Relevance Logic as an Information-Based Logic. J. Michael Dunn Indiana University Bloomington

Transcription

1 Relevance Logic as an Information-Based Logic J. Michael Dunn Indiana University Bloomington

2 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, 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 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.

3 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

4 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)

5 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.

6 Valuation clauses for compound sentences We write x ϕ rather than I(ϕ, x) = T (Routley-Meyer) The Routley-Meyer evaluation clauses: 1) x p iff v(p, x) = T 2) x A iff not x* A 3) x A B iff x A and x B 4) x A B iff x A or x B 5) x A B iff a, b, if Rxab and a A, then b B

7 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.

8 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.

9 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).

10 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)

11 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)

12 Let us imagine information states as piles of paper David Sipress

13 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.

14 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

15 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.

16 Thank you!

17 References Alexander Bochman, "Dynamic Causal Calculus,"in C. G. De Giacomo, T. Eiter (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the 14th International Conference, KR Barwise, J. (1993), "Constraints, Channels, and the Flow of Information, in S. Peters and D. Israel (eds.), Situation Theory and Its Applications, Palo Alto CA: CSLI Publications (CSLI Lecture Notes). Beall, Jc, Brady, R., Dunn, J. M., Hazen, A. P., Mares, E., Meyer, R. K., Priest, G., Restall, G., Ripley, R., Slaney, J., Sylvan (formerly Routley), R. (2012), "On the Ternary Relation and Conditionality," The Journal of Symbolic Logic, 41, van Benthem, J. (1991), Language in Action: Categories, Lambdas and Dynamic Logic, North Holland. Dunn, J. M. (2001), "The Concept of Information and the Development of Modern Logic," in Zwischen traditioneller und moderner Logik : Nichtklassische Ansatze, ed. W. Stelzner and M. Stöckler, Mentis-Verlag, Paderborn, Dunn, J. M. (2001b), "Ternary Relational Semantics and Beyond," Logical Studies, 7, Dunn, J. M. (2001c), "Representation of Relation Algebras Using Routley-Meyer Frames, in C. A. Anderson and Zeleny, M. (eds.), Logic, Meaning and Computation: Essays in Memory of Alonzo Church, Dordrecht: Kluwer. Dunn, J. M. (2014),.The Relevance of Relevance to Relevance Logic,. In Logic and its Applications, M. Banerjee and S. N. Krishna (eds.), Lecture Notes in Computer Science 8923, Proceedings of 6th Indian Conference on Logic and its Applications, ICLA 2015, Mumbai, 11-29, Springer. Fine, K. (1974), "Models for Entailment," Journal of Philosophical Logic, 3, Girard J-Y (1987),.Linear Logic,. Theoretical Computer Science, 50, Maksimova, L. (1973), "A Semantics for the System E of Entailment," Bulletin of the Section of Logic, Polish Academy of Sciences, Institute of Philosophy and Sociology, 2, Mares, E. (1996), "Relevant Logic and the Theory of Information," Synthese, 109, Peirce, C. S. (1898), Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898, K. Ketner (ed.) with an introduction by H.Putnam, Harvard University Press. Routley, R. and Meyer, R. K. (1972),.The Semantics of Entailment, II, III, Journal of Philosophical Logic, 1, and Routley, R. and Meyer, R. K. (1973), "The Semantics of Entailment," in H. Leblanc (ed.), Truth, Syntax and Modality, Proceedings of the Temple University conference on alternative semantics, Amsterdam: North Holland. Scott, D. (1974), "Completeness and Axiomatizability in Many-valued Logics." In L. Henkin et al (eds.), Proceedings of the Tarski Symposium, American Mathematical Society, 25, Urquhart, A. (1972), "Semantics for Relevant Logics," The Journal of Symbolic Logic, 37, Urquhart, A. (1973), "An Interpretation of Many-Valued Logic," Mathematical Logic Quarterly, 19, Venema, Y. (1996), "A Crash Course in Arrow Logic," in M. Marx, L. Pólos, and M. Masuch (eds.), Arrow Logic and Multi-Modal Logic, Palo Alto CA: CSLI Publications, 3-61.