is
an
adaptation
of
matrices due to Ackermann and to
tf
h
e
o
r
Belnap.)
It
is
also
the case i n E that no negated entailments entail
>
entailments. We state the corresponding fact for NR as a corols
h
o
lary to the previous theorem.
w
s
Corollary 1.1. For all A and B, NA-->NB is a non-theorem of NR.
—3
E NTA I LME NT A ND RE L E V A NT I MP L I CA TI O N
4
7
7
Proof. Consulting the previous matrix, we note that N A must
take the value — f or + 3 on all assignments, while NB must
take the value + t or —3. I n any case, the value of NA—NB
on any assignment w ill be —
3.
The semantical situation of E is unclear, as Anderson reported
in [1]. There are, however, some important partial results, and
we now ask whether these results apply to NR also.
First, Belnap and Wallace investigated in [61 the fragment
El o f E determined by its axioms whic h contain only occurrences of a n d a n d the rule of modus ponens; by providing a Gentzen consecution calculus f o r ET , they wer e
able to show this system decidable. The corresponding fragment
NR
—
NRI7,
and NRI9. I f we consider only such formulas involving f
as
I are definitional abbreviations for formulas involving negation,
we
i obtain a Gentzen system from axioms A A and the rules
s*C*, * W * , *P*, * N * , and * Y * o f formulation I I o f L K Y in
Curry's
[81. By examining a proof of a translation of a formula
t
of E l in this system, it is easy to show that the same formula
h
can be proved in the Belnap-Wallace consecution calculus for
a
ET. Since it continues to be the case that a ll theorems o f ET
t
are theorems o f NRT, i t follows that the negation-entailment
d
fragments of E and NR are identical, in the sense in which these
e
fragments have been specified above.
t A second area in which the semantical character of E is clear
e
lies i n its set o f first-degree formulas f o r m u l a s i n whic h
rno subformula o f the form C —› D contains as its proper subm
formula an item o f the for m A B , or, briefly, formulas in
i
which
n o entailments are nested w it hin entailments. Belnap
n
provided in [4] a complete semantics and a decision procedure
for
e the first-degree fragment of E and of R; in fact, the set of
first-degree
theorems o f these two systems is the same. O u r
d
second
theorem
records the fact that this fragment of E is exactly
b
translatable into NR.
y
N
Theorem
2. Let A be a first-degree formula of E. Then A is a
theorem
of E if f A is on translation a theorem of NR.
R
1
N
R
4
,
N
R
478
R
O
B
E
R
T
K . ME Y E R
Proof. Suppose that A is on translation a theorem of NR. Let
A
A,
I by deleting all occurrences of N. Noting that all axioms of
NR
, remain axioms of R when all N's are erased, and that the
rules
continue to hold after erasure, i t is clear that A„ * is a
A
theorem
of R. But A„ * with all the N's erased is just A. Hence
„
A
is
provable
in R; i f A is first-degree, it is by Belnap's result
b
also provable in E. Hence i f A is on translation a theorem of
e
NR, A is a theorem of E if it is a first degree formula of E. On
a
the other hand, all theorems of E are theorems of NR on transp
lation, as noted above, thus concluding the proof of theorem 2.
r
So far as can be presently ascertained, accordingly, E and
o
NR express the same theory of entailment. A choice between
o
the two systems is accordingly to some degree a matter of taste,
fbut it seems to me that the following considerations favor NR.
oFirst, NR separates those issues in the critique of classical logic
f
which
are generated by the classical analysis of the conditional
t
from
the issues which lead to the explicit introduction of modal
operators.
Since as Quine has rightly pointed out, it is the clash
sical
analysis
of the conditional n o t the absence of squares —
e
which
leads
to
what is felt to be strange in the classical doctrine
t
of
r entailment, one might expect a full-fledged theory o f entailment
t o include a theory of the k ind of conditional which,
a
in the vocabulary introduced here, indicates entailment. Second,
n
since t h e question, " Wh ic h axioms s hall govern the modal
s
operators ?" is notoriously in doubt, formulating one's theory
l
of entailment v ia N R leaves open the possibility o f changing
a
one's assumptions about the modalities w hile leav ing one's
t
underlying
theory of the conditional alone (
e
5those attempts to motivate modal logic which rest on the analysis
of
rather
than on classical semantic analysis seem very
)d. deduction
T h i r d
,
A
frequently
to
motivate
h o w e v e r a, system in the neighborhood of S4; one
,thinks not only of [2] but of Curry's analysis in 1181, and in a
acertain sense of McKinsey's [91. NR shows that it is possible to
n (5) I t is import ant i n this connection again t o note t hat the k ind of
conditional whic h indicates entailment i n t he Anderson-Belnap sense is
d
not the only conditional expressible in R. Cf. footnote 2 and the remarks
lto whic h it is appended.
e
t
A
,
*
r
e
s
ENTAILMENT AND RELEVANT IMPLICATION
4 7 9
preserve these insights in a semantical frame otherwise highly
non-classical.
I conclude, as I began, with an observation of Dunn. In his
dissertation, Dunn provides an algebraic semantics for R, and
he notes that axioms lik e my NR12- NR
- into a closure algebra. For those who, unlike Dunn and me,
R
consider
15 t u non-classical
r n
t h logic
e s philosophically meaningless b u t
worth
a l investigating
g e b r for
a the mathematical structures t o whic h
they
give
rise,
I
submit
NR also (9.
o
f
Bryn Mawr College
R
o
b
e
r
t
K. MEYER
(
U.S.A.,
whose partial support o f Anderson and Be'nap's studies o f en6
tailment
a t o n e t i me part ially s upport ed me as we l l . Sinc e I hav e
)
depended
throughout on Belnap's work and occasionnally on his advice,
M
any errors are probably his f ault ; errors not so traceable may perhaps
y
be blamed on my conversations wi t h Dunn; cf. his dissertation ( U. o f
t
Pittsburgh, 1966), before c redit ing me wi t h original f olly .
h
a
REFERENCES
n
k A. R. ANDERSON, "Some open problems concerning the system E of
s entailment," Acta Philosophica Fennica, fasc. 16, Helsinki, 1963.
a A. R. ANDERSON and N. D. BELNAP, Jr., " Th e pure calculus of enr tailment," The Journal o f Symbolic Logic 27 (1962), pp. 19-52.
e 1. B. BACON, Being and Existence, Two Ways o f Formal Ontology,
d Ph. D. dissertation, Yale University 1966.
u N. D. BELNAP, I r. , "I nt ens ional models f o r f irs t degree f ormulas , "
The Journal of Symbolic Logic 32 (1967) , pp. 1-22.
e
N. D. BELNAP, Jr., " A f ormal analysis o f ent ailment , " Tec hnic al
t report * 7 , Contract * S A R / Nonr-609 (16), Of f ic e o f Nav al Reo search (Group Psychology Branch) , Ne w Haven, 1960.
t N. D. BELNAP, Jr., and J. R. WALLACE, "Ent ailment wi t h negation,"
h Zeitschrift f i i r ma t hematische Logik u n d Grundlagen d e r Mat hee matik, vol. 11 (1965), pp. 277-289.
N A. CHURCH, " Th e weak theory o f implic at ion, " Kontrolliertes Dena ken, Munic h, 1951.
H. B. CURRY, Foundations of Mathematical Logic, N. Y., 1963.
t
J. C. C. MCKINSEY, " O n t he syntactical construction of systems o f
i
modal logic , " The Journal of Symbolic Logic 10 (1945). pp. 83-94.
o R. K. MEYER, "Logic s contained i n R, " (abstract), The Journal o f
n Symbolic Logic, forthcoming.
a
l
S
c
i
e
n
c
e