Relevance Logic. Hans Halvorson. December 21, New Dec 16: Updated section on possible worlds semantics.

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1 Relevance Logic Hans Halvorson December 21, 2007 New Dec 16: Updated section on possible worlds semantics 1 Structural rules Definition 11 The collection of dependencies is defined inductively (Base case) The following symbols are dependencies: T, t (Base case) Every natural number is a dependency (Inductive case) If and y are dependencies then (, y) is a dependency (Inductive case) If and y are dependencies then (; y) is a dependency (Closure clause) Nothing is a dependency that is not generated via the previous four clauses As usual, we allow ourselves to drop the outermost parentheses So, the following is a dependency: 7; ((4; 3), 2) The following is also a dependency: (7; (t, 7)), T Henceforth, we never allow empty dependency sets (as is permitted in Lemmon s system [Lem78]) Instead, the dependencies t and T will play the role that is played by the empty set in the traditional formulation In particular, the line t (n) A corresponds to the claim: 1

2 A is derivable (from no assumptions) which we also write as A We will eplain shortly how one can obtain t as a dependency by itself, even though one always begins with 1 as the first dependency What is the intended meaning of the comma and semicolon? That s not so clear For now, we use some suggestive terminology: comma corresponds to aggregation, and semicolon corresponds to fusion The key difference between the two operations is that an aggregate of two sets of sentences is logically stronger than each set individually; on the other hand, the fusion of two sets of sentences requires acting on one of the two sets with the other and so the resulting set might be logically weaker than the sets with which we started We will continue to use the rules for the comma that are implicit in Lemmon s framework: in particular, the comma is associative, commutative, and satisfies both contraction (, ) and epansion (, y) These comma rules are tacit assumptions of Lemmon s system, because it is implicitly assumed that dependencies aggregate by set union (Remind yourself that set union is associative and commutative Furthermore, contraction is smuggled into CPL natural deduction when students are instructed to write n instead of n, n Finally, epansion in CPL is a consequence of the fact that conjunction intro and elim can be used to epand the set of dependencies) We now give some meaning to the two constants t and T Formally, T acts like the additive identity (if one thinks of comma as addition) and t acts like the multiplicative identity (if one thinks of the semicolon as multiplication) Intuitively speaking, T can be thought of as the emptyset (the identity for the operation of set union), and t can be thought of as the set of all tautologies T Identity In natural deduction format: T, = =, T T, (i) A (j) A i T = 2

3 Given the structural rules for the comma (see below), we can always simply omit T Just treat it as you are wont to treat the empty set! t Identity t; = In natural deduction format: t; (i) A (j) A i t= We now list a bunch of possible postulates on the semicolon operation Rule Name System ; (y; z) = (; y); z Associativity RW ; y = y; Commutativity RW ; Contraction R ; Mingle RM ; y Epansion CPL The top two rows (associativity, commutativity) are two-sided replacement rules That means, that given either side of an equality as a dependency, we can replace it on a subsequent line with the other side of that equality For eample, commutativity allows us to replace ; y with y; The bottom three rows (contraction, mingle, epansion) allow us to replace what is on the left of the arrow with what is on the right of the arrow So, for eample, mingle would allow us to move from a line with (n) A to a line with ; (n) A If you think about it, you will realize that our practices of manipulating dependency numbers in CPL were never eplicitly justified We thought carefully about which manipulations are permitted on sentences (on the right); but we did not think carefully about which manipulations are permitted on dependencies (on the left)! Our goal now is to fi this omission 3

4 2 Inference rules 21 Rule of assumptions In words: i (i) B Assumption On any line, you are permitted to write any sentence, so long as you write the number of the line as the dependency and cite the rule A 22 Copying rules We call these copying rules because we just copy the dependencies on the left Double Negation (DN) (i) A (j) A i DN (i) A (j) A i DN Introduction (i) A (j) A B i Intro (i) A (j) B A i Intro 4

5 Elimination (i) A B (j) A i Elim (i) A B (j) B i Elim 23 Aggregation rules We call these aggregation rules because we aggregate (using the comma) the dependencies of the input sentences Introduction y (i) A (j) B, y (k) A B i, j Intro 24 Fusion rules We have to be careful now in the statement of these rules, because they depend crucially on the structure of the dependencies, and not just the set of dependencies Introduction i (i) A A ; i (j) B some rule (k) A B i, j Intro 5

6 In words: Permitted to remove dependencies that are fusioned on the right by placing the corresponding sentence as the antecedent of a conditional In practice, the modified Intro rule renders useless any making of irrelevant assumptions Indeed, suppose that we are given a derivation of B from (ie a line with dependencies and B on the right), and we want to derive A B from Then we would need a derivation of B from ; i But, roughly speaking, ; i might be logically weaker than, and so there is no guarantee that there is a derivation of B from ; i Note: Intro is only permitted for dependencies that are fusioned on the right For eample, the following step is not warranted by Intro: i (i) A i; (j) B (k) A B However, when fusion is commutative, then one additional step will transform the above into a valid inference Elimination y (i) A B (j) A ; y (k) B i, j Elim Modus Tollens (i) A B y (j) B ; y (k) A i, j MT 6

7 Elimination This rule looks much worse than it is It is just like the basic Elim rule (from CPL), ecept that each derivation of the conclusion (C) must actually use the corresponding disjunct (A and then B) where use is made precise by means of the fusion of dependencies (i) A B j (j) A A y; j (k) C l (l) B A z; l (m) C ; (y; z) (n) C i, j, k, l, m Elim 3 Some systems of relevance logic Definition 31 Consider some set of the structural and inference rules Let A 1,, A n, B be sentences Then we say that A 1,, A n B, just in case there is a correctly written proof with final line 1, 2,, n (m) B where A i is assumed on line i, for i = 1,, n Postulate All systems obey all structural rules for comma Postulate All systems have the full set of inference rules 7

8 31 DW DW is the system that assumes no structural rules for ; other than t Identity However, t Identity yields some tautologies We now show that P P in DW 1 (1) p A t; 1 (2) p 1, t = t (3) p p 1, 2, Intro In fact, it can be shown that the only tautologies in DW are of the form A A 32 Associative, commutative logic (RW) The system RW assumes associativity and commutativity for semicolon Here are some things that can be proven in RW: 1 Permutation 2 Prefiing 3 Suffiing [p (q r)] [q (p r)] (p q) [(r p) (r q)] (p q) [(q r) (p r)] 4 Contrapositive (p q) ( q p) TO DO: How is RW related to linear logic? 8

9 321 Fusion Definition 32 Consider a logic with connectives and We say that is a fusion connective just in case the following holds for all formulas A, B, C: A (B C) is logically equivalent to (A B) C In logics with Associative and Commutative ;, we can show that the connective defined in terms of: is a fusion connective 33 R A B = (A B), The system R (= RW + Contraction) is the most pure and interesting relevance logic at least according to Anderson and Belnap [AB75] It results from adding to RW the contraction rule for dependencies: In natural deduction format: ; ; (i) A (j) A i Contract Note that in the above schema, can be a comple dependency So, for eample, the following is warranted by contraction: (1, 2); (1, 2) (i) A (1, 2) (j) A i Contract In summary, R is characterized by the fact that the semicolon is assumed to be associative, commutative, and to satisfy contraction Here are some things that can be proven in R but not in RW: 1 Contraction (p (p q)) (p q) 9

10 2 Disjunctive Consequence p q p q 3 Proof by Contradiction (p q) [(p q) p] 4 Weak Reductio 5 Reductio 6 Miing 7 Dichotomy (p p) p ( p p) p {p r, q r} (q p) r ( p q) [(p q) q] 8 Ecluded Middle p p Conjecture 33 Suppose that A, B are zero-th degree formulas (ie A and B do not contain ) Then A B in R iff B is a first degree entailment (FDE) of A Thus, R is a conservative etension of FDE 34 R Mingle The system RM (= R + Mingle) is said (by Anderson and Belnap) to be slightly irrelevant It adds to R the structural rule Mingle ;, 10

11 which is the reverse of Contraction The natural deduction format of Mingle is as follows: (i) A ; (j) A i Mingle Here are some things that can be proven in RM but not in R: 1 Mingle 2 Chain order p (p p) (p q) (q p) Here s a proof of the sentence Mingle using the structural rule Mingle 1 (1) p A 1; 1 (2) p 1Mingle t; (1; 1) (3) p 2t = (t; 1); 1 (4) p 3Assoc t; 1 (5) p p 1, 4 Intro t (6) p (p p) 1, 5 Intro 35 Classical propositional logic CPL results from supplementing RM with epansion: In natural deduction format: ; y (i) A ; y (j) A i Epansion Here are some things that can be proven in CPL but not in RM: 11

12 1 Positive Parado p (q p) 2 Negative Parado p (p q) 3 Bottom (EFQ) 4 Top (q q) p p (q q) 4 Semantics This is the section where we talk about semantics We first present a simple functional semantics for RM due to Sugihara 41 Functional semantics for R-Mingle The natural deduction system RM is sound and complete relative to Sugihara s semantics, as described below Definition 41 A Sugihara valuation f is a function from sentences to integers Z = {, 2, 1, 0, 1, 2, } such that: 1 f(a B) = min{f(a), f(b)} 2 f(a B) = ma{f(a), f(b)} 3 f( A) = f(a) 4 f(a B) = sign[f(b) f(a)] ma{ f(a), f(b) }, where sign(z) = +1 when z 0, and sign(z) = 1 when z < 0 Definition 42 (Designated Values) We say that a Sugihara valuation f satisfies a sentence A just in case f(a) 0 12

13 Eample We can use the Sugihara semantics to show that negative and positive parado are not provable in RM For eample, for the sentence p (q q), every valuation assigns the consequent q q the value q But then we can assign p any value that we wish, for eample f(p) = q + 1 It then follows that f(p (q q)) = 1, and p (q q) is not a tautology of RM Similarly, for Negative Parado, for any fied value of p, we can adjust the value of q to be as low or as high as we want By doing so, we can adjust the value of p q as low as we want And since the value of p is already fied, we can make p (p q) take a negative value Hence p (p q) is not a tautology of RM 42 Possible worlds semantics We let R + denote the positive fragment of R That is, the sentences of R + are the sentences of R that contain no negation symbols Definition 43 An R + frame is a triple W, R, 1 where W is a nonemtpy set ( worlds or situations ), R is a ternary relation on W, and 1 is a distinguished element of W For brevity of epression we define: a b iff R1ab R 2 abcd just in case ( )(Rab & Rcd) R 2 a(bc)d just in case ( )(Rbc & Rad) We then require: 1 (Identity) a a; that is, R1aa; 2 (Commutativity) If Rabc then Rbac; 3 (Associativity) If R 2 abcd then R 2 a(bc)d; 4 (Monotonicity) If Rabc and a a then Ra bc; 5 (Idempotence) Raaa 13

14 The various requirements in the above definition correspond to structural rules (in the sense that if the requirements hold of the model, then the structural rule is truth preserving) In particular, Identity corresponds to t = ; Commutativity corresponds to Commutativity, Associativity corresponds to Assocativity; and Idempotence corresponds to Contraction Lemma 44 In a R + frame, is refleive and transitive Proof Refleivity is just the identity aiom To show transitivity, assume that a b and b c Then R1bc and monotonicity entails that R1ac, that is a c Definition 45 An R + model is a structure W, R, 1, where W, R, 1 is an R + frame and is a relation from elements of W to sentences of R + satisfying the following conditions: 1 (Atomic Heredity) If a p and a b then b p 2 (Valuational Clauses) ( ) a A B iff a A and a B ( ) a A B iff a A or a B ( ) a A B iff b, c W [Rabc & b A c B] ( ) a A B iff b, c W [Rbca & b A & c B] Definition 46 We say that A is verified on a model W, R, 1, just in case 1 A We say that A entails B on a model W, R, 1, if a W [a A a B] Definition 47 An R frame is a structure W, R, 1, where W, R, 1 is an R + frame, and is a unary operation on W such that: (Period two) (a ) = a; (Inversion) If Rabc then Rac b ; (Classical) 1 = 1 14

15 Definition 48 An R model is a structure W, R, 1,, where W, R, 1, is an R frame, and W, R, 1, is an R + model, and: ( ) a A iff a A Hint: R is a conservative etension of R + That is, if R A, and A contains no negation signs, then R+ A Practically speaking, this means that to refute a positive R sentence, it suffices to give a refuting R + model Eample We show that R p (p p) Let W = {1, a, b}, and let R = {111, 1bb, b1b, 1ba, b1a, 1aa, a1a, 1a1, a11, bbb, bba, bb1, aab, aaa, aa1, abb, bab, aba, baa, ab1, ba1} Define a relation from W to sentences by setting 1 p and a p, and otherwise A for atomic a It then follows that a p p, since Raab and a p, but b p But then 1 p (p p) since R1aa and a p, but a p p The central intuition behind this eample is that b a 1, and a a = b, where p is forced at a but not at b (Think of Ryz as saying that z is above the fusion y, in the sense that z forces at least as much to be true as y) Hence p p is not true at a, because a fused with a p-world (namely itself) does not result in a p-world We will see below that this maneuver is not permitted in RM models, essentially because a a is required to be above a Eample We show that R (p q) (q p) Let W = {0, 1, 2} In this eample we use to denote the standard linear order on the numbers in W ; and we use a b to denote the distinct relation on W defined as R0ab Now define Rabc iff a b c a + b (1) Commutativity of R is a consequence of commutativity of a b and a + b Identity follows from the fact that a a a, and idempotence follows from 15

16 the fact that a a = 0 a 2a = a + a Proving the associativity of R is somewhat complicated, and so we relegate that proof to Appendi A Now let = for all W Then Period Two follows automatically To see that Contrapositive holds, suppose that Rabc By using Commutativity, we may assume that b a, and hence that b a + c By the definition of Rabc, we have a b c a + b (2) Subtracting a from Eqn 2 yields b c a b, and thus a c b Therefore, a c b a + c, which means that Racb Since b = b and c = c, we have shown then that if Rabc then Rac b Now we check which constraints are imposed by heredity If a b, then b 0 c b + 0, and so b = c Hence there are no hereditary relations between distinct worlds in W In particular, we may set 1 p and 2 q, but otherwise a A for atomic A We then have 0 p q since R011 and 1 p, but 1 q Similarly, 0 q p since R022 and 2 q, but 2 p Note by the way that W, R, 0, fails to be a RM model (see below) since R220 but Possible worlds semantics for RM and CPL Definition 49 A RM frame is a R frame in which: if Rabc then a c or b c Proposition 410 If W, R, 1, is a RM frame then either 1 a or 1 a Proof If W, R, 1, is a R frame then R1aa Ra1a Raa 1 (Identity) (Commutation) (Contraposition) But if Rabc entails that a c or b c, then a 1 or a 1 Hence Contraposition and Period Two entail that 1 a or 1 a 16

17 Definition 411 A CPL frame is a R frame in which: if Rabc then a c Eample We show that p p RM q Let W = {1, a, b}, and let R = {111, aaa, bbb, abb, baa, bab, aba, 1aa, 1bb, a1a, b1b, ab1, ba1, 1b1, b11, 11a, 1ba, bb1, b1a, bba} Let 1 = 1, a = b, b = a To see that W, R, 0, satisfies the RM requirement (if Ryz then z or y z) note that the worlds are linearly ordered b 1 a Clearly terms of the form R trivially satisfy the requirement Net, note that we have Ry only in cases where y Finally, in the case where y, y z and z, the only problematic cases would be Ra1b or R1ab, neither of which occur in the table (Note, however, that this frame does not satisfy the CPL constraint, since Rab1 but a 1) We now define a relation from W to sentences: let a p, and otherwise A for atomic A (Since a only if = a, the assumption a p entails no further constraints) Since a p, it follows that a p and hence a p p But a q, and therefore p p does not entail q on this model Eample We show that RM p (q q) Let W, R, 0, be the RM frame from the previous eample Let p and a q, and otherwise let A for atomic A We claim then that 1 p (q q) First note that b q q since Rba1 and a q, but 1 q But then 1 p (q q) since R1bb and b p but b q q Note: There is a common theme in the previous two eamples: the worlds are ordered b 1 a Since 1 = 1, the world 1 forces all Boolean tautologies, and cannot force any Boolean contradiction (see below) Hence, worlds above 1 force Boolean tautologies; and worlds below 1 cannot force Boolean contradictions Furthermore, since R1y means that y, and since we require heredity, it follows that 1 forces all sentences of the form A A 17

18 But given b 1 a with a = b, it is possible for a to force classical contradictions; and it is possible for b to fail to force classical tautologies, including A A Indeed, R1bb (Identity) entails (by Commutation and Inversion) that Rba1 Thus b A A only if 1 A whenever a A But certainly when A is atomic, we can choose for a A but 1 A Lemma 412 If a = a, then a A for each Boolean tautology A, and a B for each Boolean contradiction B Proof By definition a A iff a A By assumption, a = a, hence a A iff a A Thus, a straightforward inductive argument shows that if A is a Boolean tautology, then a A Similarly, if B is a Boolean contradiction, then a B Eample We show that p q RM p q Under construction A Appendi In this appendi we prove associativity for the ternary relation R on {0, 1, 2} defined by Rabc a b c a + b Definition A1 Let, y, z be positive integers, and define δ(, y, z) = min{ γ : + γ [ y z, y + z ]} That is, δ(, y, z) is the minimum distance from the number to the interval [ y z, y + z ] Lemma A2 For positive integers, y, z, we have δ(, y, z) = δ(z, y, ) 18

19 Proof We argue in three cases depending on the relation between and the interval [ y z, y + z ] (Case 1) is in the interval [ y z, y + z ] That is y z y + z Since y z, it follows that y z Since y + z it follows that y z These two facts entail that y z Furthermore, since z y, it follows that z + y Therefore, y z y + (Case 2) < y z Then y z = + k for some positive integer k In fact, k = δ(, y, z) If z y then y z = + k It then follows that = y z k, and since z, k are positive integers, < y Furthermore, z = y k = y k Therefore, δ(z, y, ) = k If z > y then z y = + k and z = + y + k Thus, again δ(z, y, ) = k (Case 3) > y + z Then = y + z + k for some positive integer k, in fact k = δ(, y, z) But then y <, since z, k are positive, and y = z + k, therefore y = z + k Thus, δ(z, y, ) = k Proposition A3 The relation R is associative Proof We claim that and that R 2 abcd δ(c, b, a) d a + b + c, (3) R 2 a(bc)d δ(a, b, c) d a + b + c (4) Since Lemma A2 shows that δ(a, b, c) = δ(c, b, a), Eqns 3 and 4 entail that R 2 abcd R 2 a(bc)d Let γ be the smallest integer such that c+γ [ a b, a+b ]; thus, γ = δ(c, b, a) Then by the definition of R, we have Rab(c + γ) and R(c + γ)c γ Therefore, R 2 abc γ We now claim that if R 2 abcd, and d < a+b+c, then R 2 abc(d+1) Indeed, if R 2 abcd, then there is an such that Rab and Rcd If < a + b, then 19

20 also Rab( + 1), and R( + 1)c(d + 1) If = a + b, then R(a + b)cd and d < a + b + c entails that R(a + b)c(d + 1) Thus, in either case R 2 abc(d + 1) Thus, we have now established that R 2 abc γ, with γ = δ(c, b, a), and that R 2 abcd R 2 abc(d+1) whenever d < a+b+c These two facts together entail Eqn 3 Now we establish Eqn 4 Let γ be the smallest integer such that a + γ [ c b, c+b ]; thus, γ = δ(a, b, c) Then we have Rbc(a+γ) and Ra(a+γ) γ Therefore, R 2 a(bc) γ As above, it also follows that if R 2 a(bc)d and d < a+b+c, then R 2 a(bc)(d+ 1) But then Eqn 4 follows Acknowledgments: Our formulation of a natural deduction system for substructural logics is due to Read [Rea88, pp 51 71] and Slaney [Sla90] Our formulation of possible world semantics for R draws on [ABD92, pp ], see also [Rea88] References [AB75] Alan Ross Anderson and Nuel D Belnap, Jr Entailment Princeton University Press, Princeton, NJ, 1975 [ABD92] Alan Ross Anderson, Nuel D Belnap, Jr, and J Michael Dunn Entailment The logic of relevance and necessity Vol II Princeton University Press, Princeton, NJ, 1992 [Lem78] E J Lemmon Beginning logic Hackett Publishing Co, Indianapolis, Ind, revised edition, 1978 Edited by George W D Berry [Rea88] Stephen Read Relevant Logic Blackwell, 1988 [Sla90] John Slaney A general logic Australasian Journal of Philosophy, 68:74 89,