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disjunction, that answer every either-or question they raise.” In the section he
wrote in
Entailment
II, he suggests (among other names) that these may be
called “prime,” and this is the name that we use. It derives from the notion
of a prime filter in lattice theory, and Routley-Meyer used the same notion in
constructing the canonical models used in proving their semantics complete.
This suggests that we generalize the notion of an information state so that it
is not just a finite sequence of 0's and 1's, but that we also let the notation
N
(for
neither) sometimes occur to indicate that the information state is not complete.
There are also circumstances where we might have conflicting information. In
a computer network, sites are sometime cloned for various purposes, good or
bad (e.g., safety backup, facilitating access, phishing). It is entirely possible that
the cloning introduces errors so that one site has 1 in a certain position in its
current state, and a supposedly duplicate site shows a 0 at the same time in the
same position. For that we might use the notation
B
(for both). Such a 4-valued
information state was described by Dunn (2008), and clearly is an abstraction
from the so-called Belnap-Dunn 4-valued logic.
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(Tentative) Conclusion
I call this concludingsection “tentative,”becauseI hope thatthis paper hasopened
all kinds of doors to further research. You all are invited to walk through them.
All of the variations on the contextual relevance relation, handled in the right way,
give rise to different logics, some well-known, others new.
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I am tempted to follow
the example of Fermat here and say: “I have discovered a truly marvellous proof of
this, which this margin - I mean paper - is too narrow to contain.” But the truth
is I have barely begun to peak through many of the doorways.
This is an embarrassment of riches. We started out (at least according to
Meyer), with no logic of relevance, and now we have too many logics of relevance.
Will the real logic of relevance please stand up?
At any rate, we have given an interpretation of the Routley-Meyer accessibil-
ity relation
Rabc
in terms of contextual relevance that allows us to regard the
logic R of relevant implication as also a logic of relevance. Of course, we must
admit that there are various other uses of relevance outside of logic, but that
point is not relevant.
Suppose I were to claim that classical logic is the logic of the two truth-values
truth and falsity. It seems to me that this is correct even though truth and falsity
have many applications outside of logic.
References
Anderson, A., Belnap, N.: Entailment: The Logic of Relevance and Necessity, vol. I.
Princeton University Press (1975)
Anderson, A., Belnap, N., Dunn, J.M.: Entailment: The Logic of Relevance and Ne-
cessity, vol. II. Princeton University Press (1992)
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A good place to start is to consider the list in Routley and Meyer (1973) of various
properties of the ternary relation with the corresponding logical axioms they validate.