Relevance Logic as an
Information-Based Logic
J. Michael Dunn
Indiana University – Bloomington
The Founders of Relevance Logic
Ivan Orlov (1928) “Calculus of Combatibility.” Orlov’s pioneering work was
effectively lost until rediscovered in 1992 by Kosta Dosen in “The First
Axiomatization of Relevance Logic,” Journal of Philosophical Logic 21, 339 – 356.
Moh Shaw-Kwei (1950) -- Implication fragment of R
Alonzo Church (1951) “Weak Calculus of Implication” – Implication
fragment of R. The type system for his I-calculus.
Wilhelm Ackermann (1956) “Strenge Implikation.,“ Later called “Rigorous
Implication.” Equivalent to E, but proof of equivalence not obvious. Equivalence
shown by Meyer and Dunn 1969.
Alan Ross Anderson, Nuel Dinsmore Belnap, Jr. (1959) Systems E of Entailment
and R of Relevant Implication (among others). Their papers, and volumes I and II
of Entailment (1975, 1992) (co-authored with others, including various of their
former students) made relevance/relevant logic an object of serious study.
Axioms and Rules of R
A ! A Self-Implication (1)
(A ! B) ! [(C ! A) ! (C ! B)] Prefixing (2)
[A ! (A ! B)] ! (A ! B) Contraction (3)
[A ! (B ! C)] ! [B ! (A ! C)] Permutation: (4)
(A ^ B) ! A; (A ^ B) ! B Conjunction Elimination (5)
[(A ! B) ^ (A ! C)] ! [A ! (B ^ C)] Conjunction
Introduction (6)
A ! (A _ B); B ! (A _ B) Disjunction Introduction (7)
[(A ! C) ^ (B ! C)] ! [(A _ B) ! C] Disjunction
Elimination (8)
[A ^ (B _ C)] ! [(A ^ B) _ C] Distribution (9)
(A ! A) ! A Reductio (10)
(A ! B) ! (B ! A) Contraposition (11)
A ! A Double Negation (12_
• the rule modus ponens: A; A ! B ` B
• the rule of adjunction: A; B ` A ^ B
Anderson and Belnap showed that A  B is a theorem of the logic
R only if A and B share a propositional variable. This is called the
Variable Sharing Property (VSP) and has become the defining mark
of a relevance (or relevant) logic.
In particular the following are not theorems:
(p  p)  q (Explosion)
p  (q  q) (Triviality)
Routley and Meyer define a valuation v as a function
that assigns to each pair of an atom and a set up (p,
a) either the value T or F. From this, they inductively
define a function I that assigns to each pair (A, a),
where A is an arbitrary sentence, either T or F.
Valuation clauses for compound sentences
We write x ⊧ ϕ rather than I(ϕ, x) = T (Routley-Meyer)
The Routley-Meyer evaluation clauses:
1)
2)
3)
4)
5)
x ⊧ p iff v(p, x) = T
x ⊧ ∼A iff not x* ⊧ A
x ⊧ A ∧ B iff x ⊧ A and x ⊧ B
x ⊧ A ∨ B iff x ⊧ A or x ⊧ B
x ⊧ A → B iff ∀a, b, if Rxab and a ⊧ A, then b ⊧ B
But first define a  b iff R0ab.  is a quasi-order
(reflexive and transitive). They require the Hereditary
Condition for all atomic sentences p :
if a  b & v(p, a)= T, then v(p, b) = T.
They show by induction that it also holds for arbitrary
sentences A given the valuation clauses on the next
slide.
Interpretations of the Ternary Accessibility Relation
1. "Modal" Interpretation (relative relative possibility): a and b are
compossible (or compatible) relative to c (Routley-Meyer 1973
credits it to Dunn). Canonically this amounts to if A is provable in
the theory a, and B is provable in the theory b, then A∘ B =
(A→  B) is provable in the theory c.
2. Another "Modal" Interpretation (relative accessibility): if the
antecedent of a law in a is realized in b, then its consequent is
realized in c (Routley-Meyer 1972-II). Canonically this amounts to if
A → B is provable in the theory a, and A is provable in the theory b,
then B is provable in the theory c.
3. Information Combining Interpretation: the piece of
information a when combined with b equals (or is included)
in c. (Urquhart, Fine, Mares)
4. Computational Interpretation: view information state a as
"input" and view the information state b as a "program."
Information state c is a potential result of running the
program on that input. (Dunn)
5. Computation Composing interpretation: view both a and
b as programs, and view the information state c as the
program that arises when you compose the first program
with the second (Dunn 2001, 2001a).
6. Naive Interpretation: situation b is relevant to situation c
in the context of situation a (Dunn 2014).
7. Fuzzy Logic Interpretation: a, b, c are degrees of truth and c =
a + b (Scott, Urquhart).
8. "Thirdness" Interpretation: a is a relation exemplified by the
pair b and c, i.e., a is a relation between b and c. (Peirce, Dunn)
9. Causal Interpretation: Given the environment a, event b is the
cause of event c. (Peirce, Bochman)
10. Arrow Logic (Dynamic) Interpretation: a is an arrow
(transition) that includes the composition of the arrows
(transitions) b and c. (van Benthem, Venema)
11. Phase space interpretation: c is the product of a and b
(Girard).
12. Communication Interpretation: a is a channel connecting
channel b to channel c. (Barwise)
In this talk we shall focus on ways of combining
information states
Here are three natural ways of interpreting Rabc in terms of
combining information states:
1.Data Combining Interpretation: the piece of information a
combined with b equals (or is included) in c. (Urquhart, Fine).
2.Program Applied to Data Interpretation: view information state
a as "input" (static) and view the information state b as a
"program" (dynamic). Information state c is a potential result of
running that program b on that input a . (Dunn).
3.Program Combining Interpretation: view a and b both as
programs, and view the result of composing these two programs
a and b as equal to (or included in) c. (Dunn)
Let us imagine information states as piles of
paper
David
Sipress
Interpretation 1: View the pieces of paper as data. Combining two
piles of them into a single pile can be done in different ways. The
simplest being to treat them as sets and not care about the order
in which they are placed, or whether there are duplicates. Another
way might be to regard them as multisets, and disregard the order
while carefully noting the number of duplicates. Maybe the order
could matter too as with sequences. And maybe the way the are
grouped into files, say with file folders could matter.
Interpretation 2: Think of the pieces of paper in the first pile (a) as
a kind of program containing instructions about what to do with
sentences on pieces of paper, and the idea is just to apply those
instructions to the sentences in the other pile (b).
Interpretation 3: Treat the sentences in both piles as instructions,
and t compose the instructions from the first pile (a) with those in
pile (b) so as to get new instructions.
To get a complete semantics for the logic of relevance
R we use the multiset interpretation, not distinguishing
order but distinguishing number of occurrences of a piece
of paper.
Then we interpret one pile of paper as included in another
pile of paper if every piece of paper in the first pile is
duplicated in the second pile, maybe more than once.
ab≤c
Routley-Meyer Valuation Clause for Relevant Implication,
Interpreted “Relevantly”
x⊧A→B
iff
∀y, z, if Rxyz and y ⊧ A, then z ⊧ B
A relevantly implies B in the context of situation x
iff
For all situations y, z, if y is relevant to z in context
x and A holds in situation y then B holds in
situation z.
Thank you!
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